Finite Group Databasebeta

← #33112397

Subgroups — #33112397

Order 18688.

700 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22610872C211yesmaximal, characteristic, complemented
2#22610868C211yesmaximal, characteristic, complemented
3#22610655C211yesmaximal, characteristic, complemented
4#22610650C211yesmaximal, characteristic, complemented
5#22610546C211yesmaximal, characteristic
6#22610546C211yesmaximal, characteristic
7#22610539C211yesmaximal, characteristic
8#22610539C211yesmaximal, characteristic
9#22610497C211yesmaximal, characteristic, complemented
10#22610497C211yesmaximal, characteristic, complemented
11#22610490C211yesmaximal, characteristic, complemented
12#22610490C211yesmaximal, characteristic, complemented
13#22609999C211yesmaximal, characteristic
14#22609911C211yesmaximal, characteristic
15C4×C22⋊Q8×C73C211yesmaximal, characteristic
16C2×C4⋊#10596831C2211yescharacteristic
17C2×C4⋊#10596831C2211yescharacteristic
18C2×C4⋊#10596831C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×C22⋊#10596831C2211yescharacteristic
23C2×C22⋊#10596831C2211yescharacteristic
24C2×C22⋊#10596831C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×#14556502C2211yescharacteristic
27C2×#14556502C2211yescharacteristic
28C2×C4×#10596831C2211yescharacteristic
29C2×C4×#10596831C2211yescharacteristic
30#17988342C2211yescharacteristic
31#17988342C2211yescharacteristic
32#17988342C2211yescharacteristic
33#17988342C2211yescharacteristic
34#17988342C2211yescharacteristic
35#17988342C2211yescharacteristic
36#17988342C2211yescharacteristic
37#17988342C2211yescharacteristic
38#17988342C2211yescharacteristic
39#17988342C2211yescharacteristic
40#17988342C2211yescharacteristic
41#17988342C2211yescharacteristic
42#17988342C2211yescharacteristic
43#17988342C2211yescharacteristic
44C2×C22⋊Q8×C73C411yescharacteristic
45C2×C4×Q8×C73C2211yescharacteristic
46C22×C42×C73C2211yescharacteristic
47#391×C73C411yescharacteristic
48#389×C73C411yescharacteristic
49#387×C73C411yescharacteristic
50#387×C73C411yescharacteristic
51#385×C73C411yescharacteristic
52#385×C73C411yescharacteristic
53#380×C73C411yescharacteristic
54C4×C4⋊C4×C73C2211yescharacteristic
55C4×C4⋊C4×C73C2211yescharacteristic
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C22⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×#10596831C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C4⋊#10596831—41no—
68C4⋊#10596831—41no—
69C4⋊#10596831—41no—
70C4⋊#10596831—41no—
71C4⋊#10596831—41no—
72C4⋊#10596831—41no—
73C22⋊#10596831—41no—
74C22⋊#10596831—41no—
75C22⋊#10596831—41no—
76C22⋊#10596831—41no—
77#14556502—41no—
78#14556502—41no—
79C4×#10596831—41no—
80C4×#10596831—41no—
81C22×Q8×C73C2×C411yescharacteristic
82C23×C4×C73C2×C411yescharacteristic
83C23×C4×C73C2×C411yescharacteristic
84C23×C4×C73C2311yescharacteristic
85C22⋊Q8×C73—24no—
86C4×Q8×C73—41no—
87C4×Q8×C73—41no—
88C2×C4⋊C4×C73C2×C411yescharacteristic
89C2×C4⋊C4×C73C2×C411yescharacteristic
90C2×C4⋊C4×C73C2×C411yescharacteristic
91C2×C4⋊C4×C73C2×C411yescharacteristic
92C2×C4⋊C4×C73C2×C411yescharacteristic
93C2×C4⋊C4×C73C2×C411yescharacteristic
94C2×C4⋊C4×C73C2×C411yescharacteristic
95C2×C4⋊C4×C73C2×C411yescharacteristic
96C2×C4⋊C4×C73C2×C411yescharacteristic
97C2×C22⋊C4×C73C2×C411yescharacteristic
98C2×C22⋊C4×C73C2×C411yescharacteristic
99C2×C22⋊C4×C73C2×C411yescharacteristic
100C2×C22⋊C4×C73C2×C411yescharacteristic
101C2×C42×C73—41no—
102C2×C42×C73—41no—
103C2×C42×C73C2×C411yescharacteristic
104C2×C42×C73C2×C411yescharacteristic
105C2×C42×C73C2311yescharacteristic
106C2×C42×C73C2311yescharacteristic
107C2×C42×C73C2311yescharacteristic
108C2×C42×C73C2311yescharacteristic
109C2×C42×C73C2311yescharacteristic
110C2×C42×C73C2311yescharacteristic
111#95×C73C2×C411yescharacteristic
112#95×C73C2×C411yescharacteristic
113#95×C73C2×C411yescharacteristic
114#95×C73C2×C411yescharacteristic
115#95×C73C2×C411yescharacteristic
116#95×C73C2×C411yescharacteristic
117#95×C73C2×C411yescharacteristic
118#95×C73C2×C411yescharacteristic
119#95×C73C2×C411yescharacteristic
120#95×C73C2×C411yescharacteristic
121C2×#10596831—21no—
122C2×#10596831—21no—
123C2×#10596831—21no—
124C2×#10596831—21no—
125C2×#10596831—21no—
126C2×#10596831—21no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C24×C73C22×C411yescharacteristic
146C2×Q8×C73—22no—
147C2×Q8×C73—22no—
148C2×Q8×C73—41no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—21no—
157C22×C4×C73—21no—
158C22×C4×C73—41no—
159C22×C4×C73—41no—
160C22×C4×C73—41no—
161C22×C4×C73C22×C411yescharacteristic
162C22×C4×C73C22×C411yescharacteristic
163C22×C4×C73C22×C411yescharacteristic
164C22×C4×C73C22×C411yescharacteristic
165C22×C4×C73C22×C411yescharacteristic
166C22×C4×C73C22×C411yescharacteristic
167C22×C4×C73C22×C411yescharacteristic
168C22×C4×C73C22×C411yescharacteristic
169C22×C4×C73C22×C411yescharacteristic
170C22×C4×C73C22×C411yescharacteristic
171C22×C4×C73C22×C411yescharacteristic
172C22×C4×C73C22×C411yescharacteristic
173C22×C4×C73C22×C411yescharacteristic
174C22×C4×C73C2411yescharacteristic
175C4⋊C4×C73—22no—
176C4⋊C4×C73—22no—
177C4⋊C4×C73—22no—
178C4⋊C4×C73—22no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73—22no—
181C4⋊C4×C73—41no—
182C4⋊C4×C73—41no—
183C4⋊C4×C73—41no—
184C22⋊C4×C73—22no—
185C22⋊C4×C73—41no—
186C22⋊C4×C73—41no—
187C22⋊C4×C73M1614yes—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—21no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—41no—
201C42×C73—41no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73C2×M1612yes—
209C23×C73C23×C411yescharacteristic
210Q8×C73—44no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—44no—
247C2×C4×C73—44no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73C2×M1612yes—
253C2×C4×C73C2×M1612yes—
254C2×C4×C73#13112yes—
255C2×C4×C73#13112yes—
256C2×C4×C73H(5,𝔽2)11yescharacteristic
257C2×C4×C73H(5,𝔽2)11yescharacteristic
258C2×C4×C73H(5,𝔽2)11yescharacteristic
259C2×C4×C732−1+411yescharacteristic
260C2×C4×C732−1+411yescharacteristic
261C2×C4×C732−1+411yescharacteristic
262#10596831—41no—
263#10596831—41no—
264#10596831—41no—
265#10596831—41no—
266#10596831—41no—
267#10596831—41no—
268#10596831—41no—
269#10596831—41no—
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73#51811yescharacteristic
278C22×C73#51911yescharacteristic
279C22×C73#52011yescharacteristic
280C22×C73C22×M1611yescharacteristic
281C22×C73C2×#13111yescharacteristic
282C22×C73C2×#13111yescharacteristic
283C22×C73#56811yescharacteristic
284C4×C73—22no—
285C4×C73—22no—
286C4×C73—22no—
287C4×C73—22no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73—41no—
291C4×C73—41no—
292C4×C73—41no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73—41no—
296C4×C73—41no—
297C4×C73#56211yescharacteristic
298C4×C73#56211yescharacteristic
299C4×C73#56311yescharacteristic
300C4×C73#56311yescharacteristic
301#23341—731nomaximal, Sylow
302C2×C73—42no—
303C2×C73#237811yescharacteristic
304C2×C73#297411yescharacteristic
305C2×C73#297611yescharacteristic
306C2×C73#298311yescharacteristic
307C2×C73#298611yescharacteristic
308C2×C73#299011yescharacteristic
309C2×C73#299311yescharacteristic
310C4×C22⋊Q8D7311yescharacteristic
311#1985—731no—
312#1984—731no—
313#1940—731no—
314#1939—731no—
315#1924—731no—
316#1924—731no—
317#1923—731no—
318#1923—731no—
319#1917—731no—
320#1917—731no—
321#1916—731no—
322#1916—731no—
323#1817—731no—
324#1797—731no—
325C73#2334111yescharacteristic, complemented, Sylow
326C2×C22⋊Q8Dic7311yescharacteristic
327C2×C4×Q8C2×D7311yescharacteristic
328C22×C42C2×D7311yescharacteristic
329C2×C4⋊C8—731no—
330C2×C4⋊C8—731no—
331C2×C4⋊C8—731no—
332C2×C4⋊C8—731no—
333C2×C4⋊C8—731no—
334C2×C4⋊C8—731no—
335C2×C22⋊C8—731no—
336C2×C22⋊C8—731no—
337C2×C22⋊C8—731no—
338C2×C22⋊C8—731no—
339C2×#97—731no—
340C2×#97—731no—
341C2×C4×C8—731no—
342C2×C4×C8—731no—
343#391Dic7311yescharacteristic
344#389Dic7311yescharacteristic
345#387Dic7311yescharacteristic
346#387Dic7311yescharacteristic
347#385Dic7311yescharacteristic
348#385Dic7311yescharacteristic
349#380Dic7311yescharacteristic
350C4×C4⋊C4C2×D7311yescharacteristic
351C4×C4⋊C4C2×D7311yescharacteristic
352C4×C4⋊C4C2×D7311yescharacteristic
353C4×C22⋊C4C2×D7311yescharacteristic
354C4×C22⋊C4C2×D7311yescharacteristic
355#336—731no—
356#336—731no—
357#336—731no—
358#336—731no—
359#336—731no—
360#336—731no—
361#336—731no—
362#336—731no—
363#336—731no—
364#336—731no—
365#336—731no—
366#336—731no—
367#336—731no—
368#336—731no—
369C22×Q8C2×Dic7311yescharacteristic
370C23×C4C2×Dic7311yescharacteristic
371C23×C4C2×Dic7311yescharacteristic
372C23×C4C22×D7311yescharacteristic
373C22×C8—731no—
374C22×C8—731no—
375C22×C8—731no—
376C22×C8—731no—
377C22×C8—731no—
378C22×C8—731no—
379C22×C8—731no—
380C22×C8—731no—
381C22⋊Q8—24no—
382C4×Q8—41no—
383C4×Q8—41no—
384C2×C4⋊C4C2×Dic7311yescharacteristic
385C2×C4⋊C4C2×Dic7311yescharacteristic
386C2×C4⋊C4C2×Dic7311yescharacteristic
387C2×C4⋊C4C2×Dic7311yescharacteristic
388C2×C4⋊C4C2×Dic7311yescharacteristic
389C2×C4⋊C4C2×Dic7311yescharacteristic
390C2×C4⋊C4C2×Dic7311yescharacteristic
391C2×C4⋊C4C2×Dic7311yescharacteristic
392C2×C4⋊C4C2×Dic7311yescharacteristic
393C2×C22⋊C4C2×Dic7311yescharacteristic
394C2×C22⋊C4C2×Dic7311yescharacteristic
395C2×C22⋊C4C2×Dic7311yescharacteristic
396C2×C22⋊C4C2×Dic7311yescharacteristic
397C2×C42—41no—
398C2×C42—41no—
399C2×C42C2×Dic7311yescharacteristic
400C2×C42C2×Dic7311yescharacteristic
401C2×C42C22×D7311yescharacteristic
402C2×C42C22×D7311yescharacteristic
403C2×C42C22×D7311yescharacteristic
404C2×C42C22×D7311yescharacteristic
405C2×C42C22×D7311yescharacteristic
406C2×C42C22×D7311yescharacteristic
407C4⋊C8—2921no—
408C4⋊C8—2921no—
409C4⋊C8—2921no—
410C4⋊C8—2921no—
411C4⋊C8—2921no—
412C4⋊C8—2921no—
413C22⋊C8—2921no—
414C22⋊C8—2921no—
415C22⋊C8—2921no—
416C22⋊C8—2921no—
417#97—2921no—
418#97—2921no—
419C4×C8—2921no—
420C4×C8—2921no—
421#95C2×Dic7311yescharacteristic
422#95C2×Dic7311yescharacteristic
423#95C2×Dic7311yescharacteristic
424#95C2×Dic7311yescharacteristic
425#95C2×Dic7311yescharacteristic
426#95C2×Dic7311yescharacteristic
427#95C2×Dic7311yescharacteristic
428#95C2×Dic7311yescharacteristic
429#95C2×Dic7311yescharacteristic
430#95C2×Dic7311yescharacteristic
431C24C22×Dic7311yescharacteristic
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—41no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4C22×Dic7311yescharacteristic
448C22×C4C22×Dic7311yescharacteristic
449C22×C4C22×Dic7311yescharacteristic
450C22×C4C22×Dic7311yescharacteristic
451C22×C4C22×Dic7311yescharacteristic
452C22×C4C22×Dic7311yescharacteristic
453C22×C4C22×Dic7311yescharacteristic
454C22×C4C22×Dic7311yescharacteristic
455C22×C4C22×Dic7311yescharacteristic
456C22×C4C22×Dic7311yescharacteristic
457C22×C4C22×Dic7311yescharacteristic
458C22×C4C22×Dic7311yescharacteristic
459C22×C4C22×Dic7311yescharacteristic
460C22×C4C23×D7311yescharacteristic
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4#1192401214yes—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—41no—
511C42—41no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23C2×#1192401212yes—
519C23C23×Dic7311yescharacteristic
520Q8—44no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4C2×#1192401212yes—
563C2×C4C2×#1192401212yes—
564C2×C4#1455659812yes—
565C2×C4#1455659812yes—
566C2×C4#1455663611yescharacteristic
567C2×C4#1455663611yescharacteristic
568C2×C4#1455663711yescharacteristic
569C2×C4#1455663811yescharacteristic
570C2×C4#1455663911yescharacteristic
571C2×C4#1455663911yescharacteristic
572C8—2921no—
573C8—2921no—
574C8—2921no—
575C8—2921no—
576C8—2921no—
577C8—2921no—
578C8—2921no—
579C8—2921no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22#1798906211yescharacteristic
588C22#1798906511yescharacteristic
589C22#1798907011yescharacteristic
590C22C22×#1192401211yescharacteristic
591C22C2×#1455659811yescharacteristic
592C22C2×#1455659811yescharacteristic
593C22#1798935411yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4#1798933311yescharacteristic
608C4#1798933311yescharacteristic
609C4#1798933811yescharacteristic
610C4#1798933811yescharacteristic
611C2—42no—
612C2#2261309711yescharacteristic
613C2#2261896611yescharacteristic
614C2#2261897911yescharacteristic
615C2#2261901611yescharacteristic
616C2#2261903311yescharacteristic
617C2#2261905611yescharacteristic
618C2#2261908011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC4×C22⋊Q8×C73C2
prime radicalC24×C73C22×C4
cosocleC22×C4×C73C24
hypercenterC4×C22⋊Q8D73
hypocenterC73#23341
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C4×C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22610872C2semidirect
2. #22610868C2semidirect
3. #22610655C2semidirect
4. #22610650C2semidirect
9. #22610497C2semidirect
10. #22610497C2semidirect
11. #22610490C2semidirect
12. #22610490C2semidirect
325. C73#23341semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226108722
#226108682
#226106552
#226106502
#226105462
#226105392
#226104972
#226104902
#226099992
#226099112
C4×C22⋊Q8×C732
#2334173