Finite Group Databasebeta

← #10853382

Subgroups — #10853382

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2357C211yesmaximal, characteristic
2#73713C211yesmaximal, characteristic, complemented
3C4×#832C211yesmaximal, characteristic
4#71605C211yesmaximal, characteristic
5#69653C211yesmaximal, characteristic
6#69652C211yesmaximal, characteristic
7#69639C211yesmaximal, characteristic
8#69638C211yesmaximal, characteristic
9#69547C211yesmaximal, characteristic, complemented
10#69540C211yesmaximal, characteristic, complemented
11#69491C211yesmaximal, characteristic, complemented
12#69489C211yesmaximal, characteristic, complemented
13#69484C211yesmaximal, characteristic, complemented
14#69459C211yesmaximal, characteristic, complemented
15#69458C211yesmaximal, characteristic, complemented
16#26030—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#870C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C3×#390C2211yescharacteristic
21C3×#390C2211yescharacteristic
22C3×#388—21no—
23C3×#388—21no—
24C3×#382C2211yescharacteristic
25C3×#381—21no—
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C22⋊C4—21no—
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34#4982—21no—
35#4979—21no—
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4936C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40#4928C2211yescharacteristic
41#4928C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4×C3⋊C8C2211yescharacteristic
46#4647—21no—
47#4646—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2550—31no—
61#2357D311yescharacteristic
62#2067—31no—
63#2065—31no—
64#2025—31no—
65#2023—31no—
66#1989—31no—
67#1987—31no—
68#1949—31no—
69#1939—31no—
70#1919—31no—
71#1918—31no—
72#1835—31no—
73#1804—31no—
74C4×#102—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×Dic6—21no—
78C3×C4×D4—22no—
79C3×C4×D4Q814yescomplemented
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—21no—
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#870—22no—
110#870—41no—
111#869—22no—
112#869—41no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95D411yescharacteristic
118C3×#95D411yescharacteristic
119#832—22no—
120#832—22no—
121#832—41no—
122#832—41no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#808—22no—
129#807—22no—
130C4⋊(C3⋊C8)—22no—
131C4⋊(C3⋊C8)—41no—
132C4⋊(C3⋊C8)—41no—
133C4×C3⋊C8—22no—
134C2×C4⋊Q8—31no—
135C2×#125—31no—
136C2×C4×D4C2×D311yescharacteristic
137C2×#107—31no—
138C2×#106—31no—
139C2×C4⋊C8—31no—
140C2×C4⋊C8—31no—
141C2×C4⋊C8—31no—
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×C4×C8—31no—
147#398—61no—
148#395—61no—
149#391—61no—
150#390C2×D311yescharacteristic
151#390C2×D311yescharacteristic
152#389—31no—
153#389—31no—
154#389—61no—
155#388—21no—
156#388—21no—
157#382—31no—
158#382C2×D311yescharacteristic
159#381—21no—
160C4×C4⋊C4—31no—
161C4×C4⋊C4C2×D311yescharacteristic
162C4×C4⋊C4C2×D311yescharacteristic
163C4×C4⋊C4C2×D311yescharacteristic
164C4×C22⋊C4—21no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#336—31no—
176#336—31no—
177C24×C3—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4C2×Q812yes—
184C22×C3×C4—21no—
185C22×C3×C4—24no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—81no—
189C22×C3×C4—81no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C2×Dic6—42no—
202C2×Dic6—42no—
203C2×Dic6—81no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—41no—
211C3×C4⋊C4C2×Q812yes—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—81no—
216C3×C22⋊C4—81no—
217C3×C42—21no—
218C3×C42—21no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—81no—
222C3×C42—81no—
223C3×C42C2×Q812yes—
224C3×C42#4112yes—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—42no—
232C4⋊Dic3#4112yes—
233C4⋊Dic3#4112yes—
234Dic3⋊C4—42no—
235Dic3⋊C4—42no—
236Dic3⋊C4—42no—
237Dic3⋊C4—81no—
238C4×Dic3—81no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—21no—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C22×Q8—61no—
252C22×D4C22×D311yescharacteristic
253C23×C4—21no—
254C22×C8—31no—
255C22×C8—31no—
256C22×C8—31no—
257C22×C8—31no—
258C4⋊Q8—62no—
259C4⋊Q8—121no—
260#125—62no—
261#125—121no—
262C4×D4—22no—
263C4×D4Dic614yescomplemented
264C2×C4⋊C4—31no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4#6711yescharacteristic
274C2×C4⋊C4#6711yescharacteristic
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—61no—
286C2×C42C22×D311yescharacteristic
287C2×C42C22×D311yescharacteristic
288C2×C42C22×D311yescharacteristic
289#107—62no—
290#106—62no—
291C4⋊C8—62no—
292C4⋊C8—121no—
293C4⋊C8—121no—
294#102—62no—
295#102—62no—
296#102—121no—
297#102—121no—
298C4×C8—62no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95#6711yescharacteristic
309#95#6711yescharacteristic
310C23×C3—22no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3C22×D411yescharacteristic
318C3×D4—22no—
319C3×D4—22no—
320C3×D4—41no—
321C3×D4—41no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
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×C4—84no—
347C2×C3×C4C22⋊Q812yes—
348C2×C3×C4C22⋊Q812yes—
349C2×C3×C4#13512yes—
350C2×C3×C4C22×Q811yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4H(5,𝔽2)11yescharacteristic
355C2×C3×C42−1+411yescharacteristic
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368Dic6—84no—
369C3⋊C8—41no—
370C3⋊C8—41no—
371C3⋊C8—41no—
372C3⋊C8—41no—
373C24—21no—
374C2×Q8—122no—
375C2×Q8—122no—
376C2×Q8—241no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4C2×Dic612yes—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4C2×#6711yescharacteristic
394C22×C4C2×#6711yescharacteristic
395C22×C4C2×#6711yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C2×#6711yescharacteristic
398C22×C4C2×#6711yescharacteristic
399C22×C4C23×D311yescharacteristic
400C2×C8—61no—
401C2×C8—61no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—32no—
419C4⋊C4—32no—
420C4⋊C4—41no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C4⋊C4—122no—
428C4⋊C4—122no—
429C4⋊C4—122no—
430C4⋊C4—122no—
431C4⋊C4—241no—
432C4⋊C4C2×Dic612yes—
433C22⋊C4—24no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—81no—
437C22⋊C4—81no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—81no—
443C42—81no—
444C42—241no—
445C42C2×Dic612yes—
446C42#23512yes—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3C2×C22⋊Q811yescharacteristic
455C22×C3#54611yescharacteristic
456C22×C3#54811yescharacteristic
457C22×C3C2×#13511yescharacteristic
458C22×C3#57511yescharacteristic
459C22×C3#57611yescharacteristic
460C22×C3#57711yescharacteristic
461C3×C4—22no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—42no—
465C3×C4—41no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—81no—
469C3×C4—81no—
470C3×C4#53311yescharacteristic
471C3×C4#54411yescharacteristic
472C3×C4#55511yescharacteristic
473C3×C4#55611yescharacteristic
474Dic3—42no—
475Dic3—42no—
476Dic3—81no—
477Dic3—81no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23C22×#6711yescharacteristic
486Q8—244no—
487D4—22no—
488D4—22no—
489D4—41no—
490D4—41no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
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—62no—
516C2×C4—62no—
517C2×C4—84no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4#92412yes—
529C2×C4#92412yes—
530C2×C4#95012yes—
531C2×C4C22×Dic611yescharacteristic
532C2×C4C2×#23511yescharacteristic
533C2×C4D3×#4111yescharacteristic
534C2×C4D3×#4111yescharacteristic
535C2×C4#100911yescharacteristic
536C2×C4#101011yescharacteristic
537C8—121no—
538C8—121no—
539C8—121no—
540C8—121no—
541C2×C3—82no—
542C2×C3#243911yescharacteristic
543C2×C3#307611yescharacteristic
544C2×C3#308111yescharacteristic
545C2×C3#330111yescharacteristic
546C2×C3#332211yescharacteristic
547C2×C3#336111yescharacteristic
548C2×C3#337111yescharacteristic
549C22—42no—
550C22—42no—
551C22—42no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22C2×#92411yescharacteristic
557C22C2×#95011yescharacteristic
558C22#582211yescharacteristic
559C22#582911yescharacteristic
560C22#583111yescharacteristic
561C22#583511yescharacteristic
562C22#583611yescharacteristic
563C4—22no—
564C4—22no—
565C4—22no—
566C4—42no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—122no—
573C4—122no—
574C4—241no—
575C4—241no—
576C4#551711yescharacteristic
577C4#551911yescharacteristic
578C4#553211yescharacteristic
579C4#553311yescharacteristic
580C3#2603011yescharacteristic, complemented, Sylow
581C2—82no—
582C2#8224811yescharacteristic
583C2#8250711yescharacteristic
584C2#8253911yescharacteristic
585C2#8361711yescharacteristic
586C2#8361811yescharacteristic
587C2#8371311yescharacteristic
588C2#8371811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2357C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2357D3
hypocenterC3#26030
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. #73713C2semidirect
9. #69547C2semidirect
10. #69540C2semidirect
11. #69491C2semidirect
12. #69489C2semidirect
13. #69484C2semidirect
14. #69459C2semidirect
15. #69458C2semidirect
79. C3×C4×D4Q8semidirect
263. C4×D4Dic6semidirect
580. C3#26030semidirect

Maximal subgroups

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

subgroupindex
C3×#23572
#737132
C4×#8322
#716052
#696532
#696522
#696392
#696382
#695472
#695402
#694912
#694892
#694842
#694592
#694582
#260303