Finite Group Databasebeta

← #10697392

Subgroups — #10697392

Order 768.

498 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3253C211yesmaximal, characteristic
2#83319C212yesmaximal, complemented
3#71812C212yesmaximal, complemented
4#70515C211yesmaximal, characteristic, complemented
5#67716C211yesmaximal, characteristic, complemented
6#10213—31nomaximal, Sylow
7C3×#532C2212yes—
8#5470C2211yescharacteristic
9C3×#476—22no—
10C3×#469—22no—
11C3×#459—22no—
12C3×#423C2211yescharacteristic
13#5239—24no—
14#5090—22no—
15#5075—22no—
16#5060—22no—
17#5055—22no—
18#4932C2211yescharacteristic
19#4926C2212yes—
20#4555—22no—
21#4537—24no—
22#3253D311yescharacteristic
23#3216—32no—
24#1995—31no—
25#1505—32no—
26#1307—31no—
27C2×C3×#41—22no—
28C2×C3×D8—22no—
29C2×C3×M16D412yes—
30C3×C4⋊Q8D412yes—
31C3×#127D412yes—
32C3×#124—22no—
33C3×#121—22no—
34C3×#121—22no—
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C2×C3×C42C2311yescharacteristic
38C2×#216—24no—
39C2×#214—24no—
40#926—21no—
41#924—22no—
42C2×C4×Dic3—21no—
43#870—21no—
44#869D412yes—
45C4×Dic6—22no—
46C3×#106—22no—
47C3×C4⋊C8—22no—
48C3×#102—24no—
49#835—24no—
50#832—24no—
51C3⋊#95—22no—
52#830D414yes—
53C4⋊(C3⋊C8)—22no—
54#803—22no—
55#533—31no—
56#532C2×D312yes—
57#497—62no—
58#489—62no—
59#476—22no—
60#469—22no—
61#462—62no—
62#460—62no—
63#459—22no—
64#452—64no—
65#432—32no—
66#423C2×D311yescharacteristic
67#376—31no—
68#339—62no—
69#327—64no—
70C3×#41—84no—
71C2×C3×Q8—22no—
72C2×C3×D4—22no—
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—42no—
76C22×C3×C4—42no—
77C22×C3×C4C2×D412yes—
78C22×C3×C4C2×D411yescharacteristic
79C22×Dic3—41no—
80C2×Dic6—22no—
81C3×D8—82no—
82C3×M16—42no—
83C2×C3×C8—22no—
84C3×C4⋊C4—22no—
85C3×C4⋊C4—22no—
86C3×C22⋊C4—42no—
87C3×C22⋊C4—42no—
88C3×C42C2×D411yescharacteristic
89C3×C42C2×D412yes—
90C3×C42#4111yescharacteristic
91#217—41no—
92#216—84no—
93#214—84no—
94C4⋊Dic3—22no—
95C4⋊Dic3—22no—
96Dic3⋊C4—42no—
97C4×Dic3—22no—
98C4×Dic3—41no—
99C4×Dic3—41no—
100C2×C3⋊C8—24no—
101C2×#41—22no—
102C2×Q16—64no—
103C2×SD16—64no—
104C2×D8—22no—
105C2×M16#6712yes—
106C4⋊Q8—32no—
107C4⋊Q8#6712yes—
108#127#6712yes—
109#125—61no—
110#124—22no—
111C22⋊Q8—62no—
112#121—22no—
113#121—22no—
114C4×Q8—62no—
115C4×D4—22no—
116C4×D4—22no—
117#117—61no—
118C2×C42—61no—
119C2×C42C22×D311yescharacteristic
120#106—22no—
121C4⋊C8—22no—
122C4⋊C8—62no—
123#103—64no—
124#102—24no—
125#102—64no—
126C22⋊C8—34no—
127#97—62no—
128#95—62no—
129C23×C3—42no—
130C23×C3#12711yescharacteristic
131C3×Q8—44no—
132C3×D4—44no—
133C3×D4—42no—
134C3×D4—42no—
135C3×D4—82no—
136C3×D4—84no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4—84no—
142C2×C3×C4#12012yes—
143C2×C3×C4#12011yescharacteristic
144C2×C3×C4#12111yescharacteristic
145C2×C3×C4#12112yes—
146C2×Dic3—42no—
147C2×Dic3—42no—
148C2×Dic3—82no—
149Dic6—42no—
150Dic6—42no—
151C3×C8—42no—
152C3⋊C8—44no—
153#41—84no—
154C2×Q8—22no—
155C2×Q8—62no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—42no—
160C22×C4—42no—
161C22×C4—121no—
162C22×C4D3×D412yes—
163C22×C4C2×#6711yescharacteristic
164Q16—244no—
165SD16—244no—
166D8—82no—
167M16—42no—
168C2×C8—22no—
169C2×C8—64no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4—122no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—121no—
178C42—62no—
179C42—121no—
180C42—121no—
181C42D3×D411yescharacteristic
182C42#23711yescharacteristic
183C42C2×#6712yes—
184C22×C3—21no—
185C22×C3—22no—
186C22×C3—82no—
187C22×C3—82no—
188C22×C3—82no—
189C22×C3#39211yescharacteristic
190C3×C4—21no—
191C3×C4—22no—
192C3×C4—22no—
193C3×C4—82no—
194C3×C4#45511yescharacteristic
195C3×C4#45611yescharacteristic
196Dic3—82no—
197Dic3—82no—
198C23—42no—
199C23#94011yescharacteristic
200Q8—44no—
201Q8—122no—
202Q8—122no—
203D4—44no—
204D4—42no—
205D4—42no—
206D4—82no—
207D4—84no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—84no—
213C2×C4—122no—
214C2×C4—122no—
215C2×C4—242no—
216C2×C4#93712yes—
217C2×C4#93811yescharacteristic
218C2×C4#93912yes—
219C2×C4#95311yescharacteristic
220C8—42no—
221C8—124no—
222C2×C3—41no—
223C2×C3—162no—
224C2×C3#200711yescharacteristic
225C2×C3#200812yes—
226C22—22no—
227C22—21no—
228C22—82no—
229C22—82no—
230C22—82no—
231C22#522311yescharacteristic
232C4—22no—
233C4—22no—
234C4—21no—
235C4—82no—
236C4—242no—
237C4—242no—
238C4#524211yescharacteristic
239C4#524611yescharacteristic
240C3#1021311yescharacteristic, complemented, Sylow
241C2—41no—
242C2—162no—
243C2#7188111yescharacteristic
244C2#7209912yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3253C2
socleC22×C3#392
hypercenter#3253D3
hypocenterC3#10213
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3253

Complements

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

NH ≅ G/Ntype
2. #83319C2semidirect
3. #71812C2semidirect
4. #70515C2semidirect
5. #67716C2semidirect
240. C3#10213semidirect

Maximal subgroups

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

subgroupindex
C3×#32532
#833192
#718122
#705152
#677162
#102133