Finite Group Databasebeta

← #33098069

Subgroups — #33098069

Order 18688.

796 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988888C211yesmaximal, characteristic
2C4×#14556583C211yesmaximal, characteristic, complemented
3#22612395C211yesmaximal, characteristic
4#22610886C211yesmaximal, characteristic, complemented
5#22610701C211yesmaximal, characteristic, complemented
6#22610664C211yesmaximal, characteristic, complemented
7#22610659C211yesmaximal, characteristic, complemented
8#22610614C211yesmaximal, characteristic
9#22610614C211yesmaximal, characteristic
10#22610559C211yesmaximal, characteristic
11#22610552C211yesmaximal, characteristic
12#22610524C211yesmaximal, characteristic, complemented
13#22610517C211yesmaximal, characteristic, complemented
14#22609942C211yesmaximal, characteristic
15C4×C4⋊C8×C73C211yesmaximal, characteristic, complemented
16C2×#14556583C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18C2×#14556548—21no—
19#17988888C418yescomplemented
20C2×#14556525C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556513C2211yescharacteristic
25C2×#14556513C2211yescharacteristic
26C2×#14556513C2211yescharacteristic
27C2×#14556513C2211yescharacteristic
28#17988489C2211yescharacteristic
29#17988489C2211yescharacteristic
30#17988485C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988478—21no—
35#17988478—21no—
36#17988471—21no—
37#17988471—21no—
38#17988453C2211yescharacteristic
39C4×#11924007—21no—
40C4×#11924007—21no—
41C4×C4⋊Dic73C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43C4×C4⋊Dic73C2211yescharacteristic
44C4×#11924005—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53C2×C4⋊C8×C73C2211yescharacteristic
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×C4×C8×C73C2211yescharacteristic
56C2×C4×C8×C73C2211yescharacteristic
57C43×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C4×D73—21no—
62#14556583—22no—
63#14556583C2×C414yes—
64C4×D292—22no—
65C4×D292C2×C414yes—
66#14556548—44no—
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88#14556525—41no—
89#14556525C2×C414yes—
90#14556523—41no—
91#14556523C2×C414yes—
92#14556513—41no—
93#14556513—41no—
94#14556513C2×C414yes—
95#14556513C2×C414yes—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C2×C42×C73D411yescharacteristic
110C2×C42×C73D411yescharacteristic
111C2×C42×C73D411yescharacteristic
112C2×C42×C73D411yescharacteristic
113C2×C42×C73C2311yescharacteristic
114C2×C42×C73C2311yescharacteristic
115C2×C42×C73C2311yescharacteristic
116C4⋊C8×C73—22no—
117C4⋊C8×C73C2×C414yes—
118C4×C8×C73—41no—
119C4×C8×C73—41no—
120C23×D73—21no—
121C2×D292—21no—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292—21no—
126C2×D292C22×C412yes—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C2×C4×D73—24no—
135C2×C4×D73—42no—
136C2×C4×D73—42no—
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⋊Dic73C22×C412yes—
149C4⋊Dic73C22×C412yes—
150C4⋊Dic73C22×C412yes—
151#11924007—24no—
152#11924007—24no—
153#11924007—42no—
154#11924007—42no—
155#11924007—42no—
156#11924007—42no—
157#11924005—24no—
158#11924005—42no—
159#11924005—42no—
160#11924005—42no—
161#11924005—42no—
162C4×Dic73—42no—
163C4×Dic73—42no—
164C4×Dic73—42no—
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2411yescharacteristic
172C2×C8×C73—21no—
173C2×C8×C73—21no—
174C2×C8×C73—21no—
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73C22×C412yes—
183C2×C8×C73C22×C412yes—
184C42×C73—22no—
185C42×C73—22no—
186C42×C73—22no—
187C42×C73—21no—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—41no—
192C42×C73C22×C412yes—
193C42×C73#4112yes—
194C23×C73C22×D411yescharacteristic
195C22×D73—22no—
196C22×D73—41no—
197C22×D73—41no—
198C22×D73—41no—
199C22×D73—41no—
200C22×D73—41no—
201C22×D73—41no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73C4×D412yes—
216C2×C4×C73C4×D412yes—
217C2×C4×C73#13512yes—
218C2×C4×C73#13512yes—
219C2×C4×C73AGL(1,ℤ/8)12yes—
220C2×C4×C73C23×C411yescharacteristic
221C2×C4×C73C2×#4111yescharacteristic
222C2×C4×C73H(5,𝔽2)11yescharacteristic
223C2×C4×C732−1+411yescharacteristic
224C2×C4×C732−1+411yescharacteristic
225C2×C4×C732−1+411yescharacteristic
226C2×Dic73—22no—
227C2×Dic73—22no—
228C2×Dic73—22no—
229C2×Dic73—41no—
230C2×Dic73—41no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247D292—22no—
248D292—22no—
249D292—41no—
250D292—41no—
251C4×D73—48no—
252C8×C73—22no—
253C8×C73—22no—
254C8×C73—41no—
255C8×C73—41no—
256C22×C73C2×C4×D411yescharacteristic
257C22×C73#53611yescharacteristic
258C22×C73#53711yescharacteristic
259C22×C73C2×#13511yescharacteristic
260C22×C73C2×#13511yescharacteristic
261C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C73#57511yescharacteristic
263C2×D73—42no—
264C2×D73—42no—
265C2×D73—42no—
266C2×D73—81no—
267C2×D73—81no—
268C2×D73—81no—
269C2×D73—81no—
270C4×C73—22no—
271C4×C73—22no—
272C4×C73—22no—
273C4×C73—22no—
274C4×C73—22no—
275C4×C73—41no—
276C4×C73#51911yescharacteristic
277C4×C73#52011yescharacteristic
278C4×C73#54111yescharacteristic
279C4×C73#54211yescharacteristic
280Dic73—42no—
281Dic73—42no—
282Dic73—42no—
283Dic73—81no—
284Dic73—81no—
285Dic73—81no—
286Dic73—81no—
287#22114—731nomaximal, Sylow
288C2×C73#232311yescharacteristic
289C2×C73C4×#13511yescharacteristic
290C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
291C2×C73#322811yescharacteristic
292C2×C73#323211yescharacteristic
293C2×C73#324711yescharacteristic
294C2×C73#325011yescharacteristic
295D73—82no—
296C2×#478—731no—
297C4×#125—731no—
298#2293—731no—
299#1989—731no—
300#1949—731no—
301#1942—731no—
302#1941—731no—
303#1934—731no—
304#1934—731no—
305#1926—731no—
306#1925—731no—
307#1921—731no—
308#1920—731no—
309#1804—731no—
310C4×C4⋊C8D7311yescharacteristic, complemented
311C73#2211411yescharacteristic, complemented, Sylow
312C2×#125—731no—
313C2×C4×D4—731no—
314C2×#117—1461no—
315#478—738no—
316C2×#107—731no—
317C2×#107—731no—
318C2×#106—731no—
319C2×#106—731no—
320C2×C4⋊C8C2×D7311yescharacteristic
321C2×C4⋊C8C2×D7311yescharacteristic
322C2×#102—731no—
323C2×#102—731no—
324C2×#102—731no—
325C2×#102—731no—
326C2×C4×C8C2×D7311yescharacteristic
327C2×C4×C8C2×D7311yescharacteristic
328#390—731no—
329#390—731no—
330#389—731no—
331#389—731no—
332#389—731no—
333#389—731no—
334#388—1461no—
335#388—1461no—
336#387—1461no—
337#387—1461no—
338#382—731no—
339C4×C4⋊C4—731no—
340C4×C4⋊C4—731no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4—1461no—
343C4×C4⋊C4—1461no—
344C4×C22⋊C4—1461no—
345C43C2×D7311yescharacteristic
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#336C2×D7311yescharacteristic
355#336C2×D7311yescharacteristic
356C22×D4—731no—
357C23×C4—1461no—
358C22×C8C22×D7311yescharacteristic
359C22×C8C22×D7311yescharacteristic
360C22×C8C22×D7311yescharacteristic
361C22×C8C22×D7311yescharacteristic
362#125—734no—
363#125—1462no—
364C4×D4—734no—
365C4×D4—1462no—
366#117—2924no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1461no—
380C2×C22⋊C4—1461no—
381C2×C22⋊C4—1461no—
382C2×C22⋊C4—1461no—
383C2×C22⋊C4—1461no—
384C2×C22⋊C4—1461no—
385C2×C42—1461no—
386C2×C42—1461no—
387C2×C42—1461no—
388C2×C42D29211yescharacteristic
389C2×C42D29211yescharacteristic
390C2×C42D29211yescharacteristic
391C2×C42D29211yescharacteristic
392C2×C42C22×D7311yescharacteristic
393C2×C42C22×D7311yescharacteristic
394C2×C42C22×D7311yescharacteristic
395#107—2921no—
396#107—734no—
397#106—2921no—
398#106—734no—
399C4⋊C8—22no—
400C4⋊C8C4×D7314yes—
401#102—2921no—
402#102—2921no—
403#102—734no—
404#102—734no—
405C4×C8—41no—
406C4×C8—41no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416C24—1461no—
417C2×D4—732no—
418C2×D4—1461no—
419C2×D4—1461no—
420C2×D4—1461no—
421C2×D4—1461no—
422C2×D4—1461no—
423C22×C4—2922no—
424C22×C4—2922no—
425C22×C4—1461no—
426C22×C4—1461no—
427C22×C4—1461no—
428C22×C4—1461no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1464no—
433C22×C4C2×D29211yescharacteristic
434C22×C4C2×D29211yescharacteristic
435C22×C4C2×D29211yescharacteristic
436C22×C4C2×D29211yescharacteristic
437C22×C4C2×D29211yescharacteristic
438C22×C4C2×D29211yescharacteristic
439C22×C4C23×D7311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8C2×C4×D7312yes—
451C2×C8C2×C4×D7312yes—
452C4⋊C4—2922no—
453C4⋊C4—2922no—
454C4⋊C4—2922no—
455C4⋊C4—2922no—
456C4⋊C4—732no—
457C4⋊C4—732no—
458C4⋊C4—732no—
459C4⋊C4—1461no—
460C4⋊C4—1461no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1464no—
468C4⋊C4—1464no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C22⋊C4—2922no—
473C22⋊C4—2922no—
474C22⋊C4—2922no—
475C22⋊C4—2922no—
476C22⋊C4—1464no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—41no—
485C42—2922no—
486C42—2922no—
487C42—2922no—
488C42C2×C4×D7312yes—
489C42#1192402812yes—
490C23—2921no—
491C23—2921no—
492C23—2921no—
493C23—2921no—
494C23—2921no—
495C23—2921no—
496C23—1462no—
497C23C22×D29211yescharacteristic
498D4—2921no—
499D4—2921no—
500D4—1462no—
501D4—1462no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—2928no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—1462no—
535C2×C4—1462no—
536C2×C4—1462no—
537C2×C4C4×D29212yes—
538C2×C4C4×D29212yes—
539C2×C4#1455661312yes—
540C2×C4#1455661312yes—
541C2×C4#1455661912yes—
542C2×C4C22×C4×D7311yescharacteristic
543C2×C4C2×#1192402811yescharacteristic
544C2×C4#1455663611yescharacteristic
545C2×C4#1455663811yescharacteristic
546C2×C4#1455663811yescharacteristic
547C2×C4#1455663911yescharacteristic
548C8—22no—
549C8—22no—
550C8—41no—
551C8—41no—
552C22—2922no—
553C22—2922no—
554C22—2922no—
555C22—5841no—
556C22—5841no—
557C22—5841no—
558C22—5841no—
559C22C2×C4×D29211yescharacteristic
560C22#1798916311yescharacteristic
561C22#1798916811yescharacteristic
562C22C2×#1455661311yescharacteristic
563C22C2×#1455661311yescharacteristic
564C22C2×#1455661911yescharacteristic
565C22#1798938711yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—41no—
572C4—2922no—
573C4—2922no—
574C4—2922no—
575C4—5841no—
576C4—5841no—
577C4—5841no—
578C4—5841no—
579C4#1798906311yescharacteristic
580C4#1798906611yescharacteristic
581C4#1798919511yescharacteristic
582C4#1798920211yescharacteristic
583C2—5842no—
584C2#2261261411yescharacteristic
585C2C4×#1455661311yescharacteristic
586C2C4×#1455661911yescharacteristic
587C2#2262106911yescharacteristic
588C2#2262110111yescharacteristic
589C2#2262125811yescharacteristic
590C2#2262128311yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC4×C4⋊C8×C73C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenterC4×C4⋊C8D73
hypocenterC73#22114
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C4×#14556583C2semidirect
4. #22610886C2semidirect
5. #22610701C2semidirect
6. #22610664C2semidirect
7. #22610659C2semidirect
12. #22610524C2semidirect
13. #22610517C2semidirect
15. C4×C4⋊C8×C73C2semidirect
19. #17988888C4semidirect
310. C4×C4⋊C8D73semidirect
311. C73#22114semidirect

Maximal subgroups

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

subgroupindex
C2×#179888882
C4×#145565832
#226123952
#226108862
#226107012
#226106642
#226106592
#226106142
#226105592
#226105522
#226105242
#226105172
#226099422
C4×C4⋊C8×C732
#2211473