Finite Group Databasebeta

← #11774461

Subgroups — #11774461

Order 1152.

702 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 193 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×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4#10589325C212yesmaximal, complemented
5#10589322C212yesmaximal, complemented
6#69852—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#64084C411yescharacteristic
16C4×C4⋊(C3⋊D3)C2211yescharacteristic
17#64081—21no—
18#64079C411yescharacteristic
19C4×#634C2211yescharacteristic
20C32×#97—21no—
21C32×C4×C8C2212yes—
22C4⋊(C3⋊(C3⋊C8))—21no—
23C4⋊(C3⋊(C3⋊C8))C2212yes—
24#5479—34no—
25C3×C4×M16D314yescomplemented
26C2×C4⋊(C3⋊C8)—34no—
27#4459—38no—
28#4456—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)—22no—
55C2×C3⋊(C3⋊C8)—42no—
56#1483—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×#97—24no—
71C3×C4×C8C2×D318yes—
72C4⋊(C3⋊C8)—38no—
73C4⋊(C3⋊C8)—64no—
74C23×C32C22⋊C411yescharacteristic
75C22×C3⋊D3—21no—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—21no—
79C2×C32×C4—21no—
80C2×C32×C4C22⋊C411yescharacteristic
81C2×C32×C4C22⋊C411yescharacteristic
82C2×C32×C4C22⋊C411yescharacteristic
83C2×C32×C4C22×C411yescharacteristic
84C2×C32×C4C2×D411yescharacteristic
85C2×C32×C4C2×D411yescharacteristic
86#638—42no—
87C2×C3⋊Dic3—21no—
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3—21no—
90C4⋊(C3⋊D3)—22no—
91C4×C3⋊D3—41no—
92C4×C3⋊D3—41no—
93#634—22no—
94C32×C8—24no—
95C3⋊(C3⋊C8)—42no—
96#529—91no—
97C2×C4⋊C8—91no—
98C4×M16C3⋊D311yescharacteristic, complemented
99Q8⋊C8—92no—
100D4⋊C8—92no—
101C22×C3×C4C4×D314yes—
102C22×C3×C4D1214yes—
103C22×C3×C4#6714yes—
104C2×#67—64no—
105C2×D12—34no—
106C2×C4×D3—64no—
107C2×Dic6—34no—
108C3×M16—216no—
109C2×C3×C8—28no—
110C3×C42C4×D314yes—
111C3×C42D1214yes—
112C3×C42#6714yes—
113C3×C42C22×D314yes—
114#217—64no—
115(C2×D3)⋊C4—64no—
116(C2×D3)⋊C4—64no—
117(C2×D3)⋊C4—64no—
118(C2×D3)⋊C4—64no—
119C4⋊Dic3—34no—
120C4⋊Dic3—34no—
121Dic3⋊C4—64no—
122Dic3⋊C4—64no—
123Dic3⋊C4—64no—
124Dic3⋊C4—64no—
125C4×Dic3—64no—
126C2×C3⋊C8—68no—
127C2×C3⋊C8—128no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32#10412yes—
131C22×C32C2×C22⋊C411yescharacteristic
132C2×C3⋊D3—41no—
133C2×C3⋊D3—41no—
134C2×C3⋊D3—41no—
135C32×C4—21no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4—21no—
139C32×C4#13112yes—
140C32×C4AGL(1,ℤ/8)11yescharacteristic
141C32×C4#13711yescharacteristic
142C3⋊Dic3—41no—
143C3⋊Dic3—41no—
144C3⋊Dic3—41no—
145C2×M16—21no—
146C22×C8—181no—
147#126—181no—
148#125—91no—
149#124—91no—
150#123—181no—
151C22⋊Q8—91no—
152#121—91no—
153C4×Q8—91no—
154C4×D4—91no—
155C4×D4—181no—
156#117—181no—
157C2×C42C2×C3⋊D311yescharacteristic
158C4⋊C8—92no—
159C4⋊C8—181no—
160#97—21no—
161C4×C8C2×C3⋊D312yes—
162C23×C3(C2×D3)⋊C414yes—
163C22×D3—64no—
164C2×C3×C4—28no—
165C2×C3×C4—28no—
166C2×C3×C4—24no—
167C2×C3×C4—24no—
168C2×C3×C4(C2×D3)⋊C414yes—
169C2×C3×C4(C2×D3)⋊C414yes—
170C2×C3×C4(C2×D3)⋊C414yes—
171C2×C3×C4C2×C4×D314yes—
172C2×C3×C4C2×D1214yes—
173C2×C3×C4C2×#6714yes—
174#67—128no—
175C2×Dic3—64no—
176C2×Dic3—64no—
177C2×Dic3—64no—
178D12—68no—
179C4×D3—124no—
180C4×D3—124no—
181Dic6—68no—
182C3×C8—216no—
183C3⋊C8—128no—
184C2×C32—22no—
185C2×C32#40811yescharacteristic
186C2×C32#41711yescharacteristic
187C2×C32C2×#10411yescharacteristic
188C3⋊D3—81no—
189C2×Q8—91no—
190C2×D4—91no—
191C2×D4—181no—
192C22×C4—181no—
193C22×C4C4×C3⋊D311yescharacteristic
194C22×C4C4⋊(C3⋊D3)11yescharacteristic
195C22×C4#63811yescharacteristic
196M16—24no—
197C2×C8—22no—
198C2×C8—182no—
199C2×C8—362no—
200C4⋊C4—91no—
201C4⋊C4—91no—
202C4⋊C4—181no—
203C4⋊C4—181no—
204C4⋊C4—181no—
205C4⋊C4—181no—
206C22⋊C4—181no—
207C22⋊C4—181no—
208C22⋊C4—181no—
209C22⋊C4—181no—
210C22⋊C4—181no—
211C42—181no—
212C42C4×C3⋊D311yescharacteristic
213C42C4⋊(C3⋊D3)11yescharacteristic
214C42#63811yescharacteristic
215C42C22×C3⋊D311yescharacteristic
216C22×C3—24no—
217C22×C3—24no—
218C22×C3#80518yes—
219C22×C3C2×(C2×D3)⋊C414yes—
220C2×D3—124no—
221C2×D3—124no—
222C2×D3—124no—
223C3×C4—24no—
224C3×C4—24no—
225C3×C4—24no—
226C3×C4—24no—
227C3×C4#90718yes—
228C3×C4#90814yes—
229C3×C4#90914yes—
230Dic3—124no—
231Dic3—124no—
232Dic3—124no—
233C32#148311yescharacteristic, complemented, Sylow
234C23—181no—
235C23#375211yescharacteristic
236Q8—182no—
237D4—182no—
238D4—362no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—181no—
244C2×C4—181no—
245C2×C4—181no—
246C2×C4—361no—
247C2×C4—361no—
248C2×C4#375211yescharacteristic
249C2×C4#375211yescharacteristic
250C2×C4#375211yescharacteristic
251C2×C4C2×C4×C3⋊D311yescharacteristic
252C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
253C2×C4C2×#63811yescharacteristic
254C8—24no—
255C8—362no—
256C2×C3—28no—
257C2×C3C2×#80514yes—
258C2×C3#512914yes—
259C2×C3#513314yes—
260D3—244no—
261C22—21no—
262C22—21no—
263C22—361no—
264C22—361no—
265C22—361no—
266C22#6363212yes—
267C22C2×#375211yescharacteristic
268C4—21no—
269C4—21no—
270C4—21no—
271C4—21no—
272C4—361no—
273C4—361no—
274C4—361no—
275C4#6411612yes—
276C4#6411711yescharacteristic
277C4#6411811yescharacteristic
278C3#6985214yescomplemented
279C2—22no—
280C2—721no—
281C2C2×#6363211yescharacteristic
282C2#1059282811yescharacteristic
283C2#1059283211yescharacteristic
subgroupG/H
centerC2×C4C2×C4⋊(C3⋊D3)
commutatorC2×C32×C4C22×C4
FrattiniC42C22×C3⋊D3
FittingC32×C4×M16C2
socleC22×C32C2×C22⋊C4
prime radical#64139C4
cosocleC32×C42C23
hypercenterC4×M16C3⋊D3
hypocenterC32#1483
LCSG ▷ C2×C32×C4 ▷ C2×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×C4⋊(C3⋊(C3⋊C8))C2semidirect
4. #10589325C2semidirect
5. #10589322C2semidirect
25. C3×C4×M16D3semidirect
98. C4×M16C3⋊D3semidirect
233. C32#1483semidirect
278. C3#69852semidirect

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×C4⋊(C3⋊(C3⋊C8))2
#105893252
#105893222
#698523