Finite Group Databasebeta

← #10714301

Subgroups — #10714301

Order 768.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#4683C211yesmaximal, characteristic, complemented
2C2×#4518C212yesmaximal
3#69795C211yesmaximal, characteristic
4C3×#1376C211yesmaximal, characteristic, complemented
5#67816C212yesmaximal, complemented
6#13372—31nomaximal, Sylow
7C2×#904C2211yescharacteristic
8C2×#870—21no—
9C2×C3×M32C2212yes—
10C2×C3×C4×C8C2211yescharacteristic
11#5110—21no—
12C2×#819C2211yescharacteristic
13C2×#818—21no—
14C2×#817C2212yes—
15C2×#805—21no—
16#4683—24no—
17#4518—24no—
18C2×#499—31no—
19C2×#360—32no—
20#1974—31no—
21#1383—32no—
22#1376D311yescharacteristic, complemented
23C2×#235—21no—
24C22×C3×C8D411yescharacteristic
25C22×C3×C8C2311yescharacteristic
26C2×C3×C42D411yescharacteristic
27C2×C4⋊Dic3—22no—
28C2×Dic3⋊C4—42no—
29C2×#208—21no—
30C2×Dic12Q811yescharacteristic
31#904—22no—
32#904—22no—
33C2×D24Q811yescharacteristic
34C2×#204—21no—
35#870—24no—
36#870—42no—
37C3×M32—22no—
38C3×M32—22no—
39C2×C3×C16—22no—
40C3×C4×C8—22no—
41(C2×D3)⋊C8—42no—
42#819—22no—
43#818—24no—
44#817—22no—
45#817D414yes—
46#805—42no—
47C2×#135—31no—
48C2×#125—61no—
49C2×M32C2×D312yes—
50#499—64no—
51C2×#108—31no—
52C2×#107—61no—
53C2×#106—32no—
54C2×#104—61no—
55#408—61no—
56C2×C4×C8C2×D311yescharacteristic
57#360—64no—
58C22×C3×C4—21no—
59C22×C3×C4C2×D411yescharacteristic
60C2×#67—41no—
61C22×Dic3—42no—
62#235—41no—
63#235—41no—
64#235—41no—
65#235—41no—
66C2×D12—21no—
67C2×C4×D3—41no—
68C2×Dic6—21no—
69C2×C3×C8—21no—
70C2×C3×C8—21no—
71C2×C3×C8SD1612yes—
72C2×C3×C8C2×D412yes—
73C2×C3×C8C2×Q811yescharacteristic
74C2×C3×C8#4112yes—
75C2×C3×C8#4111yescharacteristic
76C3×C42—21no—
77C3×C42D812yes—
78C4⋊Dic3—24no—
79C4⋊Dic3—42no—
80Dic3⋊C4—48no—
81#208—41no—
82#208—41no—
83C2×C3⋊C8—41no—
84Dic12—22no—
85D24—22no—
86#204—42no—
87C3×C16—42no—
88C2×#41—61no—
89#135—62no—
90#135—62no—
91C2×Q16—31no—
92C2×SD16—61no—
93C2×D8—31no—
94C2×M16—61no—
95C22×C8D1211yescharacteristic
96C22×C8C22×D311yescharacteristic
97#125—64no—
98#125—122no—
99C2×C4⋊C4—62no—
100C2×C4⋊C4—122no—
101C2×C42D1211yescharacteristic
102M32—22no—
103M32—22no—
104C2×C16—22no—
105#108—62no—
106#107—64no—
107#106—34no—
108#106—62no—
109#104—122no—
110C22⋊C8—122no—
111C4×C8—22no—
112C23×C3AGL(1,ℤ/8)11yescharacteristic
113C22×D3—41no—
114C2×C3×C4—22no—
115C2×C3×C4—42no—
116C2×C3×C4#12011yescharacteristic
117C2×C3×C4C22⋊Q811yescharacteristic
118C2×C3×C4#12411yescharacteristic
119C2×C3×C4C2×D811yescharacteristic
120C2×C3×C4C2×SD1611yescharacteristic
121C2×C3×C4#13711yescharacteristic
122#67—81no—
123#67—81no—
124C2×Dic3—44no—
125C2×Dic3—41no—
126C2×Dic3—84no—
127D12—41no—
128D12—41no—
129C4×D3—81no—
130C4×D3—81no—
131Dic6—41no—
132Dic6—41no—
133C3×C8—22no—
134C3×C8C22⋊Q812yes—
135C3×C8#12412yes—
136C3⋊C8—81no—
137#41—121no—
138#41—121no—
139#41—121no—
140#41—121no—
141C2×Q8—61no—
142C2×D4—61no—
143C2×D4—121no—
144C22×C4—21no—
145C22×C4—122no—
146C22×C4—121no—
147C22×C4C2×D1211yescharacteristic
148Q16—62no—
149SD16—122no—
150D8—62no—
151M16—121no—
152M16—121no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—121no—
156C2×C8#20412yes—
157C2×C8#23512yes—
158C2×C8D3×D412yes—
159C2×C8#23711yescharacteristic
160C2×C8D3×Q811yescharacteristic
161C4⋊C4—64no—
162C4⋊C4—128no—
163C4⋊C4—122no—
164C42—21no—
165C42D2412yes—
166C16—42no—
167C22×C3—22no—
168C22×C3#44711yescharacteristic
169C22×C3#47611yescharacteristic
170C22×C3#48911yescharacteristic
171C2×D3—82no—
172C2×D3—81no—
173C3×C4—42no—
174C3×C4#39311yescharacteristic
175C3×C4#44811yescharacteristic
176C3×C4#47411yescharacteristic
177C3×C4#48611yescharacteristic
178Dic3—84no—
179Dic3—81no—
180C23—121no—
181C23#90811yescharacteristic
182Q8—121no—
183Q8—121no—
184D4—121no—
185D4—121no—
186D4—241no—
187D4—241no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—124no—
191C2×C4—121no—
192C2×C4—244no—
193C2×C4—241no—
194C2×C4—241no—
195C2×C4#87511yescharacteristic
196C2×C4#88211yescharacteristic
197C2×C4#89711yescharacteristic
198C2×C4C2×#20411yescharacteristic
199C2×C4C2×D2411yescharacteristic
200C2×C4#90911yescharacteristic
201C8—22no—
202C8—241no—
203C8#88512yes—
204C8#89612yes—
205C2×C3—21no—
206C2×C3—21no—
207C2×C3#202311yescharacteristic
208C2×C3#223112yes—
209D3—161no—
210C22—22no—
211C22—241no—
212C22—242no—
213C22#471411yescharacteristic
214C22#473211yescharacteristic
215C22#484211yescharacteristic
216C4—42no—
217C4—244no—
218C4—241no—
219C4#467311yescharacteristic
220C4#469411yescharacteristic
221C4#473811yescharacteristic
222C4#484611yescharacteristic
223C3#1337211yescharacteristic, complemented, Sylow
224C2—21no—
225C2—21no—
226C2—481no—
227C2#6867811yescharacteristic
228C2#6930812yes—
subgroupG/H
centerC22#4842
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#1376C2
socleC22×C3#476
prime radicalC2×#904C22
hypercenter#1376D3
hypocenterC3#13372
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1376

Complements

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

NH ≅ G/Ntype
1. C2×#4683C2semidirect
4. C3×#1376C2semidirect
5. #67816C2semidirect
22. #1376D3semidirect
223. C3#13372semidirect

Maximal subgroups

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

subgroupindex
C2×#46832
C2×#45182
#697952
C3×#13762
#678162
#133723