Finite Group Databasebeta

← #10854152

Subgroups — #10854152

Order 768.

714 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76302C211yesmaximal, characteristic
2#74188C211yesmaximal, characteristic, complemented
3C3×#2092C211yesmaximal, characteristic
4#70610C211yesmaximal, characteristic, complemented
5#70606C211yesmaximal, characteristic, complemented
6#70564C211yesmaximal, characteristic, complemented
7#70563C211yesmaximal, characteristic, complemented
8#70194C211yesmaximal, characteristic
9#70181C211yesmaximal, characteristic
10#70153C211yesmaximal, characteristic
11Dic3×#102C211yesmaximal, characteristic
12#69653C211yesmaximal, characteristic
13#69540C211yesmaximal, characteristic, complemented
14#68624C211yesmaximal, characteristic, complemented
15#68617C211yesmaximal, characteristic, complemented
16#26087—31nomaximal, Sylow
17C2×D4×Dic3C2211yescharacteristic
18C2×#877—21no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C3×#390C2211yescharacteristic
24C3×#382C2211yescharacteristic
25#5221C2211yescharacteristic
26C2×#832C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4984C2211yescharacteristic
33#4977C2211yescharacteristic
34#4975C2211yescharacteristic
35#4975C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38C2×#808C2211yescharacteristic
39C2×#807C2211yescharacteristic
40C2×#807C2211yescharacteristic
41#4945—21no—
42#4944—21no—
43#4942—21no—
44C4×C4⋊Dic3C2211yescharacteristic
45C4×Dic3⋊C4C2211yescharacteristic
46#4928C2211yescharacteristic
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48#4659—21no—
49#4658—21no—
50#4650—21no—
51#4647—21no—
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2571—31no—
61#2354—31no—
62#2092—31no—
63#2092D311yescharacteristic
64#2067—31no—
65#2067—31no—
66#1949—31no—
67#1949—31no—
68#1948—31no—
69#1939—31no—
70#1921—31no—
71#1921—31no—
72#1845—31no—
73#1802—31no—
74C4×#102—31no—
75C22×C3×D4C2311yescharacteristic
76C23×Dic3—21no—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42C2311yescharacteristic
85C2×#217—21no—
86C2×#217—21no—
87C2×#217—21no—
88D4×Dic3—22no—
89D4×Dic3Q814yes—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4D411yescharacteristic
101C2×Dic3⋊C4D411yescharacteristic
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#877—44no—
110C3×#107—22no—
111C3×#106—41no—
112C3×#102—22no—
113C3×#102—41no—
114C3×#95—21no—
115C3×#95—21no—
116#832—22no—
117#832—41no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95D411yescharacteristic
126C3⋊#95D411yescharacteristic
127#818—22no—
128Dic3⋊C8—22no—
129Dic3⋊C8—22no—
130C8×Dic3—22no—
131#808—41no—
132#807—41no—
133#807—41no—
134C4⋊(C3⋊C8)—22no—
135C2×C4×D4—31no—
136C2×#117—61no—
137C2×#107—31no—
138C2×#107—31no—
139C2×#107—31no—
140C2×#107C2×D311yescharacteristic
141C2×#106—31no—
142C2×#106C2×D311yescharacteristic
143C2×C4⋊C8—31no—
144C2×C4⋊C8—31no—
145C2×C4⋊C8—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102C2×D311yescharacteristic
149C2×#102C2×D311yescharacteristic
150C2×C4×C8—31no—
151#400—61no—
152#400—61no—
153#390—31no—
154#390C2×D311yescharacteristic
155#389—31no—
156#389—31no—
157#389—61no—
158#388—61no—
159#388—61no—
160#384—31no—
161#384—31no—
162#382—31no—
163#382—61no—
164#382C2×D311yescharacteristic
165#380—61no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#336—31no—
177#336C2×D311yescharacteristic
178C24×C3—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4C2×Q812yes—
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—24no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—42no—
197C22×Dic3—42no—
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8#4112yes—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—81no—
216C3×C42—21no—
217C3×C42—21no—
218#217—24no—
219#217—42no—
220#217—81no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—81no—
229C4⋊Dic3C2×Q812yes—
230Dic3⋊C4—22no—
231Dic3⋊C4—22no—
232Dic3⋊C4—42no—
233Dic3⋊C4—42no—
234Dic3⋊C4—81no—
235Dic3⋊C4—81no—
236Dic3⋊C4—81no—
237C4×Dic3—21no—
238C4×Dic3—21no—
239C4×Dic3—42no—
240C4×Dic3—42no—
241C4×Dic3—42no—
242C4×Dic3C2×Q812yes—
243C4×Dic3#4112yes—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8#4112yes—
250C22×D4C22×D311yescharacteristic
251C23×C4—61no—
252C22×C8—31no—
253C22×C8—31no—
254C22×C8C22×D311yescharacteristic
255C22×C8C22×D311yescharacteristic
256C4×D4—34no—
257C4×D4—62no—
258#117—124no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—61no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C42—31no—
280C2×C42—31no—
281C2×C42—61no—
282C2×C42—61no—
283C2×C42—61no—
284C2×C42C22×D311yescharacteristic
285#107—22no—
286#107—62no—
287#107—121no—
288#107—121no—
289#106—41no—
290#106—121no—
291C4⋊C8—62no—
292C4⋊C8—62no—
293C4⋊C8—62no—
294#102—22no—
295#102—41no—
296#102—62no—
297#102—121no—
298C4×C8—62no—
299#95—21no—
300#95—21no—
301#95—31no—
302#95—31no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
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—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4C22×Q811yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4H(5,𝔽2)11yescharacteristic
342C2×C3×C42−1+411yescharacteristic
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—22no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—48no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3C22⋊Q812yes—
365C2×Dic3C22⋊Q812yes—
366C2×Dic3#13512yes—
367C2×Dic3AGL(1,ℤ/8)12yes—
368C3×C8—22no—
369C3×C8—41no—
370C3⋊C8—22no—
371C3⋊C8—41no—
372C24—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4D3×Q812yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—64no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—122no—
392C22×C4—122no—
393C22×C4D3×D411yescharacteristic
394C22×C4D3×D411yescharacteristic
395C22×C4C23×D311yescharacteristic
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—32no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8#23712yes—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—32no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—61no—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—122no—
425C4⋊C4—122no—
426C4⋊C4—241no—
427C4⋊C4—241no—
428C4⋊C4—241no—
429C4⋊C4—241no—
430C22⋊C4—42no—
431C22⋊C4—64no—
432C22⋊C4—81no—
433C22⋊C4—122no—
434C22⋊C4—241no—
435C42—21no—
436C42—21no—
437C42—32no—
438C42—32no—
439C42—61no—
440C42—61no—
441C42—122no—
442C42—122no—
443C42—122no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3C2×C22⋊Q811yescharacteristic
452C22×C3#54711yescharacteristic
453C22×C3#55011yescharacteristic
454C22×C3C2×#13511yescharacteristic
455C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
456C22×C3#57611yescharacteristic
457C22×C3#57811yescharacteristic
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—81no—
461C3×C4—81no—
462C3×C4—81no—
463C3×C4#53311yescharacteristic
464C3×C4#54411yescharacteristic
465C3×C4#55211yescharacteristic
466C3×C4#55311yescharacteristic
467Dic3—22no—
468Dic3—42no—
469Dic3—41no—
470Dic3—41no—
471Dic3—81no—
472Dic3—81no—
473Dic3—81no—
474Dic3#47814yes—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23C2×D3×D411yescharacteristic
483D4—22no—
484D4—22no—
485D4—41no—
486D4—41no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—32no—
494C2×C4—32no—
495C2×C4—32no—
496C2×C4—32no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—62no—
515C2×C4—128no—
516C2×C4—121no—
517C2×C4—121no—
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×C4C2×#23711yescharacteristic
528C2×C4#100411yescharacteristic
529C2×C4C2×D3×Q811yescharacteristic
530C2×C4D3×#4111yescharacteristic
531C2×C4D3×#4111yescharacteristic
532C2×C4#101011yescharacteristic
533C8—22no—
534C8—41no—
535C8—62no—
536C8—121no—
537C2×C3—82no—
538C2×C3#244111yescharacteristic
539C2×C3C2×#47811yescharacteristic
540C2×C3#308311yescharacteristic
541C2×C3#334111yescharacteristic
542C2×C3#335311yescharacteristic
543C2×C3#339311yescharacteristic
544C2×C3#339911yescharacteristic
545C22—42no—
546C22—42no—
547C22—42no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22D3×C22⋊Q811yescharacteristic
553C22#565511yescharacteristic
554C22#571811yescharacteristic
555C22D3×#13511yescharacteristic
556C22#577111yescharacteristic
557C22D3×AGL(1,ℤ/8)11yescharacteristic
558C22#577411yescharacteristic
559C4—34no—
560C4—41no—
561C4—41no—
562C4—62no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—122no—
567C4—121no—
568C4—121no—
569C4—241no—
570C4—241no—
571C4—241no—
572C4#562211yescharacteristic
573C4#562311yescharacteristic
574C4#565011yescharacteristic
575C4#566111yescharacteristic
576C3#2608711yescharacteristic, complemented, Sylow
577C2—82no—
578C2#7802111yescharacteristic
579C2D3×#47811yescharacteristic
580C2#8117011yescharacteristic
581C2#8136711yescharacteristic
582C2#8138911yescharacteristic
583C2#8181311yescharacteristic
584C2#8184011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2092C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2092D3
hypocenterC3#26087
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2092

Complements

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

NH ≅ G/Ntype
2. #74188C2semidirect
4. #70610C2semidirect
5. #70606C2semidirect
6. #70564C2semidirect
7. #70563C2semidirect
13. #69540C2semidirect
14. #68624C2semidirect
15. #68617C2semidirect
576. C3#26087semidirect

Maximal subgroups

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

subgroupindex
#763022
#741882
C3×#20922
#706102
#706062
#705642
#705632
#701942
#701812
#701532
Dic3×#1022
#696532
#695402
#686242
#686172
#260873