Finite Group Databasebeta

← #11799459

Subgroups — #11799459

Order 1152.

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

normal subgroups only

This group has 319 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×#5160C211yesmaximal, characteristic
2C2×C3×#827C211yesmaximal, characteristic, complemented
3#10590930C211yesmaximal, characteristic, complemented
4#10590621C211yesmaximal, characteristic, complemented
5#10589429C211yesmaximal, characteristic, complemented
6#10589143C211yesmaximal, characteristic, complemented
7#10589129C211yesmaximal, characteristic, complemented
8#71491—31nomaximal
9#71093—31nomaximal
10C2×C32×D8C2211yescharacteristic
11C3×#932—21no—
12C2×C3×Dic12C2211yescharacteristic
13C3×#901C2211yescharacteristic
14#63827—21no—
15#63646C2211yescharacteristic
16#63639C2211yescharacteristic
17C3×#833—21no—
18C3×#827—21no—
19C3×#827—21no—
20C3×#819C2211yescharacteristic
21C2×C3×C3⋊C16C2211yescharacteristic
22#63560—21no—
23#63543—21no—
24D3×C3⋊C16—21no—
25C2×C3×SD32—31no—
26C3×#472—31no—
27#5192—31no—
28#5160D311yescharacteristic
29C2×#827D311yescharacteristic, complemented
30#5143—31no—
31#5140—31no—
32#4901—31no—
33#4565—31no—
34#4563—31no—
35#4563—31no—
36#4563—61no—
37#4506—31no—
38#4496—31no—
39#4493—31no—
40C2×C32×D4—21no—
41C3×#235D411yescharacteristic
42C2×C3×Dic6—21no—
43C32×D8—21no—
44C32×D8—21no—
45C2×C32×C8C2311yescharacteristic
46#3748—21no—
47C3×#214—41no—
48C3×#213—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#2223—91noSylow
61C3×AGL(1,ℤ/8)—61no—
62C2×C3×Q16—31no—
63C2×C3×D8—21no—
64C2×C3×D8C2×D311yescharacteristic
65C2×C3×D8C2×D311yescharacteristic
66C3×#131—31no—
67#951—61no—
68#942—61no—
69#932—21no—
70C2×Dic12C2×D311yescharacteristic
71#901C2×D311yescharacteristic
72C3×SD32—61no—
73C3×SD32—61no—
74C2×C3×C16—31no—
75C3×#108—31no—
76C3×#100—61no—
77#836—61no—
78#833—21no—
79#827—21no—
80#827—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—
94C23×C32—41no—
95C32×D4—41no—
96C32×D4—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×SD32—31no—
111#504—91no—
112#473—91no—
113#472—31no—
114#368—91no—
115#362—91no—
116#361—91no—
117C3×#41—31no—
118C2×C3×Q8—61no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—41no—
122#235#6711yescharacteristic
123C2×Dic6—21no—
124C3×Q16—61no—
125C3×Q16—61no—
126C3×SD16—121no—
127C3×D8—22no—
128C3×D8—21no—
129C3×D8—21no—
130C3×D8—21no—
131C3×D8—21no—
132C3×D8—41no—
133C3×D8—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—121no—
142#216—121no—
143#215—121no—
144#214—41no—
145#214—121no—
146#213—41no—
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—81no—
164C22×C32—161no—
165C22×C32#13511yescharacteristic
166C2×C3×D3—21no—
167C32×C4#12111yescharacteristic
168C32×C4C2×D811yescharacteristic
169C3×Dic3—21no—
170C3×Dic3—81no—
171#137—181no—
172AGL(1,ℤ/8)—61no—
173C2×Q16—31no—
174C2×D8D3211yescharacteristic
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—181no—
185#100—61no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—81no—
189C3×Q8—31no—
190C3×Q8—121no—
191C3×Q8—121no—
192C3×D4—31no—
193C3×D4—42no—
194C3×D4—41no—
195C3×D4—41no—
196C3×D4—41no—
197C3×D4—41no—
198C3×D4—61no—
199C3×D4—81no—
200C2×C3×C4—21no—
201C2×C3×C4—61no—
202C2×C3×C4—121no—
203C2×C3×C4D3×D411yescharacteristic
204C2×C3×C4C2×#6711yescharacteristic
205#67—21no—
206C2×Dic3—41no—
207D12#21311yescharacteristic
208C4×D3—21no—
209Dic6—41no—
210Dic6—41no—
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—161no—
227C2×C32#46611yescharacteristic
228C3×D3—41no—
229#41—31no—
230C2×Q8—61no—
231C2×D4—21no—
232Q16—61no—
233Q16—61no—
234Q16—361no—
235SD16—121no—
236SD16—361no—
237D8—21no—
238D8—21no—
239D8—121no—
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—82no—
252C22×C3—81no—
253C22×C3—81no—
254C22×C3—161no—
255C22×C3—161no—
256C22×C3—162no—
257C22×C3#91211yescharacteristic
258C22×C3#95011yescharacteristic
259C2×D3—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—61no—
263C3×C4—241no—
264C3×C4D3×D811yescharacteristic
265C3×C4C2×#21311yescharacteristic
266C3×C4#93811yescharacteristic
267C3×C4#93911yescharacteristic
268Dic3—21no—
269Dic3—81no—
270C32#222311yescharacteristic, complemented, Sylow
271C23—41no—
272Q8—31no—
273Q8—121no—
274Q8—121no—
275D4—31no—
276D4—41no—
277D4—41no—
278D4—61no—
279C2×C4—61no—
280C2×C4—121no—
281C2×C4D3×#6711yescharacteristic
282C8—61no—
283C8—121no—
284C8—361no—
285C8#380511yescharacteristic
286C8D3×#6711yescharacteristic
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—41no—
291C2×C3—121no—
292C2×C3—162no—
293C2×C3—161no—
294C2×C3—161no—
295C2×C3#503811yescharacteristic
296C2×C3#515711yescharacteristic
297D3—41no—
298C22—61no—
299C22—81no—
300C22—161no—
301C22#6393311yescharacteristic
302C4—61no—
303C4—241no—
304C4D3×#21311yescharacteristic
305C4#6397011yescharacteristic
306C3—21no—
307C3#7109311yescharacteristic, complemented
308C3#7149111yescharacteristic, complemented
309C2—21no—
310C2—121no—
311C2—161no—
312C2#1059149811yescharacteristic
subgroupG/H
centerC2#10591498
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×D8C22
socleC2×C32#466
prime radicalC3×#5160C2
hypercenterC2×D8D32
hypocenterC32#2223
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#827C2semidirect
3. #10590930C2semidirect
4. #10590621C2semidirect
5. #10589429C2semidirect
6. #10589143C2semidirect
7. #10589129C2semidirect
29. C2×#827D3semidirect
270. C32#2223semidirect
307. C3#71093semidirect
308. C3#71491semidirect

Maximal subgroups

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

subgroupindex
C3×#51602
C2×C3×#8272
#105909302
#105906212
#105894292
#105891432
#105891292
#714913
#710933