Finite Group Databasebeta

← #11775708

Subgroups — #11775708

Order 1152.

702 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 181 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#10595950C211yesmaximal, characteristic
2C32×C4×M16C211yesmaximal, characteristic, complemented
3C2×#63630C211yesmaximal, characteristic, complemented
4#10589331C212yesmaximal, complemented
5#10589330C212yesmaximal, complemented
6#69858—34nomaximal
7C2×C32×M16—21no—
8C2×C32×C42C2211yescharacteristic
9#64139C411yescharacteristic
10#64138—21no—
11C4×#638—21no—
12#64133C411yescharacteristic
13C22×C3⋊(C3⋊C8)—21no—
14#64085—21no—
15#64084C2211yescharacteristic
16C4×C4⋊(C3⋊D3)C411yescharacteristic
17#64081—21no—
18#64079C2211yescharacteristic
19C4×#634C411yescharacteristic
20C32×#97C2212yes—
21C32×C4×C8—21no—
22#63630—21no—
23#63630C2212yes—
24#5479—34no—
25C3×C4×M16D314yescomplemented
26C2×#803—34no—
27#4465—38no—
28#4464—38no—
29C22×C32×C4C2×C411yescharacteristic
30C22×C32×C4D411yescharacteristic
31C22×C32×C4D411yescharacteristic
32C2×#638—21no—
33C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
34C2×C4×C3⋊D3—21no—
35C2×#634C2×C411yescharacteristic
36C32×M16—24no—
37C2×C32×C8—22no—
38C32×C42C2×C411yescharacteristic
39C32×C42D411yescharacteristic
40C32×C42D411yescharacteristic
41C32×C42C2311yescharacteristic
42#3757—21no—
43#3752—21no—
44#3752—21no—
45#3752—21no—
46#3752—21no—
47C4⋊(C3⋊Dic3)C2×C411yescharacteristic
48C4⋊(C3⋊Dic3)C2×C411yescharacteristic
49#3750—21no—
50#3750—21no—
51#3750—21no—
52#3750—21no—
53C4×C3⋊Dic3—21no—
54C2×C3⋊(C3⋊C8)—24no—
55C2×C3⋊(C3⋊C8)—22no—
56#1524—91noSylow
57C2×C3×M16—24no—
58C2×C3×C42C2×D314yes—
59#930—34no—
60#929—64no—
61C4×#67—64no—
62#924—34no—
63C22×C3⋊C8—64no—
64#876—64no—
65#875—34no—
66C4×D12—34no—
67#872—64no—
68#870—34no—
69C4×Dic6—34no—
70C3×#97C2×D318yes—
71C3×C4×C8—24no—
72#803—38no—
73#803—64no—
74C23×C32C22⋊C411yescharacteristic
75C22×C3⋊D3—21no—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—41no—
79C2×C32×C4C22⋊C411yescharacteristic
80C2×C32×C4C22⋊C411yescharacteristic
81C2×C32×C4C22⋊C411yescharacteristic
82C2×C32×C4C22×C411yescharacteristic
83C2×C32×C4C2×D411yescharacteristic
84C2×C32×C4C2×D411yescharacteristic
85#638—42no—
86C2×C3⋊Dic3—21no—
87C2×C3⋊Dic3—21no—
88C2×C3⋊Dic3—21no—
89C4⋊(C3⋊D3)—41no—
90C4×C3⋊D3—42no—
91#634—41no—
92C32×C8—24no—
93C3⋊(C3⋊C8)—24no—
94#529—91no—
95C4×M16C3⋊D311yescharacteristic, complemented
96C2×#97—91no—
97#330—92no—
98#329—92no—
99C22×C3×C4C4×D314yes—
100C22×C3×C4D1214yes—
101C22×C3×C4#6714yes—
102C2×#67—64no—
103C2×D12—34no—
104C2×C4×D3—64no—
105C2×Dic6—34no—
106C3×M16—216no—
107C2×C3×C8—28no—
108C3×C42C4×D314yes—
109C3×C42D1214yes—
110C3×C42#6714yes—
111C3×C42C22×D314yes—
112#217—64no—
113(C2×D3)⋊C4—64no—
114(C2×D3)⋊C4—64no—
115(C2×D3)⋊C4—64no—
116(C2×D3)⋊C4—64no—
117C4⋊Dic3—34no—
118C4⋊Dic3—34no—
119Dic3⋊C4—64no—
120Dic3⋊C4—64no—
121Dic3⋊C4—64no—
122Dic3⋊C4—64no—
123C4×Dic3—64no—
124C2×C3⋊C8—616no—
125C2×C3⋊C8—68no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32#10412yes—
129C22×C32C2×C22⋊C411yescharacteristic
130C2×C3⋊D3—41no—
131C2×C3⋊D3—41no—
132C2×C3⋊D3—41no—
133C32×C4—21no—
134C32×C4—21no—
135C32×C4—21no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4#10412yes—
139C3⋊Dic3—41no—
140C3⋊Dic3—41no—
141C3⋊Dic3—41no—
142C2×M16—21no—
143C22×C8—181no—
144#126—181no—
145#125—91no—
146#124—91no—
147#123—181no—
148C22⋊Q8—91no—
149#121—91no—
150C4×Q8—91no—
151C4×D4—91no—
152C4×D4—181no—
153#117—181no—
154C2×C42C2×C3⋊D311yescharacteristic
155#97—92no—
156#97—181no—
157#97C2×C3⋊D312yes—
158C4×C8—21no—
159C23×C3(C2×D3)⋊C414yes—
160C22×D3—64no—
161C2×C3×C4—28no—
162C2×C3×C4—28no—
163C2×C3×C4—44no—
164C2×C3×C4(C2×D3)⋊C414yes—
165C2×C3×C4(C2×D3)⋊C414yes—
166C2×C3×C4(C2×D3)⋊C414yes—
167C2×C3×C4C2×C4×D314yes—
168C2×C3×C4C2×D1214yes—
169C2×C3×C4C2×#6714yes—
170#67—128no—
171C2×Dic3—64no—
172C2×Dic3—64no—
173C2×Dic3—64no—
174D12—124no—
175C4×D3—128no—
176Dic6—124no—
177C3×C8—216no—
178C3⋊C8—616no—
179C2×C32—22no—
180C2×C32#41311yescharacteristic
181C2×C32C2×#10411yescharacteristic
182C2×C32C2×#10411yescharacteristic
183C3⋊D3—81no—
184C2×Q8—91no—
185C2×D4—91no—
186C2×D4—181no—
187C22×C4—181no—
188C22×C4C4×C3⋊D311yescharacteristic
189C22×C4C4⋊(C3⋊D3)11yescharacteristic
190C22×C4#63811yescharacteristic
191M16—24no—
192C2×C8—22no—
193C2×C8—184no—
194C2×C8—182no—
195C4⋊C4—91no—
196C4⋊C4—91no—
197C4⋊C4—181no—
198C4⋊C4—181no—
199C4⋊C4—181no—
200C4⋊C4—181no—
201C22⋊C4—181no—
202C22⋊C4—181no—
203C22⋊C4—181no—
204C22⋊C4—181no—
205C22⋊C4—181no—
206C42—181no—
207C42C4×C3⋊D311yescharacteristic
208C42C4⋊(C3⋊D3)11yescharacteristic
209C42#63811yescharacteristic
210C42C22×C3⋊D311yescharacteristic
211C22×C3—24no—
212C22×C3—24no—
213C22×C3#80518yes—
214C22×C3C2×(C2×D3)⋊C414yes—
215C2×D3—124no—
216C2×D3—124no—
217C2×D3—124no—
218C3×C4—24no—
219C3×C4—24no—
220C3×C4—24no—
221C3×C4—24no—
222C3×C4—24no—
223C3×C4#82518yes—
224Dic3—124no—
225Dic3—124no—
226Dic3—124no—
227C32#152411yescharacteristic, complemented, Sylow
228C23—181no—
229C23#375211yescharacteristic
230Q8—361no—
231D4—361no—
232D4—362no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—181no—
237C2×C4—181no—
238C2×C4—181no—
239C2×C4—362no—
240C2×C4#375211yescharacteristic
241C2×C4#375211yescharacteristic
242C2×C4#375211yescharacteristic
243C2×C4C2×C4×C3⋊D311yescharacteristic
244C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
245C2×C4C2×#63811yescharacteristic
246C8—24no—
247C8—184no—
248C2×C3—28no—
249C2×C3C2×#80514yes—
250C2×C3#513214yes—
251C2×C3C2×#82514yes—
252D3—244no—
253C22—21no—
254C22—21no—
255C22—361no—
256C22—361no—
257C22—361no—
258C22#6363212yes—
259C22C2×#375211yescharacteristic
260C4—21no—
261C4—21no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—361no—
266C4—361no—
267C4—361no—
268C4#6365212yes—
269C3#6985814yescomplemented
270C2—22no—
271C2—721no—
272C2C2×#6363211yescharacteristic
273C2#1059283111yescharacteristic
274C2C2×#6365211yescharacteristic
subgroupG/H
centerC2×C4#3752
commutatorC2×C32×C4C22×C4
FrattiniC42C22×C3⋊D3
FittingC32×C4×M16C2
socleC22×C32C2×C22⋊C4
prime radical#64139C4
cosocleC32×C42C23
hypercenterC4×M16C3⋊D3
hypocenterC32#1524
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. C32×C4×M16C2semidirect
3. C2×#63630C2semidirect
4. #10589331C2semidirect
5. #10589330C2semidirect
25. C3×C4×M16D3semidirect
95. C4×M16C3⋊D3semidirect
227. C32#1524semidirect
269. C3#69858semidirect

Maximal subgroups

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

subgroupindex
#105959502
C32×C4×M162
C2×#636302
#105893312
#105893302
#698583