Finite Group Databasebeta

← #10697110

Subgroups — #10697110

Order 768.

488 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3209C211yesmaximal, characteristic
2#83337C211yesmaximal, characteristic
3C2×#5079C211yesmaximal, characteristic
4#71826C211yesmaximal, characteristic
5#71790C211yesmaximal, characteristic
6#70778C211yesmaximal, characteristic
7#67739C211yesmaximal, characteristic
8#10171—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#494—21no—
13C3×#487—21no—
14C3×#462—22no—
15C3×#451—22no—
16C3×#431C2211yescharacteristic
17C2×#835—22no—
18#5093—21no—
19#5093—21no—
20#5079—22no—
21#5079—22no—
22#5069—22no—
23#5004—22no—
24#4924C2211yescharacteristic
25C4×#208C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4535—22no—
28#4530—22no—
29#4486—22no—
30#3209D311yescharacteristic
31#3139—31no—
32C2×#487—31no—
33#1997—31no—
34#1546—31no—
35#1493—31no—
36#1278—31no—
37C22×C3×Q8D412yes—
38C22×Dic6—21no—
39C2×C3×Q16—24no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
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⋊Dic3—21no—
57C2×#208—21no—
58C2×#208D412yes—
59C22×C3⋊C8—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D412yes—
64C3×C4⋊C8D412yes—
65C3×#103—24no—
66C3×C22⋊C8—21no—
67C3×C4×C8—21no—
68#835—28no—
69#810—24no—
70#810—24no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73#803—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8D412yes—
77#533C2×D311yescharacteristic
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#494—21no—
81#491—62no—
82#487—21no—
83#487—62no—
84#487—62no—
85#487—61no—
86#487—61no—
87#462—22no—
88#451—22no—
89#431C2×D311yescharacteristic
90#423—31no—
91#419—62no—
92C2×#103—62no—
93C4×M16—31no—
94C2×C4×C8—31no—
95#343—62no—
96#332—62no—
97Q8⋊C8—62no—
98C2×C3×Q8—22no—
99C2×C3×Q8—22no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—44no—
103C22×C3×C4—22no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—41no—
108C2×Dic6—22no—
109C2×Dic6—42no—
110C2×Dic6—42no—
111C2×Dic6—41no—
112C3×Q16—84no—
113C2×C3×C8—22no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
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—22no—
135#208—22no—
136#208—42no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—42no—
140C2×C3⋊C8—41no—
141C22×Q8—61no—
142C22×Q8#6712yes—
143C2×Q16—24no—
144C2×M16—32no—
145C2×M16—61no—
146C22×C8—61no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—32no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8#6711yescharacteristic
156C4⋊Q8#6711yescharacteristic
157#125—21no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162#117—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C42C22×D311yescharacteristic
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8#6712yes—
171#103—24no—
172#103—64no—
173#103—64no—
174#103—68no—
175C22⋊C8—21no—
176#97—61no—
177C4×C8—21no—
178C4×C8—32no—
179C4×C8—61no—
180C4×C8—61no—
181C23×C3#12111yescharacteristic
182C3×Q8—44no—
183C3×Q8—42no—
184C3×Q8—44no—
185C3×Q8—42no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—42no—
191C2×C3×C4—82no—
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12711yescharacteristic
198C2×C3×C4#13512yes—
199C2×C3×C4#13712yes—
200C2×Dic3—42no—
201C2×Dic3—82no—
202Dic6—44no—
203Dic6—81no—
204Dic6—81no—
205C3×C8—42no—
206C3⋊C8—42no—
207C3⋊C8—42no—
208C3⋊C8—81no—
209C2×Q8—22no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—21no—
213C2×Q8—44no—
214C2×Q8—62no—
215C2×Q8—121no—
216C2×Q8—122no—
217C2×Q8—122no—
218C22×C4—22no—
219C22×C4—121no—
220C22×C4D3×D411yescharacteristic
221C22×C4#23711yescharacteristic
222C22×C4C2×#6711yescharacteristic
223Q16—84no—
224M16—62no—
225M16—62no—
226M16—122no—
227C2×C8—22no—
228C2×C8—62no—
229C2×C8—62no—
230C2×C8—122no—
231C2×C8—121no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—62no—
243C4⋊C4—62no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C22⋊C4—41no—
247C42—41no—
248C42D3×D411yescharacteristic
249C42D3×D411yescharacteristic
250C42C2×#6711yescharacteristic
251C42C2×#6711yescharacteristic
252C22×C3—21no—
253C22×C3—21no—
254C22×C3#39211yescharacteristic
255C22×C3#46412yes—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—82no—
259C3×C4—81no—
260C3×C4—81no—
261C3×C4#45212yes—
262C3×C4#45512yes—
263C3×C4#47312yes—
264C3×C4#49511yescharacteristic
265C3×C4#49711yescharacteristic
266Dic3—82no—
267C23#93911yescharacteristic
268Q8—44no—
269Q8—42no—
270Q8—42no—
271Q8—44no—
272Q8—124no—
273Q8—241no—
274Q8—241no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—122no—
282C2×C4—242no—
283C2×C4#91812yes—
284C2×C4#91912yes—
285C2×C4#93711yescharacteristic
286C2×C4#93711yescharacteristic
287C2×C4#93811yescharacteristic
288C2×C4#93911yescharacteristic
289C2×C4#94011yescharacteristic
290C2×C4#95311yescharacteristic
291C8—42no—
292C8—122no—
293C8—122no—
294C8—241no—
295C2×C3—22no—
296C2×C3#200411yescharacteristic
297C2×C3#201511yescharacteristic
298C2×C3#202011yescharacteristic
299C22—21no—
300C22—21no—
301C22#518512yes—
302C22#522311yescharacteristic
303C4—21no—
304C4—21no—
305C4—81no—
306C4—82no—
307C4—81no—
308C4—242no—
309C4#518712yes—
310C4#519011yescharacteristic
311C4#519111yescharacteristic
312C4#520412yes—
313C4#524212yes—
314C3#1017111yescharacteristic, complemented, Sylow
315C2—22no—
316C2#7185711yescharacteristic
317C2#7207211yescharacteristic
318C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3209C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3209D3
hypocenterC3#10171
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
314. C3#10171semidirect

Maximal subgroups

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

subgroupindex
C3×#32092
#833372
C2×#50792
#718262
#717902
#707782
#677392
#101713