Finite Group Databasebeta

← #11782452

Subgroups — #11782452

Order 1152.

519 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 124 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#10592392C211yesmaximal, characteristic, complemented
2C3×#4693C211yesmaximal, characteristic, complemented
3#10590350C211yesmaximal, characteristic
4#10590311C211yesmaximal, characteristic, complemented
5#10590281C211yesmaximal, characteristic, complemented
6#10590273C211yesmaximal, characteristic, complemented
7C3×C8×C3⋊C8C211yesmaximal, characteristic, complemented
8#71255—31nomaximal
9#68058—31nomaximal
10C2×#3744—22no—
11#64083C2211yescharacteristic
12#64078C2211yescharacteristic
13C2×C3×#204—22no—
14C3×#874C2211yescharacteristic
15C3×#869C2211yescharacteristic
16C2×#3714—22no—
17C32×C4×C8C2211yescharacteristic
18C3×C4×C3⋊C8C2211yescharacteristic
19C3×C4×C3⋊C8C2211yescharacteristic
20#63575—22no—
21#63564—22no—
22#63563—22no—
23C3×#492—31no—
24#5088—31no—
25#5078—31no—
26#5073—31no—
27#5070—31no—
28#4695—31no—
29#4694—31no—
30#4694—31no—
31#4693—31no—
32#4693—31no—
33#4693—61no—
34#4693D311yescharacteristic, complemented
35#4682—31no—
36C3×C82—31no—
37C8×C3⋊C8D311yescharacteristic, complemented
38C2×C4⋊(C3⋊D3)—21no—
39C2×C4⋊(C3⋊D3)—21no—
40C2×C4⋊(C3⋊D3)—21no—
41C2×#634—21no—
42C2×C3×D12—21no—
43C2×C3×D12—21no—
44C2×C3×D12—21no—
45C2×C3×Dic6—21no—
46C2×C32×C8D412yes—
47C32×C42C2311yescharacteristic
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)—21no—
50#3744—44no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C2×C3×C3⋊C8D412yes—
54C2×C3×C3⋊C8D412yes—
55C3×#204—44no—
56#3714—44no—
57#1706—91noSylow
58C2×C3×SD16—62no—
59C3×C4⋊Q8—31no—
60C3×#127—31no—
61C2×#213—62no—
62C2×D24—62no—
63C2×#204—22no—
64C2×#204—64no—
65C2×#204—62no—
66C2×#204—62no—
67#874—31no—
68#874—31no—
69#874—61no—
70#874C2×D311yescharacteristic
71#869—31no—
72#869—31no—
73#869—61no—
74#869C2×D311yescharacteristic
75C3×C4×C8—21no—
76C3×C4×C8—31no—
77C3×C4×C8—31no—
78C3×C4×C8C2×D311yescharacteristic
79C3×C4×C8C2×D311yescharacteristic
80#832—62no—
81#821—62no—
82#821—62no—
83#818—62no—
84#809—62no—
85#807—62no—
86C4×C3⋊C8C2×D311yescharacteristic
87C4×C3⋊C8C2×D311yescharacteristic
88C22×C3⋊D3—41no—
89C22×C3×D3—41no—
90C2×C32×C4C2×D411yescharacteristic
91C2×C32×C4C2×D411yescharacteristic
92C2×C32×C4C2×D411yescharacteristic
93C2×C3⋊Dic3—41no—
94C4⋊(C3⋊D3)—42no—
95C4⋊(C3⋊D3)—42no—
96C4⋊(C3⋊D3)—42no—
97#634—42no—
98C2×C3×Dic3—41no—
99C3×D12—42no—
100C3×D12—42no—
101C3×D12—42no—
102C3×Dic6—42no—
103C32×C8D814yescomplemented
104C3×C3⋊C8—22no—
105C3×C3⋊C8SD1614yescomplemented
106#500—91no—
107#493—91no—
108#492—31no—
109#492—91no—
110#486—91no—
111#486—91no—
112C82—31no—
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×D12—62no—
121C2×D12—61no—
122C2×D12—61no—
123C2×D12—61no—
124C2×D12—61no—
125C2×D12—61no—
126C2×D12—61no—
127C2×D12—121no—
128C2×D12—121no—
129C2×Dic6—21no—
130C2×Dic6—62no—
131C2×Dic6—61no—
132C2×Dic6—61no—
133C3×SD16—124no—
134C2×C3×C8—22no—
135C2×C3×C8—32no—
136C2×C3×C8—32no—
137C2×C3×C8D1212yes—
138C2×C3×C8#6712yes—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—61no—
141C3×C42—21no—
142C3×C42C22×D311yescharacteristic
143C3×C42C22×D311yescharacteristic
144#213—124no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—61no—
148C4⋊Dic3—61no—
149C4⋊Dic3—61no—
150C4⋊Dic3—61no—
151C4⋊Dic3—121no—
152C4⋊Dic3—121no—
153C2×C3⋊C8D1212yes—
154C2×C3⋊C8D1212yes—
155D24—124no—
156#204—44no—
157#204—128no—
158#204—124no—
159#204—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×C4C2×D812yes—
168C32×C4C2×SD1612yes—
169C32×C4#13512yes—
170C3⋊Dic3—81no—
171C3×Dic3—81no—
172C2×SD16—62no—
173C2×SD16—182no—
174C2×D8—182no—
175C4⋊Q8—31no—
176C4⋊Q8—91no—
177#127—31no—
178#127—91no—
179#107—182no—
180#102—182no—
181#102—182no—
182C4×C8—31no—
183C4×C8—31no—
184C4×C8D3211yescharacteristic
185C23×C3—121no—
186C22×D3—41no—
187C22×D3—122no—
188C22×D3—121no—
189C22×D3—121no—
190C3×Q8—122no—
191C3×D4—122no—
192C3×D4—122no—
193C3×D4—122no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—121no—
198C2×C3×C4C2×D1211yescharacteristic
199C2×C3×C4C2×D1211yescharacteristic
200C2×C3×C4C2×D1211yescharacteristic
201C2×C3×C4D3×D411yescharacteristic
202C2×C3×C4D3×D411yescharacteristic
203C2×C3×C4C2×#6711yescharacteristic
204C2×Dic3—41no—
205C2×Dic3—122no—
206C2×Dic3—121no—
207C2×Dic3—121no—
208D12—42no—
209D12—42no—
210D12—42no—
211D12—124no—
212D12—122no—
213D12—122no—
214D12—122no—
215D12—122no—
216D12—122no—
217D12—122no—
218D12—242no—
219D12—242no—
220Dic6—42no—
221Dic6—124no—
222Dic6—122no—
223Dic6—122no—
224C3×C8—24no—
225C3×C8—34no—
226C3×C8—62no—
227C3×C8D2414yescomplemented
228C3×C8#21314yescomplemented
229C3⋊C8—22no—
230C3⋊C8#20414yescomplemented
231C2×C32#49211yescharacteristic
232C2×C32#49311yescharacteristic
233C2×C32#49511yescharacteristic
234C3⋊D3—161no—
235C3×D3—161no—
236C2×Q8—61no—
237C2×Q8—181no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—181no—
242C2×D4—181no—
243C2×D4—181no—
244SD16—124no—
245SD16—364no—
246D8—364no—
247C2×C8—32no—
248C2×C8—32no—
249C2×C8#62612yes—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—181no—
253C4⋊C4—181no—
254C42C2×D3211yescharacteristic
255C22×C3—21no—
256C22×C3—241no—
257C22×C3—241no—
258C22×C3—241no—
259C22×C3#87411yescharacteristic
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—241no—
277C3×C4C2×#20412yes—
278C3×C4C2×D2412yes—
279C3×C4#90412yes—
280C3×C4D3×SD1612yes—
281C3×C4#91612yes—
282C3×C4C2×#21312yes—
283Dic3—81no—
284Dic3—242no—
285Dic3—241no—
286Dic3—241no—
287C32#170611yescharacteristic, complemented, Sylow
288C23—121no—
289C23—361no—
290Q8—122no—
291Q8—362no—
292D4—122no—
293D4—122no—
294D4—122no—
295D4—362no—
296D4—362no—
297D4—362no—
298C2×C4—121no—
299C2×C4—361no—
300C2×C4D3×D1211yescharacteristic
301C2×C4D3×D1211yescharacteristic
302C2×C4C2×#62611yescharacteristic
303C8—34no—
304C8—62no—
305C8#371414yescomplemented
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—481no—
310C2×C3#469311yescharacteristic
311C2×C3#469511yescharacteristic
312C2×C3#469611yescharacteristic
313C2×C3#507311yescharacteristic
314C2×C3#516811yescharacteristic
315C2×C3#517511yescharacteristic
316D3—161no—
317D3—482no—
318D3—481no—
319D3—481no—
320C22—241no—
321C22—241no—
322C22—241no—
323C22—721no—
324C22—721no—
325C22—721no—
326C22#6391111yescharacteristic
327C4—241no—
328C4—721no—
329C4D3×#20412yes—
330C4#6380612yes—
331C4C2×#371412yes—
332C3—21no—
333C3#6805811yescharacteristic, complemented
334C3#7125511yescharacteristic, complemented
335C2—481no—
336C2—1441no—
337C2#1059035011yescharacteristic
338C2#1059086211yescharacteristic
339C2#1059087711yescharacteristic
subgroupG/H
centerC22#63911
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×C8C22
socleC22×C32#127
prime radical#10590350C2
hypercenterC4×C8D32
hypocenterC32#1706
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. #10592392C2semidirect
2. C3×#4693C2semidirect
4. #10590311C2semidirect
5. #10590281C2semidirect
6. #10590273C2semidirect
7. C3×C8×C3⋊C8C2semidirect
34. #4693D3semidirect
37. C8×C3⋊C8D3semidirect
103. C32×C8D8semidirect
105. C3×C3⋊C8SD16semidirect
227. C3×C8D24semidirect
228. C3×C8#213semidirect
230. C3⋊C8#204semidirect
287. C32#1706semidirect
305. C8#3714semidirect
333. C3#68058semidirect
334. C3#71255semidirect

Maximal subgroups

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

subgroupindex
#105923922
C3×#46932
#105903502
#105903112
#105902812
#105902732
C3×C8×C3⋊C82
#712553
#680583