Finite Group Databasebeta

← #10709127

Subgroups — #10709127

Order 768.

290 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 61 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#79714C211yesmaximal, characteristic
2C3×#2227C211yesmaximal, characteristic, complemented
3#71383C211yesmaximal, characteristic
4#71379C211yesmaximal, characteristic, complemented
5#67764C211yesmaximal, characteristic, complemented
6#67562C211yesmaximal, characteristic
7#67559C211yesmaximal, characteristic, complemented
8#12391—31nomaximal, Sylow
9#5568—21no—
10C3×#500C2211yescharacteristic
11C3×C4×Q16C2211yescharacteristic
12#5190—21no—
13#5090—21no—
14#5028—21no—
15#4889—21no—
16#4819—21no—
17#4795—21no—
18#4696C2211yescharacteristic
19#4679C2211yescharacteristic
20C3×#366—21no—
21C3×C4⋊C16C2211yescharacteristic
22C3×#358—21no—
23#4547—21no—
24#4492—21no—
25#4491—21no—
26#4461C2211yescharacteristic
27#4457C2211yescharacteristic
28#3402—31no—
29#2243—31no—
30#2227D311yescharacteristic, complemented
31#1694—31no—
32#1360—31no—
33#1338—31no—
34#1334—31no—
35C2×C3×Q16D411yescharacteristic
36C3×C4⋊Q8—21no—
37C3×C4×Q8—21no—
38#947—21no—
39#944—21no—
40C2×#216—21no—
41C2×#216—41no—
42C2×#215—21no—
43C2×Dic12D411yescharacteristic
44C2×D24D411yescharacteristic
45C2×#204—21no—
46#898—41no—
47#896—41no—
48#893—21no—
49#887—21no—
50#876—41no—
51#875—21no—
52#872—41no—
53C4×Dic6D411yescharacteristic
54C2×C3×C16—21no—
55C3×#107—21no—
56C3×#107—21no—
57C3×#107D411yescharacteristic
58C3×#103—21no—
59C3×C4×C8C2311yescharacteristic
60#835—41no—
61#815—21no—
62Dic3⋊C8—21no—
63C2×C3⋊C16—21no—
64#810—21no—
65#809—21no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)D411yescharacteristic
68#556—61no—
69#500C2×D311yescharacteristic
70#497—61no—
71#495—31no—
72#464—61no—
73#463—61no—
74#462—61no—
75#462—61no—
76#446—31no—
77#440—61no—
78C4×Q16C2×D311yescharacteristic
79#366—21no—
80C4⋊C16C2×D311yescharacteristic
81#358—21no—
82#358—61no—
83#357—61no—
84#346—31no—
85#334—31no—
86#332—61no—
87C2×C3×Q8—21no—
88C2×C3×Q8—41no—
89C2×D12—21no—
90C2×C4×D3—41no—
91C2×Dic6—21no—
92C2×Dic6D811yescharacteristic
93C3×Q16—21no—
94C3×Q16—21no—
95C2×C3×C8C2×D411yescharacteristic
96C2×C3×C8C2×D411yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4—41no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—41no—
101C3×C42—41no—
102C3×C42C2×D411yescharacteristic
103#216—81no—
104#216—82no—
105#215—81no—
106(C2×D3)⋊C4—41no—
107(C2×D3)⋊C4—41no—
108(C2×D3)⋊C4—41no—
109(C2×D3)⋊C4—41no—
110(C2×D3)⋊C4—41no—
111C4⋊Dic3D811yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—41no—
114Dic3⋊C4—41no—
115Dic3⋊C4—41no—
116C4×Dic3—21no—
117C4×Dic3—41no—
118C2×C3⋊C8—21no—
119C2×C3⋊C8—41no—
120Dic12—41no—
121D24—41no—
122#204—41no—
123#204—41no—
124C3×C16—41no—
125C3⋊C16—41no—
126C2×Q16—31no—
127C2×Q16—61no—
128C2×Q16—121no—
129C2×Q16#6711yescharacteristic
130C2×SD16—61no—
131C2×SD16—61no—
132C2×D8—31no—
133C4⋊Q8—21no—
134C4⋊Q8—61no—
135#126—121no—
136#126—121no—
137#124—61no—
138#124—61no—
139C22⋊Q8—121no—
140C4×Q8—21no—
141C4×Q8—31no—
142C4×Q8—61no—
143C4×D4—61no—
144#117—121no—
145C2×C16—21no—
146C2×C16—61no—
147#107—21no—
148#107—21no—
149#107#6711yescharacteristic
150C4⋊C8—31no—
151C4⋊C8—61no—
152C4⋊C8—61no—
153#103—21no—
154#103—61no—
155#103—121no—
156#102—61no—
157#97—61no—
158C4×C8C22×D311yescharacteristic
159C22×D3—41no—
160C3×Q8—41no—
161C3×Q8—41no—
162C3×Q8—81no—
163C3×Q8—81no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4—81no—
167C2×C3×C4#12011yescharacteristic
168C2×C3×C4C2×D811yescharacteristic
169C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
170C2×Dic3—21no—
171C2×Dic3—21no—
172C2×Dic3—41no—
173D12—41no—
174D12—41no—
175C4×D3—82no—
176Dic6—41no—
177Dic6—41no—
178Dic6—41no—
179C3×C8—21no—
180C3×C8#13511yescharacteristic
181C3×C8#13711yescharacteristic
182C3⋊C8—41no—
183C3⋊C8—81no—
184C2×Q8—21no—
185C2×Q8—31no—
186C2×Q8—41no—
187C2×Q8—61no—
188C2×D4—61no—
189C22×C4—121no—
190Q16—21no—
191Q16—21no—
192Q16—121no—
193Q16—241no—
194Q16—242no—
195SD16—121no—
196SD16—121no—
197SD16—241no—
198D8—121no—
199C2×C8—61no—
200C2×C8—121no—
201C2×C8D3×D411yescharacteristic
202C2×C8C2×#6711yescharacteristic
203C4⋊C4—21no—
204C4⋊C4—31no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—61no—
209C4⋊C4—121no—
210C4⋊C4—121no—
211C4⋊C4—121no—
212C22⋊C4—121no—
213C22⋊C4—121no—
214C22⋊C4—121no—
215C22⋊C4—121no—
216C22⋊C4—121no—
217C42—41no—
218C42—61no—
219C42—121no—
220C42D3×D411yescharacteristic
221C16—41no—
222C16—121no—
223C22×C3#44711yescharacteristic
224C2×D3—81no—
225C2×D3—81no—
226C2×D3—81no—
227C3×C4—21no—
228C3×C4—21no—
229C3×C4—81no—
230C3×C4—81no—
231C3×C4—161no—
232C3×C4#45211yescharacteristic
233C3×C4#45511yescharacteristic
234Dic3—41no—
235Dic3—41no—
236Dic3—81no—
237C23—121no—
238Q8—41no—
239Q8—41no—
240Q8—81no—
241Q8—81no—
242Q8—121no—
243Q8—121no—
244Q8—121no—
245D4—121no—
246D4—121no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—61no—
250C2×C4—61no—
251C2×C4—81no—
252C2×C4—121no—
253C2×C4—242no—
254C2×C4D3×D811yescharacteristic
255C2×C4#91111yescharacteristic
256C2×C4#93711yescharacteristic
257C8—21no—
258C8—121no—
259C8—241no—
260C8#94211yescharacteristic
261C8#95011yescharacteristic
262C2×C3#164111yescharacteristic
263C2×C3#219011yescharacteristic
264C2×C3#219511yescharacteristic
265D3—161no—
266C22—241no—
267C22—241no—
268C22—241no—
269C22#515611yescharacteristic
270C4—21no—
271C4—21no—
272C4—81no—
273C4—81no—
274C4—121no—
275C4—121no—
276C4—161no—
277C4—241no—
278C4#504711yescharacteristic
279C4#520211yescharacteristic
280C3#1239111yescharacteristic, complemented, Sylow
281C2—481no—
282C2#7111211yescharacteristic
283C2#7150611yescharacteristic
284C2#7151611yescharacteristic
subgroupG/H
centerC22#5156
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#2227C2
socleC22×C3#447
prime radicalC2×D24D4
hypercenter#2227D3
hypocenterC3#12391
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2227

Complements

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

NH ≅ G/Ntype
2. C3×#2227C2semidirect
4. #71379C2semidirect
5. #67764C2semidirect
7. #67559C2semidirect
30. #2227D3semidirect
280. C3#12391semidirect

Maximal subgroups

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

subgroupindex
#797142
C3×#22272
#713832
#713792
#677642
#675622
#675592
#123913