Finite Group Databasebeta

← #10697066

Subgroups — #10697066

Order 768.

476 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 113 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×#3165C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic
3C2×#5078C211yesmaximal, characteristic
4#71826C211yesmaximal, characteristic
5#71774C211yesmaximal, characteristic
6#69879C211yesmaximal, characteristic
7#67667C211yesmaximal, characteristic
8#10165—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#501—22no—
13C3×#500—21no—
14C3×#498—21no—
15C3×#428—22no—
16C3×C4×M16C2211yescharacteristic
17#5079—21no—
18#5078—22no—
19#5078—21no—
20#5078—22no—
21#5044—22no—
22#5021—22no—
23C2×#807—22no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4554—22no—
28#4546—22no—
29#4535—22no—
30#3230—31no—
31#3165D311yescharacteristic
32C2×#500—31no—
33#1997—31no—
34#1548—31no—
35#1493—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×Dic6—21no—
39C2×C3×M16D412yes—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8D412yes—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D412yes—
57C2×#208—21no—
58C22×C3⋊C8—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63C3×#107—24no—
64C3×#106—24no—
65C3×#97—21no—
66C3×C4×C8—21no—
67#835—24no—
68#830—21no—
69#807—28no—
70#807—24no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)D412yes—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8D412yes—
77#533C2×D311yescharacteristic
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#501—22no—
81#500—21no—
82#500—62no—
83#500—62no—
84#500—61no—
85#498—21no—
86#487—61no—
87#484—62no—
88#477—62no—
89#431—31no—
90#428—22no—
91C2×#107—62no—
92#423—31no—
93C4×M16C2×D311yescharacteristic
94C2×C4×C8—31no—
95#343—62no—
96#333—62no—
97Q8⋊C8—62no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—22no—
101C2×C3×Q8—41no—
102C2×C3×Q8—82no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—22no—
108C2×Dic6—41no—
109C2×Dic6—41no—
110C2×Dic6—82no—
111C3×M16—22no—
112C3×M16Q1614yes—
113C2×C3×C8—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C22⋊C4—41no—
125C3×C42—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C4⋊Dic3—22no—
131C4⋊Dic3—22no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134#208—42no—
135C2×C3⋊C8—24no—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—41no—
139C22×Q8—21no—
140C22×Q8—61no—
141C2×M16—61no—
142C2×M16#6712yes—
143C22×C8—61no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—61no—
148C4⋊Q8—61no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8#6712yes—
152C4⋊Q8#6711yescharacteristic
153C4⋊Q8#6711yescharacteristic
154#125—21no—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157C4×Q8—21no—
158C4×Q8—21no—
159#117—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—32no—
163C2×C4⋊C4—61no—
164C2×C42C22×D311yescharacteristic
165#107—24no—
166#107—68no—
167#107—64no—
168#106—24no—
169C4⋊C8—32no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172#103—64no—
173C22⋊C8—61no—
174#97—21no—
175C4×C8—21no—
176C4×C8—32no—
177C4×C8—61no—
178C4×C8—61no—
179C23×C3#12111yescharacteristic
180C3×Q8—42no—
181C3×Q8—42no—
182C3×Q8—82no—
183C3×Q8—82no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—42no—
189C2×C3×C4—82no—
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12711yescharacteristic
196C2×C3×C4C2×Q1612yes—
197C2×C3×C4#13712yes—
198C2×Dic3—42no—
199C2×Dic3—82no—
200Dic6—82no—
201Dic6—82no—
202C3×C8—24no—
203C3⋊C8—24no—
204C3⋊C8—42no—
205C3⋊C8—81no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×Q8—41no—
210C2×Q8—82no—
211C2×Q8—121no—
212C2×Q8—121no—
213C2×Q8—242no—
214C22×C4—41no—
215C22×C4—62no—
216C22×C4D3×D411yescharacteristic
217C22×C4#23711yescharacteristic
218C22×C4C2×#6711yescharacteristic
219M16—22no—
220M16—122no—
221M16#21614yes—
222C2×C8—22no—
223C2×C8—64no—
224C2×C8—62no—
225C2×C8—62no—
226C2×C8—121no—
227C4⋊C4—22no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—22no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—62no—
238C4⋊C4—62no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C22⋊C4—41no—
242C42—41no—
243C42D3×D411yescharacteristic
244C42D3×D411yescharacteristic
245C42C2×#6711yescharacteristic
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—82no—
255C3×C4—81no—
256C3×C4#45112yes—
257C3×C4#45612yes—
258C3×C4#47212yes—
259C3×C4#49411yescharacteristic
260C3×C4#49711yescharacteristic
261Dic3—82no—
262C23#93811yescharacteristic
263Q8—42no—
264Q8—42no—
265Q8—82no—
266Q8—82no—
267Q8—242no—
268Q8—242no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—82no—
275C2×C4—122no—
276C2×C4—242no—
277C2×C4#93711yescharacteristic
278C2×C4#93711yescharacteristic
279C2×C4#93911yescharacteristic
280C2×C4#93911yescharacteristic
281C2×C4#94011yescharacteristic
282C2×C4#94212yes—
283C2×C4C2×#21612yes—
284C2×C4#95311yescharacteristic
285C8—24no—
286C8—64no—
287C8—122no—
288C8—241no—
289C2×C3—22no—
290C2×C3#200511yescharacteristic
291C2×C3#201611yescharacteristic
292C2×C3#201811yescharacteristic
293C22—21no—
294C22—21no—
295C22#509012yes—
296C22#522311yescharacteristic
297C4—21no—
298C4—21no—
299C4—82no—
300C4—81no—
301C4—81no—
302C4—242no—
303C4#508611yescharacteristic
304C4#509211yescharacteristic
305C4#519912yes—
306C4#520512yes—
307C4#522812yes—
308C3#1016511yescharacteristic, complemented, Sylow
309C2—22no—
310C2#7179311yescharacteristic
311C2#7209511yescharacteristic
312C2#7210811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3165D3
hypocenterC3#10165
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

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
C3×#31652
#833242
C2×#50782
#718262
#717742
#698792
#676672
#101653