Finite Group Databasebeta

← #11782954

Subgroups — #11782954

Order 1152.

457 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 98 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#10592791C211yesmaximal, characteristic, complemented
2C3×#5088C211yesmaximal, characteristic
3C3×#5078C211yesmaximal, characteristic, complemented
4#10590375C211yesmaximal, characteristic
5#10590311C211yesmaximal, characteristic, complemented
6#10590273C211yesmaximal, characteristic
7#10589079C211yesmaximal, characteristic, complemented
8#71366—31nomaximal
9#71195—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C2×#3756—22no—
12#64078C2211yescharacteristic
13C3×#874C2211yescharacteristic
14C3×#869C2211yescharacteristic
15C2×#3715—22no—
16C4×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×#809—22no—
18C3×#807—22no—
19C3×C4×C3⋊C8C2211yescharacteristic
20C3×C4×C3⋊C8C2211yescharacteristic
21#63575—22no—
22#63563—22no—
23C3×#500—31no—
24C3×#486—31no—
25#5092—31no—
26#5092—31no—
27#5092—61no—
28#5088D311yescharacteristic
29#5085—31no—
30#5078—31no—
31#5078D311yescharacteristic, complemented
32#5076—31no—
33#5070—31no—
34#4694—31no—
35#4682—31no—
36#4454—31no—
37#4454—31no—
38C2×C32×Q8—21no—
39C2×#634—21no—
40C2×C3×D12—21no—
41C2×C3×D12—21no—
42C2×C3×D12—21no—
43C2×C3×Dic6—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47#3756—44no—
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)—21no—
50C2×C3⋊(C3⋊C8)D412yes—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C2×C3×C3⋊C8D412yes—
54C2×C3×C3⋊C8D412yes—
55#3715—44no—
56#1722—91noSylow
57C3×C4⋊Q8—21no—
58C3×C4⋊Q8—31no—
59C3×C4⋊Q8C2×D311yescharacteristic
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×#127—31no—
62C2×#216—62no—
63C2×#216—62no—
64C2×#216—64no—
65C2×#215—62no—
66C2×#214—62no—
67#874C2×D311yescharacteristic
68#869—31no—
69#869—31no—
70#869—61no—
71#869C2×D311yescharacteristic
72C3×#107—62no—
73C3×#102—62no—
74C3×C4×C8—31no—
75C3×C4×C8—31no—
76#832—62no—
77#821—62no—
78#818—62no—
79#809—22no—
80#807—22no—
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
93C2×C3⋊Dic3—41no—
94#634—42no—
95C2×C3×Dic3—41no—
96C3×D12—42no—
97C3×D12—42no—
98C3×D12—42no—
99C3×Dic6—42no—
100C3⋊(C3⋊C8)D814yes—
101C3×C3⋊C8—22no—
102C3×C3⋊C8—22no—
103#500—31no—
104#500—91no—
105#494—91no—
106#492—91no—
107#486—31no—
108#486—91no—
109#322—91no—
110C2×C3×Q8—22no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—61no—
114C2×C3×D4—61no—
115C2×C3×D4—61no—
116C2×C3×D4—61no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×Dic6—21no—
121C2×Dic6—62no—
122C2×Dic6—61no—
123C2×Dic6—61no—
124C2×C3×C8—32no—
125C2×C3×C8—32no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—61no—
133C3×C4⋊C4—61no—
134C3×C42—21no—
135C3×C42C22×D311yescharacteristic
136C3×C42C22×D311yescharacteristic
137#216—124no—
138#216—124no—
139#216—128no—
140#215—124no—
141#214—124no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—61no—
145C4⋊Dic3—61no—
146C4⋊Dic3—61no—
147C4⋊Dic3—61no—
148C4⋊Dic3—121no—
149C4⋊Dic3—121no—
150C2×C3⋊C8—32no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—62no—
153C2×C3⋊C8#6712yes—
154C2×C3⋊C8#6712yes—
155C22×C32#12711yescharacteristic
156C2×C3×D3—81no—
157C2×C3×D3—81no—
158C2×C3×D3—81no—
159C32×C4—81no—
160C32×C4C2×D812yes—
161C32×C4#13712yes—
162C32×C4#13712yes—
163C3⋊Dic3—81no—
164C3×Dic3—81no—
165C2×Q16—182no—
166C2×SD16—182no—
167C4⋊Q8—31no—
168C4⋊Q8—91no—
169C4⋊Q8D3211yescharacteristic
170#127—31no—
171#107—62no—
172#107—182no—
173#102—62no—
174#102—182no—
175C4×C8—31no—
176C4×C8—31no—
177C4×C8—91no—
178C23×C3—121no—
179C22×D3—41no—
180C3×Q8—44no—
181C3×Q8—42no—
182C3×Q8—42no—
183C3×Q8—122no—
184C3×D4—122no—
185C3×D4—122no—
186C3×D4—122no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—121no—
194C2×C3×C4D3×D411yescharacteristic
195C2×C3×C4D3×D411yescharacteristic
196C2×C3×C4D3×D411yescharacteristic
197C2×C3×C4D3×D411yescharacteristic
198C2×C3×C4C2×#6711yescharacteristic
199C2×C3×C4C2×#6711yescharacteristic
200C2×Dic3—41no—
201C2×Dic3—122no—
202C2×Dic3—121no—
203C2×Dic3—121no—
204D12—42no—
205D12—42no—
206D12—42no—
207Dic6—42no—
208Dic6—124no—
209Dic6—122no—
210Dic6—122no—
211C3×C8—62no—
212C3×C8—62no—
213C3⋊C8—22no—
214C3⋊C8—22no—
215C3⋊C8—34no—
216C3⋊C8—34no—
217C3⋊C8—64no—
218C2×C32#49311yescharacteristic
219C2×C32#49711yescharacteristic
220C2×C32#49711yescharacteristic
221C3×D3—161no—
222C2×Q8—21no—
223C2×Q8—61no—
224C2×Q8—181no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228Q16—364no—
229SD16—364no—
230C2×C8—32no—
231C2×C8—32no—
232C2×C8—92no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—181no—
238C4⋊C4—181no—
239C42C2×D3211yescharacteristic
240C22×C3—21no—
241C22×C3—241no—
242C22×C3—241no—
243C22×C3—241no—
244C22×C3#94011yescharacteristic
245C22×C3#94011yescharacteristic
246C2×D3—81no—
247C2×D3—81no—
248C2×D3—81no—
249C3×C4—22no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—82no—
253C3×C4—81no—
254C3×C4—81no—
255C3×C4—241no—
256C3×C4D3×D812yes—
257C3×C4D3×D812yes—
258C3×C4#91512yes—
259C3×C4#91812yes—
260C3×C4#94212yes—
261C3×C4#95112yes—
262Dic3—81no—
263Dic3—242no—
264Dic3—241no—
265Dic3—241no—
266C32#172211yescharacteristic, complemented, Sylow
267C23—121no—
268Q8—42no—
269Q8—122no—
270Q8—362no—
271D4—122no—
272D4—122no—
273D4—122no—
274C2×C4—41no—
275C2×C4—121no—
276C2×C4—361no—
277C2×C4D3×#6711yescharacteristic
278C2×C4D3×#6711yescharacteristic
279C2×C4#381111yescharacteristic
280C8—62no—
281C8—62no—
282C8—94no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—481no—
287C2×C3#506011yescharacteristic
288C2×C3#508611yescharacteristic
289C2×C3#515111yescharacteristic
290C2×C3#515111yescharacteristic
291C2×C3#516711yescharacteristic
292C2×C3#519011yescharacteristic
293D3—161no—
294C22—241no—
295C22—241no—
296C22—241no—
297C22#6397911yescharacteristic
298C4—81no—
299C4—241no—
300C4—721no—
301C4#6392612yes—
302C4#6394312yes—
303C4#6394612yes—
304C3—21no—
305C3#7119511yescharacteristic, complemented
306C3#7136611yescharacteristic, complemented
307C2—481no—
308C2#1059148411yescharacteristic
309C2#1059160511yescharacteristic
310C2#1059162711yescharacteristic
subgroupG/H
centerC22#63979
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#127
prime radicalC3×#874C22
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
1. #10592791C2semidirect
3. C3×#5078C2semidirect
5. #10590311C2semidirect
7. #10589079C2semidirect
31. #5078D3semidirect
266. C32#1722semidirect
305. C3#71195semidirect
306. C3#71366semidirect

Maximal subgroups

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

subgroupindex
#105927912
C3×#50882
C3×#50782
#105903752
#105903112
#105902732
#105890792
#713663
#711953