Finite Group Databasebeta

← #10943915

Subgroups — #10943915

Order 768.

3580 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×2−1+4C211yesmaximal, characteristic
2C2×#5246C214yesmaximal, complemented
3#84084C212yesmaximal, complemented
4#84065C211yesmaximal, characteristic, complemented
5#72143C214yesmaximal, complemented
6#72138C212yesmaximal, complemented
7#72133C211yesmaximal, characteristic, complemented
8#32892—31nomaximal, Sylow
9C2×C3×2−1+4—216no—
10C2×C3×2−1+4—24no—
11C2×C3×2−1+4—22no—
12C2×C3×2−1+4C2214yes—
13C22×C3×#41—22no—
14C22×C3×#41—22no—
15C22×C3×#41C2212yes—
16C23×C3×Q8—22no—
17C23×C3×Q8C2211yescharacteristic
18C2×#951C2214yescomplemented
19C2×#944C2212yescomplemented
20C2×#924C2211yescharacteristic
21#5246—216no—
22#5239—24no—
23C2×#837C2218yescomplemented
24#5232C2214yescomplemented
25#5230—22no—
26#5229C2212yescomplemented
27#5228—22no—
28C2×#836C2214yescomplemented
29#5218—22no—
30#5210—22no—
31C4×#217C2211yescharacteristic
32#5207C2212yes—
33C22×2−1+4D311yescharacteristic
34C2×#456—34no—
35#3007—32no—
36#2434—31no—
37#1793—34no—
38#1789—32no—
39#1784—31no—
40C3×2−1+4—232no—
41C3×2−1+4—232no—
42C3×2−1+4—232no—
43C3×2−1+4—216no—
44C3×2−1+4—28no—
45C3×2−1+4D4116yes—
46C2×C3×#41—232no—
47C2×C3×#41—28no—
48C2×C3×#41—28no—
49C2×C3×#41—232no—
50C2×C3×#41—216no—
51C2×C3×#41—28no—
52C2×C3×#41—28no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41D414yes—
56C2×C3×#41C2314yes—
57C22×C3×Q8—22no—
58C22×C3×Q8—232no—
59C22×C3×Q8—28no—
60C22×C3×Q8—28no—
61C22×C3×Q8—28no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8D412yes—
65C22×C3×Q8D412yes—
66C22×C3×Q8C2312yes—
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4D412yes—
70C23×C3×C4—24no—
71C23×C3×C4—22no—
72C23×C3×C4D412yes—
73C23×C3×C4C2311yescharacteristic
74C22×Dic6C2311yescharacteristic
75C2×#217—21no—
76C2×#217—21no—
77#951—216no—
78#944—28no—
79C2×#216—24no—
80C2×#214—24no—
81C2×C4⋊Dic3C2311yescharacteristic
82#924—22no—
83#924—22no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—21no—
86C2×C4×Dic3C2312yes—
87C2×#208C2314yes—
88#837—216no—
89#836—28no—
90#835—24no—
91#832—24no—
92C3⋊#95—21no—
93C3⋊#95—24no—
94#830—22no—
95#830—22no—
96C2×2−1+4—216no—
97C2×2−1+4—24no—
98C2×2−1+4—22no—
99C2×2−1+4C2×D314yes—
100C22×#41—22no—
101C22×#41—22no—
102C22×#41C2×D312yes—
103C23×Q8—22no—
104C23×Q8C2×D311yescharacteristic
105C2×#137—34no—
106C2×C4⋊Q8—32no—
107C2×C22⋊Q8—31no—
108#456—616no—
109#452—64no—
110#451—62no—
111#450—62no—
112C2×#104—38no—
113#417—34no—
114C2×#101—34no—
115#407—32no—
116#395—62no—
117#391—32no—
118#385—62no—
119C4×C22⋊C4—31no—
120C24×C3—22no—
121C24×C3C2×D411yescharacteristic
122C3×#41—264no—
123C3×#41—264no—
124C3×#41—232no—
125C3×#41—216no—
126C3×#41—232no—
127C3×#41—216no—
128C3×#41—28no—
129C3×#41—232no—
130C3×#41—28no—
131C3×#41—232no—
132C3×#41—28no—
133C3×#41C2×D4116yes—
134C2×C3×Q8—216no—
135C2×C3×Q8—216no—
136C2×C3×Q8—28no—
137C2×C3×Q8—264no—
138C2×C3×Q8—232no—
139C2×C3×Q8—216no—
140C2×C3×Q8—28no—
141C2×C3×Q8—232no—
142C2×C3×Q8—22no—
143C2×C3×Q8—24no—
144C2×C3×Q8—28no—
145C2×C3×Q8—24no—
146C2×C3×Q8—24no—
147C2×C3×Q8C2×D414yes—
148C2×C3×Q8C2×D418yes—
149C2×C3×D4—216no—
150C2×C3×D4—216no—
151C2×C3×D4—28no—
152C2×C3×D4—24no—
153C2×C3×D4—24no—
154C2×C3×D4—28no—
155C2×C3×D4—22no—
156C2×C3×D4C2×D414yes—
157C22×C3×C4—232no—
158C22×C3×C4—216no—
159C22×C3×C4—216no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—24no—
164C22×C3×C4—28no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4C2×D414yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—22no—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C2×Dic6—22no—
176C2×Dic6—24no—
177#217—24no—
178#217—41no—
179#217—41no—
180#216—48no—
181#214—48no—
182C4⋊Dic3—22no—
183Dic3⋊C4—42no—
184Dic3⋊C4—48no—
185C4×Dic3—24no—
186#208—28no—
187C2×C3⋊C8—24no—
1882−1+4—232no—
1892−1+4—232no—
1902−1+4—232no—
1912−1+4—216no—
1922−1+4—28no—
1932−1+4#67116yes—
194C2×#41—232no—
195C2×#41—28no—
196C2×#41—28no—
197C2×#41—232no—
198C2×#41—216no—
199C2×#41—28no—
200C2×#41—28no—
201C2×#41—22no—
202C2×#41—22no—
203C2×#41#6714yes—
204C2×#41C22×D314yes—
205C22×Q8—22no—
206C22×Q8—232no—
207C22×Q8—28no—
208C22×Q8—28no—
209C22×Q8—28no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—31no—
213C22×Q8#6712yes—
214C22×Q8#6712yes—
215C22×Q8C22×D312yes—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4#6712yes—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4#6712yes—
222C23×C4C22×D311yescharacteristic
223#137—616no—
224C2×Q16—64no—
225C2×SD16—64no—
226C2×M16—34no—
227C4⋊Q8—68no—
228C22⋊Q8—62no—
229C22⋊Q8—62no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—64no—
233C2×C22⋊C4—61no—
234C2×C22⋊C4—61no—
235C2×C42—32no—
236#104—616no—
237#103—64no—
238#102—64no—
239#101—68no—
240C22⋊C8—62no—
241C22⋊C8—62no—
242#95—61no—
243#95—64no—
244C23×C3—28no—
245C23×C3—22no—
246C23×C3—24no—
247C23×C3—22no—
248C23×C3—41no—
249C23×C3—44no—
250C23×C3—44no—
251C23×C3#12011yescharacteristic
252C23×C3#12011yescharacteristic
253C23×C3C22×D411yescharacteristic
254C3×Q8—232no—
255C3×Q8—28no—
256C3×Q8—216no—
257C3×Q8—24no—
258C3×Q8—216no—
259C3×Q8#12018yes—
260C3×D4—216no—
261C3×D4—232no—
262C3×D4—28no—
263C3×D4—216no—
264C3×D4—24no—
265C3×D4#12018yes—
266C2×C3×C4—28no—
267C2×C3×C4—28no—
268C2×C3×C4—264no—
269C2×C3×C4—216no—
270C2×C3×C4—28no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—216no—
277C2×C3×C4—24no—
278C2×C3×C4#12012yes—
279C2×C3×C4#12018yes—
280C2×C3×C4C22×D412yes—
281C2×C3×C4C22×D414yes—
282C2×Dic3—24no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289Dic6—44no—
290C3⋊C8—44no—
291C24—22no—
292C24C2×#6711yescharacteristic
293#41—264no—
294#41—264no—
295#41—232no—
296#41—216no—
297#41—232no—
298#41—216no—
299#41—28no—
300#41—232no—
301#41—28no—
302#41—232no—
303#41—28no—
304#41C2×#67116yes—
305C2×Q8—216no—
306C2×Q8—216no—
307C2×Q8—28no—
308C2×Q8—264no—
309C2×Q8—232no—
310C2×Q8—216no—
311C2×Q8—28no—
312C2×Q8—232no—
313C2×Q8—22no—
314C2×Q8—24no—
315C2×Q8—28no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—62no—
319C2×Q8—64no—
320C2×Q8C2×#6714yes—
321C2×Q8C2×#6718yes—
322C2×D4—216no—
323C2×D4—216no—
324C2×D4—28no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—28no—
328C2×D4—22no—
329C2×D4C2×#6714yes—
330C22×C4—232no—
331C22×C4—216no—
332C22×C4—216no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—62no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4C2×#6714yes—
344C22×C4C2×#6711yescharacteristic
345C22×C4C2×#6712yes—
346C22×C4C2×#6712yes—
347C22×C4C23×D311yescharacteristic
348Q16—128no—
349SD16—128no—
350M16—68no—
351C2×C8—64no—
352C4⋊C4—62no—
353C4⋊C4—122no—
354C4⋊C4—128no—
355C22⋊C4—64no—
356C22⋊C4—121no—
357C22⋊C4—121no—
358C42—64no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—28no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—44no—
368C22×C3—416no—
369C22×C3—44no—
370C22×C3—44no—
371C22×C3#45612yescomplemented
372C22×C3C2×#12012yes—
373C22×C3C2×#12011yescharacteristic
374C3×C4—28no—
375C3×C4—28no—
376C3×C4—22no—
377C3×C4C2×#12014yes—
378Dic3—44no—
379Dic3—81no—
380Dic3—81no—
381C23—28no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23—41no—
386C23—44no—
387C23—44no—
388C23#95311yescharacteristic
389C23#95311yescharacteristic
390C23C22×#6711yescharacteristic
391Q8—232no—
392Q8—28no—
393Q8—216no—
394Q8—24no—
395Q8—216no—
396Q8—124no—
397Q8#95318yes—
398D4—216no—
399D4—232no—
400D4—28no—
401D4—216no—
402D4—24no—
403D4#95318yes—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—264no—
407C2×C4—216no—
408C2×C4—28no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—216no—
415C2×C4—24no—
416C2×C4—64no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4#95318yes—
424C2×C4#95312yes—
425C2×C4C22×#6714yes—
426C2×C4C22×#6712yes—
427C8—124no—
428C2×C3—22no—
429C2×C3—22no—
430C2×C3—41no—
431C2×C3—48no—
432C2×C3#284611yescharacteristic
433C2×C3C2×#45611yescharacteristic
434C2×C3#302011yescharacteristic
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—28no—
443C22—44no—
444C22—416no—
445C22—44no—
446C22—44no—
447C22#524612yescomplemented
448C22C2×#95312yes—
449C22C2×#95311yescharacteristic
450C4—28no—
451C4—28no—
452C4—22no—
453C4—124no—
454C4—241no—
455C4—241no—
456C4C2×#95314yes—
457C3#3289211yescharacteristic, complemented, Sylow
458C2—22no—
459C2—22no—
460C2—41no—
461C2—48no—
462C2#8411611yescharacteristic
463C2C2×#524611yescharacteristic
464C2#8412811yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC22×C3×2−1+4C2
socleC22×C3C2×#120
hypercenterC22×2−1+4D3
hypocenterC3#32892
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×2−1+4

Complements

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

NH ≅ G/Ntype
2. C2×#5246C2semidirect
3. #84084C2semidirect
4. #84065C2semidirect
5. #72143C2semidirect
6. #72138C2semidirect
7. #72133C2semidirect
18. C2×#951C22semidirect
19. C2×#944C22semidirect
23. C2×#837C22semidirect
24. #5232C22semidirect
26. #5229C22semidirect
28. C2×#836C22semidirect
371. C22×C3#456semidirect
447. C22#5246semidirect
457. C3#32892semidirect

Maximal subgroups

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

subgroupindex
C22×C3×2−1+42
C2×#52462
#840842
#840652
#721432
#721382
#721332
#328923