Finite Group Databasebeta

← #10708030

Subgroups — #10708030

Order 768.

380 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2811C211yesmaximal, characteristic
2#69707C212yesmaximal
3#69673C211yesmaximal, characteristic
4#67648C212yesmaximal
5#67643C211yesmaximal, characteristic
6#12224—31nomaximal, Sylow
7#5500C2211yescharacteristic
8C3×#395—22no—
9C3×#395C2212yes—
10C3×#391—21no—
11C3×#391—21no—
12C3×#391—22no—
13C3×#391C2211yescharacteristic
14#5228—22no—
15#5225—21no—
16#4530C2211yescharacteristic
17#4528C2212yes—
18#4523—21no—
19#4452—22no—
20#2811D311yescharacteristic
21#2028—31no—
22#1605—32no—
23#1354—32no—
24#1319—31no—
25C22×C3×Q8—21no—
26C22×C3×Q8D411yescharacteristic
27C22×C3×Q8D412yes—
28C2×C3×C4⋊C4—21no—
29C2×C3×C4⋊C4—21no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4C2311yescharacteristic
32C2×C3×C42—21no—
33C2×C3×C42D411yescharacteristic
34C2×#216—42no—
35#926—21no—
36#924—22no—
37C2×#208—21no—
38#890—41no—
39#888—41no—
40C3×#95—22no—
41C3×#95—22no—
42C3×#95—22no—
43C3×#95—22no—
44C3×#95D412yes—
45#835—41no—
46#835—41no—
47#835—42no—
48#830—22no—
49#544—31no—
50#451—62no—
51#419—61no—
52#395—22no—
53#395C2×D312yes—
54#391—21no—
55#391—21no—
56#391—22no—
57#391C2×D311yescharacteristic
58#343—31no—
59#339—61no—
60#328—62no—
61#324—32no—
62C2×C3×Q8—24no—
63C2×C3×Q8—21no—
64C2×C3×Q8—41no—
65C2×C3×Q8—42no—
66C2×C3×Q8—41no—
67C2×C3×Q8—82no—
68C2×C3×Q8—81no—
69C2×C3×Q8—81no—
70C2×C3×Q8Q1612yescomplemented
71C22×C3×C4—21no—
72C22×C3×C4—21no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4C2×D411yescharacteristic
76C22×C3×C4C2×D412yes—
77C2×Dic6—22no—
78C3×C4⋊C4—41no—
79C3×C4⋊C4—82no—
80C3×C4⋊C4—81no—
81C3×C4⋊C4—81no—
82C3×C42—41no—
83C3×C42—81no—
84#217—41no—
85#216—84no—
86C4⋊Dic3—21no—
87C4⋊Dic3—22no—
88C4⋊Dic3—21no—
89Dic3⋊C4—42no—
90C4×Dic3—41no—
91#208—41no—
92#208—41no—
93C2×C3⋊C8—42no—
94C2×C3⋊C8—41no—
95C22×Q8—21no—
96C22×Q8#6711yescharacteristic
97C22×Q8#6712yes—
98C2×Q16—122no—
99C2×M16—61no—
100C4⋊Q8—121no—
101#125—121no—
102C22⋊Q8—62no—
103#117—61no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4C22×D311yescharacteristic
108C2×C42—21no—
109C2×C42#6711yescharacteristic
110#103—121no—
111#103—121no—
112#103—122no—
113C22⋊C8—62no—
114#95—22no—
115#95—22no—
116#95—22no—
117#95—22no—
118#95#6712yes—
119C23×C3#12011yescharacteristic
120C3×Q8—42no—
121C3×Q8—44no—
122C3×Q8—81no—
123C3×Q8—84no—
124C3×Q8—82no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—22no—
128C2×C3×C4—24no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—41no—
132C2×C3×C4—42no—
133C2×C3×C4—41no—
134C2×C3×C4—41no—
135C2×C3×C4—42no—
136C2×C3×C4—41no—
137C2×C3×C4—42no—
138C2×C3×C4—42no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4C2×Q1611yescharacteristic
142C2×C3×C4#13711yescharacteristic
143C2×Dic3—41no—
144C2×Dic3—41no—
145Dic6—82no—
146C3⋊C8—81no—
147C3⋊C8—82no—
148C2×Q8—24no—
149C2×Q8—21no—
150C2×Q8—41no—
151C2×Q8—42no—
152C2×Q8—41no—
153C2×Q8—62no—
154C2×Q8—82no—
155C2×Q8—81no—
156C2×Q8—81no—
157C2×Q8#21612yescomplemented
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4C2×#6711yescharacteristic
163C22×C4C2×#6712yes—
164Q16—244no—
165M16—121no—
166M16—121no—
167C2×C8—122no—
168C2×C8—121no—
169C4⋊C4—41no—
170C4⋊C4—61no—
171C4⋊C4—61no—
172C4⋊C4—62no—
173C4⋊C4—82no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C4⋊C4—122no—
177C22⋊C4—121no—
178C42—41no—
179C42—81no—
180C42—121no—
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—44no—
190C3×C4—42no—
191C3×C4—42no—
192C3×C4—82no—
193Dic3—81no—
194Dic3—81no—
195C23#95311yescharacteristic
196Q8—42no—
197Q8—44no—
198Q8—81no—
199Q8—84no—
200Q8—82no—
201Q8—242no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—121no—
219C2×C4—121no—
220C2×C4#94211yescharacteristic
221C2×C4C2×#21611yescharacteristic
222C8—241no—
223C8—242no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3#160211yescharacteristic
227C2×C3#219911yescharacteristic
228C2×C3#220511yescharacteristic
229C22—21no—
230C22—21no—
231C22#522811yescharacteristic
232C22#524211yescharacteristic
233C22#524411yescharacteristic
234C4—21no—
235C4—21no—
236C4—42no—
237C4—44no—
238C4—42no—
239C4—42no—
240C4—82no—
241C4—241no—
242C4—241no—
243C3#1222411yescharacteristic, complemented, Sylow
244C2—21no—
245C2—21no—
246C2#7189311yescharacteristic
247C2#7194211yescharacteristic
248C2#7195011yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2811C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#2811D3
hypocenterC3#12224
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2811

Complements

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

NH ≅ G/Ntype
70. C2×C3×Q8Q16semidirect
157. C2×Q8#216semidirect
243. C3#12224semidirect

Maximal subgroups

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

subgroupindex
C3×#28112
#697072
#696732
#676482
#676432
#122243