Finite Group Databasebeta

← #33130442

Subgroups — #33130442

Order 18688.

796 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614239C211yesmaximal, characteristic
2#22612860C211yesmaximal, characteristic
3#22611500C211yesmaximal, characteristic, complemented
4#22611414C211yesmaximal, characteristic, complemented
5#22611364C211yesmaximal, characteristic, complemented
6#22611359C211yesmaximal, characteristic, complemented
7#22610524C211yesmaximal, characteristic, complemented
8#22610517C211yesmaximal, characteristic, complemented
9#22610132C211yesmaximal, characteristic, complemented
10#22610126C211yesmaximal, characteristic, complemented
11#22610027C211yesmaximal, characteristic, complemented
12#22610018C211yesmaximal, characteristic, complemented
13#22609960C211yesmaximal, characteristic, complemented
14C2×#17988350C211yesmaximal, characteristic, complemented
15C73×#1869C211yesmaximal, characteristic
16C2×#14556572—21no—
17C2×#14556550C2211yescharacteristic
18C22×C4⋊Dic73C2211yescharacteristic
19C22×C4⋊Dic73C2211yescharacteristic
20C2×#14556525C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24#17988517—21no—
25#17988512—21no—
26#17988485C2211yescharacteristic
27#17988485C2211yescharacteristic
28#17988480—21no—
29#17988473—21no—
30#17988467—21no—
31#17988453C2211yescharacteristic
32#17988448—21no—
33#17988447C2211yescharacteristic
34#17988447C2211yescharacteristic
35#17988447C2211yescharacteristic
36#17988447C2211yescharacteristic
37#17988447C2211yescharacteristic
38C4×C4⋊Dic73C2211yescharacteristic
39C22⋊C4×Dic73—21no—
40#17988350C2211yescharacteristic
41#17988350C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52C23×C8×C73C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54C2×C22⋊C8×C73C2211yescharacteristic
55C2×C22⋊C8×C73C2211yescharacteristic
56C2×C22⋊C8×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C23×Dic73—21no—
60C23×Dic73—21no—
61#14556572—44no—
62#14556550—22no—
63#14556550—41no—
64C2×C4⋊Dic73—22no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—22no—
68C2×C4⋊Dic73—22no—
69C2×C4⋊Dic73—41no—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88#14556525—22no—
89#14556525—41no—
90#14556523—22no—
91#14556523—41no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102C22×D4×C73D411yescharacteristic
103C23×C4×C73C2311yescharacteristic
104C23×C4×C73C2311yescharacteristic
105C22×C8×C73—22no—
106C22×C8×C73—22no—
107C22×C8×C73—41no—
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C4×D4×C73—22no—
113C4×D4×C73—22no—
114C2×C4⋊C4×C73D411yescharacteristic
115C2×C22⋊C4×C73D411yescharacteristic
116C2×C22⋊C4×C73D411yescharacteristic
117C2×C42×C73C2311yescharacteristic
118C22⋊C8×C73—22no—
119C22⋊C8×C73—22no—
120C22⋊C8×C73—41no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—42no—
130C22×Dic73—42no—
131C22×Dic73—42no—
132C22×Dic73—42no—
133C22×Dic73—42no—
134C22×Dic73—81no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
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—21no—
150C4⋊Dic73—42no—
151C4⋊Dic73—42no—
152C4⋊Dic73#4112yes—
153#11924007—42no—
154#11924007—42no—
155C22⋊Dic73—42no—
156C22⋊Dic73—42no—
157C22⋊Dic73—42no—
158C22⋊Dic73—42no—
159C22⋊Dic73—42no—
160C22⋊Dic73—81no—
161C4×Dic73—42no—
162C4×Dic73—81no—
163C24×C73C2×D411yescharacteristic
164C24×C73C2×D411yescharacteristic
165C2×D4×C73—22no—
166C2×D4×C73—22no—
167C2×D4×C73—22no—
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—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73#4112yes—
183C22×C4×C73#4112yes—
184C22×C4×C73C2411yescharacteristic
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
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—44no—
197C2×C8×C73#4112yes—
198C4⋊C4×C73—22no—
199C22⋊C4×C73—22no—
200C22⋊C4×C73—22no—
201C42×C73—21no—
202C42×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73#12312yes—
215C23×C73#13512yes—
216C23×C73AGL(1,ℤ/8)12yes—
217C23×C73C22×D411yescharacteristic
218D4×C73—44no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
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—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73#12312yes—
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×C4×C73C2×#4111yescharacteristic
243C2×C4×C73H(5,𝔽2)11yescharacteristic
244C2×C4×C732−1+411yescharacteristic
245C2×Dic73—22no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—84no—
270C2×Dic73—84no—
271C8×C73—22no—
272C8×C73—41no—
273C8×C73—41no—
274C8×C73—41no—
275C22×C73—22no—
276C22×C73—22no—
277C22×C73—22no—
278C22×C73—22no—
279C22×C73—22no—
280C22×C73—22no—
281C22×C73—41no—
282C22×C73—41no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73#48214yes—
289C22×C73C2×#12311yescharacteristic
290C22×C73#54611yescharacteristic
291C22×C73#54711yescharacteristic
292C22×C73C2×#13511yescharacteristic
293C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C73#57711yescharacteristic
295C22×C73#57811yescharacteristic
296C4×C73—22no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73—41no—
300C4×C73#52911yescharacteristic
301C4×C73#53911yescharacteristic
302C4×C73#55111yescharacteristic
303C4×C73#55211yescharacteristic
304Dic73—42no—
305Dic73—81no—
306Dic73—81no—
307Dic73—81no—
308Dic73—81no—
309Dic73—81no—
310Dic73—81no—
311Dic73—81no—
312#24941—731nomaximal, Sylow
313C2×C73—24no—
314C2×C73—81no—
315C2×C73#241911yescharacteristic
316C2×C73C2×#48211yescharacteristic
317C2×C73#309711yescharacteristic
318C2×C73#329611yescharacteristic
319C2×C73#330811yescharacteristic
320C2×C73#333611yescharacteristic
321C2×C73#334511yescharacteristic
322#2486—731no—
323#2352—731no—
324#2101—731no—
325#2085—731no—
326#2077—731no—
327#2076—731no—
328#1921—731no—
329#1920—731no—
330#1869D7311yescharacteristic
331#1854—731no—
332#1853—731no—
333#1825—731no—
334#1823—731no—
335#1808—731no—
336C2×#340—731no—
337C73#2494111yescharacteristic, complemented, Sylow
338C23×C8C2×D7311yescharacteristic
339C2×#123—1461no—
340C2×C4×D4C2×D7311yescharacteristic
341C2×#117—731no—
342C22×C4⋊C4—731no—
343C22×C4⋊C4—731no—
344C2×#107—731no—
345C2×#107—731no—
346C2×#106—731no—
347C2×#106—731no—
348C2×C22⋊C8C2×D7311yescharacteristic
349C2×C22⋊C8C2×D7311yescharacteristic
350C2×C22⋊C8C2×D7311yescharacteristic
351#398—1461no—
352#397—1461no—
353#389—731no—
354#389—731no—
355#388—1461no—
356#387—1461no—
357#386—1461no—
358#382—731no—
359#381—1461no—
360#380—731no—
361#380—731no—
362#380—731no—
363#380—731no—
364#380—731no—
365C4×C4⋊C4—731no—
366C4×C22⋊C4—1461no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#340—731no—
374#340—731no—
375#340—731no—
376#340—731no—
377#340—731no—
378#340—731no—
379#336C2×D7311yescharacteristic
380#336C2×D7311yescharacteristic
381C22×D4D29211yescharacteristic
382C23×C4—1461no—
383C23×C4—1461no—
384C23×C4C22×D7311yescharacteristic
385C23×C4C22×D7311yescharacteristic
386C22×C8—22no—
387C22×C8—22no—
388C22×C8—41no—
389C22×C8C22×D7311yescharacteristic
390C22×C8C22×D7311yescharacteristic
391C22×C8C22×D7311yescharacteristic
392C22×C8C22×D7311yescharacteristic
393#123—2924no—
394C4×D4—22no—
395C4×D4—22no—
396#117—2921no—
397#117—1462no—
398C2×C4⋊C4—2921no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—731no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4—1461no—
410C2×C4⋊C4—1461no—
411C2×C4⋊C4—1462no—
412C2×C4⋊C4—1462no—
413C2×C4⋊C4—1462no—
414C2×C4⋊C4D29211yescharacteristic
415C2×C22⋊C4—1461no—
416C2×C22⋊C4—1461no—
417C2×C22⋊C4—1461no—
418C2×C22⋊C4—1461no—
419C2×C22⋊C4—1461no—
420C2×C22⋊C4—1461no—
421C2×C22⋊C4D29211yescharacteristic
422C2×C22⋊C4D29211yescharacteristic
423C2×C42—1461no—
424C2×C42—1461no—
425C2×C42C22×D7311yescharacteristic
426#107—2921no—
427#107—1462no—
428#106—2921no—
429#106—1462no—
430C22⋊C8—22no—
431C22⋊C8—22no—
432C22⋊C8—41no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442#95—1461no—
443C24C2×D29211yescharacteristic
444C24C2×D29211yescharacteristic
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—2922no—
459C22×C4—2922no—
460C22×C4—2922no—
461C22×C4—2922no—
462C22×C4—2922no—
463C22×C4—5841no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4—1461no—
472C22×C4C2×D29211yescharacteristic
473C22×C4C2×D29211yescharacteristic
474C22×C4C2×D29211yescharacteristic
475C22×C4C2×D29211yescharacteristic
476C22×C4#1192402712yes—
477C22×C4#1192402712yes—
478C22×C4C23×D7311yescharacteristic
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—44no—
491C2×C8#1192402712yes—
492C4⋊C4—22no—
493C4⋊C4—2922no—
494C4⋊C4—2922no—
495C4⋊C4—2922no—
496C4⋊C4—2922no—
497C4⋊C4—732no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—2922no—
516C22⋊C4—2922no—
517C22⋊C4—2922no—
518C22⋊C4—2922no—
519C22⋊C4—2922no—
520C22⋊C4—5841no—
521C42—21no—
522C42—21no—
523C42—2922no—
524C42—5841no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23#1455657512yes—
537C23#1455661312yes—
538C23#1455661912yes—
539C23C22×D29211yescharacteristic
540D4—44no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
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—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—5844no—
584C2×C4—5844no—
585C2×C4—1462no—
586C2×C4#1455657512yes—
587C2×C4#41×D7311yescharacteristic
588C2×C4C2×#1192402711yescharacteristic
589C2×C4C2×#1192402711yescharacteristic
590C2×C4C2×#1192402711yescharacteristic
591C2×C4#1455663611yescharacteristic
592C2×C4#1455663811yescharacteristic
593C8—22no—
594C8—41no—
595C8—41no—
596C8—41no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22#1798891214yes—
611C22C2×#1455657511yescharacteristic
612C22#1798923211yescharacteristic
613C22#1798924111yescharacteristic
614C22C2×#1455661311yescharacteristic
615C22C2×#1455661911yescharacteristic
616C22#1798939911yescharacteristic
617C22#1798940411yescharacteristic
618C4—22no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—2922no—
623C4—5841no—
624C4—5841no—
625C4—5841no—
626C4—5841no—
627C4—5841no—
628C4—5841no—
629C4—5841no—
630C4#1798911811yescharacteristic
631C4#1798918211yescharacteristic
632C4#1798926811yescharacteristic
633C4#1798927411yescharacteristic
634C2—24no—
635C2—81no—
636C2#2261340611yescharacteristic
637C2C2×#1798891211yescharacteristic
638C2#2261990111yescharacteristic
639C2#2262176811yescharacteristic
640C2#2262192411yescharacteristic
641C2#2262230011yescharacteristic
642C2#2262240311yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1869C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenter#1869D73
hypocenterC73#24941
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
3. #22611500C2semidirect
4. #22611414C2semidirect
5. #22611364C2semidirect
6. #22611359C2semidirect
7. #22610524C2semidirect
8. #22610517C2semidirect
9. #22610132C2semidirect
10. #22610126C2semidirect
11. #22610027C2semidirect
12. #22610018C2semidirect
13. #22609960C2semidirect
14. C2×#17988350C2semidirect
337. C73#24941semidirect

Maximal subgroups

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

subgroupindex
#226142392
#226128602
#226115002
#226114142
#226113642
#226113592
#226105242
#226105172
#226101322
#226101262
#226100272
#226100182
#226099602
C2×#179883502
C73×#18692
#2494173