Finite Group Databasebeta

← #11799717

Subgroups — #11799717

Order 1152.

281 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 76 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×#5196C211yesmaximal, characteristic
2#10590889C211yesmaximal, characteristic, complemented
3#10589432C211yesmaximal, characteristic
4C3×#4566C211yesmaximal, characteristic
5#10589138C211yesmaximal, characteristic, complemented
6#10589137C211yesmaximal, characteristic, complemented
7#10589134C211yesmaximal, characteristic, complemented
8#70919—31nomaximal
9#69246—31nomaximal
10C32×#135C2211yescharacteristic
11C3×#948—21no—
12#63918—21no—
13#63644C2211yescharacteristic
14#63639C2211yescharacteristic
15C3×#837—21no—
16C3×#819C2211yescharacteristic
17C3×#817C2211yescharacteristic
18C3×C8×Dic3C2211yescharacteristic
19C3×#812C2211yescharacteristic
20#63573—21no—
21C3×#443—31no—
22#5196D311yescharacteristic
23#4855—31no—
24#4680—31no—
25C3×#360—31no—
26#4566—31no—
27#4566—31no—
28#4566—61no—
29#4566D311yescharacteristic
30#4513—31no—
31#4511—31no—
32#4494—31no—
33#4494—31no—
34#4487—31no—
35#4463—31no—
36C32×#41—21no—
37C2×#627—41no—
38C32×Q16Q811yescharacteristic, complemented
39C32×SD16—21no—
40C32×D8Q811yescharacteristic, complemented
41C2×C32×C8C2311yescharacteristic
42C4⋊(C3⋊Dic3)—21no—
43C3×C4⋊Dic3—21no—
44C3×C4×Dic3D411yescharacteristic
45C3×#208—21no—
46C3×#208—41no—
47C2×C3×C3⋊C8—41no—
48C2×C3×C3⋊C8D411yescharacteristic
49Dic3⋊Dic3—41no—
50C3⋊(C3⋊C16)—21no—
51C3×C3⋊C16—21no—
52#2230—91noSylow
53C3×#135—21no—
54C3×#135C2×D311yescharacteristic
55C3×#135C2×D311yescharacteristic
56C3×#131—61no—
57#948—21no—
58#888—61no—
59#869—61no—
60C3×M32—31no—
61C3×#108—31no—
62C3×#106—31no—
63C3×#104—61no—
64C3×C4×C8—31no—
65#837—21no—
66#819C2×D311yescharacteristic
67#817—31no—
68#817—31no—
69#817—61no—
70#817—61no—
71#817C2×D311yescharacteristic
72C8×Dic3C2×D311yescharacteristic
73#812—31no—
74#812—31no—
75#812—61no—
76#812C2×D311yescharacteristic
77#808—61no—
78C32×Q8—21no—
79C32×D4—21no—
80C32×D4—41no—
81C2×C32×C4—41no—
82C2×C32×C4C2×D411yescharacteristic
83C2×C3⋊Dic3—41no—
84C2×C3×Dic3—21no—
85C2×C3×Dic3—41no—
86#627—82no—
87C32×C8C2×Q811yescharacteristic
88C32×C8#4111yescharacteristic
89C3×C3⋊C8—41no—
90C3×C3⋊C8SD1611yescharacteristic
91C3×C3⋊C8SD1611yescharacteristic
92#498—91no—
93#443—31no—
94#365—91no—
95#365—91no—
96#364—91no—
97#360—31no—
98#360—91no—
99C3×#41—21no—
100C3×#41—21no—
101C3×#41—41no—
102C2×Dic6—121no—
103C2×Dic6—121no—
104C3×Q16—21no—
105C3×Q16Dic611yescharacteristic, complemented
106C3×Q16Dic611yescharacteristic, complemented
107C3×SD16—21no—
108C3×SD16—21no—
109C3×SD16—41no—
110C3×D8—21no—
111C3×D8Dic611yescharacteristic, complemented
112C3×D8Dic611yescharacteristic, complemented
113C3×M16—61no—
114C3×M16—121no—
115C2×C3×C8—21no—
116C2×C3×C8—31no—
117C2×C3×C8—121no—
118C2×C3×C8C22×D311yescharacteristic
119C2×C3×C8C22×D311yescharacteristic
120C3×C4⋊C4—61no—
121C3×C42—31no—
122C4⋊Dic3—21no—
123C4⋊Dic3—61no—
124C4⋊Dic3—61no—
125C4⋊Dic3—121no—
126C4⋊Dic3—121no—
127Dic3⋊C4—121no—
128C4×Dic3#6711yescharacteristic
129#208—21no—
130#208—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8#6711yescharacteristic
133C3×C16—61no—
134C3⋊C16—21no—
135C3⋊C16—61no—
136C3⋊C16—61no—
137C3⋊C16—121no—
138C22×C32—41no—
139C22×C32AGL(1,ℤ/8)11yescharacteristic
140C32×C4—41no—
141C32×C4C22⋊Q811yescharacteristic
142C32×C4C2×SD1611yescharacteristic
143C3⋊Dic3—81no—
144C3×Dic3—22no—
145C3×Dic3—81no—
146#135D3211yescharacteristic
147#131—61no—
148C4⋊Q8—181no—
149M32—31no—
150M32—91no—
151#108—31no—
152#106—31no—
153#106—91no—
154#106—181no—
155#104—61no—
156C4×C8—31no—
157C3×Q8—21no—
158C3×Q8—21no—
159C3×Q8—41no—
160C3×D4—21no—
161C3×D4—21no—
162C3×D4—41no—
163C3×D4—41no—
164C3×D4—41no—
165C3×D4—81no—
166C2×C3×C4—21no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—61no—
170C2×C3×C4—81no—
171C2×C3×C4—121no—
172C2×C3×C4D3×D411yescharacteristic
173C2×C3×C4C2×#6711yescharacteristic
174C2×Dic3—21no—
175C2×Dic3—41no—
176C2×Dic3—121no—
177C2×Dic3—121no—
178C2×Dic3—241no—
179Dic6—242no—
180Dic6—242no—
181C3×C8—21no—
182C3×C8—21no—
183C3×C8—31no—
184C3×C8—31no—
185C3×C8—121no—
186C3×C8C2×Dic611yescharacteristic
187C3×C8C2×Dic611yescharacteristic
188C3×C8#23511yescharacteristic
189C3×C8#23711yescharacteristic
190C3⋊C8—41no—
191C3⋊C8#21411yescharacteristic
192C3⋊C8#21411yescharacteristic
193C2×C32—21no—
194C2×C32—81no—
195C2×C32#47611yescharacteristic
196#41—21no—
197C2×Q8—361no—
198Q16#62711yescharacteristic, complemented
199SD16—21no—
200D8#62711yescharacteristic, complemented
201M16—61no—
202M16—121no—
203C2×C8—31no—
204C2×C8—121no—
205C2×C8C2×D3211yescharacteristic
206C4⋊C4—61no—
207C4⋊C4—181no—
208C4⋊C4—361no—
209C42—31no—
210C16—61no—
211C16—181no—
212C22×C3—21no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—81no—
216C22×C3#91111yescharacteristic
217C22×C3#94911yescharacteristic
218C3×C4—21no—
219C3×C4—21no—
220C3×C4—41no—
221C3×C4—41no—
222C3×C4—62no—
223C3×C4—81no—
224C3×C4—241no—
225C3×C4#87811yescharacteristic
226C3×C4D3×SD1611yescharacteristic
227C3×C4#92411yescharacteristic
228C3×C4C2×#21411yescharacteristic
229Dic3—22no—
230Dic3—81no—
231Dic3—241no—
232Dic3—241no—
233Dic3—481no—
234C32#223011yescharacteristic, complemented, Sylow
235Q8—21no—
236Q8—722no—
237D4—21no—
238D4—41no—
239C2×C4—41no—
240C2×C4—61no—
241C2×C4—121no—
242C2×C4—361no—
243C2×C4D3×#6711yescharacteristic
244C8—31no—
245C8—31no—
246C8—121no—
247C8#380411yescharacteristic
248C8C2×#62711yescharacteristic
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—41no—
253C2×C3—81no—
254C2×C3—81no—
255C2×C3—161no—
256C2×C3#476111yescharacteristic
257C2×C3#500811yescharacteristic
258C22—41no—
259C22#6392511yescharacteristic
260C4—41no—
261C4—62no—
262C4—241no—
263C4—721no—
264C4D3×#21411yescharacteristic
265C4#6396411yescharacteristic
266C3—21no—
267C3#6924611yescharacteristic, complemented
268C3#7091911yescharacteristic, complemented
269C2—21no—
270C2—81no—
271C2#1059140811yescharacteristic
subgroupG/H
centerC2#10591408
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
Fitting, prime radicalC32×#135C22
socleC2×C32#476
hypercenter#135D32
hypocenterC32#2230
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #135

Complements

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

NH ≅ G/Ntype
2. #10590889C2semidirect
5. #10589138C2semidirect
6. #10589137C2semidirect
7. #10589134C2semidirect
38. C32×Q16Q8semidirect
40. C32×D8Q8semidirect
105. C3×Q16Dic6semidirect
106. C3×Q16Dic6semidirect
111. C3×D8Dic6semidirect
112. C3×D8Dic6semidirect
198. Q16#627semidirect
200. D8#627semidirect
234. C32#2230semidirect
267. C3#69246semidirect
268. C3#70919semidirect

Maximal subgroups

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

subgroupindex
C3×#51962
#105908892
#105894322
C3×#45662
#105891382
#105891372
#105891342
#709193
#692463