Finite Group Databasebeta

← #10868190

Subgroups — #10868190

Order 768.

702 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76615C211yesmaximal, characteristic
2#74365C211yesmaximal, characteristic
3C3×#2072C211yesmaximal, characteristic, complemented
4#70275C211yesmaximal, characteristic
5#70275C211yesmaximal, characteristic
6#70238C211yesmaximal, characteristic, complemented
7#70220C211yesmaximal, characteristic, complemented
8#69682C211yesmaximal, characteristic, complemented
9#69567C211yesmaximal, characteristic, complemented
10#68875C211yesmaximal, characteristic
11#68871C211yesmaximal, characteristic
12#68860C211yesmaximal, characteristic, complemented
13#68787C211yesmaximal, characteristic, complemented
14#68622C211yesmaximal, characteristic, complemented
15#68616C211yesmaximal, characteristic
16#27053—31nomaximal, Sylow
17C2×#946C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×C3×#103C2211yescharacteristic
21C2×C3×#103C2211yescharacteristic
22C3×#391C2211yescharacteristic
23C3×#384C2211yescharacteristic
24C2×#835C2211yescharacteristic
25C2×#835C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×#815C2211yescharacteristic
30#4994—21no—
31#4990—21no—
32#4987C2211yescharacteristic
33#4987C2211yescharacteristic
34#4986—21no—
35#4984C2211yescharacteristic
36#4978—21no—
37Dic3×C4⋊C4C2211yescharacteristic
38#4972C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C2×#807C2211yescharacteristic
42C4×(C2×D3)⋊C4C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44#4928C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4668—21no—
47#4666—21no—
48#4652—21no—
49#4647—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2497—31no—
61#2360—31no—
62#2072D311yescharacteristic, complemented
63#2066—31no—
64#2036—31no—
65#2036—31no—
66#1947—31no—
67#1943—31no—
68#1924—31no—
69#1919—31no—
70#1918—31no—
71#1865—31no—
72#1853—31no—
73#1823—31no—
74#1763—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C4×D3—21no—
77C22×C4×D3C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86#946—22no—
87#946—22no—
88Q8×Dic3—22no—
89Q8×Dic3—41no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4D411yescharacteristic
94C2×(C2×D3)⋊C4D411yescharacteristic
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111D3×C4⋊C4—22no—
112D3×C4⋊C4—41no—
113C3×#103—22no—
114C3×#103—22no—
115C3×#95—21no—
116C3×#95—21no—
117#835—22no—
118#835—41no—
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(C2×D3)⋊C8—22no—
129(C2×D3)⋊C8—41no—
130#817—22no—
131#815—41no—
132#808—41no—
133#807—22no—
134#807—22no—
135C4⋊(C3⋊C8)—22no—
136C2×C22⋊Q8—31no—
137C2×C4×Q8—31no—
138C22×C4⋊C4—31no—
139C2×#107—31no—
140C2×#107—31no—
141C2×#106—31no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103C2×D311yescharacteristic
147C2×#103C2×D311yescharacteristic
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150C2×#97—31no—
151#400—61no—
152#396—61no—
153#391C2×D311yescharacteristic
154#389—31no—
155#389—31no—
156#389—31no—
157#389—61no—
158#388—61no—
159#387—61no—
160#387—61no—
161#385—61no—
162#384C2×D311yescharacteristic
163#380—31no—
164#380—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167C4×C22⋊C4—31no—
168C4×C22⋊C4—61no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336C2×D311yescharacteristic
179C23×D3C2×D411yescharacteristic
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8#4112yes—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×C4×D3—81no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—81no—
222C3×C4⋊C4—81no—
223C3×C42—21no—
224C3×C42—21no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—22no—
227(C2×D3)⋊C4—22no—
228(C2×D3)⋊C4—81no—
229(C2×D3)⋊C4—81no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—42no—
237C4⋊Dic3—81no—
238C4⋊Dic3#4112yes—
239C4⋊Dic3#4112yes—
240Dic3⋊C4—24no—
241Dic3⋊C4—42no—
242Dic3⋊C4—42no—
243Dic3⋊C4—81no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246C4×Dic3—42no—
247C4×Dic3—81no—
248C4×Dic3—81no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8#4112yes—
255C22×Q8C22×D311yescharacteristic
256C23×C4—31no—
257C23×C4—61no—
258C22×C8—31no—
259C22×C8—31no—
260C22×C8C22×D311yescharacteristic
261C22×C8C22×D311yescharacteristic
262C22⋊Q8—62no—
263C22⋊Q8—62no—
264C4×Q8—62no—
265C4×Q8—121no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—121no—
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C22⋊C4—31no—
284C2×C22⋊C4—31no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C42—31no—
289C2×C42—61no—
290C2×C42—61no—
291C2×C42—61no—
292C2×C42C22×D311yescharacteristic
293#107—62no—
294#107—62no—
295#106—62no—
296#106—121no—
297C4⋊C8—62no—
298#103—22no—
299#103—22no—
300#103—62no—
301#103—121no—
302C22⋊C8—62no—
303C22⋊C8—121no—
304#97—121no—
305#95—21no—
306#95—21no—
307#95—31no—
308#95—31no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316C23×C3C22×D411yescharacteristic
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—21no—
322C22×D3—21no—
323C22×D3#13712yes—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—41no—
327C3×Q8—41no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
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#13712yes—
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4H(5,𝔽2)11yescharacteristic
351C2×C3×C42−1+411yescharacteristic
352C2×Dic3—21no—
353C2×Dic3—22no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—22no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C4×D3—41no—
374C4×D3—41no—
375C4×D3—41no—
376C4×D3—41no—
377C4×D3—84no—
378C3×C8—41no—
379C3×C8—41no—
380C3⋊C8—22no—
381C3⋊C8—41no—
382C24—31no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8#23712yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4—241no—
409C22×C4D3×D411yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4C23×D311yescharacteristic
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—32no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—32no—
431C4⋊C4—32no—
432C4⋊C4—64no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—81no—
440C4⋊C4—81no—
441C4⋊C4—122no—
442C4⋊C4—122no—
443C4⋊C4—122no—
444C4⋊C4—241no—
445C4⋊C4—241no—
446C22⋊C4—62no—
447C22⋊C4—64no—
448C22⋊C4—62no—
449C22⋊C4—241no—
450C22⋊C4—241no—
451C42—21no—
452C42—21no—
453C42—61no—
454C42—61no—
455C42—122no—
456C42—241no—
457C42—241no—
458C22×C3#52511yescharacteristic
459C22×C3#54611yescharacteristic
460C22×C3#54711yescharacteristic
461C22×C3C2×#13711yescharacteristic
462C22×C3C2×#13711yescharacteristic
463C22×C3#57611yescharacteristic
464C22×C3#57711yescharacteristic
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—22no—
468C2×D3—41no—
469C2×D3—41no—
470C2×D3—41no—
471C2×D3—41no—
472C3×C4—22no—
473C3×C4—41no—
474C3×C4—42no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4#52911yescharacteristic
479C3×C4#53911yescharacteristic
480C3×C4#55311yescharacteristic
481C3×C4#55611yescharacteristic
482Dic3—41no—
483Dic3—42no—
484Dic3—41no—
485Dic3—42no—
486Dic3—41no—
487Dic3—81no—
488Dic3—81no—
489C23—32no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23—61no—
494C23—61no—
495C23—61no—
496C23C2×D3×D411yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—41no—
500Q8—41no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—62no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—244no—
544C2×C4#91812yes—
545C2×C4C2×#23711yescharacteristic
546C2×C4D3×#4111yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C2×C4D3×#4111yescharacteristic
549C2×C4#100911yescharacteristic
550C2×C4#101011yescharacteristic
551C8—41no—
552C8—41no—
553C8—62no—
554C8—121no—
555C2×C3#246611yescharacteristic
556C2×C3#311311yescharacteristic
557C2×C3#311911yescharacteristic
558C2×C3#331411yescharacteristic
559C2×C3#332911yescharacteristic
560C2×C3#334411yescharacteristic
561C2×C3#334611yescharacteristic
562D3—42no—
563C22—62no—
564C22—62no—
565C22—62no—
566C22—121no—
567C22—121no—
568C22—121no—
569C22—121no—
570C22#560611yescharacteristic
571C22#565511yescharacteristic
572C22#567611yescharacteristic
573C22#576211yescharacteristic
574C22C2×#91811yescharacteristic
575C22#577411yescharacteristic
576C22D3×#13711yescharacteristic
577C4—22no—
578C4—41no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—121no—
584C4—122no—
585C4—121no—
586C4—121no—
587C4—122no—
588C4—241no—
589C4—241no—
590C4#564911yescharacteristic
591C4#566511yescharacteristic
592C4#570211yescharacteristic
593C4#570911yescharacteristic
594C3#2705311yescharacteristic, complemented, Sylow
595C2—122no—
596C2#7747111yescharacteristic
597C2#8036411yescharacteristic
598C2#8045811yescharacteristic
599C2#8143811yescharacteristic
600C2#8147811yescharacteristic
601C2#8161711yescharacteristic
602C2#8170011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2072C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2072D3
hypocenterC3#27053
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2072

Complements

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

NH ≅ G/Ntype
3. C3×#2072C2semidirect
6. #70238C2semidirect
7. #70220C2semidirect
8. #69682C2semidirect
9. #69567C2semidirect
12. #68860C2semidirect
13. #68787C2semidirect
14. #68622C2semidirect
62. #2072D3semidirect
594. C3#27053semidirect

Maximal subgroups

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

subgroupindex
#766152
#743652
C3×#20722
#702752
#702382
#702202
#696822
#695672
#688752
#688712
#688602
#687872
#686222
#686162
#270533