Finite Group Databasebeta

← #10696883

Subgroups — #10696883

Order 768.

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

normal subgroups only

This group has 446 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
1C2×C3×#501C211yesmaximal, characteristic
2#83222C211yesmaximal, characteristic
3#83021C211yesmaximal, characteristic
4#71776C211yesmaximal, characteristic
5#71737C211yesmaximal, characteristic
6#69801C211yesmaximal, characteristic
7#67662C211yesmaximal, characteristic
8#10139—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#501—22no—
13C3×#501—22no—
14C2×C3×#107—21no—
15C2×C3×#106—21no—
16C2×C3×#97C2211yescharacteristic
17#5228—22no—
18#5092—21no—
19#5090—22no—
20#5079—21no—
21#5069—21no—
22#5064—21no—
23#5063—22no—
24#4971—21no—
25#4970—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29#4547—22no—
30#4528—22no—
31#4465—22no—
32#3236—31no—
33#3209—31no—
34C2×#501D311yescharacteristic
35#1556—31no—
36#1531—31no—
37#1470—31no—
38#1302—31no—
39C22×C3×Q8—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D412yes—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C3×#125—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42C2311yescharacteristic
56C2×#216—24no—
57#926—21no—
58#924—21no—
59#924—21no—
60#870—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×#107—24no—
66C3×#106—24no—
67C3×#97—21no—
68C3×#97D412yes—
69#835—24no—
70#830—21no—
71#830—21no—
72#830D412yes—
73#810—22no—
74#810—22no—
75#808—22no—
76#807—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80#803—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125C2×D311yescharacteristic
86#501—22no—
87#501—22no—
88#501—61no—
89#494—61no—
90#491—61no—
91#487—61no—
92#484—61no—
93#483—61no—
94#479—62no—
95#462—62no—
96#451—62no—
97#432—31no—
98#432—31no—
99#431—31no—
100C2×#107—21no—
101C2×#106—21no—
102C2×#97C2×D311yescharacteristic
103#332—62no—
104#330—62no—
105#324—62no—
106C2×C3×Q8—22no—
107C2×C3×Q8—41no—
108C2×C3×Q8—82no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×C3×C8—24no—
118C2×C3×C8—22no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135#217—41no—
136#216—84no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143C4×Dic3—41no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—41no—
147C22×Q8—21no—
148C2×Q16—64no—
149C22×C8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—31no—
154C4⋊Q8—31no—
155C4⋊Q8#6712yes—
156#125—21no—
157#125—21no—
158#125—21no—
159#125—21no—
160#125—61no—
161C22⋊Q8—61no—
162C22⋊Q8—61no—
163C4×Q8—61no—
164C4×Q8—61no—
165#117—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4#6711yescharacteristic
171C2×C4⋊C4#6711yescharacteristic
172C2×C42C22×D311yescharacteristic
173#107—24no—
174#107—62no—
175#106—24no—
176#106—62no—
177C4⋊C8—32no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180#103—62no—
181#103—62no—
182#103—64no—
183C22⋊C8—32no—
184C22⋊C8—61no—
185C22⋊C8—61no—
186#97—21no—
187#97—61no—
188#97—61no—
189#97#6712yes—
190C4×C8—61no—
191C23×C3#12111yescharacteristic
192C3×Q8—44no—
193C3×Q8—82no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—82no—
202C2×C3×C4—81no—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12711yescharacteristic
209C2×Dic3—41no—
210C2×Dic3—41no—
211Dic6—42no—
212Dic6—42no—
213C3×C8—24no—
214C3⋊C8—42no—
215C3⋊C8—42no—
216C3⋊C8—81no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—61no—
220C2×Q8—61no—
221C2×Q8—82no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4D3×D411yescharacteristic
226C22×C4#23711yescharacteristic
227C22×C4C2×#6711yescharacteristic
228Q16—244no—
229C2×C8—24no—
230C2×C8—22no—
231C2×C8—62no—
232C2×C8—62no—
233C2×C8—121no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—22no—
237C4⋊C4—21no—
238C4⋊C4—22no—
239C4⋊C4—21no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C22⋊C4—121no—
253C42—121no—
254C42D3×D411yescharacteristic
255C42D3×D411yescharacteristic
256C42C2×#6711yescharacteristic
257C42C2×#6711yescharacteristic
258C22×C3—21no—
259C22×C3—21no—
260C22×C3#39211yescharacteristic
261C22×C3#47312yes—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—81no—
268C3×C4—82no—
269C3×C4—81no—
270C3×C4#45612yes—
271Dic3—81no—
272Dic3—81no—
273C23#93911yescharacteristic
274Q8—44no—
275Q8—82no—
276Q8—122no—
277Q8—122no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—82no—
286C2×C4—81no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4#93711yescharacteristic
290C2×C4#93711yescharacteristic
291C2×C4#93811yescharacteristic
292C2×C4#93911yescharacteristic
293C2×C4#94011yescharacteristic
294C2×C4#95311yescharacteristic
295C8—24no—
296C8—122no—
297C8—122no—
298C8—241no—
299C2×C3—22no—
300C2×C3#200511yescharacteristic
301C2×C3#200811yescharacteristic
302C2×C3#202011yescharacteristic
303C22—21no—
304C22—21no—
305C22#520412yes—
306C22#522311yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—82no—
313C4—81no—
314C4—81no—
315C4—241no—
316C4—241no—
317C4#509512yes—
318C3#1013911yescharacteristic, complemented, Sylow
319C2—22no—
320C2#7179511yescharacteristic
321C2#7209911yescharacteristic
322C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#501C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenterC2×#501D3
hypocenterC3#10139
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
318. C3#10139semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5012
#832222
#830212
#717762
#717372
#698012
#676622
#101393