Finite Group Databasebeta

← #11781776

Subgroups — #11781776

Order 1152.

393 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 96 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#10592398C211yesmaximal, characteristic, complemented
2C3×#4688C211yesmaximal, characteristic
3#10590272C212yesmaximal
4C3×#4457C211yesmaximal, characteristic, complemented
5#10589094C212yesmaximal, complemented
6#70871—31nomaximal
7#68541—31nomaximal
8C2×#3746—21no—
9C2×#3746—21no—
10#64078C2212yes—
11C2×C3×ΓL(1,𝔽25)—21no—
12C3×C4×D12C2211yescharacteristic
13C32×C4×C8C2211yescharacteristic
14C3×(C2×D3)⋊C8—21no—
15C3×C4⋊(C3⋊C8)C2212yes—
16C3×C4⋊(C3⋊C8)C2211yescharacteristic
17#63575—22no—
18#63563—22no—
19C3×#436—31no—
20#5007—32no—
21#4846—32no—
22#4699—31no—
23#4699—31no—
24#4699—61no—
25#4688D311yescharacteristic
26C3×#334—31no—
27#4486—32no—
28#4482—32no—
29#4457D311yescharacteristic, complemented
30C2×#634—22no—
31C2×C3×D12D411yescharacteristic
32C2×C3×C4×D3—21no—
33C2×C32×C8D411yescharacteristic
34C2×C32×C8D411yescharacteristic
35C32×C42C2311yescharacteristic
36C4⋊(C3⋊Dic3)—22no—
37C4⋊(C3⋊Dic3)—22no—
38#3746—42no—
39#3746—42no—
40C3×(C2×D3)⋊C4—21no—
41C3×C4⋊Dic3D411yescharacteristic
42C2×C3×C3⋊C8—21no—
43C2×C3×C3⋊C8—22no—
44C3×ΓL(1,𝔽25)—42no—
45#1689—91noSylow
46C2×C3×M16—61no—
47C3×C4×D4—31no—
48C2×Dic12—62no—
49C2×Dic12—62no—
50C2×Dic12—61no—
51C2×Dic12—61no—
52C2×Dic12—61no—
53C2×Dic12—61no—
54C2×ΓL(1,𝔽25)—21no—
55C4×D12C2×D311yescharacteristic
56#869—32no—
57#869—32no—
58#869—62no—
59C3×C4⋊C8—31no—
60C3×C4⋊C8—32no—
61C3×C22⋊C8—61no—
62C3×C4×C8—21no—
63C3×C4×C8C2×D311yescharacteristic
64C3×C4×C8C2×D311yescharacteristic
65#832—62no—
66#821—62no—
67(C2×D3)⋊C8—21no—
68#818—62no—
69#807—62no—
70C4⋊(C3⋊C8)C2×D311yescharacteristic
71C4⋊(C3⋊C8)C2×D312yes—
72C22×C3×D3—21no—
73C2×C32×C4C2×D411yescharacteristic
74C2×C32×C4C2×D411yescharacteristic
75C2×C32×C4#4111yescharacteristic
76C2×C3⋊Dic3—42no—
77#634—42no—
78#634—42no—
79C2×C3×Dic3—21no—
80C3×D12—22no—
81C3×C4×D3—41no—
82C3×C4×D3—41no—
83C32×C8—21no—
84C32×C8D812yes—
85C3×C3⋊C8—41no—
86C3×C3⋊C8—42no—
87#494—91no—
88#474—92no—
89#436—31no—
90#334—31no—
91#332—92no—
92C2×C3×D4—31no—
93C22×C3×C4—61no—
94C2×D12D1211yescharacteristic
95C2×C4×D3—21no—
96C2×Dic6—64no—
97C2×Dic6—62no—
98C2×Dic6—62no—
99C3×M16—122no—
100C2×C3×C8—21no—
101C2×C3×C8—21no—
102C2×C3×C8—62no—
103C2×C3×C8—61no—
104C2×C3×C8D1211yescharacteristic
105C2×C3×C8D1211yescharacteristic
106C2×C3×C8#6711yescharacteristic
107C2×C3×C8#6711yescharacteristic
108C3×C4⋊C4—31no—
109C3×C22⋊C4—61no—
110C3×C42—21no—
111C3×C42C22×D311yescharacteristic
112C3×C42C22×D311yescharacteristic
113(C2×D3)⋊C4—21no—
114C4⋊Dic3—62no—
115C4⋊Dic3—62no—
116C4⋊Dic3—62no—
117C4⋊Dic3—62no—
118C4⋊Dic3—122no—
119C4⋊Dic3—122no—
120C4⋊Dic3D1211yescharacteristic
121C2×C3⋊C8—22no—
122C2×C3⋊C8—21no—
123Dic12—124no—
124Dic12—122no—
125Dic12—122no—
126Dic12—124no—
127Dic12—122no—
128Dic12—122no—
129ΓL(1,𝔽25)—42no—
130C22×C32#12111yescharacteristic
131C2×C3×D3—41no—
132C2×C3×D3—41no—
133C2×C3×D3—41no—
134C32×C4—21no—
135C32×C4C2×D811yescharacteristic
136C32×C4AGL(1,ℤ/8)11yescharacteristic
137C32×C4#13712yes—
138C3⋊Dic3—82no—
139C3×Dic3—41no—
140C2×Q16—181no—
141C2×Q16—181no—
142C2×M16—61no—
143C4⋊Q8—92no—
144C4×D4—31no—
145#107—182no—
146C4⋊C8—31no—
147C4⋊C8—32no—
148#102—182no—
149C22⋊C8—61no—
150C4×C8D3211yescharacteristic
151C23×C3—61no—
152C22×D3—21no—
153C3×D4—62no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—61no—
158C2×C3×C4—121no—
159C2×C3×C4—121no—
160C2×C3×C4C2×D1211yescharacteristic
161C2×C3×C4C2×D1211yescharacteristic
162C2×C3×C4#23511yescharacteristic
163C2×C3×C4D3×D411yescharacteristic
164C2×C3×C4#23911yescharacteristic
165C2×C3×C4C2×#6711yescharacteristic
166C2×Dic3—21no—
167C2×Dic3—124no—
168C2×Dic3—122no—
169C2×Dic3—122no—
170D12—22no—
171C4×D3—41no—
172C4×D3—41no—
173Dic6—124no—
174Dic6—124no—
175Dic6—122no—
176Dic6—122no—
177Dic6—122no—
178Dic6—122no—
179C3×C8—22no—
180C3×C8—21no—
181C3×C8—21no—
182C3×C8—22no—
183C3×C8—121no—
184C3×C8—122no—
185C3×C8D2412yes—
186C3×C8#21312yes—
187C3⋊C8—42no—
188C3⋊C8—41no—
189C2×C32#45911yescharacteristic
190C2×C32#47011yescharacteristic
191C2×C32#47311yescharacteristic
192C3×D3—81no—
193C2×Q8—182no—
194C2×D4—31no—
195C22×C4—61no—
196Q16—362no—
197Q16—362no—
198M16—122no—
199C2×C8—62no—
200C2×C8—61no—
201C2×C8#62611yescharacteristic
202C2×C8#62611yescharacteristic
203C4⋊C4—31no—
204C4⋊C4—182no—
205C4⋊C4—182no—
206C22⋊C4—61no—
207C42C2×D3211yescharacteristic
208C22×C3—21no—
209C22×C3—121no—
210C22×C3—121no—
211C22×C3—121no—
212C22×C3#89511yescharacteristic
213C22×C3#93011yescharacteristic
214C2×D3—41no—
215C2×D3—41no—
216C2×D3—41no—
217C3×C4—22no—
218C3×C4—21no—
219C3×C4—21no—
220C3×C4—22no—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—121no—
224C3×C4C2×D2411yescharacteristic
225C3×C4#90811yescharacteristic
226C3×C4#90912yes—
227C3×C4#91512yes—
228C3×C4C2×#21311yescharacteristic
229C3×C4#94911yescharacteristic
230Dic3—41no—
231Dic3—244no—
232Dic3—242no—
233Dic3—242no—
234C32#168911yescharacteristic, complemented, Sylow
235C23—61no—
236Q8—362no—
237Q8—362no—
238D4—62no—
239C2×C4—61no—
240C2×C4—121no—
241C2×C4—121no—
242C2×C4—362no—
243C2×C4#379911yescharacteristic
244C2×C4D3×D1211yescharacteristic
245C2×C4C2×#62611yescharacteristic
246C8—21no—
247C8—122no—
248C8—121no—
249C8#371412yes—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—241no—
254C2×C3#484311yescharacteristic
255C2×C3#486711yescharacteristic
256C2×C3#489611yescharacteristic
257C2×C3#501511yescharacteristic
258C2×C3#513711yescharacteristic
259C2×C3#514311yescharacteristic
260D3—81no—
261C22—121no—
262C22—121no—
263C22—121no—
264C22#6390911yescharacteristic
265C4—21no—
266C4—121no—
267C4—722no—
268C4#6380012yes—
269C4C2×#371411yescharacteristic
270C4#6382511yescharacteristic
271C3—21no—
272C3#6854111yescharacteristic, complemented
273C3#7087111yescharacteristic, complemented
274C2—241no—
275C2#1059034911yescharacteristic
276C2#1059085311yescharacteristic
277C2#1059093211yescharacteristic
subgroupG/H
centerC22#63909
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×C8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC4×C8D32
hypocenterC32#1689
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
1. #10592398C2semidirect
4. C3×#4457C2semidirect
5. #10589094C2semidirect
29. #4457D3semidirect
234. C32#1689semidirect
272. C3#68541semidirect
273. C3#70871semidirect

Maximal subgroups

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

subgroupindex
#105923982
C3×#46882
#105902722
C3×#44572
#105890942
#708713
#685413