Finite Group Databasebeta

← #11781858

Subgroups — #11781858

Order 1152.

352 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 88 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×#4841C211yesmaximal, characteristic
2#10590337C211yesmaximal, characteristic
3#10590303C211yesmaximal, characteristic, complemented
4#10590268C211yesmaximal, characteristic
5#10589347C211yesmaximal, characteristic, complemented
6C3×#4479C211yesmaximal, characteristic, complemented
7#10589093C211yesmaximal, characteristic, complemented
8#71040—31nomaximal
9#68583—31nomaximal
10#64078C2211yescharacteristic
11C2×C3×#204—21no—
12C3×C4×D12C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C2×#3711—21no—
15C32×C4⋊C8C2211yescharacteristic
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#816—21no—
18C3×C4⋊(C3⋊C8)C2211yescharacteristic
19C3×C4×C3⋊C8C2211yescharacteristic
20#63573—21no—
21#63573—21no—
22#63573—21no—
23#63563—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#461—31no—
26#5080—31no—
27#5012—31no—
28#5008—31no—
29#4842—31no—
30#4841D311yescharacteristic
31#4688—31no—
32#4680—31no—
33C3×#335—31no—
34#4547—31no—
35#4482—31no—
36#4481—31no—
37#4481—31no—
38#4481—61no—
39#4479D311yescharacteristic, complemented
40C2×#634—21no—
41C2×C3×D12D411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C3×Dic6—21no—
44C2×C32×C8—21no—
45C32×C42C2311yescharacteristic
46C4⋊(C3⋊Dic3)—21no—
47C4⋊(C3⋊Dic3)—21no—
48C2×C3⋊(C3⋊C8)—21no—
49C3×(C2×D3)⋊C4—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3D411yescharacteristic
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56C3×#204—42no—
57#3711—42no—
58#1691—91noSylow
59C2×C3×SD16—61no—
60C3×C4⋊Q8—31no—
61C3×C4×D4—31no—
62C2×#208—61no—
63C2×#204—21no—
64C2×ΓL(1,𝔽25)—61no—
65C4×D12C2×D311yescharacteristic
66#869—31no—
67#869—31no—
68#869—61no—
69#869C2×D311yescharacteristic
70C3×C4⋊C8—21no—
71C3×C4⋊C8—31no—
72C3×C4⋊C8C2×D311yescharacteristic
73C3×C4⋊C8C2×D311yescharacteristic
74C3×#103—61no—
75C3×C4×C8—31no—
76#832—61no—
77#830—61no—
78#821—61no—
79(C2×D3)⋊C8—61no—
80#817—61no—
81#817—61no—
82#817—61no—
83#816—21no—
84#808—61no—
85#808—61no—
86#808—61no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4×C3⋊C8C2×D311yescharacteristic
92C22×C3×D3—21no—
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4#4111yescharacteristic
96C2×C3⋊Dic3—41no—
97#634—81no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—41no—
100C3×D12—21no—
101C3×D12—21no—
102C3×C4×D3—41no—
103C3×C4×D3—41no—
104C3×Dic6—41no—
105C3×Dic6—41no—
106C32×C8—41no—
107C3⋊(C3⋊C8)—41no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—41no—
110C3×C3⋊C8SD1612yescomplemented
111#498—91no—
112#476—91no—
113#461—31no—
114#436—91no—
115#335—31no—
116#333—91no—
117#332—91no—
118C2×C3×Q8—61no—
119C2×C3×D4—31no—
120C22×C3×C4—61no—
121C2×D12#6711yescharacteristic
122C2×C4×D3—21no—
123C2×Dic6—21no—
124C2×Dic6—62no—
125C2×Dic6—61no—
126C2×Dic6—61no—
127C3×SD16—122no—
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8—31no—
131C2×C3×C8—31no—
132C2×C3×C8—41no—
133C2×C3×C8—61no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—61no—
137C3×C22⋊C4—61no—
138C3×C42—21no—
139C3×C42C22×D311yescharacteristic
140C3×C42C22×D311yescharacteristic
141(C2×D3)⋊C4—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—61no—
145C4⋊Dic3—61no—
146C4⋊Dic3—61no—
147C4⋊Dic3—61no—
148C4⋊Dic3—121no—
149C4⋊Dic3—121no—
150C4⋊Dic3#6711yescharacteristic
151#208—122no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—61no—
155C2×C3⋊C8—121no—
156C2×C3⋊C8D1211yescharacteristic
157C2×C3⋊C8D1211yescharacteristic
158#204—42no—
159ΓL(1,𝔽25)—122no—
160C22×C32#12111yescharacteristic
161C2×C3×D3—41no—
162C2×C3×D3—41no—
163C2×C3×D3—41no—
164C32×C4—21no—
165C32×C4C2×SD1611yescharacteristic
166C32×C4AGL(1,ℤ/8)11yescharacteristic
167C32×C4#13711yescharacteristic
168C32×C4#13711yescharacteristic
169C3⋊Dic3—81no—
170C3×Dic3—41no—
171C3×Dic3—81no—
172C2×SD16—61no—
173C2×M16—181no—
174C4⋊Q8—31no—
175C4⋊Q8—91no—
176C4×D4—31no—
177#106—181no—
178#106—181no—
179#106—181no—
180C4⋊C8—31no—
181C4⋊C8—91no—
182C4⋊C8D3211yescharacteristic
183#103—61no—
184#102—181no—
185C22⋊C8—181no—
186C4×C8—31no—
187C23×C3—61no—
188C22×D3—21no—
189C3×Q8—121no—
190C3×Q8—121no—
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×C4C2×D1211yescharacteristic
201C2×C3×C4D3×D411yescharacteristic
202C2×C3×C4D3×D411yescharacteristic
203C2×C3×C4#23711yescharacteristic
204C2×C3×C4#23911yescharacteristic
205C2×C3×C4C2×#6711yescharacteristic
206C2×Dic3—21no—
207C2×Dic3—41no—
208C2×Dic3—122no—
209C2×Dic3—121no—
210C2×Dic3—121no—
211D12—21no—
212D12—21no—
213C4×D3—41no—
214C4×D3—41no—
215Dic6—41no—
216Dic6—41no—
217Dic6—241no—
218Dic6—241no—
219Dic6—242no—
220C3×C8—32no—
221C3×C8—41no—
222C3×C8—41no—
223C3×C8—61no—
224C3×C8—81no—
225C3×C8—121no—
226C3⋊C8—21no—
227C3⋊C8—41no—
228C3⋊C8—121no—
229C3⋊C8—121no—
230C3⋊C8—241no—
231C3⋊C8#20412yescomplemented
232C2×C32#46111yescharacteristic
233C2×C32#46811yescharacteristic
234C2×C32#47111yescharacteristic
235C3×D3—81no—
236C2×Q8—61no—
237C2×Q8—181no—
238C2×D4—31no—
239C22×C4—61no—
240SD16—122no—
241M16—362no—
242C2×C8—21no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8—61no—
246C2×C8—181no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
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#89511yescharacteristic
259C22×C3#93911yescharacteristic
260C2×D3—41no—
261C2×D3—41no—
262C2×D3—41no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—41no—
270C3×C4—121no—
271C3×C4—241no—
272C3×C4C2×#20411yescharacteristic
273C3×C4#90911yescharacteristic
274C3×C4#91111yescharacteristic
275C3×C4D3×SD1611yescharacteristic
276C3×C4#91511yescharacteristic
277C3×C4#91811yescharacteristic
278C3×C4#93211yescharacteristic
279C3×C4#95111yescharacteristic
280Dic3—41no—
281Dic3—81no—
282Dic3—242no—
283Dic3—241no—
284Dic3—241no—
285C32#169111yescharacteristic, complemented, Sylow
286C23—61no—
287Q8—121no—
288Q8—121no—
289Q8—721no—
290D4—61no—
291D4—61no—
292C2×C4—61no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—361no—
297C2×C4#379611yescharacteristic
298C2×C4D3×D1211yescharacteristic
299C2×C4D3×#6711yescharacteristic
300C8—32no—
301C8—41no—
302C8—61no—
303C8—121no—
304C8—361no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—241no—
309C2×C3#484111yescharacteristic
310C2×C3#488611yescharacteristic
311C2×C3#489811yescharacteristic
312C2×C3#504211yescharacteristic
313C2×C3#516211yescharacteristic
314C2×C3#520311yescharacteristic
315D3—81no—
316C22—121no—
317C22—121no—
318C22—121no—
319C22#6388611yescharacteristic
320C4—21no—
321C4—121no—
322C4—241no—
323C4—721no—
324C4D3×#20411yescharacteristic
325C4#6380011yescharacteristic
326C4#6393211yescharacteristic
327C4#6394311yescharacteristic
328C3—21no—
329C3#6858311yescharacteristic, complemented
330C3#7104011yescharacteristic, complemented
331C2—241no—
332C2#1059058811yescharacteristic
333C2#1059075211yescharacteristic
334C2#1059098511yescharacteristic
subgroupG/H
centerC22#63886
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
3. #10590303C2semidirect
5. #10589347C2semidirect
6. C3×#4479C2semidirect
7. #10589093C2semidirect
39. #4479D3semidirect
110. C3×C3⋊C8SD16semidirect
231. C3⋊C8#204semidirect
285. C32#1691semidirect
329. C3#68583semidirect
330. C3#71040semidirect

Maximal subgroups

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

subgroupindex
C3×#48412
#105903372
#105903032
#105902682
#105893472
C3×#44792
#105890932
#710403
#685833