Finite Group Databasebeta

← #11799420

Subgroups — #11799420

Order 1152.

330 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 68 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#10592891C211yesmaximal, characteristic
2C2×C3×#829C211yesmaximal, characteristic, complemented
3#10590933C211yesmaximal, characteristic
4#10590624C211yesmaximal, characteristic
5C3×#4565C211yesmaximal, characteristic, complemented
6#10589143C211yesmaximal, characteristic, complemented
7#10589130C211yesmaximal, characteristic, complemented
8#71507—31nomaximal
9#69270—31nomaximal
10C2×C32×Q16C2211yescharacteristic
11#64151—21no—
12#64110C2211yescharacteristic
13C2×C3×Dic12C2211yescharacteristic
14#63833—21no—
15#63663—21no—
16#63646C2211yescharacteristic
17C3×#829—21no—
18C3×#829—21no—
19C3×#819C2211yescharacteristic
20C3×#812C2211yescharacteristic
21C2×C3×C3⋊C16C2211yescharacteristic
22#63560—21no—
23#63544—21no—
24#63542—21no—
25C2×C3×Q32—31no—
26#5192—31no—
27#5192—31no—
28#5192—31no—
29#5192—61no—
30C2×#829D311yescharacteristic, complemented
31#4907—31no—
32#4901—31no—
33#4896—31no—
34C3×#362—31no—
35#4565—31no—
36#4565D311yescharacteristic, complemented
37#4517—31no—
38#4506—31no—
39#4493—31no—
40C2×C32×Q8—21no—
41#3828D411yescharacteristic
42C2×C3×Dic6—21no—
43C32×Q16—21no—
44C32×Q16—21no—
45C2×C32×C8C2311yescharacteristic
46#3756—41no—
47#3755—41no—
48#3748—21no—
49#3748D411yescharacteristic
50#3743—21no—
51C8×C3⋊D3—21no—
52C3×#208—21no—
53C3×Dic12—21no—
54C3×Dic12—21no—
55#3719—41no—
56#3717—41no—
57C3×C3⋊C16—21no—
58C3×C3⋊C16D411yescharacteristic
59C3×C3⋊C16D411yescharacteristic
60#2222—91noSylow
61C2×C3×Q16—21no—
62C2×C3×Q16—31no—
63C2×C3×Q16C2×D311yescharacteristic
64C2×C3×Q16C2×D311yescharacteristic
65#942—61no—
66#942—61no—
67#942—61no—
68#942—62no—
69#909—61no—
70C2×Dic12C2×D311yescharacteristic
71#901—31no—
72#901—31no—
73#901—61no—
74C3×Q32—61no—
75C3×Q32—61no—
76C3×M32—31no—
77C2×C3×C16—31no—
78C3×#108—31no—
79#836—61no—
80#836—61no—
81#836—61no—
82#836—62no—
83#829—21no—
84#829—21no—
85#824—61no—
86#819—31no—
87#819—31no—
88#819—61no—
89#819C2×D311yescharacteristic
90#812C2×D311yescharacteristic
91C2×C3⋊C16C2×D311yescharacteristic
92#798—61no—
93#798—61no—
94D3×C16—61no—
95D3×C16—61no—
96C32×Q8—41no—
97C32×Q8—41no—
98C2×C32×C4—41no—
99C2×C32×C4C2×D411yescharacteristic
100#638—21no—
101C4⋊(C3⋊D3)D811yescharacteristic
102C4×C3⋊D3—21no—
103#634D811yescharacteristic
104C2×C3×Dic3—41no—
105C3×Dic6—41no—
106C3×Dic6—41no—
107C32×C8C2×D411yescharacteristic
108C32×C8#4111yescharacteristic
109C3⋊(C3⋊C8)—21no—
110C3⋊(C3⋊C8)—41no—
111C3×C3⋊C8—41no—
112C2×Q32—31no—
113#504—91no—
114#473—91no—
115#473—91no—
116#368—91no—
117#362—31no—
118#362—91no—
119C2×C3×Q8—22no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—61no—
123#235—31no—
124#235—31no—
125#235—61no—
126C2×Dic6—21no—
127C3×Q16—21no—
128C3×Q16—21no—
129C3×Q16—21no—
130C3×Q16—21no—
131C3×Q16—41no—
132C3×Q16—41no—
133C3×Q16—61no—
134C3×Q16—61no—
135C3×M16—61no—
136C2×C3×C8—21no—
137C2×C3×C8C22×D311yescharacteristic
138C2×C3×C8C22×D311yescharacteristic
139#216—121no—
140#216—121no—
141#216—121no—
142#216—122no—
143#215—121no—
144#215—121no—
145#215—121no—
146#215—122no—
147#208—21no—
148#208—31no—
149#208—31no—
150#208—61no—
151#208—62no—
152#208—61no—
153#208—61no—
154Dic12—21no—
155Dic12—21no—
156Dic12—121no—
157#204—121no—
158ΓL(1,𝔽25)—61no—
159ΓL(1,𝔽25)—61no—
160ΓL(1,𝔽25)—121no—
161D3×C8—61no—
162D3×C8—61no—
163D3×C8—121no—
164C3×C16—31no—
165C3×C16—31no—
166C3×C16—61no—
167C3⋊C16—21no—
168C3⋊C16#6711yescharacteristic
169C3⋊C16#6711yescharacteristic
170C22×C32#13511yescharacteristic
171C2×C3⋊D3—21no—
172C32×C4—81no—
173C32×C4#12111yescharacteristic
174C32×C4C2×D811yescharacteristic
175C3⋊Dic3—21no—
176C3×Dic3—81no—
177#137—181no—
178#137—181no—
179C2×Q16—31no—
180C2×Q16D3211yescharacteristic
181#131—91no—
182Q32—61no—
183Q32—61no—
184M32—31no—
185M32—181no—
186C2×C16—31no—
187C2×C16—181no—
188#108—31no—
189#108—91no—
190#101—181no—
191#101—181no—
192C3×Q8—42no—
193C3×Q8—42no—
194C3×Q8—41no—
195C3×Q8—41no—
196C3×Q8—41no—
197C3×Q8—41no—
198C3×Q8—121no—
199C3×Q8—121no—
200C2×C3×C4—21no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—121no—
205C2×C3×C4D3×D411yescharacteristic
206C2×C3×C4D3×D411yescharacteristic
207#67—61no—
208#67—61no—
209#67—121no—
210C2×Dic3—41no—
211D12—31no—
212D12—31no—
213D12—61no—
214C4×D3—61no—
215C4×D3—61no—
216C4×D3—121no—
217Dic6—31no—
218Dic6—31no—
219Dic6—41no—
220Dic6—41no—
221Dic6—61no—
222C3×C8—21no—
223C3×C8—21no—
224C3×C8—121no—
225C3×C8#23511yescharacteristic
226C3×C8D3×D411yescharacteristic
227C3×C8#23711yescharacteristic
228C3×C8C2×#6711yescharacteristic
229C3⋊C8—41no—
230C3⋊C8—61no—
231C3⋊C8—61no—
232C3⋊C8—121no—
233C3⋊C8—122no—
234C3⋊C8—121no—
235C3⋊C8—121no—
236C2×C32—21no—
237C2×C32#46611yescharacteristic
238C3⋊D3—41no—
239#41—91no—
240C2×Q8—21no—
241C2×Q8—61no—
242Q16—21no—
243Q16—21no—
244Q16—61no—
245Q16—61no—
246Q16—361no—
247Q16—361no—
248SD16—361no—
249SD16—361no—
250M16—61no—
251M16—91no—
252M16—181no—
253M16—181no—
254C2×C8—181no—
255C2×C8C2×D3211yescharacteristic
256C16—31no—
257C16—31no—
258C16—61no—
259C22×C3—21no—
260C22×C3#91211yescharacteristic
261C22×C3#91611yescharacteristic
262C2×D3—61no—
263C2×D3—61no—
264C2×D3—121no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—82no—
268C3×C4—81no—
269C3×C4—81no—
270C3×C4—241no—
271C3×C4#88411yescharacteristic
272C3×C4D3×D811yescharacteristic
273C3×C4D3×D811yescharacteristic
274C3×C4#93811yescharacteristic
275Dic3—61no—
276Dic3—61no—
277Dic3—81no—
278Dic3—121no—
279C32#222211yescharacteristic, complemented, Sylow
280Q8—41no—
281Q8—41no—
282Q8—91no—
283Q8—121no—
284Q8—121no—
285D4—91no—
286D4—181no—
287C2×C4—41no—
288C2×C4—121no—
289C2×C4—181no—
290C2×C4#381111yescharacteristic
291C8—121no—
292C8—181no—
293C8—361no—
294C8#380411yescharacteristic
295C8D3×#6711yescharacteristic
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—41no—
300C2×C3#477511yescharacteristic
301C2×C3#515711yescharacteristic
302D3—121no—
303D3—121no—
304D3—241no—
305C22—181no—
306C22#6393611yescharacteristic
307C4—81no—
308C4—181no—
309C4—241no—
310C4#6392611yescharacteristic
311C4#6395811yescharacteristic
312C3—21no—
313C3#6927011yescharacteristic, complemented
314C3#7150711yescharacteristic, complemented
315C2—21no—
316C2—361no—
317C2#1059145511yescharacteristic
subgroupG/H
centerC2#10591455
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×Q16C22
socleC2×C32#466
prime radical#3828D4
hypercenterC2×Q16D32
hypocenterC32#2222
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#829C2semidirect
5. C3×#4565C2semidirect
6. #10589143C2semidirect
7. #10589130C2semidirect
30. C2×#829D3semidirect
36. #4565D3semidirect
279. C32#2222semidirect
313. C3#69270semidirect
314. C3#71507semidirect

Maximal subgroups

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

subgroupindex
#105928912
C2×C3×#8292
#105909332
#105906242
C3×#45652
#105891432
#105891302
#715073
#692703