Finite Group Databasebeta

← #10696877

Subgroups — #10696877

Order 768.

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

normal subgroups only

This group has 468 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×#3206C211yesmaximal, characteristic
2#83273C211yesmaximal, characteristic
3C2×#5081C211yesmaximal, characteristic, complemented
4#71798C211yesmaximal, characteristic
5#71776C211yesmaximal, characteristic, complemented
6#70787C211yesmaximal, characteristic, complemented
7#67742C211yesmaximal, characteristic, complemented
8#10138—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#492—21no—
13C3×#487—21no—
14C3×#461—22no—
15C3×#448—22no—
16C3×#431C2211yescharacteristic
17#5081—22no—
18#5081—21no—
19#5081—22no—
20#5071—21no—
21#5034—21no—
22#5033—21no—
23#5009—22no—
24C2×#808—21no—
25C2×#807—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4547—22no—
30#4540—22no—
31#4469—22no—
32#3234—31no—
33#3206D311yescharacteristic
34C2×#501—31no—
35#1556—31no—
36#1530—31no—
37#1470—31no—
38#1302—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C2×C3×SD16—24no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C22×C3⋊C8—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64#870—21no—
65C3×C4⋊C8—21no—
66C3×#103—24no—
67C3×C22⋊C8D412yes—
68C3×C4×C8—21no—
69#832—22no—
70#832—22no—
71#830—21no—
72#830—21no—
73#808—24no—
74#808—22no—
75#807—24no—
76#807—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79#803—21no—
80#803—21no—
81#803—21no—
82#803D412yes—
83#532C2×D311yescharacteristic
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125—31no—
86#501—62no—
87#501—62no—
88#501—61no—
89#492—21no—
90#490—61no—
91#487—21no—
92#481—61no—
93#480—61no—
94#478—62no—
95#461—22no—
96#448—22no—
97#432—31no—
98#432—31no—
99#431C2×D311yescharacteristic
100C2×#107—61no—
101C2×#106—61no—
102C2×#97—31no—
103#332—62no—
104#329—62no—
105#324—62no—
106C3×#41—82no—
107C2×C3×Q8—22no—
108C2×C3×Q8—21no—
109C2×C3×Q8—41no—
110C2×C3×Q8—82no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—21no—
122C3×SD16—84no—
123C2×C3×C8—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144C2×C3⋊C8—24no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C2×#41—21no—
149C22×Q8—21no—
150C2×SD16—24no—
151C22×C8—61no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8#6712yes—
156C4⋊Q8#6711yescharacteristic
157#127#6711yescharacteristic
158#125—61no—
159#125—61no—
160#125—61no—
161#125—61no—
162#124—21no—
163#121—21no—
164#121—21no—
165C4×D4—21no—
166C4×D4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C42C22×D311yescharacteristic
174#107—64no—
175#107—62no—
176#106—64no—
177#106—62no—
178C4⋊C8—21no—
179C4⋊C8—32no—
180C4⋊C8—61no—
181#103—24no—
182#102—62no—
183#102—62no—
184C22⋊C8—61no—
185C22⋊C8—61no—
186C22⋊C8#6712yes—
187#97—32no—
188#97—61no—
189#97—61no—
190#97—61no—
191C4×C8—21no—
192C23×C3—41no—
193C23×C3#12111yescharacteristic
194C3×Q8—44no—
195C3×Q8—81no—
196C3×Q8—82no—
197C3×D4—42no—
198C3×D4—42no—
199C3×D4—81no—
200C3×D4—82no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—82no—
207C2×C3×C4—82no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12711yescharacteristic
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—42no—
217C2×Dic3—81no—
218C3×C8—42no—
219C3⋊C8—24no—
220C3⋊C8—42no—
221C3⋊C8—81no—
222#41—82no—
223C2×Q8—21no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×Q8—82no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4D3×D411yescharacteristic
236C22×C4#23711yescharacteristic
237C22×C4C2×#6711yescharacteristic
238SD16—84no—
239C2×C8—22no—
240C2×C8—64no—
241C2×C8—62no—
242C2×C8—62no—
243C2×C8—121no—
244C4⋊C4—22no—
245C4⋊C4—21no—
246C4⋊C4—22no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C42D3×D411yescharacteristic
261C42D3×D411yescharacteristic
262C42C2×#6711yescharacteristic
263C42C2×#6711yescharacteristic
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—81no—
267C22×C3—81no—
268C22×C3—81no—
269C22×C3#39211yescharacteristic
270C22×C3#47112yes—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—82no—
277C3×C4—81no—
278C3×C4#45612yes—
279Dic3—81no—
280Dic3—81no—
281C23—41no—
282C23#93811yescharacteristic
283Q8—44no—
284Q8—81no—
285Q8—82no—
286D4—42no—
287D4—42no—
288D4—81no—
289D4—82no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—122no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—241no—
301C2×C4#93711yescharacteristic
302C2×C4#93711yescharacteristic
303C2×C4#93911yescharacteristic
304C2×C4#93911yescharacteristic
305C2×C4#94011yescharacteristic
306C2×C4#95311yescharacteristic
307C8—42no—
308C8—64no—
309C8—122no—
310C8—241no—
311C2×C3—22no—
312C2×C3—161no—
313C2×C3#200511yescharacteristic
314C2×C3#200811yescharacteristic
315C2×C3#201911yescharacteristic
316C22—21no—
317C22—21no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22#517712yes—
322C22#522311yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—81no—
329C4—82no—
330C4—241no—
331C4—241no—
332C4#520512yes—
333C3#1013811yescharacteristic, complemented, Sylow
334C2—22no—
335C2—161no—
336C2#7185811yescharacteristic
337C2#7205911yescharacteristic
338C2#7210811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3206C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3206D3
hypocenterC3#10138
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
3. C2×#5081C2semidirect
5. #71776C2semidirect
6. #70787C2semidirect
7. #67742C2semidirect
333. C3#10138semidirect

Maximal subgroups

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

subgroupindex
C3×#32062
#832732
C2×#50812
#717982
#717762
#707872
#677422
#101383