Finite Group Databasebeta

← #11782956

Subgroups — #11782956

Order 1152.

455 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 102 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#10592787C211yesmaximal, characteristic
2C3×#5092C211yesmaximal, characteristic, complemented
3C3×#5078C211yesmaximal, characteristic, complemented
4#10590373C211yesmaximal, characteristic
5#10590312C211yesmaximal, characteristic, complemented
6#10590281C211yesmaximal, characteristic
7#10589078C211yesmaximal, characteristic, complemented
8#71366—31nomaximal
9#71301—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64083C2211yescharacteristic
12C2×C3×#216—22no—
13C3×#869C2211yescharacteristic
14C3×#869C2211yescharacteristic
15C2×#3717—22no—
16#63636—22no—
17C4×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×#807—22no—
19C3×C4×C3⋊C8C2211yescharacteristic
20C3×C4×C3⋊C8C2211yescharacteristic
21#63576—22no—
22#63564—22no—
23C3×#500—31no—
24C3×#494—31no—
25#5092D311yescharacteristic, complemented
26#5088—31no—
27#5088—31no—
28#5088—31no—
29#5088—61no—
30#5085—31no—
31#5078—31no—
32#5078—31no—
33#5078D311yescharacteristic, complemented
34#4694—31no—
35#4693—31no—
36#4455—31no—
37#4454—31no—
38C2×C32×Q8—21no—
39C2×C4⋊(C3⋊D3)—21no—
40C2×C4⋊(C3⋊D3)—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C3×Dic6—21no—
43C2×C3×Dic6—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47C2×C3⋊(C3⋊C8)D412yes—
48C3×#216—44no—
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C2×C3×C3⋊C8D412yes—
54C2×C3×C3⋊C8D412yes—
55#3717—44no—
56#1722—91noSylow
57C2×C3×Q16—62no—
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—31no—
60C3×C4⋊Q8—31no—
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×C4⋊Q8C2×D311yescharacteristic
63C2×#216—22no—
64C2×#215—62no—
65C2×#204—62no—
66#874—31no—
67#874—31no—
68#874—61no—
69#869C2×D311yescharacteristic
70#869C2×D311yescharacteristic
71C3×#107—62no—
72C3×C4×C8—31no—
73C3×C4×C8—31no—
74#821—62no—
75#809—62no—
76#809—62no—
77#809—64no—
78#809—62no—
79#807—22no—
80#807—62no—
81#807—62no—
82C4×C3⋊C8—31no—
83C4×C3⋊C8—31no—
84C4×C3⋊C8—61no—
85C4×C3⋊C8C2×D311yescharacteristic
86C4×C3⋊C8C2×D311yescharacteristic
87C22×C3⋊D3—41no—
88C32×Q8—42no—
89C2×C32×C4—41no—
90C2×C32×C4C2×D411yescharacteristic
91C2×C32×C4C2×D411yescharacteristic
92C2×C32×C4C2×D411yescharacteristic
93C4⋊(C3⋊D3)—42no—
94C4⋊(C3⋊D3)—42no—
95C4⋊(C3⋊D3)—42no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C3×Dic6—42no—
99C3×Dic6—42no—
100C3⋊(C3⋊C8)—22no—
101C3×C3⋊C8—22no—
102C3×C3⋊C8D814yes—
103#500—31no—
104#500—91no—
105#494—31no—
106#492—91no—
107#486—91no—
108#486—91no—
109#322—91no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—41no—
113C2×C3×Q8—61no—
114C2×C3×Q8—61no—
115C2×D12—62no—
116C2×D12—61no—
117C2×D12—61no—
118C2×D12—61no—
119C2×D12—61no—
120C2×D12—61no—
121C2×D12—61no—
122C2×D12—121no—
123C2×D12—121no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C3×Q16—124no—
127C2×C3×C8—32no—
128C2×C3×C8—32no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—61no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—61no—
139C3×C42—21no—
140C3×C42C22×D311yescharacteristic
141C3×C42C22×D311yescharacteristic
142#216—44no—
143#215—124no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—62no—
151C2×C3⋊C8#6712yes—
152C2×C3⋊C8#6712yes—
153#204—124no—
154C22×C32#12711yescharacteristic
155C2×C3⋊D3—81no—
156C2×C3⋊D3—81no—
157C2×C3⋊D3—81no—
158C32×C4—81no—
159C32×C4C2×D812yes—
160C32×C4#13712yes—
161C32×C4#13712yes—
162C3×Dic3—81no—
163C3×Dic3—81no—
164C2×Q16—62no—
165C2×SD16—182no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8D3211yescharacteristic
169#127—91no—
170#107—62no—
171#107—182no—
172#102—182no—
173#102—182no—
174C4×C8—31no—
175C4×C8—31no—
176C4×C8—91no—
177C22×D3—122no—
178C22×D3—121no—
179C22×D3—121no—
180C3×Q8—42no—
181C3×Q8—42no—
182C3×Q8—82no—
183C3×Q8—122no—
184C3×Q8—122no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—42no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—121no—
192C2×C3×C4—121no—
193C2×C3×C4D3×D411yescharacteristic
194C2×C3×C4D3×D411yescharacteristic
195C2×C3×C4D3×D411yescharacteristic
196C2×C3×C4D3×D411yescharacteristic
197C2×C3×C4C2×#6711yescharacteristic
198C2×C3×C4C2×#6711yescharacteristic
199C2×Dic3—41no—
200C2×Dic3—41no—
201D12—124no—
202D12—122no—
203D12—122no—
204D12—122no—
205D12—122no—
206D12—122no—
207D12—122no—
208D12—242no—
209D12—242no—
210Dic6—42no—
211Dic6—42no—
212C3×C8—34no—
213C3×C8—62no—
214C3⋊C8—22no—
215C3⋊C8—62no—
216C3⋊C8—62no—
217C3⋊C8—122no—
218C3⋊C8#21314yes—
219C2×C32#49311yescharacteristic
220C2×C32#49711yescharacteristic
221C2×C32#49711yescharacteristic
222C3⋊D3—161no—
223C2×Q8—21no—
224C2×Q8—61no—
225C2×Q8—61no—
226C2×D4—181no—
227C2×D4—181no—
228C2×D4—181no—
229Q16—124no—
230SD16—364no—
231C2×C8—32no—
232C2×C8—32no—
233C2×C8—92no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C42C2×D3211yescharacteristic
241C22×C3—21no—
242C22×C3#94011yescharacteristic
243C22×C3#94011yescharacteristic
244C2×D3—242no—
245C2×D3—242no—
246C2×D3—242no—
247C2×D3—241no—
248C2×D3—241no—
249C2×D3—241no—
250C2×D3—241no—
251C2×D3—241no—
252C2×D3—241no—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—22no—
256C3×C4—82no—
257C3×C4—81no—
258C3×C4—81no—
259C3×C4—241no—
260C3×C4—241no—
261C3×C4D3×D812yes—
262C3×C4#91812yes—
263C3×C4#91812yes—
264C3×C4#91812yes—
265C3×C4C2×#21312yes—
266C3×C4#94212yes—
267Dic3—81no—
268Dic3—81no—
269C32#172211yescharacteristic, complemented, Sylow
270C23—361no—
271Q8—42no—
272Q8—122no—
273Q8—122no—
274D4—362no—
275D4—362no—
276D4—362no—
277C2×C4—41no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4D3×#6711yescharacteristic
281C2×C4D3×#6711yescharacteristic
282C2×C4#381111yescharacteristic
283C8—34no—
284C8—62no—
285C8—182no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3#507311yescharacteristic
290C2×C3#508611yescharacteristic
291C2×C3#515111yescharacteristic
292C2×C3#519011yescharacteristic
293C2×C3#519011yescharacteristic
294C2×C3#519011yescharacteristic
295D3—482no—
296D3—481no—
297D3—481no—
298C22—721no—
299C22—721no—
300C22—721no—
301C22#6397911yescharacteristic
302C4—81no—
303C4—241no—
304C4—241no—
305C4D3×#21312yes—
306C4#6394512yes—
307C4#6394812yes—
308C3—21no—
309C3#7130111yescharacteristic, complemented
310C3#7136611yescharacteristic, complemented
311C2—1441no—
312C2#1059148211yescharacteristic
313C2#1059160811yescharacteristic
314C2#1059162911yescharacteristic
subgroupG/H
centerC22#63979
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#127
prime radical#64083C22
hypercenterC4⋊Q8D32
hypocenterC32#1722
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5092C2semidirect
3. C3×#5078C2semidirect
5. #10590312C2semidirect
7. #10589078C2semidirect
25. #5092D3semidirect
33. #5078D3semidirect
269. C32#1722semidirect
309. C3#71301semidirect
310. C3#71366semidirect

Maximal subgroups

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

subgroupindex
#105927872
C3×#50922
C3×#50782
#105903732
#105903122
#105902812
#105890782
#713663
#713013