Finite Group Databasebeta

← #11797713

Subgroups — #11797713

Order 1152.

355 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 98 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
1Q16×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×#829C211yesmaximal, characteristic
3#10590818C211yesmaximal, characteristic
4C3×#4564C211yesmaximal, characteristic
5#10589139C211yesmaximal, characteristic
6#10589118C211yesmaximal, characteristic
7#10589102C211yesmaximal, characteristic
8Dic3×Q32—31nomaximal
9#69207—31nomaximal
10C2×C32×Q16C2211yescharacteristic
11Q8×C3⋊Dic3—21no—
12C2×C3×Dic12C2211yescharacteristic
13#63843—21no—
14#63662—21no—
15#63645C2211yescharacteristic
16C8×C3⋊Dic3C2211yescharacteristic
17C3×#829C414yescomplemented
18C3×#818C2211yescharacteristic
19C2×C3×C3⋊C16C2211yescharacteristic
20C2×C3×C3⋊C16C2211yescharacteristic
21#63565—21no—
22C2×C3×Q32—31no—
23Dic3×Q16—31no—
24Dic3×Q16—31no—
25Dic3×Q16—31no—
26Dic3×Q16—61no—
27C2×#829D311yescharacteristic
28#4873—31no—
29C3×#358—31no—
30#4564—31no—
31#4564D311yescharacteristic
32#4504—31no—
33#4503—31no—
34Dic3×C16—31no—
35Dic3×C16—31no—
36#4489—31no—
37C2×C32×Q8—21no—
38C2×C3×Dic6—21no—
39C32×Q16—21no—
40C32×Q16C2×C412yes—
41C2×C32×C8C2311yescharacteristic
42C4⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊Dic3)—41no—
44C4×C3⋊Dic3—41no—
45C4×C3⋊Dic3D411yescharacteristic
46C2×C3⋊(C3⋊C8)D411yescharacteristic
47C3×C4⋊Dic3—21no—
48C3×Dic12—21no—
49C3×Dic12C2×C412yes—
50Dic3⋊Dic3—41no—
51Dic32—41no—
52C3×C3⋊C16—21no—
53C3×C3⋊C16C2×C412yes—
54C4×Q32—91noSylow
55C2×C3×Q16—21no—
56C2×C3×Q16—31no—
57C2×C3×Q16C2×D311yescharacteristic
58C2×C3×Q16C2×D311yescharacteristic
59Q8×Dic3—61no—
60Q8×Dic3—61no—
61Q8×Dic3—61no—
62Q8×Dic3—62no—
63C2×Dic12C2×D311yescharacteristic
64#887—61no—
65C3×Q32—34no—
66C2×C3×C16—31no—
67C2×C3×C16—31no—
68C3×#107—31no—
69#835—61no—
70#835—61no—
71#835—61no—
72#835—62no—
73#829Dic314yescomplemented
74#818—31no—
75#818—31no—
76#818—61no—
77#818C2×D311yescharacteristic
78C8×Dic3—31no—
79C8×Dic3—31no—
80C8×Dic3—61no—
81C2×C3⋊C16C2×D311yescharacteristic
82C2×C3⋊C16C2×D311yescharacteristic
83#810—61no—
84C32×Q8—22no—
85C32×Q8—41no—
86C2×C32×C4—41no—
87C2×C32×C4C2×D411yescharacteristic
88C2×C3⋊Dic3—41no—
89C2×C3⋊Dic3D811yescharacteristic
90C2×C3⋊Dic3D811yescharacteristic
91C2×C3×Dic3—41no—
92C2×C3×Dic3—41no—
93C3×Dic6—22no—
94C3×Dic6—41no—
95C32×C8C22×C411yescharacteristic
96C32×C8#4111yescharacteristic
97C3⋊(C3⋊C8)—21no—
98C2×Q32—31no—
99C4×Q16—91no—
100C4×Q16—91no—
101#366—91no—
102#358—31no—
103#358—91no—
104C4×C16—91no—
105C2×C3×Q8—22no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—61no—
109C2×Dic6—21no—
110C3×Q16—22no—
111C3×Q16—21no—
112C3×Q16—21no—
113C3×Q16—22no—
114C3×Q16—32no—
115C3×Q16—61no—
116C3×Q16C4×D312yes—
117C3×Q16C2×Dic312yes—
118C2×C3×C8—21no—
119C2×C3×C8C22×D311yescharacteristic
120C2×C3×C8C22×D311yescharacteristic
121C3×C4⋊C4—61no—
122C4⋊Dic3—21no—
123C4⋊Dic3—62no—
124C4⋊Dic3—61no—
125C4⋊Dic3—61no—
126C4⋊Dic3—121no—
127C4⋊Dic3—121no—
128C4⋊Dic3—121no—
129C4⋊Dic3—122no—
130Dic3⋊C4—121no—
131C4×Dic3—31no—
132C4×Dic3—31no—
133C4×Dic3—61no—
134C4×Dic3—121no—
135C4×Dic3—121no—
136C4×Dic3—121no—
137C4×Dic3—121no—
138C4×Dic3—122no—
139C2×C3⋊C8—31no—
140C2×C3⋊C8—31no—
141C2×C3⋊C8—61no—
142Dic12—21no—
143Dic12C2×Dic312yes—
144C3×C16—32no—
145C3×C16—61no—
146C3⋊C16—21no—
147C3⋊C16C2×Dic312yes—
148C22×C32C2×D811yescharacteristic
149C32×C4—42no—
150C32×C4C4×D411yescharacteristic
151C32×C4#13511yescharacteristic
152C3⋊Dic3—21no—
153C3⋊Dic3—81no—
154C3⋊Dic3Q3212yescomplemented
155C3×Dic3—42no—
156C3×Dic3—81no—
157C2×Q16—31no—
158C2×Q16D3211yescharacteristic
159C4×Q8—181no—
160C4×Q8—181no—
161Q32—34no—
162C2×C16—31no—
163C2×C16—31no—
164#107—31no—
165#107—91no—
166#103—181no—
167#103—181no—
168C4×C8—91no—
169C3×Q8—24no—
170C3×Q8—22no—
171C3×Q8—22no—
172C3×Q8—42no—
173C3×Q8—41no—
174C3×Q8—41no—
175C3×Q8—62no—
176C3×Q8—121no—
177C2×C3×C4—21no—
178C2×C3×C4—42no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—121no—
182C2×C3×C4—121no—
183C2×C3×C4D3×D411yescharacteristic
184C2×C3×C4D3×D411yescharacteristic
185C2×Dic3—31no—
186C2×Dic3—31no—
187C2×Dic3—31no—
188C2×Dic3—31no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—61no—
192C2×Dic3—61no—
193C2×Dic3—122no—
194C2×Dic3—121no—
195C2×Dic3—121no—
196Dic6—22no—
197Dic6—41no—
198C3×C8—21no—
199C3×C8—21no—
200C3×C8C2×C4×D311yescharacteristic
201C3×C8#23711yescharacteristic
202C3×C8#23711yescharacteristic
203C3×C8C22×Dic311yescharacteristic
204C3⋊C8—61no—
205C3⋊C8—61no—
206C3⋊C8—121no—
207C2×C32C4×D811yescharacteristic
208C2×C32C2×Q3211yescharacteristic
209C2×C32#50811yescharacteristic
210C2×Q8—21no—
211C2×Q8—61no—
212Q16—21no—
213Q16—32no—
214Q16—61no—
215Q16D3×Dic312yes—
216C2×C8—91no—
217C2×C8C2×D3211yescharacteristic
218C4⋊C4—61no—
219C4⋊C4—181no—
220C4⋊C4—361no—
221C4⋊C4—361no—
222C42—91no—
223C42—361no—
224C42—361no—
225C16—32no—
226C16—61no—
227C22×C3—21no—
228C22×C3D3×D811yescharacteristic
229C22×C3D3×D811yescharacteristic
230C3×C4—21no—
231C3×C4—21no—
232C3×C4—44no—
233C3×C4—42no—
234C3×C4—42no—
235C3×C4—122no—
236C3×C4—241no—
237C3×C4#67⋊C411yescharacteristic
238C3×C4#91211yescharacteristic
239C3×C4#91611yescharacteristic
240C3×C4D4×Dic311yescharacteristic
241Dic3—32no—
242Dic3—32no—
243Dic3—42no—
244Dic3—62no—
245Dic3—61no—
246Dic3—61no—
247Dic3—81no—
248Dic3—121no—
249Dic3—242no—
250Dic3—241no—
251Dic3—241no—
252C32C4×Q3211yescharacteristic, complemented, Sylow
253Q8—22no—
254Q8—41no—
255Q8—62no—
256Q8—121no—
257C2×C4—41no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—121no—
261C2×C4—121no—
262C2×C4—361no—
263C2×C4#381111yescharacteristic
264C8—181no—
265C8C2×D3×Dic311yescharacteristic
266C8#380511yescharacteristic
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3#475611yescharacteristic
271C2×C3#491611yescharacteristic
272C2×C3D3×Q3211yescharacteristic
273C2×C3D3×Q3211yescharacteristic
274C2×C3#491911yescharacteristic
275C2×C3Dic3×D811yescharacteristic
276C22#6392611yescharacteristic
277C4—42no—
278C4—92no—
279C4—122no—
280C4—181no—
281C4—241no—
282C4—721no—
283C4#6393611yescharacteristic
284C4#67⋊Dic311yescharacteristic
285C3—21no—
286C3#6920711yescharacteristic, complemented
287C3Dic3×Q3211yescharacteristic, complemented
288C2#1059104311yescharacteristic
289C2#1059104911yescharacteristic
290C2#1059143511yescharacteristic
subgroupG/H
centerC22#63926
commutatorC32×C8C22×C4
FrattiniC2×C8C2×D32
FittingC2×C32×Q16C22
socle, prime radicalC22×C32C2×D8
cosocleC2×C32×C8C23
hypercenterC2×Q16D32
hypocenterC32C4×Q32
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
17. C3×#829C4semidirect
73. #829Dic3semidirect
154. C3⋊Dic3Q32semidirect
252. C32C4×Q32semidirect
286. C3#69207semidirect
287. C3Dic3×Q32semidirect

Maximal subgroups

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

subgroupindex
Q16×C3⋊Dic32
C2×C3×#8292
#105908182
C3×#45642
#105891392
#105891182
#105891022
Dic3×Q323
#692073