Finite Group Databasebeta

← #33130348

Subgroups — #33130348

Order 18688.

834 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
1#22614248C211yesmaximal, characteristic
2#22611488C211yesmaximal, characteristic, complemented
3#22611408C211yesmaximal, characteristic, complemented
4#22611367C211yesmaximal, characteristic, complemented
5#22611367C211yesmaximal, characteristic, complemented
6#22610516C211yesmaximal, characteristic, complemented
7#22610516C211yesmaximal, characteristic, complemented
8#22610224C211yesmaximal, characteristic
9#22610124C211yesmaximal, characteristic, complemented
10#22610124C211yesmaximal, characteristic, complemented
11#22610019C211yesmaximal, characteristic, complemented
12#22610017C211yesmaximal, characteristic, complemented
13#22609954C211yesmaximal, characteristic, complemented
14C2×#17988349C211yesmaximal, characteristic, complemented
15D4×C4⋊C4×C73C211yesmaximal, characteristic
16C23×#10596831C2211yescharacteristic
17C22×C4⋊Dic73C2211yescharacteristic
18C2×#14556522C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×C22⋊#10596831C2211yescharacteristic
23C2×C22⋊#10596831C2211yescharacteristic
24C2×C22⋊#10596831C2211yescharacteristic
25#17988526—21no—
26#17988523—21no—
27#17988485C2211yescharacteristic
28#17988485C2211yescharacteristic
29#17988467—21no—
30#17988459—21no—
31#17988447C2211yescharacteristic
32#17988447C2211yescharacteristic
33#17988447C2211yescharacteristic
34#17988349C2211yescharacteristic
35#17988349C2211yescharacteristic
36#17988349C2211yescharacteristic
37#17988349C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C4×D4×C73—21no—
49C2×C4×D4×C73C2211yescharacteristic
50C22×C4⋊C4×C73C2211yescharacteristic
51C22×C4⋊C4×C73C2211yescharacteristic
52#389×C73—21no—
53#385×C73—21no—
54#385×C73—21no—
55#384×C73C2211yescharacteristic
56#380×C73C2211yescharacteristic
57#380×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59C23×Dic73—21no—
60C22×#10596831—22no—
61C22×#10596831—22no—
62C22×#10596831—41no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—22no—
69C2×C4⋊Dic73—22no—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79#14556522—22no—
80#14556522—22no—
81#14556522—41no—
82#14556522—41no—
83C22⋊#10596831—22no—
84C22⋊#10596831—22no—
85C22⋊#10596831—41no—
86#14556498—21no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91C22×D4×C73D411yescharacteristic
92C23×C4×C73—21no—
93C23×C4×C73—21no—
94C23×C4×C73C2311yescharacteristic
95C23×C4×C73C2311yescharacteristic
96C4×D4×C73—22no—
97C4×D4×C73—22no—
98C4×D4×C73—44no—
99C2×C4⋊C4×C73—22no—
100C2×C4⋊C4×C73—22no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—41no—
106C2×C4⋊C4×C73—41no—
107C2×C4⋊C4×C73D411yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73D411yescharacteristic
115C2×C22⋊C4×C73D411yescharacteristic
116C2×C42×C73—21no—
117C2×C42×C73C2311yescharacteristic
118#95×C73—21no—
119#95×C73—21no—
120#95×C73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—42no—
126C22×Dic73—81no—
127C22×Dic73—81no—
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—44no—
140C2×#10596831#4112yes—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—42no—
150#11924007—81no—
151#11924007—81no—
152C22⋊Dic73—42no—
153C22⋊Dic73—81no—
154C22⋊Dic73—81no—
155C24×C73C2×D411yescharacteristic
156C24×C73C2×D411yescharacteristic
157C2×D4×C73—22no—
158C2×D4×C73—22no—
159C2×D4×C73SD1614yes—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—42no—
175C22×C4×C73—42no—
176C22×C4×C73—42no—
177C22×C4×C73—42no—
178C22×C4×C73—81no—
179C22×C4×C73—81no—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73#4112yes—
185C22×C4×C73#4112yes—
186C22×C4×C73C2411yescharacteristic
187C4⋊C4×C73—22no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—42no—
196C4⋊C4×C73—42no—
197C4⋊C4×C73—42no—
198C4⋊C4×C73—81no—
199C4⋊C4×C73#4112yes—
200C22⋊C4×C73—22no—
201C22⋊C4×C73—22no—
202C22⋊C4×C73—42no—
203C22⋊C4×C73—42no—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—42no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73#12312yes—
218C23×C73C2×SD1612yes—
219C23×C73C2×SD1612yes—
220C23×C73#13712yes—
221C23×C73C22×D411yescharacteristic
222D4×C73—28no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—22no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
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—84no—
254C2×C4×C73—84no—
255C2×C4×C73#12312yes—
256C2×C4×C73C2×SD1612yes—
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C2×C4×C73C2×#4111yescharacteristic
261C2×C4×C73H(5,𝔽2)11yescharacteristic
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—
275C2×Dic73—84no—
276#10596831—22no—
277#10596831—41no—
278#10596831—41no—
279#10596831—41no—
280C22×C73—22no—
281C22×C73—22no—
282C22×C73—22no—
283C22×C73—22no—
284C22×C73—22no—
285C22×C73—22no—
286C22×C73—22no—
287C22×C73—22no—
288C22×C73—22no—
289C22×C73—41no—
290C22×C73—41no—
291C22×C73—41no—
292C22×C73—41no—
293C22×C73#48314yes—
294C22×C73C2×#12311yescharacteristic
295C22×C73#54611yescharacteristic
296C22×C73#54711yescharacteristic
297C22×C73C22×SD1611yescharacteristic
298C22×C73C2×#13711yescharacteristic
299C22×C73#57611yescharacteristic
300C22×C73#57711yescharacteristic
301C4×C73—22no—
302C4×C73—22no—
303C4×C73—41no—
304C4×C73—41no—
305C4×C73—42no—
306C4×C73—81no—
307C4×C73—81no—
308C4×C73—81no—
309C4×C73#52911yescharacteristic
310C4×C73#53911yescharacteristic
311C4×C73#55111yescharacteristic
312C4×C73#55211yescharacteristic
313Dic73—81no—
314Dic73—81no—
315Dic73—81no—
316Dic73—81no—
317#24935—731nomaximal, Sylow
318C2×C73—24no—
319C2×C73—42no—
320C2×C73#241911yescharacteristic
321C2×C73C2×#48311yescharacteristic
322C2×C73#309611yescharacteristic
323C2×C73#329511yescharacteristic
324C2×C73#332011yescharacteristic
325C2×C73#333511yescharacteristic
326C2×C73#334611yescharacteristic
327#2487—731no—
328D4×C4⋊C4D7311yescharacteristic
329#2099—731no—
330#2084—731no—
331#2077—731no—
332#2077—731no—
333#1920—731no—
334#1920—731no—
335#1869—731no—
336#1853—731no—
337#1853—731no—
338#1824—731no—
339#1823—731no—
340#1807—731no—
341C2×#340—731no—
342C73#2493511yescharacteristic, complemented, Sylow
343C23×C8—731no—
344C2×C4×D4—21no—
345C2×C4×D4C2×D7311yescharacteristic
346C22×C4⋊C4—731no—
347C22×C4⋊C4C2×D7311yescharacteristic
348C22×C4⋊C4C2×D7311yescharacteristic
349C2×#106—731no—
350C2×#106—731no—
351C2×#106—731no—
352C2×#106—731no—
353C2×C22⋊C8—731no—
354C2×C22⋊C8—731no—
355C2×C22⋊C8—731no—
356#400—1461no—
357#399—1461no—
358#389—21no—
359#389—731no—
360#389—731no—
361#386—1461no—
362#385—21no—
363#385—21no—
364#385—1461no—
365#384C2×D7311yescharacteristic
366#380—731no—
367#380—731no—
368#380—731no—
369#380C2×D7311yescharacteristic
370#380C2×D7311yescharacteristic
371C4×C4⋊C4C2×D7311yescharacteristic
372#340—731no—
373#340—731no—
374#340—731no—
375#340—731no—
376#340—731no—
377#340—731no—
378#340—731no—
379#340—731no—
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#336—731no—
385#336—731no—
386C22×D4C22⋊D7311yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4—1461no—
390C23×C4C22×D7311yescharacteristic
391C23×C4C22×D7311yescharacteristic
392C22×C8—2921no—
393C22×C8—731no—
394C22×C8—731no—
395C22×C8—731no—
396C22×C8—731no—
397C22×C8—1462no—
398C22×C8—1462no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—44no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—41no—
409C2×C4⋊C4—41no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4—1462no—
416C2×C4⋊C4—1462no—
417C2×C4⋊C4—1461no—
418C2×C4⋊C4—1461no—
419C2×C4⋊C4C22⋊D7311yescharacteristic
420C2×C4⋊C4C22×D7311yescharacteristic
421C2×C4⋊C4C22×D7311yescharacteristic
422C2×C4⋊C4C22×D7311yescharacteristic
423C2×C4⋊C4C22×D7311yescharacteristic
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—1461no—
427C2×C22⋊C4—1461no—
428C2×C22⋊C4—1461no—
429C2×C22⋊C4C22⋊D7311yescharacteristic
430C2×C22⋊C4C22⋊D7311yescharacteristic
431C2×C42—21no—
432C2×C42C22×D7311yescharacteristic
433#106—2921no—
434#106—2921no—
435#106—1462no—
436#106—1462no—
437C22⋊C8—2921no—
438C22⋊C8—1462no—
439C22⋊C8—1462no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—1461no—
444#95—1461no—
445#95—1461no—
446#95—1461no—
447#95—1461no—
448C24C2×C22⋊D7311yescharacteristic
449C24C2×C22⋊D7311yescharacteristic
450C2×D4—22no—
451C2×D4—22no—
452C2×D4#1192401614yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—2922no—
474C22×C4—5841no—
475C22×C4—5841no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4C2×C22⋊D7311yescharacteristic
481C22×C4C2×C22⋊D7311yescharacteristic
482C22×C4C2×C22⋊D7311yescharacteristic
483C22×C4C2×C22⋊D7311yescharacteristic
484C22×C4#1192402812yes—
485C22×C4#1192402812yes—
486C22×C4C23×D7311yescharacteristic
487C2×C8—2924no—
488C2×C8—732no—
489C2×C8—1461no—
490C2×C8—1461no—
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—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—2922no—
513C4⋊C4—5841no—
514C4⋊C4—5841no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4—1461no—
520C4⋊C4—1461no—
521C4⋊C4—1461no—
522C4⋊C4—1461no—
523C4⋊C4#1192402712yes—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—2922no—
529C22⋊C4—5841no—
530C22⋊C4—5841no—
531C42—21no—
532C42—21no—
533C42—42no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23#1455657412yes—
545C23C2×#1192401612yes—
546C23C2×#1192401612yes—
547C23#1455662212yes—
548C23C22×C22⋊D7311yescharacteristic
549D4—28no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
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—84no—
581C2×C4—84no—
582C2×C4—2921no—
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—5844no—
595C2×C4#1455657412yes—
596C2×C4C2×#1192401612yes—
597C2×C4#41×D7311yescharacteristic
598C2×C4C2×#1192402711yescharacteristic
599C2×C4C2×#1192402811yescharacteristic
600C2×C4C2×#1192402811yescharacteristic
601C2×C4#1455663611yescharacteristic
602C2×C4#1455663811yescharacteristic
603C8—2921no—
604C8—2921no—
605C8—2921no—
606C8—1462no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22#1798891714yes—
621C22C2×#1455657411yescharacteristic
622C22#1798923111yescharacteristic
623C22#1798923611yescharacteristic
624C22C22×#1192401611yescharacteristic
625C22C2×#1455662211yescharacteristic
626C22#1798939011yescharacteristic
627C22#1798939511yescharacteristic
628C4—22no—
629C4—22no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—5841no—
637C4—5841no—
638C4—5841no—
639C4—5841no—
640C4#1798911711yescharacteristic
641C4#1798917911yescharacteristic
642C4#1798926611yescharacteristic
643C4#1798927711yescharacteristic
644C2—24no—
645C2—42no—
646C2#2261340211yescharacteristic
647C2C2×#1798891711yescharacteristic
648C2#2261988911yescharacteristic
649C2#2262175511yescharacteristic
650C2#2262210211yescharacteristic
651C2#2262228811yescharacteristic
652C2#2262241011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingD4×C4⋊C4×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenterD4×C4⋊C4D73
hypocenterC73#24935
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22611488C2semidirect
3. #22611408C2semidirect
4. #22611367C2semidirect
5. #22611367C2semidirect
6. #22610516C2semidirect
7. #22610516C2semidirect
9. #22610124C2semidirect
10. #22610124C2semidirect
11. #22610019C2semidirect
12. #22610017C2semidirect
13. #22609954C2semidirect
14. C2×#17988349C2semidirect
342. C73#24935semidirect

Maximal subgroups

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

subgroupindex
#226142482
#226114882
#226114082
#226113672
#226105162
#226102242
#226101242
#226100192
#226100172
#226099542
C2×#179883492
D4×C4⋊C4×C732
#2493573