Finite Group Databasebeta

← #34028267

Subgroups — #34028267

Order 18688.

824 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
1C2×#17989638C211yesmaximal, characteristic, complemented
2C2×#17989604C211yesmaximal, characteristic
3#22626105C211yesmaximal, characteristic, complemented
4#22626033C211yesmaximal, characteristic, complemented
5#22626005C211yesmaximal, characteristic
6#22626005C211yesmaximal, characteristic
7#22626001C211yesmaximal, characteristic
8#22626001C211yesmaximal, characteristic
9#22625987C211yesmaximal, characteristic, complemented
10#22625984C211yesmaximal, characteristic, complemented
11#22625731C211yesmaximal, characteristic, complemented
12#22625731C211yesmaximal, characteristic, complemented
13C4⋊C4×#10596833C211yesmaximal, characteristic, complemented
14#22625657C211yesmaximal, characteristic
15C22⋊Q8×Dic73C211yesmaximal, characteristic
16#17989638—24no—
17#17989604—24no—
18C2×#14556658C2211yescharacteristic
19C2×#14556658C2211yescharacteristic
20C2×#14556658C2211yescharacteristic
21C2×#14556658C2211yescharacteristic
22C2×C4⋊#10596833C2211yescharacteristic
23C2×C4⋊#10596833C2211yescharacteristic
24C2×C22⋊#10596833C2211yescharacteristic
25C2×C22⋊#10596833C2211yescharacteristic
26C2×C22⋊#10596833C2211yescharacteristic
27C2×C22⋊#10596833C2211yescharacteristic
28C2×#14556646C2211yescharacteristic
29C2×#14556646C2211yescharacteristic
30C2×#14556646C2211yescharacteristic
31C2×#14556646C2211yescharacteristic
32C2×C4×#10596833C2211yescharacteristic
33C2×C4×#10596833C2211yescharacteristic
34C2×C4×#10596833C2211yescharacteristic
35C2×C4×#10596833C2211yescharacteristic
36#17989453C2211yescharacteristic
37#17989453C2211yescharacteristic
38#17989453C2211yescharacteristic
39#17989453C2211yescharacteristic
40#17989453C2211yescharacteristic
41#17989453C2211yescharacteristic
42#17989453C2211yescharacteristic
43#17989453C2211yescharacteristic
44#17989453C2211yescharacteristic
45#17989453C2211yescharacteristic
46C2×Q8×Dic73C2211yescharacteristic
47C22×C4×Dic73C2211yescharacteristic
48#17988494C411yescharacteristic
49#17988486C411yescharacteristic
50#17988473C411yescharacteristic
51#17988473C411yescharacteristic
52#17988462C411yescharacteristic
53#17988462C411yescharacteristic
54#17988447C411yescharacteristic
55C4⋊C4×Dic73C2211yescharacteristic
56C4⋊C4×Dic73C2211yescharacteristic
57C4⋊C4×Dic73C2211yescharacteristic
58C22⋊C4×Dic73C2211yescharacteristic
59C22⋊C4×Dic73C2211yescharacteristic
60C2×C22⋊Q8×C73C411yescharacteristic
61C22×#10596833C2311yescharacteristic
62C22×#10596833C2311yescharacteristic
63C22×#10596833C2311yescharacteristic
64C22×#10596833C2311yescharacteristic
65C22×#10596833C2311yescharacteristic
66C22×#10596833C2311yescharacteristic
67C22×#10596833C2311yescharacteristic
68C22×#10596833C2311yescharacteristic
69#14556658—22no—
70#14556658—22no—
71#14556658—22no—
72#14556658D414yes—
73C4⋊#10596833—22no—
74C4⋊#10596833—22no—
75C22⋊#10596833—22no—
76C22⋊#10596833—22no—
77C22⋊#10596833—22no—
78C22⋊#10596833—22no—
79#14556646—22no—
80#14556646—22no—
81#14556646—22no—
82#14556646—22no—
83C4×#10596833—22no—
84C4×#10596833—22no—
85C4×#10596833—22no—
86C4×#10596833—22no—
87C23×Dic73C2×C411yescharacteristic
88C23×Dic73C2311yescharacteristic
89Q8×Dic73—22no—
90Q8×Dic73—22no—
91C2×C4⋊Dic73C2×C411yescharacteristic
92C2×C4⋊Dic73C2×C411yescharacteristic
93C2×C4⋊Dic73C2×C411yescharacteristic
94C2×C4⋊Dic73C2×C411yescharacteristic
95C2×C4⋊Dic73C2×C411yescharacteristic
96C2×C4⋊Dic73C2×C411yescharacteristic
97C2×C22⋊Dic73C2×C411yescharacteristic
98C2×C22⋊Dic73C2×C411yescharacteristic
99C2×C4×Dic73—22no—
100C2×C4×Dic73—22no—
101C2×C4×Dic73C2×C411yescharacteristic
102C2×C4×Dic73C2×C411yescharacteristic
103C2×C4×Dic73C2311yescharacteristic
104C2×C4×Dic73C2311yescharacteristic
105C2×C4×Dic73C2311yescharacteristic
106C2×C4×Dic73C2311yescharacteristic
107C2×C4×Dic73C2311yescharacteristic
108C2×C4×Dic73C2311yescharacteristic
109#14556498C2×C411yescharacteristic
110#14556498C2×C411yescharacteristic
111#14556498C2×C411yescharacteristic
112#14556498C2×C411yescharacteristic
113#14556498C2×C411yescharacteristic
114#14556498C2×C411yescharacteristic
115#14556498C2×C411yescharacteristic
116#14556498C2×C411yescharacteristic
117#14556498C2×C411yescharacteristic
118#14556498C2×C411yescharacteristic
119C22×Q8×C73C2×C411yescharacteristic
120C23×C4×C73C2×C411yescharacteristic
121C22⋊Q8×C73—24no—
122C2×C4⋊C4×C73C2×C411yescharacteristic
123C2×C4⋊C4×C73C2×C411yescharacteristic
124C2×C4⋊C4×C73C2×C411yescharacteristic
125C2×C22⋊C4×C73C2×C411yescharacteristic
126C2×C22⋊C4×C73C2×C411yescharacteristic
127C2×#10596833—21no—
128C2×#10596833—21no—
129C2×#10596833—21no—
130C2×#10596833—21no—
131C2×#10596833—21no—
132C2×#10596833—21no—
133C2×#10596833—21no—
134C2×#10596833—21no—
135C2×#10596833—21no—
136C2×#10596833—21no—
137C2×#10596833—21no—
138C2×#10596833—21no—
139C2×#10596833—21no—
140C2×#10596833—21no—
141C2×#10596833—21no—
142C2×#10596833—21no—
143C2×#10596833—21no—
144C2×#10596833—21no—
145C2×#10596833—21no—
146C2×#10596833—21no—
147C2×#10596833C2×D412yes—
148C2×#10596833C2×D412yes—
149C2×#10596833#4112yes—
150C2×#10596833#4112yes—
151C22×Dic73—22no—
152C22×Dic73—22no—
153C22×Dic73—22no—
154C22×Dic73—21no—
155C22×Dic73—21no—
156C22×Dic73—21no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C22×Dic73—21no—
160C22×Dic73C22×C411yescharacteristic
161C22×Dic73C22×C411yescharacteristic
162C22×Dic73C22×C411yescharacteristic
163C22×Dic73C22×C411yescharacteristic
164C22×Dic73C22×C411yescharacteristic
165C22×Dic73C22×C411yescharacteristic
166C22×Dic73C22×C411yescharacteristic
167C22×Dic73C2411yescharacteristic
168C4⋊Dic73—22no—
169C4⋊Dic73—22no—
170C4⋊Dic73—22no—
171C4⋊Dic73—22no—
172C4⋊Dic73—22no—
173C4⋊Dic73—41no—
174C22⋊Dic73—22no—
175C22⋊Dic73—22no—
176C4×Dic73—22no—
177C4×Dic73—22no—
178C4×Dic73—21no—
179C4×Dic73—21no—
180C4×Dic73—21no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73C2×D412yes—
189C4×Dic73#4112yes—
190C24×C73C22×C411yescharacteristic
191C2×Q8×C73—22no—
192C2×Q8×C73—22no—
193C2×Q8×C73—41no—
194C22×C4×C73—22no—
195C22×C4×C73—22no—
196C22×C4×C73—41no—
197C22×C4×C73C22×C411yescharacteristic
198C22×C4×C73C22×C411yescharacteristic
199C22×C4×C73C22×C411yescharacteristic
200C22×C4×C73C22×C411yescharacteristic
201C22×C4×C73C22×C411yescharacteristic
202C22×C4×C73C22×C411yescharacteristic
203C4⋊C4×C73—22no—
204C4⋊C4×C73—22no—
205C4⋊C4×C73M1614yes—
206C22⋊C4×C73—22no—
207C22⋊C4×C73—22no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73#13112yes—
215C23×C73C23×C411yescharacteristic
216Q8×C73—44no—
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—44no—
232C2×C4×C73C4×D412yes—
233C2×C4×C73C2×M1612yes—
234C2×C4×C73C2×M1612yes—
235C2×C4×C73C2×M1612yes—
236C2×Dic73—28no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
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—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73C4×D412yes—
261C2×Dic73#13112yes—
262C2×Dic73C22×D411yescharacteristic
263C2×Dic73C2×#4111yescharacteristic
264C2×Dic73C2×#4111yescharacteristic
265C2×Dic73C2×#4111yescharacteristic
266C2×Dic732−1+411yescharacteristic
267C2×Dic732−1+411yescharacteristic
268#10596833—22no—
269#10596833—22no—
270#10596833—22no—
271#10596833—22no—
272#10596833—41no—
273#10596833—41no—
274#10596833—41no—
275#10596833—41no—
276C22×C73—22no—
277C22×C73—22no—
278C22×C73—22no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73—41no—
283C22×C73C2×C4×D411yescharacteristic
284C22×C73C4×#4111yescharacteristic
285C22×C73#51911yescharacteristic
286C22×C73C22×M1611yescharacteristic
287C22×C73C2×#13111yescharacteristic
288C22×C73C2×#13111yescharacteristic
289C22×C73#56811yescharacteristic
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—22no—
293C4×C73—22no—
294C4×C73—41no—
295C4×C73—41no—
296Dic73—22no—
297Dic73—22no—
298Dic73—41no—
299Dic73—41no—
300Dic73—41no—
301Dic73—41no—
302Dic73—41no—
303Dic73#54711yescharacteristic
304Dic73#54811yescharacteristic
305Dic73#55211yescharacteristic
306Dic73#55611yescharacteristic
307#23052—731nomaximal, Sylow
308C2×C73—42no—
309C2×C73#236211yescharacteristic
310C2×C73#292911yescharacteristic
311C2×C73D4×M1611yescharacteristic
312C2×C73#296511yescharacteristic
313C2×C73#296811yescharacteristic
314C2×C73#298411yescharacteristic
315C2×C73#298511yescharacteristic
316C2×#446—731no—
317C2×#431—731no—
318C4×C22⋊Q8—731no—
319#1985—731no—
320#1940—731no—
321#1924—731no—
322#1924—731no—
323#1923—731no—
324#1923—731no—
325#1916—731no—
326#1916—731no—
327#1771—731no—
328#1771—731no—
329C8×C4⋊C4—731no—
330#1749—731no—
331C73#2305211yescharacteristic, complemented, Sylow
332C2×C22⋊Q8#6440811yescharacteristic
333C2×C4×Q8—731no—
334C22×C42—731no—
335#446—1464no—
336#431—1464no—
337C2×C4⋊C8—731no—
338C2×C4⋊C8—731no—
339C2×C4⋊C8—731no—
340C2×C4⋊C8—731no—
341C2×C4⋊C8—731no—
342C2×C4⋊C8—731no—
343C2×C22⋊C8—731no—
344C2×C22⋊C8—731no—
345C2×C22⋊C8—731no—
346C2×C22⋊C8—731no—
347C2×#97—731no—
348C2×#97—731no—
349C2×#97—731no—
350C2×#97—731no—
351C2×C4×C8—731no—
352C2×C4×C8—731no—
353C2×C4×C8—731no—
354C2×C4×C8—731no—
355#391—731no—
356#389—731no—
357#387—731no—
358#387—731no—
359#385—731no—
360#385—731no—
361#380—731no—
362C4×C4⋊C4—731no—
363C4×C4⋊C4—731no—
364C4×C4⋊C4—731no—
365C4×C22⋊C4—731no—
366C4×C22⋊C4—731no—
367#336—731no—
368#336—731no—
369#336—731no—
370#336—731no—
371#336—731no—
372#336—731no—
373#336—731no—
374#336—731no—
375#336—731no—
376#336—731no—
377C22×Q8C2×#6440811yescharacteristic
378C23×C4—731no—
379C23×C4—731no—
380C23×C4C2×#6440811yescharacteristic
381C22×C8—731no—
382C22×C8—731no—
383C22×C8—731no—
384C22×C8—731no—
385C22×C8—731no—
386C22×C8—731no—
387C22×C8—731no—
388C22×C8—731no—
389C22⋊Q8—24no—
390C4×Q8—1462no—
391C4×Q8—1462no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4C2×#6440811yescharacteristic
399C2×C4⋊C4C2×#6440811yescharacteristic
400C2×C4⋊C4C2×#6440811yescharacteristic
401C2×C22⋊C4—731no—
402C2×C22⋊C4—731no—
403C2×C22⋊C4C2×#6440811yescharacteristic
404C2×C22⋊C4C2×#6440811yescharacteristic
405C2×C42—731no—
406C2×C42—731no—
407C2×C42—731no—
408C2×C42—731no—
409C2×C42—731no—
410C2×C42—731no—
411C2×C42—731no—
412C2×C42—731no—
413C2×C42—1462no—
414C2×C42—1462no—
415C4⋊C8—734no—
416C4⋊C8—1462no—
417C4⋊C8—1462no—
418C4⋊C8—1462no—
419C4⋊C8—1462no—
420C4⋊C8—1462no—
421C22⋊C8—1462no—
422C22⋊C8—1462no—
423C22⋊C8—1462no—
424C22⋊C8—1462no—
425#97—1462no—
426#97—1462no—
427#97—1462no—
428#97—1462no—
429C4×C8—1462no—
430C4×C8—1462no—
431C4×C8—1462no—
432C4×C8—1462no—
433#95—731no—
434#95—731no—
435#95—731no—
436#95—731no—
437#95—731no—
438#95—731no—
439#95—731no—
440#95—731no—
441#95—731no—
442#95—731no—
443C24C22×#6440811yescharacteristic
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—41no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—41no—
450C22×C4—731no—
451C22×C4—731no—
452C22×C4—731no—
453C22×C4—731no—
454C22×C4—731no—
455C22×C4—731no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4C22×#6440811yescharacteristic
468C22×C4C22×#6440811yescharacteristic
469C22×C4C22×#6440811yescharacteristic
470C22×C4C22×#6440811yescharacteristic
471C22×C4C22×#6440811yescharacteristic
472C22×C4C22×#6440811yescharacteristic
473C2×C8—732no—
474C2×C8—732no—
475C2×C8—732no—
476C2×C8—732no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
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×C8—1461no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—2921no—
500C4⋊C4—1462no—
501C4⋊C4—1462no—
502C4⋊C4—1462no—
503C4⋊C4—1462no—
504C4⋊C4—1462no—
505C4⋊C4#1192403714yes—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—1462no—
509C22⋊C4—1462no—
510C42—732no—
511C42—732no—
512C42—1461no—
513C42—1461no—
514C42—1461no—
515C42—1461no—
516C42—1461no—
517C42—1461no—
518C42—1461no—
519C42—1461no—
520C42—1462no—
521C42—1461no—
522C42—1461no—
523C42—1462no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1455667912yes—
531C23C23×#6440811yescharacteristic
532Q8—44no—
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—44no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—731no—
553C2×C4—731no—
554C2×C4—731no—
555C2×C4—731no—
556C2×C4—731no—
557C2×C4—731no—
558C2×C4—732no—
559C2×C4—732no—
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—1461no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1468no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4—1461no—
579C2×C4—1461no—
580C2×C4D4×#6440812yes—
581C2×C4C2×#1192403712yes—
582C2×C4C2×#1192403712yes—
583C2×C4C2×#1192403712yes—
584C8—2921no—
585C8—2921no—
586C8—2921no—
587C8—2921no—
588C8—1462no—
589C8—1462no—
590C8—1462no—
591C8—1462no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22C2×D4×#6440811yescharacteristic
600C22#41×#6440811yescharacteristic
601C22#1798965511yescharacteristic
602C22C22×#1192403711yescharacteristic
603C22D73⋊#13111yescharacteristic
604C22C2×#1455667911yescharacteristic
605C22#1798966511yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—41no—
611C4—41no—
612C4—2921no—
613C4—2921no—
614C4—2921no—
615C4—2921no—
616C4—2921no—
617C4—731no—
618C4—731no—
619C4—731no—
620C4—731no—
621C4—1462no—
622C4—1462no—
623C2—42no—
624C2#2262645711yescharacteristic
625C2#2262664711yescharacteristic
626C2D4×#1192403711yescharacteristic
627C2#2262674011yescharacteristic
628C2#2262675911yescharacteristic
629C2#2262683411yescharacteristic
630C2#2262683711yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
FittingC2×C22⋊Q8×C73C4
prime radicalC24×C73C22×C4
cosocleC22×Dic73C24
hypercenterC2×C22⋊Q8#64408
hypocenterC73#23052
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#17989638C2semidirect
3. #22626105C2semidirect
4. #22626033C2semidirect
9. #22625987C2semidirect
10. #22625984C2semidirect
11. #22625731C2semidirect
12. #22625731C2semidirect
13. C4⋊C4×#10596833C2semidirect
331. C73#23052semidirect

Maximal subgroups

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

subgroupindex
C2×#179896382
C2×#179896042
#226261052
#226260332
#226260052
#226260012
#226259872
#226259842
#226257312
C4⋊C4×#105968332
#226256572
C22⋊Q8×Dic732
#2305273