Finite Group Databasebeta

← #10826868

Subgroups — #10826868

Order 768.

1738 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3486C211yesmaximal, characteristic
2C22×#944C211yesmaximal, characteristic, complemented
3C2×#5246C212yesmaximal, complemented
4#82933C212yesmaximal, complemented
5#72133C211yesmaximal, characteristic, complemented
6#71868C214yesmaximal, complemented
7#71711C212yesmaximal, complemented
8#71660C212yesmaximal, complemented
9#23911—31nomaximal, Sylow
10C2×C3×2−1+4C2211yescharacteristic
11C23×C3×Q8C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C3×D4×Q8—24no—
14C3×#548—24no—
15C3×#536—22no—
16C3×#536C2212yes—
17C3×#525—22no—
18C3×#525C2212yes—
19C3×#519C2211yescharacteristic
20C2×#951C2214yes—
21C2×#944—22no—
22C2×#944—28no—
23C2×#944C2212yes—
24C2×#944C2212yes—
25C22×Dic3⋊C4—22no—
26C22×C4×Dic3C2211yescharacteristic
27#5246—28no—
28C2×#837C2214yes—
29C2×#836C2212yes—
30C2×#836C2212yes—
31#5207C2212yes—
32#5050—24no—
33#5040—24no—
34#5004C2214yes—
35#4532C2214yes—
36#3486D311yescharacteristic
37C22×C4⋊Q8—31no—
38C2×#456—32no—
39#3004—32no—
40#2005—34no—
41#1896—32no—
42#1886—32no—
43#1784—31no—
44C3×2−1+4—24no—
45C3×2−1+4—42no—
46C2×C3×#41—22no—
47C2×C3×#41—22no—
48C2×C3×#41C2312yes—
49C22×C3×Q8—24no—
50C22×C3×Q8—22no—
51C22×C3×Q8—24no—
52C22×C3×Q8—42no—
53C22×C3×Q8D412yes—
54C22×C3×Q8C2311yescharacteristic
55C22×C3×Q8C2311yescharacteristic
56C23×C3×C4D412yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—21no—
59C23×Dic3—21no—
60C22×Dic6—21no—
61C22×Dic6—24no—
62C22×Dic6—24no—
63C22×Dic6—42no—
64C22×Dic6C2312yes—
65C3×C4⋊Q8—24no—
66C3×C4⋊Q8—22no—
67C3×#124—24no—
68C3×#124—22no—
69C3×#123—24no—
70C3×#123—22no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—28no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8D414yes—
75C3×#121—24no—
76C3×#121D414yes—
77C3×#120—22no—
78C3×C4×Q8—24no—
79C3×C4×D4—28no—
80C3×C4×D4—24no—
81C3×#117—22no—
82C3×#117C2312yes—
83#951—28no—
84#951—44no—
85#944—216no—
86#944—24no—
87#944—24no—
88#944—24no—
89#944D418yescomplemented
90C2×#216—24no—
91C2×#214—24no—
92C2×Dic3⋊C4—216no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4—24no—
95C2×C4×Dic3—21no—
96C2×C4×Dic3—24no—
97C2×C4×Dic3C2312yes—
98C2×#208C2314yes—
99#837—28no—
100#836—24no—
101#836—42no—
102#830—24no—
103#810—28no—
104C2×2−1+4C2×D311yescharacteristic
105C23×Q8—31no—
106C23×Q8C2×D311yescharacteristic
107C2×#137—34no—
108D4×Q8—24no—
109#548—24no—
110#536—22no—
111#536C2×D312yes—
112C2×C4⋊Q8—32no—
113C2×C4⋊Q8—32no—
114C2×C4⋊Q8—68no—
115C2×C4⋊Q8—62no—
116#525—22no—
117#525C2×D312yes—
118#519C2×D311yescharacteristic
119C22×C4⋊C4—62no—
120C22×C42—31no—
121#456—68no—
122#451—64no—
123#448—64no—
124C2×#104—34no—
125#419—34no—
126C2×#101—32no—
127C2×#101—32no—
128#407—32no—
129#339—34no—
130C24×C3C2×D411yescharacteristic
131C3×#41—24no—
132C3×#41—24no—
133C3×#41—44no—
134C3×#41—48no—
135C3×#41—44no—
136C2×C3×Q8—22no—
137C2×C3×Q8—24no—
138C2×C3×Q8—24no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—42no—
145C2×C3×Q8—416no—
146C2×C3×Q8—48no—
147C2×C3×Q8C2×D414yes—
148C2×C3×Q8C2×D412yes—
149C2×C3×D4—22no—
150C2×C3×D4—24no—
151C2×C3×D4—22no—
152C2×C3×D4C2×D412yes—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—24no—
156C22×C3×C4—24no—
157C22×C3×C4—24no—
158C22×C3×C4—22no—
159C22×C3×C4—42no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D414yes—
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—24no—
166C22×Dic3—28no—
167C22×Dic3—22no—
168C22×Dic3—24no—
169C22×Dic3—22no—
170C22×Dic3—42no—
171C22×Dic3—42no—
172C2×Dic6—24no—
173C2×Dic6—24no—
174C2×Dic6—24no—
175C2×Dic6—22no—
176C2×Dic6—416no—
177C2×Dic6—42no—
178C2×Dic6—42no—
179C2×Dic6C2×D414yes—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—28no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4C2×D414yes—
184C3×C22⋊C4—28no—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—24no—
188C3×C42—24no—
189C3×C42—22no—
190#216—48no—
191#214—48no—
192Dic3⋊C4—216no—
193Dic3⋊C4—216no—
194C4×Dic3—24no—
195C4×Dic3—22no—
196C4×Dic3C2×D414yes—
197#208—24no—
198#208—24no—
199C2×C3⋊C8—24no—
2002−1+4—24no—
2012−1+4—42no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41C22×D312yes—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—24no—
208C22×Q8—32no—
209C22×Q8—42no—
210C22×Q8—64no—
211C22×Q8—61no—
212C22×Q8—64no—
213C22×Q8—122no—
214C22×Q8#6712yes—
215C22×Q8C22×D311yescharacteristic
216C22×Q8C22×D311yescharacteristic
217C23×C4—61no—
218C23×C4—61no—
219C23×C4#6712yes—
220C23×C4C22×D311yescharacteristic
221#137—68no—
222#137—124no—
223C2×Q16—64no—
224C2×SD16—64no—
225C2×M16—34no—
226C4⋊Q8—24no—
227C4⋊Q8—22no—
228C4⋊Q8—38no—
229C4⋊Q8—64no—
230C4⋊Q8—64no—
231C4⋊Q8—616no—
232C4⋊Q8—64no—
233#124—24no—
234#124—22no—
235#123—24no—
236#123—22no—
237C22⋊Q8—24no—
238C22⋊Q8—28no—
239C22⋊Q8—24no—
240C22⋊Q8#6714yes—
241#121—24no—
242#121#6714yes—
243#120—22no—
244C4×Q8—24no—
245C4×D4—28no—
246C4×D4—24no—
247#117—22no—
248#117C22×D312yes—
249C2×C4⋊C4—616no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—64no—
252C2×C42—32no—
253C2×C42—64no—
254C2×C42—61no—
255#104—68no—
256#103—68no—
257#101—64no—
258#101—122no—
259C22⋊C8—64no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3#12012yes—
265C23×C3C22×D411yescharacteristic
266C3×Q8—28no—
267C3×Q8—24no—
268C3×Q8—48no—
269C3×Q8—44no—
270C3×Q8—44no—
271C3×D4—24no—
272C3×D4—44no—
273C3×D4—42no—
274C3×D4—42no—
275C3×D4—48no—
276C2×C3×C4—21no—
277C2×C3×C4—22no—
278C2×C3×C4—21no—
279C2×C3×C4—22no—
280C2×C3×C4—28no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—48no—
287C2×C3×C4—48no—
288C2×C3×C4—44no—
289C2×C3×C4#12014yes—
290C2×C3×C4#12012yes—
291C2×C3×C4C22×D412yes—
292C2×C3×C4C22×D412yes—
293C2×C3×C4H(5,𝔽2)12yes—
294C2×Dic3—216no—
295C2×Dic3—28no—
296C2×Dic3—24no—
297C2×Dic3—24no—
298C2×Dic3—48no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301Dic6—28no—
302Dic6—48no—
303Dic6—44no—
304C3⋊C8—44no—
305C24C2×#6711yescharacteristic
306#41—24no—
307#41—24no—
308#41—44no—
309#41—48no—
310#41—44no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—34no—
320C2×Q8—42no—
321C2×Q8—416no—
322C2×Q8—48no—
323C2×Q8—64no—
324C2×Q8—62no—
325C2×Q8—64no—
326C2×Q8—64no—
327C2×Q8—1216no—
328C2×Q8—122no—
329C2×Q8—122no—
330C2×Q8D3×D414yes—
331C2×Q8C2×#6712yes—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4C2×#6712yes—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—64no—
344C22×C4—68no—
345C22×C4—62no—
346C22×C4—64no—
347C22×C4—62no—
348C22×C4—122no—
349C22×C4—122no—
350C22×C4C2×#6712yes—
351C22×C4C2×#6712yes—
352C22×C4C2×#6711yescharacteristic
353C22×C4C2×#6714yes—
354C22×C4C23×D311yescharacteristic
355Q16—128no—
356SD16—128no—
357M16—64no—
358M16—64no—
359C2×C8—64no—
360C4⋊C4—24no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—616no—
364C4⋊C4—616no—
365C4⋊C4C2×#6714yes—
366C22⋊C4—28no—
367C22⋊C4—24no—
368C22⋊C4—22no—
369C22⋊C4—24no—
370C42—24no—
371C42—22no—
372C42—34no—
373C42—64no—
374C42—62no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—42no—
381C22×C3—44no—
382C22×C3#45614yescomplemented
383C22×C3C2×#12011yescharacteristic
384C22×C3C2×#12011yescharacteristic
385C22×C3#53411yescharacteristic
386C3×C4—22no—
387C3×C4—24no—
388C3×C4—42no—
389C3×C4—42no—
390C3×C4—44no—
391C3×C4D4214yes—
392Dic3—28no—
393Dic3—44no—
394C23—24no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23#95312yes—
399C23C22×#6711yescharacteristic
400Q8—28no—
401Q8—24no—
402Q8—48no—
403Q8—44no—
404Q8—44no—
405Q8—68no—
406Q8—128no—
407Q8—124no—
408D4—24no—
409D4—44no—
410D4—48no—
411D4—42no—
412D4—42no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—44no—
426C2×C4—616no—
427C2×C4—68no—
428C2×C4—64no—
429C2×C4—64no—
430C2×C4—128no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4#93714yes—
434C2×C4#95312yes—
435C2×C4C2×D3×D412yes—
436C2×C4#100412yes—
437C2×C4C22×#6712yes—
438C8—124no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—24no—
442C2×C3—82no—
443C2×C3#240311yescharacteristic
444C2×C3C2×#45612yes—
445C22—24no—
446C22—24no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—42no—
451C22—44no—
452C22#524614yescomplemented
453C22C2×#93711yescharacteristic
454C22#580411yescharacteristic
455C22C2×#95311yescharacteristic
456C4—22no—
457C4—24no—
458C4—42no—
459C4—42no—
460C4—44no—
461C4—68no—
462C4—124no—
463C4D4×#6714yes—
464C3#2391111yescharacteristic, complemented, Sylow
465C2—21no—
466C2—24no—
467C2—21no—
468C2—82no—
469C2#8405211yescharacteristic
470C2C2×#524612yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3486C2
socleC22×C3C2×#120
hypercenter#3486D3
hypocenterC3#23911
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3486

Complements

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

NH ≅ G/Ntype
2. C22×#944C2semidirect
3. C2×#5246C2semidirect
4. #82933C2semidirect
5. #72133C2semidirect
6. #71868C2semidirect
7. #71711C2semidirect
8. #71660C2semidirect
89. #944D4semidirect
382. C22×C3#456semidirect
452. C22#5246semidirect
464. C3#23911semidirect

Maximal subgroups

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

subgroupindex
C3×#34862
C22×#9442
C2×#52462
#829332
#721332
#718682
#717112
#716602
#239113