Finite Group Databasebeta

← #10708032

Subgroups — #10708032

Order 768.

380 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 59 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#76604C211yesmaximal, characteristic
2#69673C211yesmaximal, characteristic
3#68139C212yesmaximal
4C3×#1319C211yesmaximal, characteristic
5#67457C212yesmaximal
6#12224—31nomaximal, Sylow
7#5500C2211yescharacteristic
8C3×#391C2211yescharacteristic
9#5229—21no—
10#5225—21no—
11#4979C2212yes—
12#4929—21no—
13#4739—22no—
14#4656—22no—
15#4646—22no—
16C3×#324C2212yes—
17#4530C2211yescharacteristic
18#4523—21no—
19#4467—22no—
20#2811—31no—
21#2028—31no—
22#1605—32no—
23#1354—32no—
24#1319D311yescharacteristic
25C22×C3×Q8D411yescharacteristic
26C22×Dic6—21no—
27C22×Dic6D412yes—
28C2×C3×C4⋊C4C2311yescharacteristic
29C2×C3×C42D411yescharacteristic
30#926—21no—
31C2×C4⋊Dic3—21no—
32C2×C4⋊Dic3—21no—
33#924—22no—
34C2×Dic3⋊C4—22no—
35C2×C4×Dic3—21no—
36C2×#208—21no—
37C2×Dic12—42no—
38#890—41no—
39#888—41no—
40C3×C22⋊C8—22no—
41C3×#95—22no—
42#835—41no—
43#835—41no—
44C3⋊#95—22no—
45C3⋊#95—22no—
46C3⋊#95—22no—
47C3⋊#95D412yes—
48#816—42no—
49#544—31no—
50#451—62no—
51#419—61no—
52#395—32no—
53#395—62no—
54#391—62no—
55#391—61no—
56#391—61no—
57#391C2×D311yescharacteristic
58#343—31no—
59#339—61no—
60#328—62no—
61#324C2×D312yes—
62C2×C3×Q8—21no—
63C2×C3×Q8—41no—
64C2×C3×Q8—41no—
65C2×C3×Q8Q1612yescomplemented
66C22×C3×C4—21no—
67C22×C3×C4—21no—
68C22×C3×C4C2×D411yescharacteristic
69C22×C3×C4C2×D412yes—
70C22×Dic3—22no—
71C22×Dic3—22no—
72C2×Dic6—24no—
73C2×Dic6—22no—
74C2×Dic6—42no—
75C2×Dic6—82no—
76C2×Dic6—81no—
77C2×Dic6—81no—
78C2×C3×C8—42no—
79C3×C4⋊C4—41no—
80C3×C42—41no—
81#217—41no—
82C4⋊Dic3—22no—
83C4⋊Dic3—21no—
84C4⋊Dic3—21no—
85C4⋊Dic3—81no—
86C4⋊Dic3—81no—
87Dic3⋊C4—42no—
88Dic3⋊C4—82no—
89C4×Dic3—41no—
90C4×Dic3—81no—
91#208—41no—
92#208—41no—
93C2×C3⋊C8—41no—
94Dic12—84no—
95C22×Q8—32no—
96C22×Q8—61no—
97C22×Q8D1211yescharacteristic
98C2×Q16—122no—
99C2×M16—61no—
100C4⋊Q8—121no—
101#125—121no—
102C22⋊Q8—62no—
103#117—61no—
104C2×C4⋊C4—61no—
105C2×C4⋊C4—61no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4C22×D311yescharacteristic
108C2×C42—61no—
109C2×C42D1211yescharacteristic
110#103—122no—
111#103—121no—
112#103—121no—
113C22⋊C8—22no—
114#95—22no—
115#95—32no—
116#95—62no—
117#95—62no—
118#95—62no—
119C23×C3#12011yescharacteristic
120C3×Q8—42no—
121C3×Q8—81no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—41no—
128C2×C3×C4—41no—
129C2×C3×C4—41no—
130C2×C3×C4—41no—
131C2×C3×C4—42no—
132C2×C3×C4C2×Q1611yescharacteristic
133C2×C3×C4#13711yescharacteristic
134C2×Dic3—24no—
135C2×Dic3—42no—
136C2×Dic3—42no—
137C2×Dic3—42no—
138C2×Dic3—42no—
139C2×Dic3—41no—
140C2×Dic3—41no—
141C2×Dic3—42no—
142Dic6—44no—
143Dic6—84no—
144Dic6—82no—
145Dic6—82no—
146C3×C8—82no—
147C3⋊C8—81no—
148C2×Q8—21no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—64no—
152C2×Q8—62no—
153C2×Q8—122no—
154C2×Q8—242no—
155C2×Q8—241no—
156C2×Q8—241no—
157C2×Q8Dic1212yescomplemented
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—62no—
161C22×C4—62no—
162C22×C4C2×D1211yescharacteristic
163C22×C4D3×D412yes—
164Q16—244no—
165M16—121no—
166M16—121no—
167C2×C8—42no—
168C2×C8—121no—
169C4⋊C4—41no—
170C4⋊C4—61no—
171C4⋊C4—61no—
172C4⋊C4—62no—
173C4⋊C4—122no—
174C4⋊C4—242no—
175C4⋊C4—241no—
176C4⋊C4—241no—
177C22⋊C4—121no—
178C42—41no—
179C42—121no—
180C42—241no—
181C22×C3—21no—
182C22×C3—21no—
183C22×C3#45111yescharacteristic
184C22×C3#45511yescharacteristic
185C22×C3B(4,𝔽2)11yescharacteristic
186C3×C4—21no—
187C3×C4—21no—
188C3×C4—42no—
189C3×C4—42no—
190C3×C4—42no—
191Dic3—44no—
192Dic3—82no—
193Dic3—81no—
194Dic3—81no—
195C23#88211yescharacteristic
196Q8—42no—
197Q8—81no—
198Q8—124no—
199Q8—244no—
200Q8—242no—
201Q8—242no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—64no—
213C2×C4—122no—
214C2×C4—122no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—121no—
218C2×C4—121no—
219C2×C4—122no—
220C2×C4C2×Dic1211yescharacteristic
221C2×C4#90911yescharacteristic
222C8—82no—
223C8—241no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3#160211yescharacteristic
227C2×C3#219911yescharacteristic
228C2×C3#220511yescharacteristic
229C22—21no—
230C22—21no—
231C22#473911yescharacteristic
232C22#474111yescharacteristic
233C22#474611yescharacteristic
234C4—21no—
235C4—21no—
236C4—42no—
237C4—42no—
238C4—42no—
239C4—124no—
240C4—242no—
241C4—241no—
242C4—241no—
243C3#1222411yescharacteristic, complemented, Sylow
244C2—21no—
245C2—21no—
246C2#6811511yescharacteristic
247C2#6915411yescharacteristic
248C2#6916911yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1319C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1319D3
hypocenterC3#12224
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
65. C2×C3×Q8Q16semidirect
157. C2×Q8Dic12semidirect
243. C3#12224semidirect

Maximal subgroups

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

subgroupindex
#766042
#696732
#681392
C3×#13192
#674572
#122243