Finite Group Databasebeta

← #10826864

Subgroups — #10826864

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
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2#87251C211yesmaximal, characteristic
3#84059C212yesmaximal, complemented
4C2×#5095C212yesmaximal, complemented
5#72133C211yesmaximal, characteristic, complemented
6#71795C214yesmaximal, complemented
7#71784C212yesmaximal, complemented
8#71532C212yesmaximal, complemented
9#23911—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C23×C3×Q8C2211yescharacteristic
12C2×#1007C2211yescharacteristic
13C2×C3×C4⋊Q8—22no—
14C2×C3×C4⋊Q8—28no—
15C2×C3×C4⋊Q8C2212yes—
16C2×C3×C4⋊Q8C2212yes—
17C22×C3×C4⋊C4—22no—
18C22×C3×C42C2211yescharacteristic
19#5817—24no—
20#5816—22no—
21#5816C2212yes—
22Q8×#67—24no—
23#5812C2211yescharacteristic
24C2×#942C2214yes—
25#5791—22no—
26#5791C2212yes—
27#5228—24no—
28#5227—24no—
29C2×#836C2212yes—
30C2×#836C2212yes—
31#5225C2214yes—
32#5207C2212yes—
33#5095—28no—
34C2×#805C2214yes—
35#4523C2214yes—
36#3486—31no—
37C22×C4⋊Q8D311yescharacteristic, complemented
38C2×#456—32no—
39#3004—32no—
40#2005—34no—
41#1896—32no—
42#1886—32no—
43#1784—31no—
44C22×C3×Q8—24no—
45C22×C3×Q8—24no—
46C22×C3×Q8—24no—
47C22×C3×Q8—21no—
48C22×C3×Q8—24no—
49C22×C3×Q8—42no—
50C22×C3×Q8—42no—
51C22×C3×Q8D412yes—
52C22×C3×Q8C2312yes—
53C22×C3×Q8C2311yescharacteristic
54C22×C3×Q8C2311yescharacteristic
55C23×C3×C4—21no—
56C23×C3×C4—21no—
57C23×C3×C4D412yes—
58C23×C3×C4C2311yescharacteristic
59#1007—24no—
60#1007—42no—
61C2×#239—22no—
62C2×D3×Q8—22no—
63C2×#235—22no—
64C2×#235C2312yes—
65C3×C4⋊Q8—216no—
66C3×C4⋊Q8—24no—
67C3×C4⋊Q8—24no—
68C3×C4⋊Q8—24no—
69C3×C4⋊Q8D418yescomplemented
70C2×C3×C4⋊C4—216no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C42—21no—
74C2×C3×C42—24no—
75C2×C3×C42C2312yes—
76#953—22no—
77#947—24no—
78#947—22no—
79#946—28no—
80#946—24no—
81Q8×Dic3—24no—
82#944—24no—
83#944—22no—
84C2×#216—24no—
85#942—28no—
86#942—44no—
87C2×#215—24no—
88#930—24no—
89#930D414yes—
90#929—24no—
91#929—22no—
92C4×#67—28no—
93C4×#67—24no—
94#926—22no—
95#926C2312yes—
96#924—24no—
97#924D414yes—
98C2×#208C2314yes—
99#836—24no—
100#836—42no—
101#835—28no—
102#830—24no—
103#805—28no—
104C2×2−1+4—31no—
105C23×Q8C2×D311yescharacteristic
106C23×Q8C2×D311yescharacteristic
107C2×#137—34no—
108D4×Q8—64no—
109#548—64no—
110#536—32no—
111#536—62no—
112C2×C4⋊Q8—22no—
113C2×C4⋊Q8—28no—
114C2×C4⋊Q8C2×D312yes—
115C2×C4⋊Q8C2×D312yes—
116#525—32no—
117#525—62no—
118#519—31no—
119C22×C4⋊C4—22no—
120C22×C42C2×D311yescharacteristic
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
131C2×C3×Q8—24no—
132C2×C3×Q8—24no—
133C2×C3×Q8—24no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—24no—
141C2×C3×Q8—24no—
142C2×C3×Q8—42no—
143C2×C3×Q8—416no—
144C2×C3×Q8—416no—
145C2×C3×Q8—42no—
146C2×C3×Q8—42no—
147C2×C3×Q8C2×D414yes—
148C2×C3×Q8C2×D414yes—
149C22×C3×C4—24no—
150C22×C3×C4—28no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—24no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4C2×D414yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C2×#67—24no—
167C2×#67—22no—
168#239—48no—
169D3×Q8—48no—
170#235—24no—
171#235—24no—
172#235—44no—
173#235—44no—
174C2×D12—22no—
175C2×D12C2×D412yes—
176C2×C4×D3—24no—
177C2×C4×D3—22no—
178C2×Dic6—22no—
179C2×Dic6C2×D412yes—
180C3×C4⋊C4—216no—
181C3×C4⋊C4—216no—
182C3×C42—24no—
183C3×C42—22no—
184C3×C42C2×D414yes—
185#217—24no—
186#217—22no—
187#216—48no—
188#215—48no—
189(C2×D3)⋊C4—28no—
190(C2×D3)⋊C4—24no—
191C4⋊Dic3—24no—
192C4⋊Dic3C2×D414yes—
193Dic3⋊C4—28no—
194Dic3⋊C4—24no—
195C4×Dic3—24no—
196C4×Dic3—22no—
197#208—24no—
198#208—24no—
199C2×C3⋊C8—24no—
2002−1+4—64no—
2012−1+4—122no—
202C2×#41—32no—
203C2×#41—62no—
204C2×#41—62no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—21no—
209C22×Q8—24no—
210C22×Q8—42no—
211C22×Q8—42no—
212C22×Q8—62no—
213C22×Q8#6712yes—
214C22×Q8C22×D312yes—
215C22×Q8C22×D311yescharacteristic
216C22×Q8C22×D311yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4#6712yes—
220C23×C4C22×D311yescharacteristic
221#137—68no—
222#137—124no—
223C2×Q16—64no—
224C2×SD16—64no—
225C2×M16—34no—
226C4⋊Q8—216no—
227C4⋊Q8—24no—
228C4⋊Q8—24no—
229C4⋊Q8—24no—
230C4⋊Q8—62no—
231C4⋊Q8—64no—
232C4⋊Q8#6718yescomplemented
233#124—62no—
234#124—64no—
235#123—62no—
236#123—64no—
237C22⋊Q8—34no—
238C22⋊Q8—64no—
239C22⋊Q8—64no—
240C22⋊Q8—68no—
241#121—34no—
242#121—64no—
243#120—62no—
244C4×Q8—64no—
245C4×D4—68no—
246C4×D4—64no—
247#117—32no—
248#117—62no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C42—21no—
253C2×C42—24no—
254C2×C42C22×D312yes—
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#12012yes—
264C23×C3C22×D411yescharacteristic
265C22×D3—22no—
266C3×Q8—28no—
267C3×Q8—28no—
268C3×Q8—48no—
269C3×Q8—48no—
270C3×Q8—44no—
271C3×Q8—44no—
272C2×C3×C4—216no—
273C2×C3×C4—28no—
274C2×C3×C4—22no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—22no—
278C2×C3×C4—21no—
279C2×C3×C4—28no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—48no—
285C2×C3×C4—48no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4#12012yes—
289C2×C3×C4#12014yes—
290C2×C3×C4C22×D412yes—
291C2×C3×C4C22×D412yes—
292C2×C3×C4H(5,𝔽2)12yes—
293#67—42no—
294#67—42no—
295#67—48no—
296C2×Dic3—24no—
297C2×Dic3—22no—
298D12—24no—
299D12—44no—
300C4×D3—48no—
301C4×D3—44no—
302Dic6—24no—
303Dic6—44no—
304C3⋊C8—44no—
305C24C2×#6711yescharacteristic
306#41—64no—
307#41—64no—
308#41—124no—
309#41—128no—
310#41—124no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—32no—
323C2×Q8—42no—
324C2×Q8—416no—
325C2×Q8—416no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—62no—
329C2×Q8—128no—
330C2×Q8C2×#6714yes—
331C2×Q8C2×#6714yes—
332C2×D4—32no—
333C2×D4—62no—
334C2×D4—64no—
335C2×D4—62no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—64no—
349C22×C4—62no—
350C22×C4D3×D414yes—
351C22×C4C2×#6712yes—
352C22×C4C2×#6711yescharacteristic
353C22×C4C2×#6712yes—
354C22×C4C23×D311yescharacteristic
355Q16—128no—
356SD16—128no—
357M16—64no—
358M16—64no—
359C2×C8—64no—
360C4⋊C4—216no—
361C4⋊C4—216no—
362C4⋊C4—34no—
363C4⋊C4—64no—
364C4⋊C4—68no—
365C4⋊C4—64no—
366C22⋊C4—68no—
367C22⋊C4—64no—
368C22⋊C4—62no—
369C22⋊C4—64no—
370C42—24no—
371C42—22no—
372C42—64no—
373C42—62no—
374C42C2×#6714yes—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3#45614yescomplemented
381C22×C3C2×#12011yescharacteristic
382C22×C3C2×#12011yescharacteristic
383C22×C3#53411yescharacteristic
384C2×D3—42no—
385C2×D3—44no—
386C3×C4—28no—
387C3×C4—24no—
388C3×C4—22no—
389C3×C4—42no—
390C3×C4—44no—
391C3×C4D4214yes—
392Dic3—42no—
393Dic3—44no—
394C23—21no—
395C23—21no—
396C23—24no—
397C23—62no—
398C23#93712yes—
399C23C22×#6711yescharacteristic
400Q8—28no—
401Q8—28no—
402Q8—48no—
403Q8—48no—
404Q8—44no—
405Q8—44no—
406Q8—64no—
407Q8—124no—
408D4—64no—
409D4—124no—
410D4—122no—
411D4—122no—
412D4—128no—
413C2×C4—216no—
414C2×C4—28no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—28no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—64no—
430C2×C4—62no—
431C2×C4—128no—
432C2×C4—124no—
433C2×C4#93712yes—
434C2×C4#95314yes—
435C2×C4C2×D3×D412yes—
436C2×C4#100412yes—
437C2×C4C22×#6712yes—
438C8—124no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—24no—
442C2×C3#240311yescharacteristic
443C2×C3C2×#45612yes—
444D3—82no—
445C22—24no—
446C22—24no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—124no—
451C22—122no—
452C22#509514yescomplemented
453C22C2×#93711yescharacteristic
454C22#580411yescharacteristic
455C22C2×#95311yescharacteristic
456C4—28no—
457C4—22no—
458C4—24no—
459C4—42no—
460C4—44no—
461C4—124no—
462C4—122no—
463C4D4×#6714yes—
464C3#2391111yescharacteristic, complemented, Sylow
465C2—21no—
466C2—21no—
467C2—24no—
468C2—242no—
469C2C2×#509512yes—
470C2#8405211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×C4⋊Q8C2
socleC22×C3C2×#120
prime radical#87251C2
hypercenterC22×C4⋊Q8D3
hypocenterC3#23911
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊Q8C2semidirect
3. #84059C2semidirect
4. C2×#5095C2semidirect
5. #72133C2semidirect
6. #71795C2semidirect
7. #71784C2semidirect
8. #71532C2semidirect
37. C22×C4⋊Q8D3semidirect
69. C3×C4⋊Q8D4semidirect
232. C4⋊Q8#67semidirect
380. C22×C3#456semidirect
452. C22#5095semidirect
464. C3#23911semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
#872512
#840592
C2×#50952
#721332
#717952
#717842
#715322
#239113