Finite Group Databasebeta

← #11799468

Subgroups — #11799468

Order 1152.

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

normal subgroups only

This group has 315 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, complemented
2C2×C3×#828C211yesmaximal, characteristic, complemented
3#10590926C211yesmaximal, characteristic
4#10590621C211yesmaximal, characteristic, complemented
5#10589431C211yesmaximal, characteristic, complemented
6#10589143C211yesmaximal, characteristic, complemented
7#10589126C211yesmaximal, characteristic, complemented
8#71491—31nomaximal
9#71097—31nomaximal
10C2×C32×Q16C2211yescharacteristic
11C3×#942—21no—
12C2×C3×D24C2211yescharacteristic
13C3×#901C2211yescharacteristic
14#63823—21no—
15#63646C2211yescharacteristic
16#63639C2211yescharacteristic
17C3×#836—21no—
18C3×#828—21no—
19C3×#828—21no—
20C3×#819C2211yescharacteristic
21C2×C3×C3⋊C16C2211yescharacteristic
22#63558—21no—
23#63543—21no—
24D3×C3⋊C16—21no—
25C2×C3×SD32—31no—
26C3×#473—31no—
27#5192D311yescharacteristic, complemented
28C2×#828D311yescharacteristic, complemented
29#5160—31no—
30#5142—31no—
31#5140—31no—
32#4901—31no—
33#4565—31no—
34#4565—31no—
35#4565—61no—
36#4563—31no—
37#4506—31no—
38#4495—31no—
39#4493—31no—
40C2×C32×Q8—21no—
41C3×#235D411yescharacteristic
42C2×C3×D12—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×D24—21no—
52C3×D24—21no—
53C3×ΓL(1,𝔽25)—21no—
54C3×D3×C8—21no—
55#3715—41no—
56#3713—41no—
57C3⋊(C3⋊C16)—21no—
58C3×C3⋊C16D411yescharacteristic
59C3×C3⋊C16D411yescharacteristic
60#2223—91noSylow
61C3×#137—61no—
62C2×C3×Q16—21no—
63C2×C3×Q16C2×D311yescharacteristic
64C2×C3×Q16C2×D311yescharacteristic
65C2×C3×D8—31no—
66C3×#131—31no—
67#949—61no—
68#942—21no—
69#932—61no—
70C2×D24C2×D311yescharacteristic
71#901C2×D311yescharacteristic
72C3×SD32—61no—
73C3×SD32—61no—
74C2×C3×C16—31no—
75C3×#108—31no—
76C3×#101—61no—
77#836—21no—
78#833—61no—
79#828—21no—
80#828—21no—
81#823—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—
94C22×C3×D3—41no—
95C32×Q8—41no—
96C32×Q8—41no—
97C2×C32×C4—41no—
98C2×C32×C4C2×D411yescharacteristic
99C3×#67—21no—
100C3×D12—41no—
101C3×D12—41no—
102C3×D12D811yescharacteristic
103C3×C4×D3—21no—
104C3×Dic6D811yescharacteristic
105C32×C8C2×D411yescharacteristic
106C32×C8#4111yescharacteristic
107C3⋊(C3⋊C8)—41no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—41no—
110C2×SD32—31no—
111#504—91no—
112#473—31no—
113#472—91no—
114#368—91no—
115#362—91no—
116#361—91no—
117C3×#41—31no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—41no—
121C2×C3×D4—61no—
122#235#6711yescharacteristic
123C2×D12—21no—
124C3×Q16—22no—
125C3×Q16—21no—
126C3×Q16—21no—
127C3×Q16—21no—
128C3×Q16—21no—
129C3×Q16—41no—
130C3×Q16—121no—
131C3×SD16—121no—
132C3×D8—61no—
133C3×D8—61no—
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#215—41no—
143#215—121no—
144#214—121no—
145#213—121no—
146#213—121no—
147#208—21no—
148#208—61no—
149#208—61no—
150#208—121no—
151#208#6711yescharacteristic
152D24—21no—
153D24—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—
165C2×C3×D3—81no—
166C2×C3×D3—161no—
167C32×C4—81no—
168C32×C4#12111yescharacteristic
169C32×C4C2×D811yescharacteristic
170C3×Dic3—21no—
171#137—61no—
172AGL(1,ℤ/8)—181no—
173C2×Q16D3211yescharacteristic
174C2×D8—31no—
175#131—31no—
176SD32—61no—
177SD32—61no—
178M32—91no—
179M32—181no—
180C2×C16—31no—
181C2×C16—181no—
182#108—31no—
183#108—91no—
184#101—61no—
185#100—181no—
186C23×C3—121no—
187C22×D3—41no—
188C3×Q8—31no—
189C3×Q8—42no—
190C3×Q8—41no—
191C3×Q8—41no—
192C3×Q8—41no—
193C3×Q8—41no—
194C3×Q8—81no—
195C3×D4—31no—
196C3×D4—61no—
197C3×D4—121no—
198C3×D4—121no—
199C2×C3×C4—21no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—61no—
203C2×C3×C4—81no—
204C2×C3×C4D3×D411yescharacteristic
205C2×C3×C4C2×#6711yescharacteristic
206#67—21no—
207D12—41no—
208D12—41no—
209D12#21311yescharacteristic
210C4×D3—21no—
211Dic6#21311yescharacteristic
212C3×C8—21no—
213C3×C8—21no—
214C3×C8—61no—
215C3×C8—121no—
216C3×C8D3×D411yescharacteristic
217C3×C8#23711yescharacteristic
218C3×C8#23711yescharacteristic
219C3×C8C2×#6711yescharacteristic
220C3⋊C8—21no—
221C3⋊C8—41no—
222C3⋊C8—121no—
223C3⋊C8—121no—
224C3⋊C8—241no—
225C2×C32—21no—
226C2×C32#46611yescharacteristic
227C3×D3—41no—
228C3×D3—161no—
229#41—31no—
230C2×Q8—21no—
231C2×D4—61no—
232Q16—21no—
233Q16—21no—
234Q16—121no—
235SD16—121no—
236SD16—361no—
237D8—61no—
238D8—61no—
239D8—361no—
240M16—31no—
241M16—61no—
242M16—61no—
243M16—181no—
244C2×C8—61no—
245C2×C8C2×D3211yescharacteristic
246C16—31no—
247C16—31no—
248C16—181no—
249C22×C3—21no—
250C22×C3—61no—
251C22×C3—241no—
252C22×C3—481no—
253C22×C3#91211yescharacteristic
254C22×C3#95011yescharacteristic
255C2×D3—21no—
256C2×D3—81no—
257C2×D3—161no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—61no—
261C3×C4—82no—
262C3×C4—81no—
263C3×C4—81no—
264C3×C4D3×D811yescharacteristic
265C3×C4C2×#21311yescharacteristic
266C3×C4#93811yescharacteristic
267C3×C4#93911yescharacteristic
268Dic3—21no—
269C32#222311yescharacteristic, complemented, Sylow
270C23—121no—
271Q8—31no—
272Q8—41no—
273Q8—41no—
274D4—31no—
275D4—61no—
276D4—121no—
277D4—121no—
278C2×C4—41no—
279C2×C4—61no—
280C2×C4D3×#6711yescharacteristic
281C8—61no—
282C8—121no—
283C8—361no—
284C8#380511yescharacteristic
285C8D3×#6711yescharacteristic
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—41no—
290C2×C3—121no—
291C2×C3—481no—
292C2×C3#503811yescharacteristic
293C2×C3#515711yescharacteristic
294D3—41no—
295D3—161no—
296C22—61no—
297C22—241no—
298C22—481no—
299C22#6393311yescharacteristic
300C4—61no—
301C4—81no—
302C4D3×#21311yescharacteristic
303C4#6397011yescharacteristic
304C3—21no—
305C3#7109711yescharacteristic, complemented
306C3#7149111yescharacteristic, complemented
307C2—21no—
308C2—121no—
309C2—481no—
310C2#1059149811yescharacteristic
subgroupG/H
centerC2#10591498
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×Q16C22
socleC2×C32#466
prime radical#10590926C2
hypercenterC2×Q16D32
hypocenterC32#2223
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
1. C3×#5192C2semidirect
2. C2×C3×#828C2semidirect
4. #10590621C2semidirect
5. #10589431C2semidirect
6. #10589143C2semidirect
7. #10589126C2semidirect
27. #5192D3semidirect
28. C2×#828D3semidirect
269. C32#2223semidirect
305. C3#71097semidirect
306. C3#71491semidirect

Maximal subgroups

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

subgroupindex
C3×#51922
C2×C3×#8282
#105909262
#105906212
#105894312
#105891432
#105891262
#714913
#710973