Finite Group Databasebeta

← #10817112

Subgroups — #10817112

Order 768.

834 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83025C211yesmaximal, characteristic, complemented
2C2×#5066C211yesmaximal, characteristic
3#74228C211yesmaximal, characteristic
4C3×#1991C211yesmaximal, characteristic, complemented
5C4×#821C211yesmaximal, characteristic
6#69757C211yesmaximal, characteristic, complemented
7#69540C211yesmaximal, characteristic, complemented
8#68895C211yesmaximal, characteristic
9#68894C211yesmaximal, characteristic
10#68861C211yesmaximal, characteristic, complemented
11#68860C211yesmaximal, characteristic, complemented
12#68788C211yesmaximal, characteristic, complemented
13#68787C211yesmaximal, characteristic, complemented
14#68782C211yesmaximal, characteristic
15#68782C211yesmaximal, characteristic
16#22972—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22C3×#382C2211yescharacteristic
23C2×#821C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#815C2211yescharacteristic
26C2×#815C2211yescharacteristic
27#5066C418yescomplemented
28#4984—21no—
29#4984C2211yescharacteristic
30#4984C2211yescharacteristic
31#4980—21no—
32#4980—21no—
33Dic3×C4⋊C4—21no—
34Dic3×C4⋊C4C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4972C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47#4668—21no—
48#4668—21no—
49#4667—21no—
50D3×#95—21no—
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#478—31no—
62#2370—31no—
63#2349—31no—
64#1991D311yescharacteristic, complemented
65#1949—31no—
66#1944—31no—
67#1943—31no—
68#1919—31no—
69#1918—31no—
70#1864—31no—
71#1862—31no—
72#1770—31no—
73#1762—31no—
74#1762—31no—
75C4×#102—31no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#125—22no—
83C3×#125C2×C414yescomplemented
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3D411yescharacteristic
97C2×C4⋊Dic3D411yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110#893—22no—
111#893—22no—
112#893—41no—
113#893—41no—
114C4×D12—22no—
115C4×D12C2×C414yes—
116C3×C4×C8—22no—
117C3×#95—21no—
118C3×#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128#821—22no—
129#821Q814yes—
130#815—41no—
131#815—41no—
132#809C2×C414yes—
133#809C2×C414yes—
134#808—41no—
135#808C2×C414yescomplemented
136#807—41no—
137#807C2×C414yescomplemented
138C4⋊(C3⋊C8)C2×C414yescomplemented
139C2×#125C2×D311yescharacteristic
140C2×C4×D4—31no—
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143#478—38no—
144C2×#107—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#97—31no—
154C2×#97—31no—
155C2×C4×C8C2×D311yescharacteristic
156#389—31no—
157#389—31no—
158#389—31no—
159#389—31no—
160#389—61no—
161#387—61no—
162#387—61no—
163#385—61no—
164#385—61no—
165#382C2×D311yescharacteristic
166#380—61no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170C4×C4⋊C4—31no—
171C4×C4⋊C4—61no—
172C2×#95—61no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340C2×D311yescharacteristic
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183C23×D3—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3C2×D411yescharacteristic
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12C22×C412yes—
202C2×D12C2×Q812yes—
203C2×D12#4112yes—
204C2×C4×D3—24no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×C4×D3—42no—
209C2×C4×D3—81no—
210C2×C4×D3—81no—
211C2×C4×D3—81no—
212C2×C4×D3—81no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8C2×Q812yes—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4C22×C412yes—
228C3×C4⋊C4C22×C412yes—
229C3×C42C22×C412yes—
230C3×C42#4112yes—
231(C2×D3)⋊C4—24no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4—42no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—41no—
239C4⋊Dic3—41no—
240C4⋊Dic3—42no—
241C4⋊Dic3—81no—
242C4⋊Dic3C22×C412yes—
243C4⋊Dic3C2×Q812yes—
244Dic3⋊C4—81no—
245Dic3⋊C4—81no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—42no—
251C4×Dic3—81no—
252C4×Dic3—81no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8C22×C412yes—
258C2×C3⋊C8C22×C412yes—
259C22×D4—31no—
260C23×C4—61no—
261C23×C4—61no—
262C23×C4—61no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#125—22no—
268#125C4×D314yescomplemented
269C4×D4—34no—
270C4×D4—62no—
271C4×D4—62no—
272C4×D4—62no—
273C4×D4—121no—
274C4×D4—121no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C42—31no—
294C2×C42—31no—
295C2×C42—61no—
296C2×C42—61no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299#107—34no—
300#107—121no—
301#106—34no—
302#106—121no—
303C4⋊C8—34no—
304#102—34no—
305#102—34no—
306#102—34no—
307#102—62no—
308#97—121no—
309#97—121no—
310C4×C8—22no—
311#95—21no—
312#95—21no—
313#95—31no—
314#95—31no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322C23×C3C22×D411yescharacteristic
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4C4×D412yes—
347C2×C3×C4C4×D412yes—
348C2×C3×C4C23×C411yescharacteristic
349C2×C3×C4C22×Q811yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4H(5,𝔽2)11yescharacteristic
353C2×C3×C42−1+411yescharacteristic
354C2×Dic3—22no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—22no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375D12—22no—
376D12—22no—
377D12—22no—
378D12C4×Q814yes—
379C4×D3—84no—
380C4×D3—84no—
381C4×D3—84no—
382C3×C8—22no—
383C3×C8—41no—
384C3⋊C8—22no—
385C3⋊C8—22no—
386C24—61no—
387C2×D4—32no—
388C2×D4—32no—
389C2×D4—32no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—64no—
404C22×C4—61no—
405C22×C4—122no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4—122no—
409C22×C4—241no—
410C22×C4—241no—
411C22×C4—241no—
412C22×C4—241no—
413C22×C4D3×D411yescharacteristic
414C22×C4D3×D411yescharacteristic
415C22×C4C23×D311yescharacteristic
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—32no—
422C2×C8—32no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8D3×Q812yes—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—32no—
437C4⋊C4—32no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—121no—
443C4⋊C4—121no—
444C4⋊C4—122no—
445C4⋊C4—241no—
446C4⋊C4—241no—
447C4⋊C4—241no—
448C4⋊C4C2×C4×D312yes—
449C4⋊C4C2×C4×D312yes—
450C22⋊C4—64no—
451C22⋊C4—122no—
452C22⋊C4—122no—
453C42—61no—
454C42—61no—
455C42—61no—
456C42—61no—
457C42—122no—
458C42—241no—
459C42—241no—
460C42C2×C4×D312yes—
461C42#23712yes—
462C22×C3C2×C4×D411yescharacteristic
463C22×C3#54611yescharacteristic
464C22×C3D4×Q811yescharacteristic
465C22×C3#57511yescharacteristic
466C22×C3#57511yescharacteristic
467C22×C3#57611yescharacteristic
468C22×C3#57711yescharacteristic
469C2×D3—24no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—42no—
475C2×D3—42no—
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—81no—
481C3×C4—81no—
482C3×C4C2×C4×Q811yescharacteristic
483C3×C4#52011yescharacteristic
484C3×C4#55211yescharacteristic
485C3×C4#55411yescharacteristic
486Dic3—42no—
487Dic3—42no—
488Dic3—41no—
489Dic3—41no—
490Dic3—41no—
491Dic3—41no—
492Dic3—81no—
493C23—62no—
494C23—62no—
495C23—62no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23C2×D3×D411yescharacteristic
501D4—34no—
502D4—62no—
503D4—62no—
504D4—62no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—244no—
543C2×C4—244no—
544C2×C4—244no—
545C2×C4#67⋊C412yes—
546C2×C4#67⋊C412yes—
547C2×C4C22×C4×D311yescharacteristic
548C2×C4C2×#23711yescharacteristic
549C2×C4C2×D3×Q811yescharacteristic
550C2×C4#100711yescharacteristic
551C2×C4D3×#4111yescharacteristic
552C2×C4#100911yescharacteristic
553C8—22no—
554C8—41no—
555C8—62no—
556C8—62no—
557C2×C3#236011yescharacteristic
558C2×C3#294311yescharacteristic
559C2×C3#294611yescharacteristic
560C2×C3#332211yescharacteristic
561C2×C3#332311yescharacteristic
562C2×C3#338811yescharacteristic
563C2×C3#338911yescharacteristic
564D3—44no—
565C22—64no—
566C22—122no—
567C22—122no—
568C22—122no—
569C22—122no—
570C22—122no—
571C22—122no—
572C22C2×#67⋊C411yescharacteristic
573C22#567611yescharacteristic
574C22#572711yescharacteristic
575C22#576811yescharacteristic
576C22#576811yescharacteristic
577C22#576911yescharacteristic
578C22#577011yescharacteristic
579C4—22no—
580C4—22no—
581C4—42no—
582C4—42no—
583C4—81no—
584C4—81no—
585C4—122no—
586C4—122no—
587C4—121no—
588C4—121no—
589C4—121no—
590C4—121no—
591C4—241no—
592C4C4×D3×Q811yescharacteristic
593C4#557411yescharacteristic
594C4#563511yescharacteristic
595C4#563911yescharacteristic
596C3#2297211yescharacteristic, complemented, Sylow
597C2—124no—
598C2#7661511yescharacteristic
599C2#7881911yescharacteristic
600C2#7882011yescharacteristic
601C2#8165611yescharacteristic
602C2#8165711yescharacteristic
603C2#8193011yescharacteristic
604C2#8193111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1991C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1991D3
hypocenterC3#22972
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
1. #83025C2semidirect
4. C3×#1991C2semidirect
6. #69757C2semidirect
7. #69540C2semidirect
10. #68861C2semidirect
11. #68860C2semidirect
12. #68788C2semidirect
13. #68787C2semidirect
27. #5066C4semidirect
64. #1991D3semidirect
83. C3×#125C2×C4semidirect
135. #808C2×C4semidirect
137. #807C2×C4semidirect
138. C4⋊(C3⋊C8)C2×C4semidirect
268. #125C4×D3semidirect
596. C3#22972semidirect

Maximal subgroups

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

subgroupindex
#830252
C2×#50662
#742282
C3×#19912
C4×#8212
#697572
#695402
#688952
#688942
#688612
#688602
#687882
#687872
#687822
#229723