Finite Group Databasebeta

← #11799421

Subgroups — #11799421

Order 1152.

307 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 72 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×#5192C211yesmaximal, characteristic
2C2×C3×#829C211yesmaximal, characteristic, complemented
3#10590930C211yesmaximal, characteristic
4#10590621C211yesmaximal, characteristic
5#10589431C211yesmaximal, characteristic, complemented
6#10589143C211yesmaximal, characteristic, complemented
7#10589129C211yesmaximal, characteristic, complemented
8#71507—31nomaximal
9#71096—31nomaximal
10C2×C32×Q16C2211yescharacteristic
11C3×#942—21no—
12C2×C3×Dic12C2211yescharacteristic
13C3×#901C2211yescharacteristic
14#63827—21no—
15#63646C2211yescharacteristic
16#63639C2211yescharacteristic
17C3×#836—21no—
18C3×#829—21no—
19C3×#829—21no—
20C3×#819C2211yescharacteristic
21C2×C3×C3⋊C16C2211yescharacteristic
22#63560—21no—
23#63543—21no—
24D3×C3⋊C16—21no—
25C2×C3×Q32—31no—
26C3×#473—31no—
27#5192—31no—
28#5192D311yescharacteristic
29C2×#829D311yescharacteristic, complemented
30#5143—31no—
31#5140—31no—
32#4901—31no—
33#4565—31no—
34#4565—31no—
35#4565—31no—
36#4565—61no—
37#4506—31no—
38#4496—31no—
39#4493—31no—
40C2×C32×Q8—21no—
41C3×#235D411yescharacteristic
42C2×C3×Dic6—21no—
43C32×Q16—21no—
44C32×Q16—21no—
45C2×C32×C8C2311yescharacteristic
46#3748—21no—
47C3×#216—41no—
48C3×#215—41no—
49C3×#208—21no—
50C3×#208D411yescharacteristic
51C3×Dic12—21no—
52C3×Dic12—21no—
53C3×ΓL(1,𝔽25)—21no—
54C3×D3×C8—21no—
55#3718—41no—
56#3715—41no—
57C3⋊(C3⋊C16)—21no—
58C3×C3⋊C16D411yescharacteristic
59C3×C3⋊C16D411yescharacteristic
60#2222—91noSylow
61C3×#137—61no—
62C2×C3×Q16—21no—
63C2×C3×Q16—31no—
64C2×C3×Q16C2×D311yescharacteristic
65C2×C3×Q16C2×D311yescharacteristic
66C3×#131—31no—
67#951—61no—
68#942—21no—
69#942—61no—
70C2×Dic12C2×D311yescharacteristic
71#901C2×D311yescharacteristic
72C3×Q32—61no—
73C3×Q32—61no—
74C2×C3×C16—31no—
75C3×#108—31no—
76C3×#101—61no—
77#836—21no—
78#836—61no—
79#829—21no—
80#829—21no—
81#824—61no—
82#819—31no—
83#819—31no—
84#819—61no—
85#819C2×D311yescharacteristic
86#812—31no—
87#812—31no—
88#812—61no—
89#812—61no—
90C2×C3⋊C16—61no—
91C2×C3⋊C16C2×D311yescharacteristic
92#798—61no—
93D3×C16—61no—
94C32×Q8—41no—
95C32×Q8—41no—
96C2×C32×C4—41no—
97C2×C32×C4C2×D411yescharacteristic
98C3×#67—21no—
99C2×C3×Dic3—41no—
100C3×D12D811yescharacteristic
101C3×C4×D3—21no—
102C3×Dic6—41no—
103C3×Dic6—41no—
104C3×Dic6D811yescharacteristic
105C32×C8C2×D411yescharacteristic
106C32×C8#4111yescharacteristic
107C3⋊(C3⋊C8)—41no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—41no—
110C2×Q32—31no—
111#504—91no—
112#473—31no—
113#473—91no—
114#368—91no—
115#362—91no—
116#362—91no—
117C3×#41—31no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—41no—
121C2×C3×Q8—61no—
122#235#6711yescharacteristic
123C2×Dic6—21no—
124C3×Q16—22no—
125C3×Q16—21no—
126C3×Q16—21no—
127C3×Q16—21no—
128C3×Q16—21no—
129C3×Q16—41no—
130C3×Q16—61no—
131C3×Q16—61no—
132C3×Q16—121no—
133C3×SD16—121no—
134C3×M16—31no—
135C3×M16—61no—
136C3×M16—61no—
137C2×C3×C8—21no—
138C2×C3×C8—61no—
139C2×C3×C8C22×D311yescharacteristic
140C2×C3×C8C22×D311yescharacteristic
141#216—41no—
142#216—121no—
143#216—121no—
144#215—41no—
145#215—121no—
146#214—121no—
147#208—21no—
148#208—61no—
149#208—61no—
150#208—121no—
151#208#6711yescharacteristic
152Dic12—21no—
153Dic12—21no—
154ΓL(1,𝔽25)—21no—
155D3×C8—21no—
156C3×C16—31no—
157C3×C16—31no—
158C3⋊C16—61no—
159C3⋊C16—61no—
160C3⋊C16—121no—
161C3⋊C16#6711yescharacteristic
162C3⋊C16#6711yescharacteristic
163C22×C32#13511yescharacteristic
164C2×C3×D3—21no—
165C32×C4—81no—
166C32×C4#12111yescharacteristic
167C32×C4C2×D811yescharacteristic
168C3×Dic3—21no—
169C3×Dic3—81no—
170#137—61no—
171#137—181no—
172C2×Q16—31no—
173C2×Q16D3211yescharacteristic
174#131—31no—
175Q32—61no—
176Q32—61no—
177M32—91no—
178M32—181no—
179C2×C16—31no—
180C2×C16—181no—
181#108—31no—
182#108—91no—
183#101—61no—
184#101—181no—
185C3×Q8—31no—
186C3×Q8—42no—
187C3×Q8—41no—
188C3×Q8—41no—
189C3×Q8—41no—
190C3×Q8—41no—
191C3×Q8—81no—
192C3×Q8—121no—
193C3×Q8—121no—
194C3×D4—31no—
195C3×D4—61no—
196C2×C3×C4—21no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—61no—
200C2×C3×C4—81no—
201C2×C3×C4—121no—
202C2×C3×C4D3×D411yescharacteristic
203C2×C3×C4C2×#6711yescharacteristic
204#67—21no—
205C2×Dic3—41no—
206D12#21311yescharacteristic
207C4×D3—21no—
208Dic6—41no—
209Dic6—41no—
210Dic6#21311yescharacteristic
211C3×C8—21no—
212C3×C8—21no—
213C3×C8—61no—
214C3×C8—121no—
215C3×C8D3×D411yescharacteristic
216C3×C8#23711yescharacteristic
217C3×C8#23711yescharacteristic
218C3×C8C2×#6711yescharacteristic
219C3⋊C8—21no—
220C3⋊C8—41no—
221C3⋊C8—121no—
222C3⋊C8—121no—
223C3⋊C8—241no—
224C2×C32—21no—
225C2×C32#46611yescharacteristic
226C3×D3—41no—
227#41—31no—
228C2×Q8—21no—
229C2×Q8—61no—
230Q16—21no—
231Q16—21no—
232Q16—61no—
233Q16—61no—
234Q16—121no—
235Q16—361no—
236SD16—121no—
237SD16—361no—
238M16—31no—
239M16—61no—
240M16—61no—
241M16—181no—
242C2×C8—61no—
243C2×C8C2×D3211yescharacteristic
244C16—31no—
245C16—31no—
246C16—181no—
247C22×C3—21no—
248C22×C3—61no—
249C22×C3#91211yescharacteristic
250C22×C3#95011yescharacteristic
251C2×D3—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—61no—
255C3×C4—82no—
256C3×C4—81no—
257C3×C4—81no—
258C3×C4—241no—
259C3×C4D3×D811yescharacteristic
260C3×C4C2×#21311yescharacteristic
261C3×C4#93811yescharacteristic
262C3×C4#93911yescharacteristic
263Dic3—21no—
264Dic3—81no—
265C32#222211yescharacteristic, complemented, Sylow
266Q8—31no—
267Q8—41no—
268Q8—41no—
269Q8—121no—
270Q8—121no—
271D4—31no—
272D4—61no—
273C2×C4—41no—
274C2×C4—61no—
275C2×C4—121no—
276C2×C4D3×#6711yescharacteristic
277C8—61no—
278C8—121no—
279C8—361no—
280C8#380511yescharacteristic
281C8D3×#6711yescharacteristic
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—41no—
286C2×C3—121no—
287C2×C3#503811yescharacteristic
288C2×C3#515711yescharacteristic
289D3—41no—
290C22—61no—
291C22#6393311yescharacteristic
292C4—61no—
293C4—81no—
294C4—241no—
295C4D3×#21311yescharacteristic
296C4#6397011yescharacteristic
297C3—21no—
298C3#7109611yescharacteristic, complemented
299C3#7150711yescharacteristic, complemented
300C2—21no—
301C2—121no—
302C2#1059149811yescharacteristic
subgroupG/H
centerC2#10591498
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×Q16C22
socleC2×C32#466
prime radicalC3×#235D4
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. #10589431C2semidirect
6. #10589143C2semidirect
7. #10589129C2semidirect
29. C2×#829D3semidirect
265. C32#2222semidirect
298. C3#71096semidirect
299. C3#71507semidirect

Maximal subgroups

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

subgroupindex
C3×#51922
C2×C3×#8292
#105909302
#105906212
#105894312
#105891432
#105891292
#715073
#710963