Finite Group Databasebeta

← #11781856

Subgroups — #11781856

Order 1152.

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

normal subgroups only

This group has 347 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
1C3×#4841C211yesmaximal, characteristic
2#10590339C211yesmaximal, characteristic
3#10590309C211yesmaximal, characteristic, complemented
4#10590266C211yesmaximal, characteristic
5#10589345C211yesmaximal, characteristic, complemented
6C3×#4481C211yesmaximal, characteristic, complemented
7#10589092C211yesmaximal, characteristic, complemented
8#71040—31nomaximal
9#68500—31nomaximal
10C2×C3×#204—21no—
11C3×C4×D12C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C2×#3711—21no—
15C32×C4⋊C8C2211yescharacteristic
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C4×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×#816—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20#63574—21no—
21#63574—21no—
22#63574—21no—
23#63562—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#461—31no—
26#5084—31no—
27#5080—31no—
28#5080—31no—
29#5012—31no—
30#5008—31no—
31#4841D311yescharacteristic
32#4838—31no—
33C3×#333—31no—
34#4547—31no—
35#4486—31no—
36#4481D311yescharacteristic, complemented
37#4479—31no—
38#4479—31no—
39#4479—61no—
40C2×C3×D12D411yescharacteristic
41C2×C3×C4×D3—21no—
42C2×C3×Dic6—21no—
43C2×C3×Dic6—21no—
44C2×C32×C8—21no—
45C32×C42C2311yescharacteristic
46C2×C3⋊(C3⋊C8)—21no—
47C2×C3⋊(C3⋊C8)D411yescharacteristic
48C2×C3⋊(C3⋊C8)D411yescharacteristic
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×#204—42no—
57#3711—42no—
58#1691—91noSylow
59C2×C3×SD16—61no—
60C3×C4⋊Q8—31no—
61C3×C4⋊Q8—31no—
62C3×C4×D4—31no—
63C2×#208—61no—
64C2×#204—21no—
65C2×ΓL(1,𝔽25)—61no—
66C4×D12C2×D311yescharacteristic
67#869C2×D311yescharacteristic
68#869C2×D311yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8—31no—
71C3×C4⋊C8C2×D311yescharacteristic
72C3×C4⋊C8C2×D311yescharacteristic
73C3×#103—61no—
74#832—61no—
75#830—61no—
76(C2×D3)⋊C8—61no—
77#816—21no—
78#809—61no—
79#808—61no—
80#808—61no—
81#808—61no—
82#808—61no—
83#808—61no—
84#808—61no—
85C4⋊(C3⋊C8)—31no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—61no—
88C4⋊(C3⋊C8)C2×D311yescharacteristic
89C4×C3⋊C8—31no—
90C4×C3⋊C8—31no—
91C4×C3⋊C8—61no—
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—41no—
107C3⋊(C3⋊C8)—21no—
108C3⋊(C3⋊C8)—41no—
109C3⋊(C3⋊C8)SD1612yescomplemented
110C3×C3⋊C8—41no—
111#498—91no—
112#476—91no—
113#461—31no—
114#436—91no—
115#335—91no—
116#333—31no—
117#332—91no—
118C2×C3×Q8—61no—
119C2×C3×Q8—61no—
120C2×C3×D4—31no—
121C22×C3×C4—61no—
122C2×D12#6711yescharacteristic
123C2×C4×D3—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C3×SD16—122no—
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C2×C3×C8—41no—
130C2×C3×C8—61no—
131C3×C4⋊C4—31no—
132C3×C4⋊C4—61no—
133C3×C4⋊C4—61no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—61no—
136C3×C22⋊C4—61no—
137C3×C42—21no—
138C3×C42C22×D311yescharacteristic
139C3×C42C22×D311yescharacteristic
140(C2×D3)⋊C4—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3#6711yescharacteristic
146#208—122no—
147C2×C3⋊C8—21no—
148C2×C3⋊C8—31no—
149C2×C3⋊C8—31no—
150C2×C3⋊C8—31no—
151C2×C3⋊C8—31no—
152C2×C3⋊C8—61no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—61no—
155C2×C3⋊C8—61no—
156C2×C3⋊C8—121no—
157#204—42no—
158ΓL(1,𝔽25)—122no—
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—61no—
172C2×M16—181no—
173C4⋊Q8—31no—
174C4⋊Q8—31no—
175C4×D4—31no—
176#106—181no—
177#106—181no—
178#106—181no—
179C4⋊C8—31no—
180C4⋊C8—91no—
181C4⋊C8D3211yescharacteristic
182#103—61no—
183#102—181no—
184C22⋊C8—181no—
185C4×C8—91no—
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×C4#23511yescharacteristic
202C2×C3×C4D3×D411yescharacteristic
203C2×C3×C4D3×D411yescharacteristic
204C2×C3×C4D3×D411yescharacteristic
205C2×C3×C4#23711yescharacteristic
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—41no—
218C3×C8—41no—
219C3×C8—81no—
220C3×C8—121no—
221C3⋊C8—32no—
222C3⋊C8—32no—
223C3⋊C8—41no—
224C3⋊C8—62no—
225C3⋊C8—61no—
226C3⋊C8—61no—
227C3⋊C8—121no—
228C3⋊C8—121no—
229C3⋊C8—121no—
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—122no—
240M16—362no—
241C2×C8—21no—
242C2×C8—61no—
243C2×C8—91no—
244C2×C8—91no—
245C2×C8—181no—
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#88411yescharacteristic
258C22×C3#93911yescharacteristic
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×C4D3×SD1611yescharacteristic
273C3×C4D3×SD1611yescharacteristic
274C3×C4#91411yescharacteristic
275C3×C4#91511yescharacteristic
276C3×C4#91811yescharacteristic
277C3×C4#91811yescharacteristic
278C3×C4#93211yescharacteristic
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×C4#380211yescharacteristic
297C2×C4D3×#6711yescharacteristic
298C8—41no—
299C8—92no—
300C8—121no—
301C8—181no—
302C8—361no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—241no—
307C2×C3#474811yescharacteristic
308C2×C3#481211yescharacteristic
309C2×C3#481811yescharacteristic
310C2×C3#504211yescharacteristic
311C2×C3#516211yescharacteristic
312C2×C3#520311yescharacteristic
313D3—81no—
314C22—121no—
315C22—121no—
316C22—121no—
317C22#6388511yescharacteristic
318C4—21no—
319C4—121no—
320C4—241no—
321C4—241no—
322C4#6379611yescharacteristic
323C4#6381111yescharacteristic
324C4#6393411yescharacteristic
325C4#6394311yescharacteristic
326C3—21no—
327C3#6850011yescharacteristic, complemented
328C3#7104011yescharacteristic, complemented
329C2—241no—
330C2#1059051511yescharacteristic
331C2#1059074811yescharacteristic
332C2#1059097911yescharacteristic
subgroupG/H
centerC22#63885
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. #10590309C2semidirect
5. #10589345C2semidirect
6. C3×#4481C2semidirect
7. #10589092C2semidirect
36. #4481D3semidirect
109. C3⋊(C3⋊C8)SD16semidirect
283. C32#1691semidirect
327. C3#68500semidirect
328. C3#71040semidirect

Maximal subgroups

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

subgroupindex
C3×#48412
#105903392
#105903092
#105902662
#105893452
C3×#44812
#105890922
#710403
#685003