Finite Group Databasebeta

← #11782455

Subgroups — #11782455

Order 1152.

519 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 106 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#10592784C211yesmaximal, characteristic, complemented
2C3×#5088C211yesmaximal, characteristic, complemented
3C3×#5078C211yesmaximal, characteristic, complemented
4#10590375C211yesmaximal, characteristic, complemented
5#10590350C211yesmaximal, characteristic
6#10590281C211yesmaximal, characteristic, complemented
7C3⋊C82C211yesmaximal, characteristic, complemented
8#71359—31nomaximal
9#71180—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C2×#3755—22no—
12#64083C2211yescharacteristic
13C3×#874C2211yescharacteristic
14C3×#869C2211yescharacteristic
15C2×#3715—22no—
16C2×#3714—22no—
17C4×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×#809—22no—
19C3×#807—22no—
20C3×C4×C3⋊C8C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63564—22no—
23C3×#500—31no—
24C3×#486—31no—
25#5088—31no—
26#5088D311yescharacteristic, complemented
27#5085—31no—
28#5085—31no—
29#5085—31no—
30#5085—61no—
31#5078D311yescharacteristic, complemented
32#5076—31no—
33#5073—31no—
34#4695—31no—
35#4694—31no—
36C8×C3⋊C8—31no—
37C8×C3⋊C8—31no—
38C2×C32×Q8—21no—
39C2×C4⋊(C3⋊D3)—21no—
40C2×C4⋊(C3⋊D3)—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C3×D12—21no—
43C2×C3×D12—21no—
44C2×C3×D12—21no—
45C2×C3×Dic6—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C42C2311yescharacteristic
49#3755—44no—
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#3714—44no—
57#1706—91noSylow
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—31no—
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×#127—31no—
63C2×#215—62no—
64C2×#215—62no—
65C2×#215—62no—
66C2×#215—64no—
67C2×#214—62no—
68C2×#213—62no—
69C2×D24—62no—
70#874—31no—
71#874—31no—
72#874—61no—
73#874C2×D311yescharacteristic
74#869C2×D311yescharacteristic
75C3×#107—62no—
76C3×#102—62no—
77C3×C4×C8—31no—
78C3×C4×C8—31no—
79#821—62no—
80#809—22no—
81#809—62no—
82#807—22no—
83C4×C3⋊C8—31no—
84C4×C3⋊C8—31no—
85C4×C3⋊C8—61no—
86C4×C3⋊C8C2×D311yescharacteristic
87C4×C3⋊C8C2×D311yescharacteristic
88C22×C3⋊D3—41no—
89C22×C3×D3—41no—
90C32×Q8—42no—
91C2×C32×C4—41no—
92C2×C32×C4C2×D411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C4⋊(C3⋊D3)—42no—
96C4⋊(C3⋊D3)—42no—
97C4⋊(C3⋊D3)—42no—
98C2×C3×Dic3—41no—
99C3×D12—42no—
100C3×D12—42no—
101C3×D12—42no—
102C3×Dic6—42no—
103C3⋊(C3⋊C8)D814yescomplemented
104C3×C3⋊C8—22no—
105C3×C3⋊C8SD1614yescomplemented
106#500—31no—
107#493—91no—
108#492—91no—
109#492—91no—
110#486—31no—
111#486—91no—
112C82—91no—
113C2×C3×Q8—22no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—61no—
117C2×C3×D4—61no—
118C2×C3×D4—61no—
119C2×C3×D4—61no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—62no—
124C2×D12—61no—
125C2×D12—61no—
126C2×D12—61no—
127C2×D12—61no—
128C2×D12—61no—
129C2×D12—61no—
130C2×D12—121no—
131C2×D12—121no—
132C2×Dic6—21no—
133C2×C3×C8—32no—
134C2×C3×C8—32no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—61no—
142C3×C4⋊C4—61no—
143C3×C42—21no—
144C3×C42C22×D311yescharacteristic
145C3×C42C22×D311yescharacteristic
146#215—124no—
147#215—124no—
148#215—124no—
149#215—128no—
150#214—124no—
151#213—124no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C2×C3⋊C8—32no—
155C2×C3⋊C8—32no—
156C2×C3⋊C8—62no—
157C2×C3⋊C8#6712yes—
158C2×C3⋊C8#6712yes—
159D24—124no—
160C22×C32#12711yescharacteristic
161C2×C3⋊D3—81no—
162C2×C3⋊D3—81no—
163C2×C3⋊D3—81no—
164C2×C3×D3—81no—
165C2×C3×D3—81no—
166C2×C3×D3—81no—
167C32×C4—81no—
168C32×C4C2×D812yes—
169C32×C4C2×SD1612yes—
170C32×C4#13512yes—
171C3×Dic3—81no—
172C2×SD16—182no—
173C2×SD16—182no—
174C2×D8—182no—
175C4⋊Q8—31no—
176C4⋊Q8D3211yescharacteristic
177#127—31no—
178#127—91no—
179#107—62no—
180#102—62no—
181#102—182no—
182C4×C8—31no—
183C4×C8—31no—
184C4×C8—91no—
185C23×C3—121no—
186C22×D3—41no—
187C22×D3—122no—
188C22×D3—121no—
189C22×D3—121no—
190C3×Q8—44no—
191C3×Q8—42no—
192C3×Q8—42no—
193C3×Q8—122no—
194C3×D4—122no—
195C3×D4—122no—
196C3×D4—122no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—121no—
204C2×C3×C4D3×D411yescharacteristic
205C2×C3×C4D3×D411yescharacteristic
206C2×C3×C4D3×D411yescharacteristic
207C2×C3×C4D3×D411yescharacteristic
208C2×C3×C4C2×#6711yescharacteristic
209C2×C3×C4C2×#6711yescharacteristic
210C2×Dic3—41no—
211D12—42no—
212D12—42no—
213D12—42no—
214D12—124no—
215D12—122no—
216D12—122no—
217D12—122no—
218D12—122no—
219D12—122no—
220D12—122no—
221D12—242no—
222D12—242no—
223Dic6—42no—
224C3×C8—34no—
225C3×C8—62no—
226C3⋊C8—22no—
227C3⋊C8—34no—
228C3⋊C8—34no—
229C3⋊C8—64no—
230C3⋊C8#21514yescomplemented
231C2×C32#49211yescharacteristic
232C2×C32#49311yescharacteristic
233C2×C32#49511yescharacteristic
234C3⋊D3—161no—
235C3×D3—161no—
236C2×Q8—21no—
237C2×Q8—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—181no—
242C2×D4—181no—
243C2×D4—181no—
244SD16—364no—
245SD16—364no—
246D8—364no—
247C2×C8—32no—
248C2×C8—32no—
249C2×C8—92no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C42C2×D3211yescharacteristic
255C22×C3—21no—
256C22×C3—241no—
257C22×C3—241no—
258C22×C3—241no—
259C22×C3#94011yescharacteristic
260C22×C3#94011yescharacteristic
261C2×D3—81no—
262C2×D3—81no—
263C2×D3—81no—
264C2×D3—242no—
265C2×D3—242no—
266C2×D3—242no—
267C2×D3—241no—
268C2×D3—241no—
269C2×D3—241no—
270C2×D3—241no—
271C2×D3—241no—
272C2×D3—241no—
273C3×C4—22no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—82no—
277C3×C4—81no—
278C3×C4—81no—
279C3×C4—241no—
280C3×C4D3×D812yes—
281C3×C4D3×D812yes—
282C3×C4D3×SD1612yes—
283C3×C4#91912yes—
284C3×C4C2×#21512yes—
285C3×C4#95012yes—
286Dic3—81no—
287C32#170611yescharacteristic, complemented, Sylow
288C23—121no—
289C23—361no—
290Q8—42no—
291Q8—122no—
292D4—122no—
293D4—122no—
294D4—122no—
295D4—362no—
296D4—362no—
297D4—362no—
298C2×C4—41no—
299C2×C4—121no—
300C2×C4D3×#6711yescharacteristic
301C2×C4D3×#6711yescharacteristic
302C2×C4#381111yescharacteristic
303C8—34no—
304C8—62no—
305C8—94no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—481no—
310C2×C3#505911yescharacteristic
311C2×C3#508511yescharacteristic
312C2×C3#515111yescharacteristic
313C2×C3#515111yescharacteristic
314C2×C3#517511yescharacteristic
315C2×C3#519111yescharacteristic
316D3—161no—
317D3—482no—
318D3—481no—
319D3—481no—
320C22—241no—
321C22—241no—
322C22—241no—
323C22—721no—
324C22—721no—
325C22—721no—
326C22#6397911yescharacteristic
327C4—81no—
328C4—241no—
329C4#6392612yes—
330C4D3×#21512yes—
331C4#6394912yes—
332C3—21no—
333C3#7118011yescharacteristic, complemented
334C3#7135911yescharacteristic, complemented
335C2—481no—
336C2—1441no—
337C2#1059148411yescharacteristic
338C2#1059160911yescharacteristic
339C2#1059161811yescharacteristic
subgroupG/H
centerC22#63979
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#127
prime radical#10590350C2
hypercenterC4⋊Q8D32
hypocenterC32#1706
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. #10592784C2semidirect
2. C3×#5088C2semidirect
3. C3×#5078C2semidirect
4. #10590375C2semidirect
6. #10590281C2semidirect
7. C3⋊C82C2semidirect
26. #5088D3semidirect
31. #5078D3semidirect
103. C3⋊(C3⋊C8)D8semidirect
105. C3×C3⋊C8SD16semidirect
230. C3⋊C8#215semidirect
287. C32#1706semidirect
333. C3#71180semidirect
334. C3#71359semidirect

Maximal subgroups

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

subgroupindex
#105927842
C3×#50882
C3×#50782
#105903752
#105903502
#105902812
C3⋊C822
#713593
#711803