Finite Group Databasebeta

← #10828820

Subgroups — #10828820

Order 768.

1100 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#552C211yesmaximal, characteristic
2C2×#5043C211yesmaximal, characteristic, complemented
3#82373C211yesmaximal, characteristic
4C22×#807C211yesmaximal, characteristic, complemented
5#71739C212yesmaximal, complemented
6#71696C211yesmaximal, characteristic, complemented
7#71695C212yesmaximal, complemented
8#71628C212yesmaximal, complemented
9#71610C211yesmaximal, characteristic
10#71576C212yesmaximal, complemented
11#71553C211yesmaximal, characteristic, complemented
12#24094—31nomaximal, Sylow
13C3×#552—24no—
14C3×#552—24no—
15C2×C3×#126—21no—
16C2×C3×#125—21no—
17C2×C3×#125C2211yescharacteristic
18C2×C3×#123—21no—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×#117—21no—
24C2×C3×#117C2211yescharacteristic
25C22×C3×C4⋊C4C2211yescharacteristic
26C2×#935—21no—
27C22×C4⋊Dic3C2211yescharacteristic
28C23×C3⋊C8C2211yescharacteristic
29C2×#835C2212yes—
30C2×#835C2212yes—
31#5212C2212yes—
32#5212C2211yescharacteristic
33C2×#830C2212yes—
34C2×#830C2211yescharacteristic
35#5043—24no—
36#4981—21no—
37C2×#808C2212yes—
38C2×#807—24no—
39C2×#807C2212yes—
40C2×#807C2212yes—
41#4953—21no—
42#4943—21no—
43#4928C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#552D311yescharacteristic
49C2×#483—31no—
50C22×#107—31no—
51#2486—31no—
52#2059—32no—
53#2036—32no—
54#1892—31no—
55#1869—31no—
56#1863—32no—
57#1854—32no—
58#1823—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4D411yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×Dic3—21no—
65C3×#126—28no—
66C3×#125—28no—
67C3×#125—22no—
68C3×#125—22no—
69C3×#123—28no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8D414yes—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C3×#117—28no—
79C3×#117—22no—
80C3×#117—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D414yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—22no—
103#935—44no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—22no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2312yes—
108C2×C4⋊Dic3C2312yes—
109C2×Dic3⋊C4—22no—
110C22×C3⋊C8—22no—
111C22×C3⋊C8—24no—
112C22×C3⋊C8C2312yes—
113C22×C3⋊C8C2312yes—
114#835—24no—
115#835—24no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118C3⋊#95—21no—
119#830—22no—
120#830—24no—
121#808—24no—
122#807—24no—
123#807—24no—
124C23×C8—31no—
125#552—24no—
126#552—24no—
127C2×#126—21no—
128C2×#125—21no—
129C2×#125C2×D311yescharacteristic
130C2×#123—21no—
131C2×#123—61no—
132C2×C22⋊Q8C2×D311yescharacteristic
133C2×C22⋊Q8C2×D311yescharacteristic
134C2×C4×Q8C2×D311yescharacteristic
135C2×C4×D4C2×D311yescharacteristic
136C2×#117—21no—
137C2×#117C2×D311yescharacteristic
138C22×C4⋊C4—31no—
139C22×C4⋊C4C2×D311yescharacteristic
140#483—64no—
141C2×#107—32no—
142C2×#107—32no—
143C2×#107—64no—
144C2×#106—32no—
145C2×#103—32no—
146C2×#103—32no—
147C2×C22⋊C8—31no—
148C2×C22⋊C8—32no—
149#398—61no—
150#397—61no—
151#389—32no—
152#381—61no—
153#380—31no—
154#380—32no—
155#340—32no—
156#340—32no—
157#340—32no—
158#336—32no—
159C24×C3C2×D411yescharacteristic
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8C2×D412yes—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—42no—
191C22×Dic3—42no—
192C22×Dic3—42no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4C2×D412yes—
204C3×C4⋊C4C2×D412yes—
205C3×C4⋊C4C2×D412yes—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—22no—
211C3×C42—24no—
212C3×C42—24no—
213C3×C42—21no—
214C3×C42—21no—
215#217—44no—
216#217—42no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—42no—
222Dic3⋊C4—44no—
223C2×C3⋊C8—28no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8#4114yes—
230C22×Q8C22×D311yescharacteristic
231C22×D4#6711yescharacteristic
232C23×C4—21no—
233C23×C4—61no—
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D311yescharacteristic
236C22×C8—32no—
237C22×C8—32no—
238C22×C8—64no—
239C22×C8—62no—
240#126—28no—
241#125—28no—
242#125—22no—
243#125—22no—
244#123—28no—
245#123—124no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8#6714yes—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×D4—22no—
253C4×D4—22no—
254#117—28no—
255#117—22no—
256#117—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4#6711yescharacteristic
271C2×C4⋊C4#6714yes—
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4#6711yescharacteristic
281C2×C22⋊C4#6711yescharacteristic
282C2×C42—21no—
283C2×C42—21no—
284C2×C42C22×D311yescharacteristic
285#107—64no—
286#107—64no—
287#106—64no—
288#103—64no—
289#103—64no—
290C22⋊C8—64no—
291C22⋊C8—62no—
292#95—62no—
293#95—61no—
294#95—62no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3#12012yes—
303C23×C3#13512yes—
304C23×C3AGL(1,ℤ/8)12yes—
305C23×C3C22×D411yescharacteristic
306C3×Q8—24no—
307C3×Q8—41no—
308C3×Q8—41no—
309C3×D4—44no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—48no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4#12012yes—
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C2×#4112yes—
339C2×C3×C42−1+412yes—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—84no—
347C3⋊C8—24no—
348C3⋊C8—42no—
349C24C2×#6711yescharacteristic
350C24C2×#6711yescharacteristic
351C2×Q8—21no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8C2×#6712yes—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—122no—
374C22×C4—122no—
375C22×C4—122no—
376C22×C4C2×#6711yescharacteristic
377C22×C4C2×#6711yescharacteristic
378C22×C4C2×#6711yescharacteristic
379C22×C4C2×#6711yescharacteristic
380C22×C4C2×#6712yes—
381C22×C4C2×#6712yes—
382C22×C4C23×D311yescharacteristic
383C2×C8—34no—
384C2×C8—68no—
385C2×C8—62no—
386C2×C8—62no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—62no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—124no—
405C4⋊C4—122no—
406C4⋊C4C2×#6712yes—
407C4⋊C4C2×#6712yes—
408C4⋊C4C2×#6712yes—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—124no—
415C22⋊C4—122no—
416C42—24no—
417C42—24no—
418C42—21no—
419C42—21no—
420C22×C3—24no—
421C22×C3—24no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3#44914yes—
430C22×C3C2×#12011yescharacteristic
431C22×C3#54712yes—
432C22×C3C2×#13511yescharacteristic
433C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
434C22×C3#57812yes—
435C3×C4—22no—
436C3×C4—41no—
437C3×C4—42no—
438C3×C4—41no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—41no—
442C3×C4—42no—
443C3×C4#52511yescharacteristic
444C3×C4#53611yescharacteristic
445C3×C4#54712yes—
446Dic3—82no—
447Dic3—82no—
448C23—22no—
449C23—22no—
450C23—21no—
451C23—21no—
452C23—22no—
453C23—22no—
454C23—21no—
455C23#94912yes—
456C23#95012yes—
457C23#95312yes—
458C23C22×#6711yescharacteristic
459Q8—24no—
460Q8—41no—
461Q8—41no—
462D4—44no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—48no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—244no—
495C2×C4#95312yes—
496C2×C4D3×#4112yes—
497C2×C4#101012yes—
498C2×C4C22×#6711yescharacteristic
499C2×C4C22×#6711yescharacteristic
500C8—64no—
501C8—122no—
502C2×C3—24no—
503C2×C3—81no—
504C2×C3#240611yescharacteristic
505C2×C3C2×#44911yescharacteristic
506C2×C3#301111yescharacteristic
507C2×C3#333612yes—
508C2×C3#334312yes—
509C22—22no—
510C22—24no—
511C22—24no—
512C22—22no—
513C22—41no—
514C22—41no—
515C22—42no—
516C22—42no—
517C22—41no—
518C22#523814yes—
519C22C2×#94911yescharacteristic
520C22C2×#95011yescharacteristic
521C22#582412yes—
522C22#583812yes—
523C22C2×#95311yescharacteristic
524C4—22no—
525C4—41no—
526C4—42no—
527C4—42no—
528C4—41no—
529C4—42no—
530C4—41no—
531C4—42no—
532C4—242no—
533C4—242no—
534C4#582412yes—
535C4#582511yescharacteristic
536C4#582811yescharacteristic
537C3#2409411yescharacteristic, complemented, Sylow
538C2—24no—
539C2—81no—
540C2#8359012yes—
541C2#8359212yes—
542C2C2×#523811yescharacteristic
543C2#8409011yescharacteristic
544C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#552C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenterC2×#552D3
hypocenterC3#24094
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
2. C2×#5043C2semidirect
4. C22×#807C2semidirect
5. #71739C2semidirect
6. #71696C2semidirect
7. #71695C2semidirect
8. #71628C2semidirect
10. #71576C2semidirect
11. #71553C2semidirect
537. C3#24094semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5522
C2×#50432
#823732
C22×#8072
#717392
#716962
#716952
#716282
#716102
#715762
#715532
#240943