Finite Group Databasebeta

← #11781849

Subgroups — #11781849

Order 1152.

347 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C3×#4688C211yesmaximal, characteristic
2C3×#4680C211yesmaximal, characteristic, complemented
3#10590363C211yesmaximal, characteristic
4#10590266C211yesmaximal, characteristic
5#10589324C211yesmaximal, characteristic, complemented
6#10589095C211yesmaximal, characteristic, complemented
7#10589092C211yesmaximal, characteristic, complemented
8#71345—31nomaximal
9#70872—31nomaximal
10C2×C3×ΓL(1,𝔽25)—21no—
11C3×C4×D12C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C2×#3715—21no—
15C32×C4×C8C2211yescharacteristic
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×(C2×D3)⋊C8—21no—
19C3×#817—21no—
20C3×#817—21no—
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22#63574—21no—
23#63565—21no—
24#63562—21no—
25C3×#498—31no—
26C3×#436—31no—
27#5084—31no—
28#5083—31no—
29#5019—31no—
30#5008—31no—
31#4688D311yescharacteristic
32#4680D311yescharacteristic, complemented
33#4547—31no—
34#4546—31no—
35#4486—31no—
36#4481—31no—
37#4458—31no—
38#4458—31no—
39#4458—61no—
40C2×C3×D12D411yescharacteristic
41C2×C3×C4×D3—21no—
42C2×C3×Dic6—21no—
43C2×C3×Dic6—21no—
44C2×C32×C8D411yescharacteristic
45C2×C32×C8D411yescharacteristic
46C32×C42C2311yescharacteristic
47C2×C3⋊(C3⋊C8)—21no—
48C2×C3⋊(C3⋊C8)—21no—
49C3×(C2×D3)⋊C4—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—21no—
54C3×C4⋊Dic3D411yescharacteristic
55C2×C3×C3⋊C8—21no—
56C3×ΓL(1,𝔽25)—42no—
57#3715—42no—
58#1691—91noSylow
59C2×C3×M16—61no—
60C3×C4⋊Q8—31no—
61C3×C4⋊Q8—31no—
62C3×C4×D4—31no—
63C2×#215—61no—
64C2×#214—61no—
65C2×ΓL(1,𝔽25)—21no—
66C4×D12C2×D311yescharacteristic
67#869C2×D311yescharacteristic
68#869C2×D311yescharacteristic
69C3×#106—61no—
70C3×#106—61no—
71C3×C4⋊C8—31no—
72C3×C22⋊C8—61no—
73C3×C4×C8—21no—
74C3×C4×C8C2×D311yescharacteristic
75C3×C4×C8C2×D311yescharacteristic
76#835—61no—
77#832—61no—
78(C2×D3)⋊C8—21no—
79#817—21no—
80#817—21no—
81#810—61no—
82#809—61no—
83#808—61no—
84#808—61no—
85C4⋊(C3⋊C8)—31no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)—61no—
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C22×C3×D3—21no—
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4#4111yescharacteristic
96C2×C3×Dic3—21no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—41no—
99C3×D12—21no—
100C3×D12—21no—
101C3×C4×D3—41no—
102C3×C4×D3—41no—
103C3×Dic6—41no—
104C3×Dic6—41no—
105C3×Dic6—81no—
106C32×C8—21no—
107C32×C8SD1612yescomplemented
108C3⋊(C3⋊C8)—41no—
109C3⋊(C3⋊C8)—41no—
110C3×C3⋊C8—41no—
111#498—31no—
112#476—91no—
113#461—91no—
114#436—31no—
115#335—91no—
116#333—91no—
117#332—91no—
118C2×C3×Q8—61no—
119C2×C3×Q8—61no—
120C2×C3×D4—31no—
121C22×C3×C4—61no—
122C2×D12D1211yescharacteristic
123C2×C4×D3—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C3×M16—122no—
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C2×C3×C8—61no—
130C2×C3×C8#6711yescharacteristic
131C2×C3×C8#6711yescharacteristic
132C2×C3×C8#6711yescharacteristic
133C2×C3×C8#6711yescharacteristic
134C3×C4⋊C4—31no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—61no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—61no—
139C3×C22⋊C4—61no—
140C3×C42—21no—
141C3×C42C22×D311yescharacteristic
142C3×C42C22×D311yescharacteristic
143#215—122no—
144#214—122no—
145(C2×D3)⋊C4—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3D1211yescharacteristic
151C2×C3⋊C8—21no—
152C2×C3⋊C8—61no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—61no—
155C2×C3⋊C8—61no—
156C2×C3⋊C8—121no—
157C2×C3⋊C8—121no—
158ΓL(1,𝔽25)—42no—
159C22×C32#12111yescharacteristic
160C2×C3×D3—41no—
161C2×C3×D3—41no—
162C2×C3×D3—41no—
163C32×C4—21no—
164C32×C4C2×SD1611yescharacteristic
165C32×C4AGL(1,ℤ/8)11yescharacteristic
166C32×C4#13711yescharacteristic
167C32×C4#13711yescharacteristic
168C3×Dic3—41no—
169C3×Dic3—81no—
170C3×Dic3—81no—
171C2×SD16—181no—
172C2×M16—61no—
173C4⋊Q8—31no—
174C4⋊Q8—31no—
175C4×D4—31no—
176#106—61no—
177#106—61no—
178#106—181no—
179C4⋊C8—31no—
180C4⋊C8—91no—
181C4⋊C8—91no—
182#103—181no—
183#102—181no—
184C22⋊C8—61no—
185C4×C8D3211yescharacteristic
186C23×C3—61no—
187C22×D3—21no—
188C3×Q8—121no—
189C3×Q8—121no—
190C3×Q8—241no—
191C3×D4—61no—
192C3×D4—61no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—61no—
197C2×C3×C4—121no—
198C2×C3×C4—121no—
199C2×C3×C4—121no—
200C2×C3×C4—121no—
201C2×C3×C4C2×D1211yescharacteristic
202C2×C3×C4#23511yescharacteristic
203C2×C3×C4D3×D411yescharacteristic
204C2×C3×C4#23711yescharacteristic
205C2×C3×C4C2×#6711yescharacteristic
206C2×C3×C4C2×#6711yescharacteristic
207C2×Dic3—21no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210D12—21no—
211D12—21no—
212C4×D3—41no—
213C4×D3—41no—
214Dic6—41no—
215Dic6—41no—
216Dic6—81no—
217C3×C8—21no—
218C3×C8—21no—
219C3×C8—22no—
220C3×C8—41no—
221C3×C8—121no—
222C3×C8#21412yescomplemented
223C3×C8#21512yescomplemented
224C3⋊C8—41no—
225C3⋊C8—121no—
226C3⋊C8—121no—
227C3⋊C8—121no—
228C3⋊C8—121no—
229C3⋊C8—241no—
230C3⋊C8—241no—
231C2×C32#46111yescharacteristic
232C2×C32#46811yescharacteristic
233C2×C32#47111yescharacteristic
234C3×D3—81no—
235C2×Q8—61no—
236C2×Q8—61no—
237C2×D4—31no—
238C22×C4—61no—
239SD16—362no—
240M16—122no—
241C2×C8—61no—
242C2×C8—181no—
243C2×C8—181no—
244C2×C8#62511yescharacteristic
245C2×C8#62511yescharacteristic
246C4⋊C4—31no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C22⋊C4—61no—
252C42C2×D3211yescharacteristic
253C22×C3—21no—
254C22×C3—121no—
255C22×C3—121no—
256C22×C3—121no—
257C22×C3#93011yescharacteristic
258C22×C3#93811yescharacteristic
259C2×D3—41no—
260C2×D3—41no—
261C2×D3—41no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—41no—
269C3×C4—121no—
270C3×C4—241no—
271C3×C4—241no—
272C3×C4#90811yescharacteristic
273C3×C4#90911yescharacteristic
274C3×C4#91411yescharacteristic
275C3×C4#91511yescharacteristic
276C3×C4C2×#21411yescharacteristic
277C3×C4C2×#21511yescharacteristic
278C3×C4#94211yescharacteristic
279C3×C4#95111yescharacteristic
280Dic3—41no—
281Dic3—81no—
282Dic3—81no—
283C32#169111yescharacteristic, complemented, Sylow
284C23—61no—
285Q8—121no—
286Q8—121no—
287Q8—241no—
288D4—61no—
289D4—61no—
290C2×C4—61no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4#379511yescharacteristic
296C2×C4D3×D1211yescharacteristic
297C2×C4C2×#62511yescharacteristic
298C8—21no—
299C8—121no—
300C8—361no—
301C8—361no—
302C8#371512yescomplemented
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—241no—
307C2×C3#501911yescharacteristic
308C2×C3#508311yescharacteristic
309C2×C3#513411yescharacteristic
310C2×C3#514111yescharacteristic
311C2×C3#517411yescharacteristic
312C2×C3#517711yescharacteristic
313D3—81no—
314C22—121no—
315C22—121no—
316C22—121no—
317C22#6390811yescharacteristic
318C4—21no—
319C4—121no—
320C4—241no—
321C4—241no—
322C4#6379411yescharacteristic
323C4#6380011yescharacteristic
324C4C2×#371511yescharacteristic
325C4#6382711yescharacteristic
326C3—21no—
327C3#7087211yescharacteristic, complemented
328C3#7134511yescharacteristic, complemented
329C2—241no—
330C2#1059036311yescharacteristic
331C2#1059083311yescharacteristic
332C2#1059092511yescharacteristic
subgroupG/H
centerC22#63908
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×C8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC4×C8D32
hypocenterC32#1691
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
2. C3×#4680C2semidirect
5. #10589324C2semidirect
6. #10589095C2semidirect
7. #10589092C2semidirect
32. #4680D3semidirect
107. C32×C8SD16semidirect
222. C3×C8#214semidirect
223. C3×C8#215semidirect
283. C32#1691semidirect
302. C8#3715semidirect
327. C3#70872semidirect
328. C3#71345semidirect

Maximal subgroups

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

subgroupindex
C3×#46882
C3×#46802
#105903632
#105902662
#105893242
#105890952
#105890922
#713453
#708723