Finite Group Databasebeta

← #11797935

Subgroups — #11797935

Order 1152.

305 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 78 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×#4894C211yesmaximal, characteristic
2#10590812C211yesmaximal, characteristic, complemented
3#10590636C211yesmaximal, characteristic
4#10589360C211yesmaximal, characteristic, complemented
5C3×#4496C211yesmaximal, characteristic, complemented
6#10589124C211yesmaximal, characteristic
7#10589105C211yesmaximal, characteristic, complemented
8#70928—31nomaximal
9#69264—31nomaximal
10C3×#907—21no—
11C2×C3×Dic12C2211yescharacteristic
12C3×#904C2211yescharacteristic
13Dic3×Dic6—21no—
14C32×#108C2211yescharacteristic
15#63644C2211yescharacteristic
16#63639C2211yescharacteristic
17C3×#825—21no—
18C3×#815C2211yescharacteristic
19C3×#812C2211yescharacteristic
20#63566—21no—
21#63550C412yescomplemented
22#63547C412yescomplemented
23C3×#444—31no—
24#5195—31no—
25#5184—31no—
26#5179—31no—
27#4894D311yescharacteristic
28#4716—31no—
29C3×#362—31no—
30#4566—31no—
31#4518—31no—
32#4512—31no—
33#4496D311yescharacteristic, complemented
34#4494—31no—
35#4494—31no—
36#4494—61no—
37#4466—31no—
38C3×#235—21no—
39C2×C3×Dic6—21no—
40C32×M16—21no—
41C2×C32×C8C2311yescharacteristic
42C4⋊(C3⋊Dic3)—21no—
43C3×C4×Dic3D411yescharacteristic
44C2×C3×C3⋊C8D411yescharacteristic
45C3×Dic12—21no—
46C3×Dic12C2×C411yescharacteristic
47C3×Dic12C2×C412yes—
48C3×D24C2×C411yescharacteristic
49C3×#204—21no—
50C3×ΓL(1,𝔽25)—41no—
51C3×D3×C8—41no—
52Dic3⋊Dic3—41no—
53Dic32—41no—
54C3⋊(C3⋊C16)C2×C412yes—
55C3×C3⋊C16—21no—
56#2180—91noSylow
57C3×#135—31no—
58C2×C3×Q16—31no—
59C3×#131—61no—
60Q8×Dic3—61no—
61#907—21no—
62C2×Dic12C2×D311yescharacteristic
63#904C2×D311yescharacteristic
64C4×Dic6—61no—
65C3×M32—31no—
66C3×#108—21no—
67C3×#108C2×D311yescharacteristic
68C3×#108C2×D311yescharacteristic
69C3×#104—61no—
70C3×#97—31no—
71#835—61no—
72#829—32no—
73#829—32no—
74#828—32no—
75#827—32no—
76#825—21no—
77#817—31no—
78#817—31no—
79#817—61no—
80#816—61no—
81#815C2×D311yescharacteristic
82#812—31no—
83#812—31no—
84#812—61no—
85#812C2×D311yescharacteristic
86C2×C32×C4C2×D411yescharacteristic
87C2×C3⋊Dic3—41no—
88C3×#67—41no—
89C2×C3×Dic3—21no—
90C2×C3×Dic3—41no—
91C3×D12—21no—
92C3×C4×D3—41no—
93C3×Dic6—21no—
94C3×Dic6—22no—
95C3×Dic6—41no—
96C32×C8—41no—
97C32×C8C22×C411yescharacteristic
98C32×C8#4111yescharacteristic
99C3×C3⋊C8—21no—
100#510—91no—
101#444—31no—
102#441—91no—
103#365—91no—
104#362—31no—
105#360—91no—
106#347—91no—
107C3×#41—61no—
108C2×C3×Q8—61no—
109#235—21no—
110C2×Dic6—21no—
111C3×Q16—32no—
112C3×Q16—31no—
113C3×Q16—61no—
114C3×SD16—61no—
115C3×D8—31no—
116C3×M16—21no—
117C3×M16—21no—
118C3×M16—41no—
119C3×M16—121no—
120C2×C3×C8—21no—
121C2×C3×C8—31no—
122C2×C3×C8—121no—
123C2×C3×C8C22×D311yescharacteristic
124C2×C3×C8C22×D311yescharacteristic
125C3×C42—31no—
126C4⋊Dic3—61no—
127C4⋊Dic3—61no—
128C4⋊Dic3—121no—
129C4⋊Dic3—121no—
130Dic3⋊C4—121no—
131C4×Dic3—121no—
132C4×Dic3—121no—
133C4×Dic3#6711yescharacteristic
134C2×C3⋊C8#6711yescharacteristic
135Dic12—21no—
136Dic12C4×D312yes—
137Dic12C4×D311yescharacteristic
138D24C4×D311yescharacteristic
139#204—21no—
140ΓL(1,𝔽25)—41no—
141D3×C8—41no—
142C3×C16—61no—
143C3⋊C16—21no—
144C3⋊C16—32no—
145C3⋊C16—32no—
146C3⋊C16—62no—
147C22×C32AGL(1,ℤ/8)11yescharacteristic
148C2×C3×D3—41no—
149C32×C4C4×D411yescharacteristic
150C32×C4AGL(1,ℤ/8)11yescharacteristic
151C3⋊Dic3—81no—
152C3×Dic3—41no—
153C3×Dic3—42no—
154C3×Dic3—41no—
155#135—31no—
156C2×Q16—31no—
157#131—61no—
158C4×Q8—181no—
159Q32—92no—
160SD32—92no—
161M32—31no—
162M32—91no—
163#108D3211yescharacteristic
164#106—91no—
165#104—61no—
166#103—181no—
167#97—31no—
168C3×Q8—61no—
169C3×Q8—62no—
170C3×Q8—121no—
171C3×D4—61no—
172C3×D4—121no—
173C2×C3×C4—21no—
174C2×C3×C4—61no—
175C2×C3×C4—121no—
176C2×C3×C4—121no—
177C2×C3×C4D3×D411yescharacteristic
178C2×C3×C4C2×#6711yescharacteristic
179#67—41no—
180C2×Dic3—21no—
181C2×Dic3—41no—
182C2×Dic3—121no—
183C2×Dic3—121no—
184C2×Dic3—241no—
185D12—21no—
186C4×D3—41no—
187Dic6—22no—
188Dic6—21no—
189Dic6—41no—
190C3×C8—21no—
191C3×C8—21no—
192C3×C8—41no—
193C3×C8—41no—
194C3×C8—61no—
195C3×C8—81no—
196C3×C8C2×C4×D311yescharacteristic
197C3×C8C2×C4×D311yescharacteristic
198C3×C8#23511yescharacteristic
199C3×C8#23711yescharacteristic
200C3⋊C8—21no—
201C2×C32—21no—
202C2×C32#44211yescharacteristic
203C3×D3—81no—
204#41—61no—
205C2×Q8—61no—
206Q16—32no—
207Q16—31no—
208Q16—61no—
209SD16—61no—
210D8—31no—
211M16—21no—
212M16—121no—
213C2×C8—31no—
214C2×C8—121no—
215C2×C8C2×D3211yescharacteristic
216C4⋊C4—181no—
217C4⋊C4—361no—
218C42—31no—
219C42—361no—
220C16—61no—
221C16—92no—
222C22×C3—21no—
223C22×C3—121no—
224C22×C3#91411yescharacteristic
225C22×C3#94911yescharacteristic
226C2×D3—41no—
227C3×C4—21no—
228C3×C4—21no—
229C3×C4—121no—
230C3×C4—122no—
231C3×C4—121no—
232C3×C4#67⋊C411yescharacteristic
233C3×C4#91111yescharacteristic
234C3×C4C4×#6711yescharacteristic
235C3×C4#93211yescharacteristic
236Dic3—41no—
237Dic3—41no—
238Dic3—42no—
239Dic3—241no—
240Dic3—241no—
241Dic3—481no—
242C32#218011yescharacteristic, complemented, Sylow
243Q8—61no—
244Q8—62no—
245Q8—121no—
246D4—61no—
247D4—121no—
248C2×C4—61no—
249C2×C4—121no—
250C2×C4—121no—
251C2×C4—361no—
252C2×C4D3×#6711yescharacteristic
253C8—41no—
254C8—61no—
255C8#379511yescharacteristic
256C8C4×D3211yescharacteristic
257C2×C3—21no—
258C2×C3—21no—
259C2×C3—21no—
260C2×C3—41no—
261C2×C3—241no—
262C2×C3#478511yescharacteristic
263C2×C3#501411yescharacteristic
264D3—81no—
265C22—121no—
266C22#6393911yescharacteristic
267C4—121no—
268C4—121no—
269C4—122no—
270C4—721no—
271C4#6387911yescharacteristic
272C4#6393211yescharacteristic
273C3—21no—
274C3#6926411yescharacteristic, complemented
275C3#7092811yescharacteristic, complemented
276C2—21no—
277C2—241no—
278C2#1059045511yescharacteristic
subgroupG/H
centerC2#10590455
commutatorC32×C8C22×C4
FrattiniC2×C8C2×D32
FittingC32×#108C22
socleC2×C32#442
prime radicalC3×#904C22
cosocleC2×C32×C8C23
hypercenter#108D32
hypocenterC32#2180
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #108

Complements

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

NH ≅ G/Ntype
2. #10590812C2semidirect
4. #10589360C2semidirect
5. C3×#4496C2semidirect
7. #10589105C2semidirect
21. #63550C4semidirect
22. #63547C4semidirect
33. #4496D3semidirect
242. C32#2180semidirect
274. C3#69264semidirect
275. C3#70928semidirect

Maximal subgroups

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

subgroupindex
C3×#48942
#105908122
#105906362
#105893602
C3×#44962
#105891242
#105891052
#709283
#692643