Finite Group Databasebeta

← #10801161

Subgroups — #10801161

Order 768.

1282 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87296C211yesmaximal, characteristic, complemented
2C3×#3018C211yesmaximal, characteristic
3#83999C212yesmaximal, complemented
4#83276C211yesmaximal, characteristic, complemented
5#72146C211yesmaximal, characteristic, complemented
6#72116C212yesmaximal, complemented
7#71817C211yesmaximal, characteristic, complemented
8#70375C212yesmaximal, complemented
9#70370C212yesmaximal, complemented
10#70135C211yesmaximal, characteristic, complemented
11#70133C211yesmaximal, characteristic, complemented
12#21478—31nomaximal, Sylow
13C3×#585C2211yescharacteristic
14C3×#575C2212yes—
15C3×#532C2211yescharacteristic
16#5819—24no—
17#5819—22no—
18#5819C2211yescharacteristic
19#5819C2212yes—
20C2×#948—24no—
21C2×#948—22no—
22C2×#948C2212yes—
23C2×#948C2211yescharacteristic
24C2×#948C2211yescharacteristic, complemented
25C3×#456C412yescomplemented
26C3×#455C412yescomplemented
27C3×#454C412yescomplemented
28C3×#453C412yescomplemented
29C2×C3×#104C2212yes—
30C3×#413C2211yescharacteristic
31#5240C2211yescharacteristic
32#5236C2212yes—
33#5236C2212yescomplemented
34#5234C2211yescharacteristic
35#5234C2211yescharacteristic, complemented
36#5233C2212yes—
37#5197—24no—
38#5196—24no—
39#5125C2212yescomplemented
40#5122C2212yescomplemented
41#5011—22no—
42#5011—22no—
43#4924C2211yescharacteristic, complemented
44C2×#803C2211yescharacteristic, complemented
45#4559C2212yescomplemented
46#4558C2212yescomplemented
47#4523C2212yescomplemented
48#3573—31no—
49#3018D311yescharacteristic
50#2955—32no—
51#2932—31no—
52#1884—31no—
53#1881—32no—
54#1861—32no—
55#1858—31no—
56#1794—31no—
57#1765—32no—
58#1757—31no—
59C3×2−1+4—22no—
60C3×2−1+4C2×C412yes—
61C3×H(5,𝔽2)—22no—
62C3×H(5,𝔽2)—22no—
63C3×H(5,𝔽2)C2×C412yes—
64C2×C3×#41—24no—
65C2×C3×#41—22no—
66C2×C3×#41C2311yescharacteristic
67C2×C3×#41C2312yes—
68C2×C3×#41C2311yescharacteristic
69C3×#137—22no—
70C3×#137C2×C414yes—
71C3×AGL(1,ℤ/8)—22no—
72C3×AGL(1,ℤ/8)C2×C414yes—
73C3×#135—24no—
74C2×C3×M16C2312yes—
75C3×C4⋊Q8C2×C411yescharacteristic
76C3×#127C2×C411yescharacteristic
77C3×#124C2×C411yescharacteristic
78C3×#124C2×C411yescharacteristic
79C3×#121—22no—
80C3×C4×D4—22no—
81C2×C3×C42C2311yescharacteristic
82#948—24no—
83#948—28no—
84#948—24no—
85#948—24no—
86#948—24no—
87#948—22no—
88#948—22no—
89#948—24no—
90#948—22no—
91#948—22no—
92#948—22no—
93#948—22no—
94#948D414yescomplemented
95#948D414yescomplemented
96#926C2312yes—
97C2×#208—28no—
98C2×#208—24no—
99C2×#208—22no—
100C2×#208—24no—
101C2×#208—22no—
102C2×#208—21no—
103C2×#208D412yes—
104C2×#208C2311yescharacteristic
105C2×#208C2312yes—
106C2×#208C2311yescharacteristic
107C22×C3⋊C8—24no—
108C22×C3⋊C8—22no—
109C22×C3⋊C8—21no—
110C22×C3⋊C8D412yes—
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C3×#104C2×C414yes—
114C3×#104C2×C414yes—
115C3×#101C2×C412yes—
116C3×#100C2×C412yes—
117#837—24no—
118#837—24no—
119#835—22no—
120#834—22no—
121#832—22no—
122#830—22no—
123#830—22no—
124#819—24no—
125#815—22no—
126C8×Dic3—22no—
127C4⋊(C3⋊C8)—22no—
128#803—21no—
129#803—21no—
130#585C2×D311yescharacteristic
131#575C2×D312yes—
132#568—31no—
133#568—32no—
134#568—62no—
135#568—64no—
136C2×#131—31no—
137C2×#131—32no—
138C2×#131—31no—
139C2×#131—62no—
140C2×#131—64no—
141#532C2×D311yescharacteristic
142#456Dic312yescomplemented
143#455Dic312yescomplemented
144#454Dic312yescomplemented
145#453Dic312yescomplemented
146#444—64no—
147#443—64no—
148#435—62no—
149#435—62no—
150#430—32no—
151#423—31no—
152#421—32no—
153#421—32no—
154C2×#104C2×D312yes—
155#416—32no—
156#413C2×D311yescharacteristic
157#410—31no—
158#408—31no—
159#408—31no—
160#405—32no—
161C2×#97—31no—
162#344—32no—
163#341—32no—
164#339—32no—
165C3×#41—24no—
166C3×#41—28no—
167C3×#41—24no—
168C3×#41—24no—
169C3×#41—24no—
170C3×#41—24no—
171C3×#41—22no—
172C3×#41—22no—
173C3×#41—22no—
174C3×#41—22no—
175C3×#41—22no—
176C3×#41—22no—
177C3×#41C22×C414yes—
178C3×#41C2×D414yes—
179C2×C3×Q8—22no—
180C2×C3×Q8—24no—
181C2×C3×Q8C22×C411yescharacteristic
182C2×C3×Q8C22×C411yescharacteristic
183C2×C3×Q8#4112yes—
184C2×C3×D4—28no—
185C2×C3×D4—22no—
186C2×C3×D4—24no—
187C2×C3×D4—22no—
188C2×C3×D4—24no—
189C2×C3×D4—21no—
190C2×C3×D4—22no—
191C2×C3×D4C22×C411yescharacteristic
192C2×C3×D4C22×C411yescharacteristic
193C2×C3×D4#4112yes—
194C2×C3×D4#4112yes—
195C22×C3×C4—24no—
196C22×C3×C4—22no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2411yescharacteristic
201C3×Q16—24no—
202C3×SD16—24no—
203C3×SD16—24no—
204C3×D8—24no—
205C3×M16—22no—
206C3×M16C22×C414yes—
207C2×C3×C8—22no—
208C3×C4⋊C4—22no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—22no—
211C3×C42—21no—
212C3×C42C22×C411yescharacteristic
213C3×C42C22×C411yescharacteristic
214#217—22no—
215C4⋊Dic3—22no—
216C4×Dic3—22no—
217#208—24no—
218#208—28no—
219#208—22no—
220#208—22no—
221#208—24no—
222#208—24no—
223#208—24no—
224#208—22no—
225#208—41no—
226#208—41no—
227#208C2×D414yes—
228C2×C3⋊C8—24no—
229C2×C3⋊C8—28no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—24no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—24no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—42no—
239C2×C3⋊C8C2×D412yes—
240C2×C3⋊C8C2×D411yescharacteristic
241C2×C3⋊C8C2×D411yescharacteristic
242C2×C3⋊C8C2×D412yes—
243C2×C3⋊C8C2×D412yes—
2442−1+4—22no—
2452−1+4C2×Dic312yes—
246H(5,𝔽2)—22no—
247H(5,𝔽2)—22no—
248H(5,𝔽2)C2×Dic312yes—
249C2×#41—24no—
250C2×#41—22no—
251C2×#41C22×D312yes—
252C2×#41C22×D311yescharacteristic
253C2×#41C22×D311yescharacteristic
254#137—22no—
255#137C2×Dic314yes—
256AGL(1,ℤ/8)—22no—
257AGL(1,ℤ/8)C2×Dic314yes—
258#135—24no—
259#131—34no—
260#131—34no—
261#131—62no—
262#131—62no—
263#131—62no—
264#131—62no—
265#131—64no—
266#131—62no—
267#131—62no—
268#131—64no—
269#131—64no—
270#131—64no—
271#131—68no—
272#131—64no—
273C2×M16—32no—
274C2×M16—31no—
275C2×M16—32no—
276C2×M16—31no—
277C2×M16—62no—
278C2×M16—61no—
279C2×M16—64no—
280C2×M16—64no—
281C2×M16—68no—
282C2×M16—62no—
283C2×M16C22×D312yes—
284C22×C8—31no—
285C22×C8—32no—
286C22×C8—31no—
287C22×C8—62no—
288C22×C8—61no—
289C22×C8—64no—
290C4⋊Q8C2×Dic311yescharacteristic
291#127C2×Dic311yescharacteristic
292#124C2×Dic311yescharacteristic
293#124C2×Dic311yescharacteristic
294#121—22no—
295C4×D4—22no—
296#117—32no—
297C2×C42C22×D311yescharacteristic
298#108—64no—
299C4⋊C8—62no—
300#104—64no—
301#104—64no—
302#104C2×Dic314yes—
303#104C2×Dic314yes—
304#103—62no—
305#102—62no—
306#101C2×Dic312yes—
307#100C2×Dic312yes—
308#99—62no—
309C22⋊C8—62no—
310C22⋊C8—62no—
311#97—61no—
312#97—61no—
313#97—62no—
314C4×C8—62no—
315C23×C3—24no—
316C23×C3—22no—
317C23×C3—21no—
318C23×C3C4×D412yes—
319C23×C3C2×#4111yescharacteristic
320C3×Q8—24no—
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8C4×D414yes—
325C3×D4—28no—
326C3×D4—24no—
327C3×D4—24no—
328C3×D4—24no—
329C3×D4—24no—
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4—42no—
334C3×D4—82no—
335C3×D4C4×D414yes—
336C2×C3×C4—24no—
337C2×C3×C4—28no—
338C2×C3×C4—22no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—81no—
349C2×C3×C4C4×D412yes—
350C2×C3×C4C23×C411yescharacteristic
351C2×C3×C4C22×D412yes—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C2×#4112yes—
354C2×Dic3—42no—
355C3×C8—24no—
356C3⋊C8—24no—
357C3⋊C8—22no—
358C3⋊C8—41no—
359C3⋊C8—41no—
360C3⋊C8#12011yescharacteristic
361C3⋊C8#12012yes—
362C3⋊C8#12011yescharacteristic
363#41—24no—
364#41—28no—
365#41—24no—
366#41—24no—
367#41—24no—
368#41—24no—
369#41—22no—
370#41—22no—
371#41—22no—
372#41—22no—
373#41—22no—
374#41—22no—
375#41D3×D414yes—
376#41C22×Dic314yes—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8#23712yes—
380C2×Q8C22×Dic311yescharacteristic
381C2×Q8C22×Dic311yescharacteristic
382C2×D4—28no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—24no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4#23712yes—
390C2×D4#23712yes—
391C2×D4C22×Dic311yescharacteristic
392C2×D4C22×Dic311yescharacteristic
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4D3×D412yes—
398C22×C4C23×D311yescharacteristic
399Q16—24no—
400SD16—24no—
401SD16—24no—
402D8—24no—
403M16—22no—
404M16—34no—
405M16—64no—
406M16—64no—
407M16—64no—
408M16—68no—
409M16—64no—
410M16—62no—
411M16—62no—
412M16—62no—
413M16—121no—
414M16—121no—
415M16C22×Dic314yes—
416C2×C8—22no—
417C2×C8—32no—
418C2×C8—31no—
419C2×C8—31no—
420C2×C8—32no—
421C2×C8—32no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—64no—
425C2×C8—64no—
426C2×C8—68no—
427C2×C8—64no—
428C2×C8—62no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—62no—
432C2×C8—122no—
433C4⋊C4—22no—
434C4⋊C4—62no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—62no—
438C42—21no—
439C42—62no—
440C42C22×Dic311yescharacteristic
441C42C22×Dic311yescharacteristic
442C22×C3—22no—
443C22×C3—24no—
444C22×C3—41no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—41no—
448C22×C3—82no—
449C22×C3—84no—
450C22×C3C2×C4×D411yescharacteristic
451C22×C3#52512yes—
452C3×C4—24no—
453C3×C4—22no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—41no—
457C3×C4C2×C4×D412yes—
458C3×C4C2×#12011yescharacteristic
459C3×C4#52511yescharacteristic
460Dic3—82no—
461C23—24no—
462C23—22no—
463C23—21no—
464C23D4×Dic312yes—
465C23C2×#23711yescharacteristic
466Q8—24no—
467Q8—22no—
468Q8—22no—
469Q8—22no—
470Q8D4×Dic314yes—
471D4—28no—
472D4—24no—
473D4—24no—
474D4—24no—
475D4—24no—
476D4—22no—
477D4—22no—
478D4—22no—
479D4—42no—
480D4—82no—
481D4D4×Dic314yes—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—81no—
495C2×C4—122no—
496C2×C4D4×Dic312yes—
497C2×C4C2×D3×D412yes—
498C2×C4C2×D3×D411yescharacteristic
499C2×C4C2×#23712yes—
500C2×C4C23×Dic311yescharacteristic
501C8—24no—
502C8—31no—
503C8—32no—
504C8—31no—
505C8—62no—
506C8—64no—
507C8—121no—
508C8—121no—
509C2×C3—22no—
510C2×C3—21no—
511C2×C3—42no—
512C2×C3—44no—
513C2×C3—81no—
514C2×C3C4×#12011yescharacteristic
515C22—22no—
516C22—24no—
517C22—41no—
518C22—42no—
519C22—42no—
520C22—41no—
521C22—82no—
522C22—84no—
523C22#558612yes—
524C22C2×D4×Dic311yescharacteristic
525C4—24no—
526C4—22no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—242no—
531C4#558611yescharacteristic
532C4D3×#12011yescharacteristic
533C4C2×D4×Dic312yes—
534C3#2147811yescharacteristic, complemented, Sylow
535C2—21no—
536C2—22no—
537C2—42no—
538C2—44no—
539C2—81no—
540C2Dic3×#12011yescharacteristic
subgroupG/H
centerC4D3×#120
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3018C2
socleC2×C3C4×#120
cosocleC22×C3×C4C24
hypercenter#3018D3
hypocenterC3#21478
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3018

Complements

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

NH ≅ G/Ntype
1. #87296C2semidirect
3. #83999C2semidirect
4. #83276C2semidirect
5. #72146C2semidirect
6. #72116C2semidirect
7. #71817C2semidirect
8. #70375C2semidirect
9. #70370C2semidirect
10. #70135C2semidirect
11. #70133C2semidirect
24. C2×#948C22semidirect
25. C3×#456C4semidirect
26. C3×#455C4semidirect
27. C3×#454C4semidirect
28. C3×#453C4semidirect
33. #5236C22semidirect
35. #5234C22semidirect
39. #5125C22semidirect
40. #5122C22semidirect
43. #4924C22semidirect
44. C2×#803C22semidirect
45. #4559C22semidirect
46. #4558C22semidirect
47. #4523C22semidirect
94. #948D4semidirect
95. #948D4semidirect
142. #456Dic3semidirect
143. #455Dic3semidirect
144. #454Dic3semidirect
145. #453Dic3semidirect
534. C3#21478semidirect

Maximal subgroups

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

subgroupindex
#872962
C3×#30182
#839992
#832762
#721462
#721162
#718172
#703752
#703702
#701352
#701332
#214783