Finite Group Databasebeta

← #33127321

Subgroups — #33127321

Order 18688.

802 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988867C211yesmaximal, characteristic, complemented
2C2×#17988648C211yesmaximal, characteristic, complemented
3#22615752C211yesmaximal, characteristic, complemented
4#22611362C211yesmaximal, characteristic, complemented
5#22611357C211yesmaximal, characteristic, complemented
6#22611117C211yesmaximal, characteristic
7#22611110C211yesmaximal, characteristic
8#22610879C211yesmaximal, characteristic, complemented
9#22610699C211yesmaximal, characteristic, complemented
10#22610671C211yesmaximal, characteristic, complemented
11#22610661C211yesmaximal, characteristic, complemented
12#22610631C211yesmaximal, characteristic, complemented
13#22610617C211yesmaximal, characteristic, complemented
14#22609941C211yesmaximal, characteristic, complemented
15C73×#2324C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17#17988867—24no—
18#17988648—24no—
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556512C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×C22⋊#10596831C2211yescharacteristic
31C2×#14556502C2211yescharacteristic
32C2×C4×#10596831C2211yescharacteristic
33#17988526—21no—
34#17988523—21no—
35#17988507—21no—
36#17988499—21no—
37#17988485C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988453C2211yescharacteristic
40#17988447C2211yescharacteristic
41#17988447C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50C2×#121×C73C2211yescharacteristic
51C2×C4×D4×C73C2211yescharacteristic
52C2×#117×C73C2211yescharacteristic
53#390×C73—21no—
54#390×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#388×C73—21no—
57#386×C73—21no—
58#385×C73—21no—
59#380×C73C2211yescharacteristic
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×Dic146—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556591—22no—
67#14556591—22no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73—21no—
77#14556524—22no—
78#14556524—22no—
79#14556524—22no—
80#14556522—22no—
81C4⋊#10596831—22no—
82C4⋊#10596831D414yescomplemented
83#14556512—22no—
84#14556512—22no—
85#14556512—22no—
86#14556512—22no—
87C22⋊#10596831—22no—
88C22⋊#10596831—22no—
89#14556502—22no—
90C4×#10596831—22no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C22×D4×C73—21no—
98C22×D4×C73C2311yescharacteristic
99C23×C4×C73—21no—
100C23×C4×C73C2311yescharacteristic
101#121×C73—22no—
102#121×C73—22no—
103C4×D4×C73—22no—
104C4×D4×C73—22no—
105#117×C73—22no—
106#117×C73—22no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73D411yescharacteristic
109C2×C4⋊C4×C73D411yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73D411yescharacteristic
119C2×C22⋊C4×C73D411yescharacteristic
120C2×C42×C73—21no—
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123#95×C73—21no—
124#95×C73—21no—
125#95×C73—21no—
126C2×Dic146—42no—
127C2×Dic146—81no—
128C2×Dic146—81no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—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×#10596831C2×D412yes—
144C2×#10596831C2×D412yes—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—42no—
154#11924007—81no—
155#11924007—81no—
156C22⋊Dic73—81no—
157C22⋊Dic73—81no—
158C24×C73—21no—
159C24×C73C2×D411yescharacteristic
160C2×D4×C73—24no—
161C2×D4×C73—21no—
162C2×D4×C73—21no—
163C2×D4×C73—21no—
164C2×D4×C73—21no—
165C2×D4×C73—21no—
166C2×D4×C73—81no—
167C2×D4×C73—81no—
168C2×D4×C73#4112yes—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—42no—
179C22×C4×C73—81no—
180C22×C4×C73—81no—
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2411yescharacteristic
187C4⋊C4×C73—22no—
188C4⋊C4×C73—22no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—81no—
195C4⋊C4×C73#4112yes—
196C22⋊C4×C73—24no—
197C22⋊C4×C73—22no—
198C22⋊C4×C73—22no—
199C22⋊C4×C73—42no—
200C22⋊C4×C73—81no—
201C22⋊C4×C73—81no—
202C22⋊C4×C73—81no—
203C42×C73—21no—
204C42×C73—21no—
205C42×C73—81no—
206C42×C73C2×D412yes—
207C42×C73#4112yes—
208C23×C73—21no—
209C23×C73—22no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73C22×D411yescharacteristic
223D4×C73—22no—
224D4×C73—22no—
225D4×C73—41no—
226D4×C73—41no—
227D4×C73—84no—
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—22no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—84no—
255C2×C4×C73#12112yes—
256C2×C4×C73#12112yes—
257C2×C4×C73C22×D411yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C2×C4×C73C2×#4111yescharacteristic
261C2×C4×C732−1+411yescharacteristic
262C2×C4×C732−1+411yescharacteristic
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275Dic146—84no—
276#10596831—22no—
277#10596831—22no—
278#10596831—41no—
279#10596831—41no—
280C22×C73—41no—
281C22×C73—42no—
282C22×C73—41no—
283C22×C73—42no—
284C22×C73—41no—
285C22×C73—42no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73—41no—
290C22×C73—81no—
291C22×C73—81no—
292C22×C73—81no—
293C22×C73—81no—
294C22×C73C2×#12111yescharacteristic
295C22×C73#52511yescharacteristic
296C22×C73#53611yescharacteristic
297C22×C73#57611yescharacteristic
298C22×C73#57611yescharacteristic
299C22×C73#57711yescharacteristic
300C22×C73#57811yescharacteristic
301C4×C73—22no—
302C4×C73—22no—
303C4×C73—42no—
304C4×C73—41no—
305C4×C73—41no—
306C4×C73—41no—
307C4×C73—81no—
308C4×C73—81no—
309C4×C73#54711yescharacteristic
310C4×C73#54811yescharacteristic
311C4×C73#55211yescharacteristic
312C4×C73#55611yescharacteristic
313Dic73—81no—
314Dic73—81no—
315Dic73—81no—
316Dic73—81no—
317#24712—731nomaximal, Sylow
318C2×C73—81no—
319C2×C73—82no—
320C2×C73#241511yescharacteristic
321C2×C73#305811yescharacteristic
322C2×C73#306311yescharacteristic
323C2×C73#311811yescharacteristic
324C2×C73#312411yescharacteristic
325C2×C73#324111yescharacteristic
326C2×C73#324311yescharacteristic
327C2×#474—731no—
328C2×#433—731no—
329#2628—731no—
330#2324D7311yescharacteristic
331#2076—731no—
332#2075—731no—
333#2034—731no—
334#2033—731no—
335#1987—731no—
336#1949—731no—
337#1944—731no—
338#1942—731no—
339#1936—731no—
340#1934—731no—
341#1804—731no—
342C73#2471211yescharacteristic, complemented, Sylow
343C2×C4⋊Q8—731no—
344C2×#121C2×D7311yescharacteristic
345C2×C4×D4C2×D7311yescharacteristic
346C2×#117C2×D7311yescharacteristic
347#474—1464no—
348#433—1464no—
349C2×#107—731no—
350C2×#107—731no—
351C2×#107—731no—
352C2×#106—731no—
353C2×C4⋊C8—731no—
354C2×C4⋊C8—731no—
355C2×#102—731no—
356C2×#102—731no—
357C2×#102—731no—
358C2×#102—731no—
359C2×C22⋊C8—731no—
360C2×C22⋊C8—731no—
361C2×#97—731no—
362C2×C4×C8—731no—
363#400—1461no—
364#399—1461no—
365#395—1461no—
366#393—1461no—
367#390—21no—
368#390C2×D7311yescharacteristic
369#389—731no—
370#389—731no—
371#389C2×D7311yescharacteristic
372#388—21no—
373#386—21no—
374#385—21no—
375#382—731no—
376#380—731no—
377#380—731no—
378#380C2×D7311yescharacteristic
379C4×C4⋊C4C2×D7311yescharacteristic
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340—731no—
385#340—731no—
386#340—731no—
387#340—731no—
388C22×Q8—1461no—
389C22×D4—21no—
390C22×D4C22×D7311yescharacteristic
391C23×C4—21no—
392C23×C4C22×D7311yescharacteristic
393C22×C8—731no—
394C22×C8—731no—
395C22×C8—731no—
396C22×C8—731no—
397C4⋊Q8—1462no—
398C4⋊Q8—1462no—
399#121—22no—
400#121—22no—
401C4×D4—22no—
402C4×D4—22no—
403#117—22no—
404#117—22no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—1461no—
411C2×C4⋊C4—1461no—
412C2×C4⋊C4—1461no—
413C2×C4⋊C4C22⋊D7311yescharacteristic
414C2×C4⋊C4C22⋊D7311yescharacteristic
415C2×C4⋊C4C22×D7311yescharacteristic
416C2×C4⋊C4C22×D7311yescharacteristic
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1461no—
424C2×C22⋊C4—1461no—
425C2×C22⋊C4C22⋊D7311yescharacteristic
426C2×C22⋊C4C22⋊D7311yescharacteristic
427C2×C42—21no—
428C2×C42C22×D7311yescharacteristic
429C2×C42C22×D7311yescharacteristic
430#107—1462no—
431#107—1462no—
432#107—1462no—
433#106—1462no—
434C4⋊C8—734no—
435C4⋊C8—1462no—
436#102—1462no—
437#102—1462no—
438#102—1462no—
439#102—1462no—
440C22⋊C8—1462no—
441C22⋊C8—1462no—
442#97—1462no—
443C4×C8—1462no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1461no—
448#95—1461no—
449#95—1461no—
450#95—1461no—
451#95—1461no—
452#95—1461no—
453C24—21no—
454C24C2×C22⋊D7311yescharacteristic
455C2×Q8—2922no—
456C2×Q8—5841no—
457C2×Q8—5841no—
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—81no—
465C2×D4—81no—
466C2×D4#1192402812yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4C2×C22⋊D7311yescharacteristic
484C22×C4C2×C22⋊D7311yescharacteristic
485C22×C4C2×C22⋊D7311yescharacteristic
486C22×C4C2×C22⋊D7311yescharacteristic
487C22×C4C2×C22⋊D7311yescharacteristic
488C22×C4C23×D7311yescharacteristic
489C2×C8—732no—
490C2×C8—732no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1461no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8—1461no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—81no—
509C4⋊C4—2922no—
510C4⋊C4—5841no—
511C4⋊C4—5841no—
512C4⋊C4—1461no—
513C4⋊C4—1461no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4—1461no—
520C4⋊C4#1192402812yes—
521C22⋊C4—24no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—81no—
528C22⋊C4—5841no—
529C22⋊C4—5841no—
530C42—21no—
531C42—21no—
532C42—81no—
533C42D4×D7312yes—
534C42#1192402712yes—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C22×C22⋊D7311yescharacteristic
550Q8—5844no—
551D4—22no—
552D4—22no—
553D4—41no—
554D4—41no—
555D4—84no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4#1455656312yes—
596C2×C4#1455656312yes—
597C2×C4C2×D4×D7311yescharacteristic
598C2×C4C2×#1192402711yescharacteristic
599C2×C4C2×#1192402811yescharacteristic
600C2×C4C2×#1192402811yescharacteristic
601C2×C4#1455663811yescharacteristic
602C2×C4#1455663811yescharacteristic
603C8—2921no—
604C8—2921no—
605C8—1462no—
606C8—1462no—
607C22—41no—
608C22—41no—
609C22—42no—
610C22—41no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C2×#1455656311yescharacteristic
622C22#1798909811yescharacteristic
623C22#1798915811yescharacteristic
624C22#1798939011yescharacteristic
625C22#1798939011yescharacteristic
626C22#1798939811yescharacteristic
627C22#1798940211yescharacteristic
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—5841no—
637C4—5841no—
638C4—5841no—
639C4—5841no—
640C4#1798924211yescharacteristic
641C4#1798924611yescharacteristic
642C4#1798927711yescharacteristic
643C4#1798930711yescharacteristic
644C2—81no—
645C2—82no—
646C2#2261336011yescharacteristic
647C2#2261957611yescharacteristic
648C2#2261962411yescharacteristic
649C2#2262011511yescharacteristic
650C2#2262019311yescharacteristic
651C2#2262117811yescharacteristic
652C2#2262120811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2324C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenter#2324D73
hypocenterC73#24712
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
1. C2×#17988867C2semidirect
2. C2×#17988648C2semidirect
3. #22615752C2semidirect
4. #22611362C2semidirect
5. #22611357C2semidirect
8. #22610879C2semidirect
9. #22610699C2semidirect
10. #22610671C2semidirect
11. #22610661C2semidirect
12. #22610631C2semidirect
13. #22610617C2semidirect
14. #22609941C2semidirect
82. C4⋊#10596831D4semidirect
342. C73#24712semidirect

Maximal subgroups

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

subgroupindex
C2×#179888672
C2×#179886482
#226157522
#226113622
#226113572
#226111172
#226111102
#226108792
#226106992
#226106712
#226106612
#226106312
#226106172
#226099412
C73×#23242
#2471273