Finite Group Databasebeta

← #33099919

Subgroups — #33099919

Order 18688.

814 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556569C211yesmaximal, characteristic
2#22610867C211yesmaximal, characteristic, complemented
3#22610865C211yesmaximal, characteristic, complemented
4#22610650C211yesmaximal, characteristic, complemented
5#22610648C211yesmaximal, characteristic, complemented
6#22610491C211yesmaximal, characteristic, complemented
7#22610486C211yesmaximal, characteristic, complemented
8#22610485C211yesmaximal, characteristic, complemented
9#22610110C211yesmaximal, characteristic
10#22610110C211yesmaximal, characteristic
11#22609908C211yesmaximal, characteristic
12C8×#11924005C211yesmaximal, characteristic
13#22609701C211yesmaximal, characteristic
14#336×D73C211yesmaximal, characteristic
15C73×#1916C211yesmaximal, characteristic, complemented
16C22×C8×D73C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18C2×#14556569C411yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20C2×C42×D73C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556520C2211yescharacteristic
26C2×#14556503C2211yescharacteristic
27C2×#14556503C2211yescharacteristic
28C2×C4×#10596831C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31#17988491C411yescharacteristic
32#17988482C411yescharacteristic
33#17988473C411yescharacteristic
34#17988469C411yescharacteristic
35#17988461C411yescharacteristic
36#17988458C411yescharacteristic
37#17988446C411yescharacteristic
38C4×#11924007C2211yescharacteristic
39C4×#11924007C2211yescharacteristic
40C4×#11924005C2211yescharacteristic
41C4×#11924005C2211yescharacteristic
42#17988342C2211yescharacteristic
43#17988342C2211yescharacteristic
44#17988341C2211yescharacteristic
45#17988341C2211yescharacteristic
46#17988341C2211yescharacteristic
47#17988341C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×C4⋊C8×C73C2211yescharacteristic
53C2×C4×C8×C73C2211yescharacteristic
54C4×C4⋊C4×C73C2211yescharacteristic
55#336×C73C2211yescharacteristic
56#336×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146C2×C411yescharacteristic
60C22×C4×D73C2×C411yescharacteristic
61C22×C4×D73C2×C411yescharacteristic
62C22×C4×D73C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C2×C8×D73—22no—
68C2×C8×D73—41no—
69C2×C8×D73—41no—
70C2×C8×D73—41no—
71C2×C8×D73—41no—
72C2×C8×D73Q814yescomplemented
73#14556569C818yescomplemented
74C4×Dic146—41no—
75C4×Dic146—41no—
76C2×C4⋊Dic73C2×C411yescharacteristic
77C2×#11924007C2×C411yescharacteristic
78C2×#11924007C2×C411yescharacteristic
79C2×#11924007C2×C411yescharacteristic
80C2×#11924007C2×C411yescharacteristic
81C2×#11924007C2×C411yescharacteristic
82C2×#11924007C2×C411yescharacteristic
83C2×#11924005C2×C411yescharacteristic
84C2×#11924005C2×C411yescharacteristic
85C2×#11924005C2×C411yescharacteristic
86C2×#11924005C2×C411yescharacteristic
87C2×C4×Dic73C2×C411yescharacteristic
88C2×C4×Dic73C2×C411yescharacteristic
89C2×C4×Dic73C2311yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92C42×D73—41no—
93C42×D73—41no—
94C4⋊#10596831—41no—
95C4⋊#10596831—41no—
96C4⋊#10596831—41no—
97#14556520—41no—
98#14556520—41no—
99#14556503—22no—
100#14556503—22no—
101C4×#10596831—41no—
102C8×Dic73—41no—
103C8×Dic73—41no—
104#14556498C2×C411yescharacteristic
105#14556498C2×C411yescharacteristic
106#14556498C2×C411yescharacteristic
107#14556498C2×C411yescharacteristic
108#14556498C2×C411yescharacteristic
109#14556498C2×C411yescharacteristic
110#14556498C2×C411yescharacteristic
111#14556498C2×C411yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2×C411yescharacteristic
117C2×C4⋊C4×C73C2×C411yescharacteristic
118C2×C42×C73C2311yescharacteristic
119C2×C42×C73C2311yescharacteristic
120C2×C42×C73C2311yescharacteristic
121C4⋊C8×C73—41no—
122C4×C8×C73—41no—
123#95×C73C2×C411yescharacteristic
124#95×C73C2×C411yescharacteristic
125C23×D73C22×C411yescharacteristic
126C2×Dic146—41no—
127C2×Dic146—41no—
128C2×Dic146C2×C814yescomplemented
129C22×Dic73C22×C411yescharacteristic
130C22×Dic73C22×C411yescharacteristic
131C22×Dic73C22×C411yescharacteristic
132C22×Dic73C22×C411yescharacteristic
133C22×Dic73C22×C411yescharacteristic
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C2×C4×D73—21no—
137C2×C4×D73—21no—
138C2×C4×D73—21no—
139C2×C4×D73—21no—
140C2×C4×D73—22no—
141C2×C4×D73—41no—
142C2×C4×D73—41no—
143C2×C4×D73—41no—
144C2×C4×D73—41no—
145C2×C4×D73C2×C814yes—
146C2×C4×D73C2×Q812yes—
147C2×C4×D73#4112yes—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831C2×Q812yes—
160C8×D73—44no—
161C8×D73—44no—
162C4⋊Dic73—41no—
163#11924007—41no—
164#11924007—41no—
165#11924007—41no—
166#11924007—41no—
167#11924007C2×C814yescomplemented
168#11924007C2×C814yescomplemented
169#11924005—22no—
170#11924005—22no—
171#11924005C2×C814yes—
172#11924005C2×C814yes—
173C4×Dic73—21no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C4×Dic73—21no—
177C4×Dic73—21no—
178C4×Dic73—21no—
179C4×Dic73—41no—
180C4×Dic73—41no—
181C22×C4×C73C22×C411yescharacteristic
182C22×C4×C73C22×C411yescharacteristic
183C22×C4×C73C22×C411yescharacteristic
184C22×C4×C73C22×C411yescharacteristic
185C22×C4×C73C22×C411yescharacteristic
186C22×C4×C73C22×C411yescharacteristic
187C22×C4×C73C2411yescharacteristic
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73C2×Q812yes—
200C4⋊C4×C73—41no—
201C4⋊C4×C73C2×C814yescomplemented
202C42×C73—21no—
203C42×C73—21no—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—21no—
207C42×C73—21no—
208C23×C73C23×C411yescharacteristic
209C22×D73—21no—
210C22×D73—21no—
211C22×D73—21no—
212C22×D73—21no—
213C22×D73C4×Q812yes—
214C22×D73C22×C812yes—
215C22×D73#13112yes—
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×C73C22×C812yes—
232C2×C4×C73C22×C812yes—
233C2×C4×C73C22×C812yes—
234C2×C4×C73C22×Q811yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73H(5,𝔽2)11yescharacteristic
237C2×C4×C73H(5,𝔽2)11yescharacteristic
238C2×C4×C73H(5,𝔽2)11yescharacteristic
239C2×C4×C732−1+411yescharacteristic
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73C4×Q812yes—
258C2×Dic73C22×C812yes—
259C2×Dic73C22×C812yes—
260C2×Dic73C22×C812yes—
261C4×D73—22no—
262C4×D73—22no—
263C4×D73—22no—
264C4×D73—22no—
265C4×D73—44no—
266C4×D73—44no—
267Dic146—44no—
268C8×C73—22no—
269C8×C73—41no—
270C8×C73—41no—
271C8×C73—41no—
272#10596831—22no—
273#10596831—41no—
274#10596831—41no—
275#10596831—41no—
276C22×C73C2×C4×Q811yescharacteristic
277C22×C73#51811yescharacteristic
278C22×C73#52011yescharacteristic
279C22×C73C23×C811yescharacteristic
280C22×C73C2×#13111yescharacteristic
281C22×C73#56811yescharacteristic
282C22×C73#56811yescharacteristic
283C2×D73—22no—
284C2×D73—22no—
285C2×D73—22no—
286C2×D73—22no—
287C2×D73—22no—
288C2×D73—22no—
289C2×D73C8×Q814yes—
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—22no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73—41no—
296C4×C73#53811yescharacteristic
297C4×C73#53911yescharacteristic
298C4×C73#54311yescharacteristic
299C4×C73#54411yescharacteristic
300Dic73—22no—
301Dic73—22no—
302Dic73—22no—
303Dic73—22no—
304Dic73—41no—
305Dic73—41no—
306Dic73—41no—
307#22287—731nomaximal, Sylow
308C2×C73#232911yescharacteristic
309C2×C73C2×C8×Q811yescharacteristic
310C2×C73#296111yescharacteristic
311C2×C73#297211yescharacteristic
312C2×C73#297611yescharacteristic
313C2×C73#298811yescharacteristic
314C2×C73#299211yescharacteristic
315D73—24no—
316C4×C22⋊Q8—731no—
317#1984—731no—
318#1984—731no—
319#1939—731no—
320#1939—731no—
321#1916—731no—
322#1916—731no—
323#1916—731no—
324#1916D7311yescharacteristic, complemented
325#1850—731no—
326#1850—731no—
327#1797—731no—
328C8×C22⋊C4—731no—
329C8×C22⋊C4—731no—
330C2×#336—731no—
331C73#2228711yescharacteristic, complemented, Sylow
332C23×C8—731no—
333C23×C8—731no—
334C2×C22⋊Q8—731no—
335C2×C4×Q8—731no—
336C22×C42—731no—
337C2×C4⋊C8—731no—
338C2×C4⋊C8—731no—
339C2×C4⋊C8—731no—
340C2×C4⋊C8—731no—
341C2×C4⋊C8—731no—
342C2×C4⋊C8C2×D7311yescharacteristic
343C2×C22⋊C8—731no—
344C2×C22⋊C8—731no—
345C2×C4×C8—731no—
346C2×C4×C8—731no—
347C2×C4×C8—731no—
348C2×C4×C8C2×D7311yescharacteristic
349#391—731no—
350#389—731no—
351#387—731no—
352#387—731no—
353#385—731no—
354#385—731no—
355#380—731no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4—731no—
358C4×C4⋊C4C2×D7311yescharacteristic
359C4×C22⋊C4—731no—
360C4×C22⋊C4—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—
369#336—731no—
370#336—731no—
371#336C2×D7311yescharacteristic
372#336C2×D7311yescharacteristic
373#336C2×D7311yescharacteristic
374#336C2×D7311yescharacteristic
375C22×Q8—731no—
376C23×C4—731no—
377C23×C4—731no—
378C23×C4—731no—
379C22×C8—2921no—
380C22×C8—2921no—
381C22×C8—2921no—
382C22×C8—2921no—
383C22×C8—734no—
384C22×C8—731no—
385C22×C8—731no—
386C22×C8—731no—
387C22×C8—731no—
388C22×C8—1462no—
389C22×C8C22×D7311yescharacteristic
390C22×C8C22×D7311yescharacteristic
391C22×C8C22×D7311yescharacteristic
392C22×C8C22×D7311yescharacteristic
393C22⋊Q8—738no—
394C4×Q8—2921no—
395C4×Q8—2921no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4C4×D7311yescharacteristic
404C2×C4⋊C4C4×D7311yescharacteristic
405C2×C22⋊C4—731no—
406C2×C22⋊C4—731no—
407C2×C22⋊C4—731no—
408C2×C22⋊C4—731no—
409C2×C42—2921no—
410C2×C42—2921no—
411C2×C42—731no—
412C2×C42—731no—
413C2×C42—731no—
414C2×C42—731no—
415C2×C42—731no—
416C2×C42C22×D7311yescharacteristic
417C2×C42C22×D7311yescharacteristic
418C2×C42C22×D7311yescharacteristic
419C4⋊C8—41no—
420C4⋊C8—2921no—
421C4⋊C8—2921no—
422C4⋊C8—2921no—
423C4⋊C8—2921no—
424C4⋊C8—2921no—
425C22⋊C8—1462no—
426C22⋊C8—1462no—
427C4×C8—41no—
428C4×C8—2921no—
429C4×C8—2921no—
430C4×C8—2921no—
431#95—731no—
432#95—731no—
433#95—731no—
434#95—731no—
435#95—731no—
436#95—731no—
437#95—731no—
438#95—731no—
439#95C4×D7311yescharacteristic
440#95C4×D7311yescharacteristic
441C24—731no—
442C2×Q8—2921no—
443C2×Q8—2921no—
444C2×Q8—734no—
445C22×C4—2921no—
446C22×C4—2921no—
447C22×C4—2921no—
448C22×C4—2921no—
449C22×C4—734no—
450C22×C4—731no—
451C22×C4—731no—
452C22×C4—731no—
453C22×C4—731no—
454C22×C4—732no—
455C22×C4—732no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—731no—
459C22×C4—1462no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4C2×C4×D7311yescharacteristic
465C22×C4C2×C4×D7311yescharacteristic
466C22×C4C2×C4×D7311yescharacteristic
467C22×C4C2×C4×D7311yescharacteristic
468C22×C4C2×C4×D7311yescharacteristic
469C22×C4C2×C4×D7311yescharacteristic
470C22×C4C23×D7311yescharacteristic
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—2924no—
483C2×C8—2924no—
484C2×C8—732no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8Q8×D7312yes—
497C4⋊C4—41no—
498C4⋊C4—2921no—
499C4⋊C4—2921no—
500C4⋊C4—2921no—
501C4⋊C4—2921no—
502C4⋊C4—2921no—
503C4⋊C4—734no—
504C4⋊C4—734no—
505C4⋊C4C8×D7314yescomplemented
506C22⋊C4—734no—
507C22⋊C4—734no—
508C22⋊C4—1462no—
509C22⋊C4—1462no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—2921no—
517C42—2921no—
518C42—1461no—
519C42—1461no—
520C42—1461no—
521C42—1461no—
522C42—1461no—
523C42—1461no—
524C23—732no—
525C23—732no—
526C23—732no—
527C23—1461no—
528C23—1461no—
529C23—1461no—
530C23—1461no—
531C23C22×C4×D7311yescharacteristic
532Q8—2924no—
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—2924no—
549C2×C4—2924no—
550C2×C4—732no—
551C2×C4—732no—
552C2×C4—732no—
553C2×C4—732no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1462no—
572C2×C4—1462no—
573C2×C4—1462no—
574C2×C4—1462no—
575C2×C4C2×C8×D7312yes—
576C2×C4C2×C8×D7312yes—
577C2×C4C2×C8×D7312yes—
578C2×C4C2×Q8×D7311yescharacteristic
579C2×C4#41×D7311yescharacteristic
580C2×C4#1455663611yescharacteristic
581C2×C4#1455663711yescharacteristic
582C2×C4#1455663711yescharacteristic
583C2×C4#1455663811yescharacteristic
584C8—22no—
585C8—41no—
586C8—41no—
587C8—41no—
588C8—2921no—
589C8—2921no—
590C8—2921no—
591C8—1462no—
592C22—734no—
593C22—1462no—
594C22—1462no—
595C22—1462no—
596C22—1462no—
597C22—1462no—
598C22—1462no—
599C22C4×Q8×D7311yescharacteristic
600C22#1798905911yescharacteristic
601C22#1798906711yescharacteristic
602C22C22×C8×D7311yescharacteristic
603C22#131×D7311yescharacteristic
604C22#1798935211yescharacteristic
605C22#1798935311yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—2921no—
613C4—2921no—
614C4—2921no—
615C4—1462no—
616C4—1462no—
617C4—1462no—
618C4—1462no—
619C4#1798917611yescharacteristic
620C4#1798918011yescharacteristic
621C4#1798921211yescharacteristic
622C4#1798921711yescharacteristic
623C2—1464no—
624C2#2261265811yescharacteristic
625C2C8×Q8×D7311yescharacteristic
626C2#2261886711yescharacteristic
627C2#2261895211yescharacteristic
628C2#2261897811yescharacteristic
629C2#2261903911yescharacteristic
630C2#2261906511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutatorC22×C73C23×C8
FittingC73×#1916C2
socleC23×C73C23×C4
prime radicalC23×D73C22×C4
cosocleC22×C4×C73C24
hypercenter#1916D73
hypocenterC73#22287
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
2. #22610867C2semidirect
3. #22610865C2semidirect
4. #22610650C2semidirect
5. #22610648C2semidirect
6. #22610491C2semidirect
7. #22610486C2semidirect
8. #22610485C2semidirect
15. C73×#1916C2semidirect
72. C2×C8×D73Q8semidirect
73. #14556569C8semidirect
128. C2×Dic146C2×C8semidirect
167. #11924007C2×C8semidirect
168. #11924007C2×C8semidirect
201. C4⋊C4×C73C2×C8semidirect
324. #1916D73semidirect
331. C73#22287semidirect
505. C4⋊C4C8×D73semidirect

Maximal subgroups

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

subgroupindex
C4×#145565692
#226108672
#226108652
#226106502
#226106482
#226104912
#226104862
#226104852
#226101102
#226099082
C8×#119240052
#226097012
#336×D732
C73×#19162
#2228773