Finite Group Databasebeta

← #11781996

Subgroups — #11781996

Order 1152.

357 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 84 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#10592777C211yesmaximal, characteristic, complemented
2C3×#5084C211yesmaximal, characteristic
3#10590339C211yesmaximal, characteristic
4#10590268C211yesmaximal, characteristic
5C3×#4546C211yesmaximal, characteristic, complemented
6#10589098C211yesmaximal, characteristic, complemented
7#10589096C211yesmaximal, characteristic, complemented
8#71137—31nomaximal
9#68501—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64078C2211yescharacteristic
12C3×C4×D12C2211yescharacteristic
13C2×#3711—21no—
14#63634—21no—
15#63634—21no—
16C4×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×#809—21no—
18C3×#808—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22#63573—21no—
23#63563—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#476—31no—
26#5084D311yescharacteristic
27#5078—31no—
28#5078—31no—
29#5078—61no—
30#5012—31no—
31#5008—31no—
32#4842—31no—
33#4838—31no—
34C3×#333—31no—
35#4546D311yescharacteristic, complemented
36#4481—31no—
37#4481—31no—
38#4480—31no—
39#4480—31no—
40C2×C32×Q8—21no—
41C2×#634—21no—
42C2×C3×D12D411yescharacteristic
43C2×C3×C4×D3—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47C4⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊C8)D411yescharacteristic
50C2×C3⋊(C3⋊C8)D411yescharacteristic
51C3×(C2×D3)⋊C4—21no—
52C3×C4⋊Dic3D411yescharacteristic
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—21no—
56#3711—42no—
57#1695—91noSylow
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8C2×D311yescharacteristic
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×C4×D4—31no—
62C2×#208—61no—
63C2×ΓL(1,𝔽25)—61no—
64C4×D12C2×D311yescharacteristic
65#869—31no—
66#869—31no—
67#869—61no—
68C3×#106—61no—
69C3×C4⋊C8—31no—
70C3×C4⋊C8—31no—
71C3×C4⋊C8—31no—
72C3×#102—61no—
73#832—61no—
74#830—61no—
75#821—61no—
76(C2×D3)⋊C8—61no—
77#817—61no—
78#809—21no—
79#808—21no—
80#808—61no—
81#807—61no—
82#807—61no—
83#807—61no—
84#807—61no—
85#807—62no—
86#807—62no—
87C4⋊(C3⋊C8)C2×D311yescharacteristic
88C4⋊(C3⋊C8)C2×D311yescharacteristic
89C4⋊(C3⋊C8)C2×D311yescharacteristic
90C4×C3⋊C8—31no—
91C4×C3⋊C8—31no—
92C4×C3⋊C8—61no—
93C22×C3×D3—21no—
94C32×Q8—81no—
95C2×C32×C4—41no—
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4#4111yescharacteristic
99C2×C3⋊Dic3—41no—
100#634—81no—
101C2×C3×Dic3—21no—
102C3×D12—21no—
103C3×D12—21no—
104C3×C4×D3—41no—
105C3×C4×D3—41no—
106C3⋊(C3⋊C8)—21no—
107C3⋊(C3⋊C8)SD1612yes—
108C3×C3⋊C8—41no—
109C3×C3⋊C8—41no—
110C3×C3⋊C8—41no—
111#500—91no—
112#476—31no—
113#476—91no—
114#436—91no—
115#334—91no—
116#333—31no—
117#333—91no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—41no—
121C2×C3×D4—31no—
122C22×C3×C4—61no—
123C2×D12#6711yescharacteristic
124C2×C4×D3—21no—
125C2×Dic6—61no—
126C2×Dic6—61no—
127C2×Dic6—121no—
128C2×C3×C8—61no—
129C2×C3×C8—61no—
130C2×C3×C8—61no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—41no—
138C3×C22⋊C4—61no—
139C3×C42—21no—
140C3×C42C22×D311yescharacteristic
141C3×C42C22×D311yescharacteristic
142(C2×D3)⋊C4—21no—
143C4⋊Dic3—62no—
144C4⋊Dic3—61no—
145C4⋊Dic3—61no—
146C4⋊Dic3—61no—
147C4⋊Dic3—61no—
148C4⋊Dic3—121no—
149C4⋊Dic3#6711yescharacteristic
150#208—122no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8—31no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8—31no—
158C2×C3⋊C8—61no—
159C2×C3⋊C8—61no—
160ΓL(1,𝔽25)—122no—
161C22×C32#12111yescharacteristic
162C2×C3×D3—41no—
163C2×C3×D3—41no—
164C2×C3×D3—41no—
165C32×C4—21no—
166C32×C4—81no—
167C32×C4C2×SD1611yescharacteristic
168C32×C4#13711yescharacteristic
169C32×C4#13711yescharacteristic
170C32×C4#13711yescharacteristic
171C3⋊Dic3—81no—
172C3×Dic3—41no—
173C2×M16—181no—
174C4⋊Q8—91no—
175C4⋊Q8D3211yescharacteristic
176C4×D4—31no—
177#107—181no—
178#107—181no—
179#106—61no—
180#106—181no—
181C4⋊C8—31no—
182C4⋊C8—31no—
183C4⋊C8—31no—
184#102—61no—
185#102—181no—
186C22⋊C8—181no—
187C4×C8—91no—
188C23×C3—61no—
189C22×D3—21no—
190C3×Q8—81no—
191C3×Q8—81no—
192C3×Q8—82no—
193C3×D4—61no—
194C3×D4—61no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—42no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—61no—
202C2×C3×C4—121no—
203C2×C3×C4—121no—
204C2×C3×C4#23511yescharacteristic
205C2×C3×C4D3×D411yescharacteristic
206C2×C3×C4D3×D411yescharacteristic
207C2×C3×C4D3×D411yescharacteristic
208C2×C3×C4#23711yescharacteristic
209C2×C3×C4C2×#6711yescharacteristic
210C2×Dic3—21no—
211C2×Dic3—122no—
212C2×Dic3—121no—
213C2×Dic3—121no—
214D12—21no—
215D12—21no—
216C4×D3—41no—
217C4×D3—41no—
218Dic6—242no—
219Dic6—241no—
220Dic6—241no—
221C3×C8—121no—
222C3×C8—121no—
223C3×C8—121no—
224C3⋊C8—32no—
225C3⋊C8—32no—
226C3⋊C8—41no—
227C3⋊C8—41no—
228C3⋊C8—41no—
229C3⋊C8—62no—
230C3⋊C8—62no—
231C3⋊C8—61no—
232C3⋊C8—61no—
233C2×C32#46111yescharacteristic
234C2×C32#47011yescharacteristic
235C2×C32#47211yescharacteristic
236C3×D3—81no—
237C2×Q8—21no—
238C2×Q8—181no—
239C2×D4—31no—
240C22×C4—61no—
241M16—362no—
242C2×C8—61no—
243C2×C8—61no—
244C2×C8—61no—
245C2×C8—91no—
246C2×C8—91no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—31no—
250C4⋊C4—181no—
251C4⋊C4—181no—
252C22⋊C4—61no—
253C42C2×D3211yescharacteristic
254C22×C3—21no—
255C22×C3—121no—
256C22×C3—121no—
257C22×C3—121no—
258C22×C3#88411yescharacteristic
259C22×C3#93911yescharacteristic
260C2×D3—41no—
261C2×D3—41no—
262C2×D3—41no—
263C3×C4—22no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—82no—
271C3×C4—81no—
272C3×C4—81no—
273C3×C4—121no—
274C3×C4D3×SD1611yescharacteristic
275C3×C4D3×SD1611yescharacteristic
276C3×C4#91511yescharacteristic
277C3×C4#91511yescharacteristic
278C3×C4#91811yescharacteristic
279C3×C4#91811yescharacteristic
280C3×C4#94211yescharacteristic
281C3×C4#95111yescharacteristic
282Dic3—41no—
283Dic3—242no—
284Dic3—241no—
285Dic3—241no—
286C32#169511yescharacteristic, complemented, Sylow
287C23—61no—
288Q8—81no—
289Q8—721no—
290D4—61no—
291D4—61no—
292C2×C4—41no—
293C2×C4—61no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—361no—
297C2×C4#380411yescharacteristic
298C2×C4D3×#6711yescharacteristic
299C2×C4#381111yescharacteristic
300C8—92no—
301C8—121no—
302C8—121no—
303C8—121no—
304C8—181no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—241no—
309C2×C3#474911yescharacteristic
310C2×C3#481611yescharacteristic
311C2×C3#481811yescharacteristic
312C2×C3#505211yescharacteristic
313C2×C3#516211yescharacteristic
314C2×C3#519911yescharacteristic
315D3—81no—
316C22—121no—
317C22—121no—
318C22—121no—
319C22#6397111yescharacteristic
320C4—21no—
321C4—81no—
322C4—121no—
323C4—721no—
324C4#6394011yescharacteristic
325C4#6394311yescharacteristic
326C4#6394611yescharacteristic
327C4#6395111yescharacteristic
328C3—21no—
329C3#6850111yescharacteristic, complemented
330C3#7113711yescharacteristic, complemented
331C2—241no—
332C2#1059150811yescharacteristic
333C2#1059156911yescharacteristic
334C2#1059159811yescharacteristic
subgroupG/H
centerC22#63971
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC4⋊Q8D32
hypocenterC32#1695
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. #10592777C2semidirect
5. C3×#4546C2semidirect
6. #10589098C2semidirect
7. #10589096C2semidirect
35. #4546D3semidirect
286. C32#1695semidirect
329. C3#68501semidirect
330. C3#71137semidirect

Maximal subgroups

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

subgroupindex
#105927772
C3×#50842
#105903392
#105902682
C3×#45462
#105890982
#105890962
#711373
#685013