Finite Group Databasebeta

← #10696872

Subgroups — #10696872

Order 768.

484 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3215C211yesmaximal, characteristic
2#83370C211yesmaximal, characteristic, complemented
3C2×#5055C211yesmaximal, characteristic
4#71822C211yesmaximal, characteristic, complemented
5#71720C211yesmaximal, characteristic
6#70822C211yesmaximal, characteristic, complemented
7#67751C211yesmaximal, characteristic, complemented
8#10137—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#497—21no—
13C3×#489—21no—
14C3×#464—22no—
15C3×#451—21no—
16C3×#450—21no—
17C3×#432C2211yescharacteristic
18C2×#835—21no—
19C2×#832—21no—
20#5093—21no—
21#5092—21no—
22#5055—22no—
23#5055—22no—
24#5050—22no—
25#5031—22no—
26#4926C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4543—22no—
30#4525—22no—
31#4486—22no—
32#3215D311yescharacteristic
33#3209—31no—
34C2×#489—31no—
35#1556—31no—
36#1531—31no—
37#1469—31no—
38#1302—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C22×Dic6—21no—
42C2×C3×Q16—22no—
43C2×C3×SD16—22no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×#117—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#216—24no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C22×C3⋊C8—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869D412yes—
67C3×C4⋊C8D412yes—
68C3×#103—22no—
69C3×#102—22no—
70C3×C22⋊C8—21no—
71C3×#97—21no—
72#835—24no—
73#832—24no—
74#830—21no—
75#830D412yes—
76#810—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79#803—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8—31no—
85C2×#124C2×D311yescharacteristic
86#497—21no—
87#494—61no—
88#489—21no—
89#489—62no—
90#489—62no—
91#487—61no—
92#464—22no—
93#462—62no—
94#451—21no—
95#451—62no—
96#450—21no—
97#432—31no—
98#432C2×D311yescharacteristic
99#431—31no—
100C2×#103—61no—
101C2×#102—61no—
102C2×#97—31no—
103#332—62no—
104#330—62no—
105#323—62no—
106C24×C3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—42no—
115C22×C3×C4—21no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—41no—
120C2×Dic6—22no—
121C2×Dic6—41no—
122C2×Dic6—82no—
123C3×Q16—82no—
124C3×SD16—82no—
125C2×C3×C8—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142#216—84no—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—42no—
150C2×C3⋊C8—41no—
151C22×Q8—61no—
152C22×Q8#6711yescharacteristic
153C22×D4#6711yescharacteristic
154C2×Q16—22no—
155C2×Q16—64no—
156C2×SD16—22no—
157C22×C8—61no—
158C4⋊Q8—32no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8#6711yescharacteristic
163C4⋊Q8#6711yescharacteristic
164#125—21no—
165#124—21no—
166#124—21no—
167#124—21no—
168#124—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173#117—21no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C42C22×D311yescharacteristic
179C4⋊C8—61no—
180C4⋊C8—61no—
181C4⋊C8#6712yes—
182#103—22no—
183#103—64no—
184#103—64no—
185#102—22no—
186#102—64no—
187C22⋊C8—21no—
188C22⋊C8—32no—
189C22⋊C8—61no—
190#97—21no—
191#97—32no—
192#97—61no—
193#97—61no—
194C4×C8—61no—
195C23×C3—41no—
196C23×C3—42no—
197C23×C3—81no—
198C23×C3—81no—
199C23×C3#12111yescharacteristic
200C3×Q8—42no—
201C3×Q8—42no—
202C3×Q8—42no—
203C3×Q8—42no—
204C3×D4—42no—
205C3×D4—42no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—81no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12711yescharacteristic
219C2×Dic3—42no—
220C2×Dic3—82no—
221Dic6—44no—
222Dic6—82no—
223C3×C8—42no—
224C3⋊C8—42no—
225C3⋊C8—42no—
226C3⋊C8—81no—
227C24—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—42no—
233C2×Q8—62no—
234C2×Q8—121no—
235C2×Q8—242no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—42no—
239C22×C4—21no—
240C22×C4—121no—
241C22×C4D3×D411yescharacteristic
242C22×C4#23711yescharacteristic
243C22×C4C2×#6711yescharacteristic
244Q16—82no—
245Q16—244no—
246SD16—82no—
247C2×C8—22no—
248C2×C8—62no—
249C2×C8—62no—
250C2×C8—122no—
251C2×C8—121no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—62no—
259C4⋊C4—62no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C42—41no—
268C42D3×D411yescharacteristic
269C42D3×D411yescharacteristic
270C42C2×#6711yescharacteristic
271C42C2×#6711yescharacteristic
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3—162no—
278C22×C3#39211yescharacteristic
279C22×C3#47312yes—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—81no—
286C3×C4—81no—
287C3×C4—81no—
288C3×C4#45512yes—
289Dic3—82no—
290C23—41no—
291C23—42no—
292C23—81no—
293C23—81no—
294C23#93911yescharacteristic
295Q8—42no—
296Q8—42no—
297Q8—42no—
298Q8—42no—
299Q8—124no—
300Q8—242no—
301D4—42no—
302D4—42no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—122no—
311C2×C4—242no—
312C2×C4#93711yescharacteristic
313C2×C4#93711yescharacteristic
314C2×C4#93811yescharacteristic
315C2×C4#93911yescharacteristic
316C2×C4#94011yescharacteristic
317C2×C4#95311yescharacteristic
318C8—42no—
319C8—122no—
320C8—122no—
321C8—241no—
322C2×C3—22no—
323C2×C3—161no—
324C2×C3#200411yescharacteristic
325C2×C3#200811yescharacteristic
326C2×C3#202011yescharacteristic
327C22—21no—
328C22—21no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—162no—
333C22#520412yes—
334C22#522311yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—242no—
344C4#524212yes—
345C3#1013711yescharacteristic, complemented, Sylow
346C2—22no—
347C2—161no—
348C2#7185711yescharacteristic
349C2#7210911yescharacteristic
350C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3215C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#3215D3
hypocenterC3#10137
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3215

Complements

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

NH ≅ G/Ntype
2. #83370C2semidirect
4. #71822C2semidirect
6. #70822C2semidirect
7. #67751C2semidirect
345. C3#10137semidirect

Maximal subgroups

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

subgroupindex
C3×#32152
#833702
C2×#50552
#718222
#717202
#708222
#677512
#101373