Finite Group Databasebeta

← #10690506

Subgroups — #10690506

Order 768.

530 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 103 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#83285C211yesmaximal, characteristic
2#83278C211yesmaximal, characteristic, complemented
3C3×#1525C211yesmaximal, characteristic, complemented
4#69853C212yesmaximal, complemented
5#67667C212yesmaximal, complemented
6#9069—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8C2×#950—22no—
9#5477C2211yescharacteristic
10C3×C4×M16C2212yes—
11#5234—22no—
12#5092C411yescharacteristic
13#5085C411yescharacteristic
14#5076C411yescharacteristic
15#5073C411yescharacteristic
16#5059—22no—
17#5012—22no—
18D4×C3⋊C8—22no—
19#4925C2212yes—
20C2×C4×C3⋊C8C2211yescharacteristic
21C3×#329—22no—
22#4459—22no—
23#4459—22no—
24#3147—31no—
25#2979—31no—
26#1525D311yescharacteristic, complemented
27#1484—32no—
28#1278—32no—
29C2×C3×#41D412yes—
30C2×#235—21no—
31C2×C3×M16—22no—
32C3×C4⋊Q8C2×C411yescharacteristic
33C3×#127C2×C411yescharacteristic
34C3×#124D412yes—
35C3×#121—22no—
36C3×C4×D4—22no—
37C2×C3×C42C2311yescharacteristic
38#950—44no—
39#950—44no—
40C2×#216—22no—
41C2×#215—22no—
42C2×#214—22no—
43C2×#213—22no—
44#930—22no—
45C2×#208—22no—
46C22×C3⋊C8—21no—
47C22×C3⋊C8—22no—
48#875—21no—
49#874C2×C411yescharacteristic
50C4×D12—22no—
51#869C2×C411yescharacteristic
52C3×#97—22no—
53C3×C4×C8—22no—
54#830—24no—
55#830—22no—
56C4⋊(C3⋊C8)D412yes—
57C4⋊(C3⋊C8)D412yes—
58C4×C3⋊C8—21no—
59C4×C3⋊C8—22no—
60C4×C3⋊C8C2×C411yescharacteristic
61C4×C3⋊C8C2×C411yescharacteristic
62C2×#135—62no—
63#532—31no—
64#532C2×D311yescharacteristic
65#495—62no—
66#494—31no—
67#493—31no—
68#492—31no—
69#492—31no—
70#436—62no—
71C8×D4—62no—
72#431—32no—
73#408—62no—
74C4×M16C2×D312yes—
75C2×C4×C8—31no—
76#329—22no—
77D4⋊C8—62no—
78D4⋊C8—62no—
79C3×#41—42no—
80C3×#41—42no—
81C2×C3×Q8C22⋊C412yes—
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4C22⋊C412yes—
85C22×C3×C4—22no—
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4#4112yes—
88C2×#67—41no—
89#235—42no—
90#235—42no—
91C2×D12—21no—
92C2×D12—22no—
93C2×C4×D3—41no—
94C2×Dic6—21no—
95C3×M16—24no—
96C3×M16—42no—
97C2×C3×C8—42no—
98C3×C4⋊C4—22no—
99C3×C22⋊C4—24no—
100C3×C42C22×C411yescharacteristic
101C3×C42C2×D412yes—
102C3×C42C2×D411yescharacteristic
103#216—44no—
104#215—44no—
105#214—44no—
106#213—44no—
107(C2×D3)⋊C4—42no—
108C4⋊Dic3—22no—
109#208—82no—
110C2×C3⋊C8—22no—
111C2×C3⋊C8—22no—
112C2×C3⋊C8—21no—
113C2×C3⋊C8—21no—
114C2×C3⋊C8—22no—
115C2×C3⋊C8—22no—
116C2×C3⋊C8—82no—
117C2×#41—61no—
118C2×#41D1212yes—
119#135—124no—
120#135—124no—
121C2×Q16—62no—
122C2×SD16—62no—
123C2×SD16—62no—
124C2×D8—62no—
125C2×M16—22no—
126C2×M16—62no—
127C22×C8—61no—
128C22×C8—62no—
129C4⋊Q8—31no—
130C4⋊Q8C4×D311yescharacteristic
131#127—31no—
132#127C4×D311yescharacteristic
133#124—61no—
134#124#6712yes—
135#121—22no—
136#121—62no—
137C4×D4—22no—
138C4×D4—62no—
139C2×C42C22×D311yescharacteristic
140C4⋊C8—32no—
141C4⋊C8—32no—
142C22⋊C8—64no—
143C22⋊C8—62no—
144#97—22no—
145C4×C8—22no—
146C4×C8—31no—
147C4×C8—31no—
148C4×C8—62no—
149C4×C8—61no—
150C23×C3—22no—
151C23×C3#12311yescharacteristic
152C22×D3—41no—
153C3×Q8—42no—
154C3×D4—42no—
155C3×D4—42no—
156C3×D4—42no—
157C3×D4—82no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—22no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4C2×C22⋊C411yescharacteristic
166C2×C3×C4C4×D412yes—
167C2×C3×C4#12011yescharacteristic
168C2×C3×C4#12112yes—
169#67—82no—
170C2×Dic3—41no—
171D12—42no—
172D12—42no—
173D12—42no—
174C4×D3—81no—
175C4×D3—81no—
176Dic6—42no—
177C3×C8—44no—
178C3⋊C8—22no—
179C3⋊C8—22no—
180C3⋊C8—42no—
181C3⋊C8—42no—
182#41—42no—
183#41—42no—
184#41—122no—
185#41—122no—
186C2×Q8—61no—
187C2×Q8(C2×D3)⋊C412yes—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—62no—
191C2×D4—61no—
192C2×D4—121no—
193C2×D4(C2×D3)⋊C412yes—
194C22×C4—22no—
195C22×C4—121no—
196C22×C4C2×D1211yescharacteristic
197C22×C4#23712yes—
198Q16—124no—
199SD16—124no—
200SD16—124no—
201D8—124no—
202M16—24no—
203M16—42no—
204M16—242no—
205C2×C8—42no—
206C2×C8—62no—
207C2×C8—62no—
208C2×C8—61no—
209C2×C8—61no—
210C2×C8—62no—
211C2×C8—62no—
212C2×C8—242no—
213C4⋊C4—22no—
214C4⋊C4—62no—
215C22⋊C4—24no—
216C22⋊C4—122no—
217C42C2×C4×D311yescharacteristic
218C42D3×D412yes—
219C42C2×#6711yescharacteristic
220C22×C3—22no—
221C22×C3—21no—
222C22×C3—44no—
223C22×C3—42no—
224C22×C3#38611yescharacteristic
225C2×D3—82no—
226C2×D3—81no—
227C3×C4—22no—
228C3×C4—21no—
229C3×C4—42no—
230C3×C4#44312yes—
231C3×C4#44412yes—
232C3×C4#45311yescharacteristic
233C3×C4#45311yescharacteristic
234Dic3—81no—
235C23—22no—
236C23—121no—
237C23#88611yescharacteristic
238Q8—42no—
239Q8—122no—
240D4—42no—
241D4—42no—
242D4—42no—
243D4—82no—
244D4—122no—
245D4—122no—
246D4—122no—
247D4—242no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—121no—
256C2×C4—241no—
257C2×C4—241no—
258C2×C4#67⋊C412yes—
259C2×C4#88211yescharacteristic
260C2×C4C2×(C2×D3)⋊C411yescharacteristic
261C2×C4#93912yes—
262C8—44no—
263C8—62no—
264C8—62no—
265C8—122no—
266C8—122no—
267C2×C3—41no—
268C2×C3—82no—
269C2×C3#187612yes—
270C2×C3#188711yescharacteristic
271D3—161no—
272C22—21no—
273C22—22no—
274C22—42no—
275C22—44no—
276C22—241no—
277C22—242no—
278C22#495811yescharacteristic
279C4—21no—
280C4—22no—
281C4—42no—
282C4—241no—
283C4#474711yescharacteristic
284C4#474711yescharacteristic
285C4#482312yes—
286C4#482412yes—
287C3#906911yescharacteristic, complemented, Sylow
288C2—41no—
289C2—82no—
290C2—481no—
291C2#7007011yescharacteristic
292C2#7037712yes—
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1525C2
socleC22×C3#386
prime radical#83285C2
cosocleC2×C3×C42C23
hypercenter#1525D3
hypocenterC3#9069
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1525

Complements

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

NH ≅ G/Ntype
2. #83278C2semidirect
3. C3×#1525C2semidirect
4. #69853C2semidirect
5. #67667C2semidirect
26. #1525D3semidirect
287. C3#9069semidirect

Maximal subgroups

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

subgroupindex
#832852
#832782
C3×#15252
#698532
#676672
#90693