Finite Group Databasebeta

← #10831767

Subgroups — #10831767

Order 768.

988 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5685C211yesmaximal, characteristic
2C2×#4875C211yesmaximal, characteristic
3C2×#4853C211yesmaximal, characteristic
4#76076C211yesmaximal, characteristic
5C3×#2059C211yesmaximal, characteristic
6#71744C211yesmaximal, characteristic
7#70642C211yesmaximal, characteristic
8#70566C211yesmaximal, characteristic
9#69631C211yesmaximal, characteristic
10#68911C211yesmaximal, characteristic
11#68908C211yesmaximal, characteristic
12#68905C211yesmaximal, characteristic
13#68806C211yesmaximal, characteristic
14#68685C211yesmaximal, characteristic
15#68620C211yesmaximal, characteristic
16#24344—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C22×Dic12C2211yescharacteristic
19#5685—24no—
20#5685—24no—
21C2×#890C2211yescharacteristic
22C2×#889—21no—
23C2×#889—21no—
24C2×#888—21no—
25C2×#888C2211yescharacteristic
26C2×#887—21no—
27C2×#887C2211yescharacteristic
28C2×#887C2211yescharacteristic
29C2×#887C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#107C2211yescharacteristic
33C2×C3×#106C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4980—21no—
40#4979—21no—
41#4977C2211yescharacteristic
42#4973C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4929C2211yescharacteristic
50#4875—24no—
51#4853—24no—
52#4658—21no—
53#4652—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#557—31no—
65C2×#477—31no—
66C2×#441—31no—
67#2581—31no—
68#2059—31no—
69#2059D311yescharacteristic
70#2026—31no—
71#2026—31no—
72#1993—31no—
73#1985—31no—
74#1878—31no—
75#1878—31no—
76#1863—31no—
77#1863—31no—
78#1805—31no—
79C22×Dic6—21no—
80C22×Dic6D411yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#125—22no—
86C3×#125—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3D411yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4D411yescharacteristic
107C2×Dic3⋊C4D411yescharacteristic
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C2×Dic12—22no—
117C2×Dic12—22no—
118C2×Dic12Q814yescomplemented
119#890—22no—
120#890—22no—
121#889—28no—
122#889—28no—
123#888—28no—
124#888—22no—
125#888D414yes—
126#887—28no—
127#887—22no—
128#887—22no—
129#887—22no—
130#887—22no—
131#887—22no—
132#887D414yes—
133C4×Dic6—22no—
134C4×Dic6—22no—
135C4×Dic6—22no—
136C4×Dic6—22no—
137C3×#107—22no—
138C3×#106—22no—
139C3×#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147#816—22no—
148#816—22no—
149#815—22no—
150Dic3⋊C8—22no—
151#810—22no—
152#810—22no—
153#810—22no—
154#810—22no—
155#807—22no—
156C22×Q16—31no—
157#557—64no—
158#557—64no—
159C2×C4⋊Q8—31no—
160C2×C4⋊Q8—61no—
161C2×#125—31no—
162C2×#125—61no—
163C2×#125—61no—
164C2×#125C2×D311yescharacteristic
165C2×C4×Q8—31no—
166C2×C4×Q8—31no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×Q8—61no—
171#477—64no—
172#441—64no—
173C2×#107—31no—
174C2×#107C2×D311yescharacteristic
175C2×#106C2×D311yescharacteristic
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#97—31no—
184#400—61no—
185#395—61no—
186#391—31no—
187#391—31no—
188#389—31no—
189#389C2×D311yescharacteristic
190#387—61no—
191#387—61no—
192#382—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#340C2×D311yescharacteristic
199#336—31no—
200#336—31no—
201#336—31no—
202#336C2×D311yescharacteristic
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—22no—
227C2×Dic6—24no—
228C2×Dic6—41no—
229C2×Dic6—41no—
230C2×Dic6—42no—
231C2×Dic6—42no—
232C2×Dic6C2×D412yes—
233C2×Dic6C2×Q812yes—
234C2×Dic6C2×Q812yes—
235C2×Dic6#4112yes—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8C2×Q812yes—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4C2×D412yes—
252C3×C4⋊C4C2×D412yes—
253C3×C42—21no—
254C3×C42—21no—
255C4⋊Dic3—24no—
256C4⋊Dic3—21no—
257C4⋊Dic3—22no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3C2×D412yes—
261Dic3⋊C4—24no—
262Dic3⋊C4—24no—
263Dic3⋊C4—24no—
264Dic3⋊C4—22no—
265Dic3⋊C4—22no—
266Dic3⋊C4—24no—
267Dic3⋊C4—24no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270C4×Dic3—24no—
271C4×Dic3—24no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—24no—
275C4×Dic3—42no—
276C4×Dic3C2×D412yes—
277C4×Dic3C2×D412yes—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284Dic12—28no—
285C22×Q8—31no—
286C22×Q8—31no—
287C22×Q8—31no—
288C22×Q8—61no—
289C2×Q16—34no—
290C2×Q16—62no—
291C2×Q16—62no—
292C22×C8—31no—
293C22×C8—31no—
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296C4⋊Q8—34no—
297C4⋊Q8—62no—
298C4⋊Q8—68no—
299#125—22no—
300#125—22no—
301#125—62no—
302#125—68no—
303#125—68no—
304#125—62no—
305C4×Q8—34no—
306C4×Q8—62no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×Q8—62no—
311C4×Q8—62no—
312C4×Q8—62no—
313C4×Q8—62no—
314C4×Q8—68no—
315C4×Q8—62no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C4⋊C4C22×D311yescharacteristic
334C2×C4⋊C4C22×D311yescharacteristic
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—61no—
339C2×C42—61no—
340C2×C42—61no—
341C2×C42—61no—
342C2×C42C22×D311yescharacteristic
343#107—22no—
344#107—62no—
345#106—22no—
346C4⋊C8—62no—
347#103—62no—
348#103—62no—
349#103—62no—
350#103—62no—
351#103—62no—
352#103—62no—
353#97—62no—
354#95—21no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362C23×C3C22×D411yescharacteristic
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×Q811yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C42−1+411yescharacteristic
386C2×C3×C42−1+411yescharacteristic
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3#12012yes—
410C2×Dic3#12012yes—
411Dic6—22no—
412Dic6—22no—
413Dic6—22no—
414Dic6—22no—
415Dic6—22no—
416Dic6—44no—
417Dic6—48no—
418Dic6—41no—
419Dic6—41no—
420Dic6C22⋊Q814yes—
421C3×C8—22no—
422C3×C8—41no—
423C3⋊C8—41no—
424C3⋊C8—41no—
425C2×Q8—32no—
426C2×Q8—32no—
427C2×Q8—32no—
428C2×Q8—32no—
429C2×Q8—64no—
430C2×Q8—62no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—121no—
440C2×Q8—121no—
441C2×Q8—122no—
442C2×Q8—122no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4D3×D411yescharacteristic
456C22×C4D3×D411yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—68no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8D3×Q812yes—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—32no—
480C4⋊C4—42no—
481C4⋊C4—62no—
482C4⋊C4—62no—
483C4⋊C4—62no—
484C4⋊C4—64no—
485C4⋊C4—64no—
486C4⋊C4—64no—
487C4⋊C4—64no—
488C4⋊C4—64no—
489C4⋊C4—64no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—122no—
494C4⋊C4—122no—
495C4⋊C4D3×D412yes—
496C4⋊C4D3×D412yes—
497C42—21no—
498C42—21no—
499C42—32no—
500C42—32no—
501C42—64no—
502C42—64no—
503C42—64no—
504C42—61no—
505C42—61no—
506C42—122no—
507C22×C3C2×#12011yescharacteristic
508C22×C3#54811yescharacteristic
509C22×C3D4×Q811yescharacteristic
510C22×C3#57511yescharacteristic
511C22×C3#57511yescharacteristic
512C22×C3#57811yescharacteristic
513C22×C3#57811yescharacteristic
514C3×C4—41no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—42no—
518C3×C4—81no—
519C3×C4—81no—
520C3×C4C2×C22⋊Q811yescharacteristic
521C3×C4#53611yescharacteristic
522C3×C4#54811yescharacteristic
523C3×C4D4×Q811yescharacteristic
524Dic3—24no—
525Dic3—22no—
526Dic3—22no—
527Dic3—42no—
528Dic3—41no—
529Dic3—42no—
530Dic3—41no—
531Dic3—81no—
532C23C2×D3×D411yescharacteristic
533Q8—34no—
534Q8—62no—
535Q8—62no—
536Q8—62no—
537Q8—62no—
538Q8—62no—
539Q8—124no—
540Q8—128no—
541Q8—121no—
542Q8—121no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—62no—
577C2×C4—61no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4C2×D3×D411yescharacteristic
586C2×C4C2×D3×D411yescharacteristic
587C2×C4C2×D3×Q811yescharacteristic
588C2×C4#100711yescharacteristic
589C2×C4#100711yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C8—22no—
592C8—41no—
593C8—121no—
594C8—121no—
595C2×C3#240911yescharacteristic
596C2×C3#300511yescharacteristic
597C2×C3#301111yescharacteristic
598C2×C3#337711yescharacteristic
599C2×C3#337711yescharacteristic
600C2×C3#339011yescharacteristic
601C2×C3#339011yescharacteristic
602C22D3×#12011yescharacteristic
603C22#563311yescharacteristic
604C22#572711yescharacteristic
605C22#576811yescharacteristic
606C22#577111yescharacteristic
607C22#577311yescharacteristic
608C22#577911yescharacteristic
609C4—41no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—62no—
614C4—64no—
615C4—62no—
616C4—81no—
617C4—81no—
618C4—121no—
619C4—122no—
620C4—121no—
621C4—122no—
622C4—241no—
623C4D3×C22⋊Q811yescharacteristic
624C4#563311yescharacteristic
625C4#572411yescharacteristic
626C4#572711yescharacteristic
627C3#2434411yescharacteristic, complemented, Sylow
628C2#7717211yescharacteristic
629C2#8008311yescharacteristic
630C2#8023011yescharacteristic
631C2#8115511yescharacteristic
632C2#8126711yescharacteristic
633C2#8193211yescharacteristic
634C2#8197111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2059C2
socle, prime radicalC23×C3C22×D4
hypercenter#2059D3
hypocenterC3#24344
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
118. C2×Dic12Q8semidirect
627. C3#24344semidirect

Maximal subgroups

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

subgroupindex
C2×#56852
C2×#48752
C2×#48532
#760762
C3×#20592
#717442
#706422
#705662
#696312
#689112
#689082
#689052
#688062
#686852
#686202
#243443