Finite Group Databasebeta

← #10697070

Subgroups — #10697070

Order 768.

476 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#83341C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic
3C2×#4682C211yesmaximal, characteristic
4C3×#1493C211yesmaximal, characteristic
5#70524C211yesmaximal, characteristic
6#70507C211yesmaximal, characteristic
7#67656C211yesmaximal, characteristic
8#10165—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C2×#818—22no—
15#5080—21no—
16#5079—21no—
17#5078—21no—
18#5078—21no—
19#5064—22no—
20#5044—22no—
21#5021—22no—
22#4995—22no—
23#4925C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4682—22no—
26#4682—22no—
27C3×Q8⋊C8—22no—
28#4554—22no—
29#4481—22no—
30#3230—31no—
31#3165—31no—
32C2×#500—31no—
33#1997—31no—
34#1548—31no—
35#1493D311yescharacteristic
36#1278—31no—
37C22×Dic6—21no—
38C22×Dic6—21no—
39C2×C3×M16—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8D411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117—21no—
49C2×C3×C42C2311yescharacteristic
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D412yes—
54C2×#208D412yes—
55#869—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869D412yes—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4×C8—21no—
66C3×C4×C8D412yes—
67#835—24no—
68#830—21no—
69#818—28no—
70#808—24no—
71#807—24no—
72#807—24no—
73C4⋊(C3⋊C8)D412yes—
74#803—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77#533C2×D311yescharacteristic
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8—31no—
80#501—62no—
81#500—62no—
82#500—61no—
83#500—61no—
84#500—62no—
85#498—61no—
86#487—61no—
87#484—62no—
88#477—62no—
89#431—31no—
90#428—62no—
91C2×#107—62no—
92#423C2×D311yescharacteristic
93C4×M16—31no—
94C2×C4×C8C2×D311yescharacteristic
95#343—62no—
96#333—62no—
97Q8⋊C8—22no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×Dic3—22no—
104C22×Dic3—41no—
105C2×Dic6—22no—
106C2×Dic6—41no—
107C2×Dic6—41no—
108C2×Dic6—41no—
109C2×Dic6—82no—
110C2×Dic6—82no—
111C3×M16—42no—
112C2×C3×C8—24no—
113C2×C3×C8—22no—
114C2×C3×C8—41no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C42—41no—
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C4⋊Dic3—22no—
128C4⋊Dic3—22no—
129C4⋊Dic3—22no—
130C4⋊Dic3—22no—
131C4⋊Dic3—41no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135#208—22no—
136#208Q1614yes—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—22no—
139C22×Q8—61no—
140C22×Q8—61no—
141C2×M16—21no—
142C2×M16—32no—
143C22×C8—21no—
144C4⋊Q8—32no—
145C4⋊Q8—61no—
146C4⋊Q8—61no—
147C4⋊Q8—61no—
148C4⋊Q8—61no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8D1211yescharacteristic
153C4⋊Q8D1211yescharacteristic
154#125—21no—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157C4×Q8—21no—
158C4×Q8—21no—
159#117—21no—
160C2×C4⋊C4—32no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C42C22×D311yescharacteristic
165#107—64no—
166#107—64no—
167#107—68no—
168#106—64no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—32no—
172#103—64no—
173C22⋊C8—61no—
174#97—61no—
175C4×C8—21no—
176C4×C8—61no—
177C4×C8—61no—
178C4×C8#6712yes—
179C23×C3#12111yescharacteristic
180C3×Q8—42no—
181C3×Q8—42no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12711yescharacteristic
192C2×C3×C4C2×Q1612yes—
193C2×C3×C4#13712yes—
194C2×Dic3—42no—
195C2×Dic3—42no—
196C2×Dic3—82no—
197C2×Dic3—82no—
198Dic6—82no—
199Dic6—82no—
200Dic6—82no—
201Dic6—82no—
202C3×C8—24no—
203C3×C8—81no—
204C3⋊C8—24no—
205C3⋊C8—42no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—62no—
209C2×Q8—121no—
210C2×Q8—121no—
211C2×Q8—121no—
212C2×Q8—242no—
213C2×Q8—242no—
214C22×C4—62no—
215C22×C4—121no—
216C22×C4#23511yescharacteristic
217C22×C4D3×D411yescharacteristic
218C22×C4D3×D411yescharacteristic
219M16—34no—
220M16—42no—
221M16—62no—
222C2×C8—24no—
223C2×C8—22no—
224C2×C8—41no—
225C2×C8—62no—
226C2×C8—62no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—62no—
234C4⋊C4—62no—
235C4⋊C4—62no—
236C4⋊C4—62no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C22⋊C4—41no—
242C42—41no—
243C42C2×D1211yescharacteristic
244C42D3×D411yescharacteristic
245C42D3×D411yescharacteristic
246C42C2×#6711yescharacteristic
247C22×C3—21no—
248C22×C3—21no—
249C22×C3#39211yescharacteristic
250C22×C3#46212yes—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—81no—
254C3×C4—81no—
255C3×C4#45112yes—
256C3×C4#45612yes—
257C3×C4#47212yes—
258C3×C4#49411yescharacteristic
259C3×C4#49711yescharacteristic
260Dic3—82no—
261Dic3—82no—
262C23#88411yescharacteristic
263Q8—42no—
264Q8—42no—
265Q8—242no—
266Q8—242no—
267Q8—242no—
268Q8—242no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—122no—
274C2×C4—122no—
275C2×C4—242no—
276C2×C4—242no—
277C2×C4#88211yescharacteristic
278C2×C4#88211yescharacteristic
279C2×C4#88411yescharacteristic
280C2×C4#91512yes—
281C2×C4D3×Q1612yes—
282C2×C4#93011yescharacteristic
283C2×C4#93711yescharacteristic
284C2×C4#94011yescharacteristic
285C8—24no—
286C8—64no—
287C8—81no—
288C8—122no—
289C2×C3—22no—
290C2×C3#200511yescharacteristic
291C2×C3#201611yescharacteristic
292C2×C3#201811yescharacteristic
293C22—21no—
294C22—21no—
295C22#479512yes—
296C22#496011yescharacteristic
297C4—21no—
298C4—21no—
299C4—81no—
300C4—81no—
301C4—242no—
302C4—242no—
303C4#480912yes—
304C4#482612yes—
305C4#514212yes—
306C4#516711yescharacteristic
307C4#518311yescharacteristic
308C3#1016511yescharacteristic, complemented, Sylow
309C2—22no—
310C2#7000611yescharacteristic
311C2#7028311yescharacteristic
312C2#7038911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1493C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1493D3
hypocenterC3#10165
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
308. C3#10165semidirect

Maximal subgroups

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

subgroupindex
#833412
#833242
C2×#46822
C3×#14932
#705242
#705072
#676562
#101653