Finite Group Databasebeta

← #10697684

Subgroups — #10697684

Order 768.

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

normal subgroups only

This group has 526 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#79059C212yesmaximal
2#79037C211yesmaximal, characteristic, complemented
3#70501C212yesmaximal, complemented
4#70470C211yesmaximal, characteristic
5C3×#1292C211yesmaximal, characteristic, complemented
6#10258—31nomaximal, Sylow
7C2×C4×D12C2211yescharacteristic
8#5470C2212yes—
9C3×#423C2212yes—
10C2×C3×C4⋊C8C2211yescharacteristic
11#5131—22no—
12#5121—22no—
13#5116—22no—
14C2×(C2×D3)⋊C8—22no—
15C2×C4×C3⋊C8C2211yescharacteristic
16#4850—22no—
17#4849—22no—
18#4847—24no—
19#4846—22no—
20#4842—22no—
21#4484—24no—
22#3249—31no—
23#3077—32no—
24#1954—31no—
25#1493—32no—
26#1292D311yescharacteristic, complemented
27C22×D12D411yescharacteristic
28C22×C4×D3—21no—
29C2×C3×M16—22no—
30C22×C3×C8D412yes—
31C2×C3×C42C2311yescharacteristic
32C2×(C2×D3)⋊C4—21no—
33#926—22no—
34C2×C4⋊Dic3D411yescharacteristic
35#924—22no—
36#924—22no—
37C22×C3⋊C8—21no—
38C2×Dic12—22no—
39C4×D12—22no—
40C4×D12—22no—
41#870—22no—
42#869D412yes—
43#869D412yes—
44C4×Dic6—22no—
45C4×Dic6—22no—
46C3×C4⋊C8—22no—
47C3×C4⋊C8—22no—
48C3×C4⋊C8D414yes—
49#821—24no—
50#821—24no—
51(C2×D3)⋊C8—44no—
52#818—24no—
53#818—22no—
54#817—24no—
55#816—24no—
56C4×C3⋊C8—22no—
57#533—32no—
58C2×C4×D4—31no—
59#478—64no—
60#477—62no—
61#476—62no—
62#474—62no—
63#467—62no—
64#462—62no—
65#428—62no—
66#423C2×D312yes—
67C2×C4⋊C8C2×D311yescharacteristic
68#418—62no—
69C2×C22⋊C8—62no—
70C2×C4×C8—31no—
71Q8⋊C8—64no—
72C23×D3—21no—
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D411yescharacteristic
75C22×C3×C4#4111yescharacteristic
76C22×Dic3—21no—
77C2×D12—22no—
78C2×D12—22no—
79C2×D12—21no—
80C2×D12Q1612yescomplemented
81C2×C4×D3—42no—
82C2×C4×D3—42no—
83C2×C4×D3—41no—
84C2×C4×D3—41no—
85C2×Dic6—22no—
86C2×Dic6—22no—
87C3×M16—42no—
88C3×M16—42no—
89C2×C3×C8—22no—
90C2×C3×C8—24no—
91C2×C3×C8—42no—
92C3×C42C2×D412yes—
93C3×C42C2×D412yes—
94#217—42no—
95(C2×D3)⋊C4—42no—
96C4⋊Dic3—22no—
97C4⋊Dic3—22no—
98C4⋊Dic3—22no—
99C4⋊Dic3—22no—
100C4⋊Dic3—22no—
101Dic3⋊C4—42no—
102Dic3⋊C4—42no—
103C4×Dic3—42no—
104C2×C3⋊C8—22no—
105C2×C3⋊C8—41no—
106C2×C3⋊C8—41no—
107Dic12—82no—
108C22×D4—31no—
109C23×C4—61no—
110C2×Q16—62no—
111C2×M16—22no—
112C22×C8—61no—
113C22×C8D1212yes—
114C4⋊Q8—32no—
115C4⋊Q8—32no—
116#125—62no—
117C22⋊Q8—62no—
118C22⋊Q8—62no—
119C4×Q8—62no—
120C4×Q8—62no—
121C4×D4—62no—
122C4×D4—62no—
123#117—62no—
124C2×C4⋊C4—31no—
125C2×C22⋊C4—61no—
126C2×C42C22×D311yescharacteristic
127#107—64no—
128#107—62no—
129#106—64no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8D1214yes—
133#103—64no—
134#102—64no—
135#102—64no—
136C22⋊C8—124no—
137C4×C8—62no—
138C23×C3#12111yescharacteristic
139C22×D3—22no—
140C22×D3—42no—
141C22×D3—42no—
142C22×D3—41no—
143C22×D3—41no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—41no—
147C2×C3×C4#12012yes—
148C2×C3×C4#12011yescharacteristic
149C2×C3×C4#12112yes—
150C2×C3×C4#12711yescharacteristic
151C2×C3×C4C2×Q1611yescharacteristic
152C2×C3×C4#13511yescharacteristic
153C2×C3×C4#13712yes—
154C2×Dic3—42no—
155C2×Dic3—42no—
156C2×Dic3—42no—
157C2×Dic3—41no—
158D12—24no—
159D12—42no—
160C4×D3—82no—
161C4×D3—81no—
162C4×D3—81no—
163Dic6—44no—
164Dic6—42no—
165Dic6—42no—
166C3×C8—44no—
167C3×C8—42no—
168C3⋊C8—42no—
169C24—61no—
170C2×Q8—62no—
171C2×Q8—62no—
172C2×D4—32no—
173C2×D4—62no—
174C2×D4—61no—
175C2×D4—62no—
176C22×C4—61no—
177C22×C4—122no—
178C22×C4—122no—
179C22×C4—121no—
180C22×C4—121no—
181C22×C4C2×D1211yescharacteristic
182C22×C4D3×D411yescharacteristic
183C22×C4#23911yescharacteristic
184Q16—242no—
185M16—42no—
186M16—42no—
187C2×C8—24no—
188C2×C8—22no—
189C2×C8—42no—
190C2×C8—62no—
191C2×C8—121no—
192C2×C8—121no—
193C4⋊C4—62no—
194C4⋊C4—62no—
195C4⋊C4—62no—
196C4⋊C4—62no—
197C4⋊C4—62no—
198C4⋊C4—122no—
199C4⋊C4—122no—
200C22⋊C4—122no—
201C22⋊C4—122no—
202C42—122no—
203C42C2×D1212yes—
204C42D3×D412yes—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3#39211yescharacteristic
208C22×C3#46711yescharacteristic
209C22×C3#47011yescharacteristic
210C2×D3—44no—
211C2×D3—42no—
212C2×D3—82no—
213C2×D3—81no—
214C2×D3—81no—
215C3×C4—21no—
216C3×C4—22no—
217C3×C4#45112yes—
218C3×C4#45212yes—
219C3×C4#45311yescharacteristic
220C3×C4#45411yescharacteristic
221Dic3—82no—
222Dic3—81no—
223Dic3—82no—
224C23—62no—
225C23—122no—
226C23—122no—
227C23—121no—
228C23—121no—
229C23#89511yescharacteristic
230Q8—124no—
231Q8—122no—
232Q8—122no—
233D4—64no—
234D4—122no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—122no—
239C2×C4—122no—
240C2×C4—122no—
241C2×C4—121no—
242C2×C4—242no—
243C2×C4—241no—
244C2×C4—241no—
245C2×C4#87411yescharacteristic
246C2×C4#88211yescharacteristic
247C2×C4#88212yes—
248C2×C4#89512yes—
249C2×C4#91211yescharacteristic
250C2×C4#91512yes—
251C2×C4D3×Q1611yescharacteristic
252C8—42no—
253C8—44no—
254C8—122no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3#200111yescharacteristic
258C2×C3#200611yescharacteristic
259C2×C3#201711yescharacteristic
260D3—82no—
261C22—21no—
262C22—21no—
263C22—124no—
264C22—122no—
265C22—242no—
266C22—241no—
267C22—241no—
268C22#467111yescharacteristic
269C22#486711yescharacteristic
270C22#488711yescharacteristic
271C4—22no—
272C4—21no—
273C4—242no—
274C4—241no—
275C4—242no—
276C4#474711yescharacteristic
277C4#475311yescharacteristic
278C4#478312yes—
279C4#480912yes—
280C3#1025811yescharacteristic, complemented, Sylow
281C2—21no—
282C2—21no—
283C2—242no—
284C2#6872711yescharacteristic
285C2#6888811yescharacteristic
286C2#6900811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1292C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1292D3
hypocenterC3#10258
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
2. #79037C2semidirect
3. #70501C2semidirect
5. C3×#1292C2semidirect
26. #1292D3semidirect
80. C2×D12Q16semidirect
280. C3#10258semidirect

Maximal subgroups

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

subgroupindex
#790592
#790372
#705012
#704702
C3×#12922
#102583