Finite Group Databasebeta

← #10696716

Subgroups — #10696716

Order 768.

466 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 89 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×#3255C211yesmaximal, characteristic
2#83370C211yesmaximal, characteristic
3#83337C211yesmaximal, characteristic
4#71825C211yesmaximal, characteristic
5#71821C211yesmaximal, characteristic
6#70524C211yesmaximal, characteristic
7#67710C211yesmaximal, characteristic
8#10115—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#533C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#475—21no—
13C3×#475—21no—
14C3×#470—22no—
15C3×#462—21no—
16C3×#462—21no—
17C3×#423C2211yescharacteristic
18#5093—21no—
19#5093—21no—
20#5093—21no—
21#5092—21no—
22#5069—22no—
23#5050—22no—
24#5031—22no—
25#5004—22no—
26#4925C2211yescharacteristic
27C4×#208C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4554—22no—
30#4543—22no—
31#4535—22no—
32#3255D311yescharacteristic
33#3209—31no—
34#3139—31no—
35#1997—31no—
36#1526—31no—
37#1485—31no—
38#1299—31no—
39C22×Dic6D412yes—
40C2×C3×Q16—22no—
41C2×C3×M16D412yes—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×#117—21no—
57C3×#117—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#216—24no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×#208—21no—
63C2×#208—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869—21no—
68C3×#106—22no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×#103—22no—
72C3×#103—22no—
73#830—21no—
74#810—24no—
75#810—24no—
76#810—24no—
77C4⋊(C3⋊C8)D412yes—
78#803—21no—
79#803D412yes—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84#533C2×D311yescharacteristic
85C2×C4⋊Q8—31no—
86#494—61no—
87#491—62no—
88#487—61no—
89#487—61no—
90#487—61no—
91#475—21no—
92#475—21no—
93#470—22no—
94#462—21no—
95#462—21no—
96#462—62no—
97#451—62no—
98#431—31no—
99#423C2×D311yescharacteristic
100#419—62no—
101C4×M16—31no—
102C4×M16—31no—
103#343—62no—
104#330—62no—
105Q8⋊C8—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×Dic3—22no—
114C2×Dic6—22no—
115C2×Dic6—22no—
116C2×Dic6—44no—
117C3×Q16—82no—
118C3×M16—42no—
119C2×C3×C8—22no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C42—41no—
135C3×C42—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140#216—84no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145#208—22no—
146#208—42no—
147#208—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C22×Q8—32no—
152C2×Q16—22no—
153C2×Q16—64no—
154C2×M16—61no—
155C2×M16—61no—
156C2×M16#6712yes—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8#6711yescharacteristic
162C4⋊Q8#6711yescharacteristic
163C4⋊Q8#6711yescharacteristic
164C4⋊Q8#6711yescharacteristic
165#125—21no—
166#125—21no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175#117—21no—
176#117—21no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C42C22×D311yescharacteristic
180#106—22no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—32no—
184#103—22no—
185#103—22no—
186#103—64no—
187#103—64no—
188#103—64no—
189C22⋊C8—61no—
190#97—32no—
191#97—61no—
192C4×C8—61no—
193C4×C8—61no—
194C4×C8—61no—
195C23×C3#12111yescharacteristic
196C3×Q8—42no—
197C3×Q8—41no—
198C3×Q8—41no—
199C3×Q8—41no—
200C3×Q8—41no—
201C3×Q8—42no—
202C2×C3×C4—21no—
203C2×C3×C4—22no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12711yescharacteristic
216C2×Dic3—42no—
217C2×Dic3—82no—
218Dic6—44no—
219Dic6—44no—
220C3×C8—42no—
221C3⋊C8—42no—
222C3⋊C8—42no—
223C3⋊C8—42no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—62no—
229C2×Q8—62no—
230C2×Q8—124no—
231C22×C4—62no—
232C22×C4D3×D411yescharacteristic
233C22×C4#23711yescharacteristic
234C22×C4C2×#6711yescharacteristic
235Q16—82no—
236Q16—244no—
237M16—42no—
238M16—62no—
239M16—122no—
240M16—121no—
241C2×C8—22no—
242C2×C8—62no—
243C2×C8—62no—
244C2×C8—121no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C42—41no—
264C42—41no—
265C42D3×D411yescharacteristic
266C42D3×D411yescharacteristic
267C42C2×#6711yescharacteristic
268C42C2×#6711yescharacteristic
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3#39211yescharacteristic
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—81no—
277C3×C4—81no—
278C3×C4—81no—
279C3×C4—81no—
280C3×C4#45512yes—
281C3×C4#45511yescharacteristic
282C3×C4#45611yescharacteristic
283C3×C4#47312yes—
284Dic3—82no—
285C23#93811yescharacteristic
286Q8—42no—
287Q8—42no—
288Q8—41no—
289Q8—41no—
290Q8—41no—
291Q8—41no—
292Q8—124no—
293Q8—124no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—122no—
303C2×C4—242no—
304C2×C4#93711yescharacteristic
305C2×C4#93711yescharacteristic
306C2×C4#93911yescharacteristic
307C2×C4#93911yescharacteristic
308C2×C4#94011yescharacteristic
309C2×C4#95311yescharacteristic
310C8—42no—
311C8—122no—
312C8—122no—
313C8—122no—
314C2×C3—41no—
315C2×C3#200411yescharacteristic
316C2×C3#200711yescharacteristic
317C2×C3#202011yescharacteristic
318C22—21no—
319C22—21no—
320C22—21no—
321C22#522311yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—242no—
330C4#520212yes—
331C4#520412yes—
332C4#524211yescharacteristic
333C4#524611yescharacteristic
334C3#1011511yescharacteristic, complemented, Sylow
335C2—41no—
336C2#7188111yescharacteristic
337C2#7210611yescharacteristic
338C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3255C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3255D3
hypocenterC3#10115
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3255

Complements

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

NH ≅ G/Ntype
334. C3#10115semidirect

Maximal subgroups

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

subgroupindex
C3×#32552
#833702
#833372
#718252
#718212
#705242
#677102
#101153