Finite Group Databasebeta

← #5593

Subgroups — #5593

Order 192.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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×#120C211yesmaximal, characteristic, complemented
2#953C211yesmaximal, characteristic, complemented
3C2×#217C211yesmaximal, characteristic, complemented
4#939C211yesmaximal, characteristic, complemented
5#939C211yesmaximal, characteristic, complemented
6#938C211yesmaximal, characteristic, complemented
7#936C211yesmaximal, characteristic, complemented
8#935C211yesmaximal, characteristic, complemented
9D4×Dic3C211yesmaximal, characteristic, complemented
10#885C211yesmaximal, characteristic, complemented
11#883C211yesmaximal, characteristic, complemented
12#67⋊C4C211yesmaximal, characteristic, complemented
13#879C211yesmaximal, characteristic, complemented
14#879C211yesmaximal, characteristic, complemented
15#878C211yesmaximal, characteristic, complemented
16#538—31nomaximal, Sylow
17C24×C3C2211yescharacteristic
18C2×C3×D4C2211yescharacteristic
19C2×C3×D4C2211yescharacteristic
20C2×C3×D4C2211yescharacteristic
21C2×#67C2211yescharacteristic
22C2×#67C2211yescharacteristic
23C2×#67C2211yescharacteristic
24C2×#67C2211yescharacteristic, complemented
25C22×Dic3C2211yescharacteristic
26C22×Dic3C2211yescharacteristic
27C22×Dic3C2211yescharacteristic, complemented
28C2×C4×D3C2211yescharacteristic, complemented
29C2×Dic6C2211yescharacteristic, complemented
30C3×C22⋊C4C2211yescharacteristic
31C3×C22⋊C4C2211yescharacteristic
32C3×C22⋊C4C2211yescharacteristic
33#217C2211yescharacteristic
34#217C2211yescharacteristic
35#217C2211yescharacteristic
36#217C2211yescharacteristic
37#217C2211yescharacteristic
38#217C2211yescharacteristic
39#217C2211yescharacteristic
40#217C2211yescharacteristic, complemented
41#217C2211yescharacteristic, complemented
42(C2×D3)⋊C4C2211yescharacteristic, complemented
43(C2×D3)⋊C4C2211yescharacteristic, complemented
44C4⋊Dic3C2211yescharacteristic, complemented
45C4⋊Dic3C2211yescharacteristic, complemented
46Dic3⋊C4C2211yescharacteristic, complemented
47Dic3⋊C4C2211yescharacteristic, complemented
48Dic3⋊C4C2211yescharacteristic, complemented
49Dic3⋊C4C2211yescharacteristic, complemented
50C4×Dic3C2211yescharacteristic, complemented
51C4×Dic3C2211yescharacteristic, complemented
52#126—31no—
53#126—31no—
54#124—31no—
55#124—31no—
56#123—31no—
57#123—31no—
58C22⋊Q8—31no—
59#121—31no—
60#121—31no—
61#121—31no—
62#120—31no—
63#120D311yescharacteristic, complemented
64C4×D4—31no—
65C4×D4—31no—
66C2×C22⋊C4—31no—
67C23×C3—22no—
68C23×C3—41no—
69C23×C3—41no—
70C23×C3C2311yescharacteristic
71C23×C3C2311yescharacteristic
72C23×C3C2311yescharacteristic
73C23×C3C2311yescharacteristic
74C22×D3C2311yescharacteristic
75C3×D4—22no—
76C3×D4—41no—
77C3×D4—41no—
78C2×C3×C4C2311yescharacteristic
79C2×C3×C4C2311yescharacteristic
80C2×C3×C4C2311yescharacteristic
81#67—22no—
82#67—41no—
83#67—41no—
84#67—41no—
85C2×Dic3—41no—
86C2×Dic3—41no—
87C2×Dic3—41no—
88C2×Dic3C2311yescharacteristic
89C2×Dic3C2311yescharacteristic
90C2×Dic3C2311yescharacteristic
91C2×Dic3C2311yescharacteristic
92C2×Dic3C2311yescharacteristic
93C2×Dic3C2311yescharacteristic
94C2×Dic3C2311yescharacteristic
95C4×D3—41no—
96Dic6—41no—
97C24C2×D311yescharacteristic
98C2×Q8—31no—
99C2×D4—31no—
100C2×D4—31no—
101C2×D4—31no—
102C2×D4—31no—
103C2×D4C2×D311yescharacteristic
104C2×D4C2×D311yescharacteristic
105C2×D4C2×D311yescharacteristic
106C22×C4—31no—
107C22×C4—31no—
108C22×C4—31no—
109C22×C4—31no—
110C4⋊C4—31no—
111C4⋊C4—31no—
112C4⋊C4—31no—
113C4⋊C4—31no—
114C4⋊C4—31no—
115C4⋊C4—31no—
116C22⋊C4—31no—
117C22⋊C4—31no—
118C22⋊C4—31no—
119C22⋊C4—31no—
120C22⋊C4—31no—
121C22⋊C4—31no—
122C22⋊C4—31no—
123C22⋊C4—31no—
124C22⋊C4—31no—
125C22⋊C4—31no—
126C22⋊C4—31no—
127C22⋊C4C2×D311yescharacteristic
128C22⋊C4C2×D311yescharacteristic
129C22⋊C4C2×D311yescharacteristic
130C42—31no—
131C42—31no—
132C22×C3—21no—
133C22×C3—21no—
134C22×C3—21no—
135C22×C3—21no—
136C22×C3—21no—
137C22×C3—21no—
138C22×C3—21no—
139C22×C3—21no—
140C22×C3—21no—
141C22×C3—21no—
142C22×C3—21no—
143C22×C3—44no—
144C22×C3#4112yes—
145C22×C3C2411yescharacteristic
146C2×D3—21no—
147C2×D3—21no—
148C2×D3—21no—
149C3×C4—21no—
150C3×C4—21no—
151C3×C4—21no—
152Dic3—21no—
153Dic3—21no—
154Dic3—21no—
155Dic3—21no—
156Dic3—21no—
157Dic3—21no—
158Dic3—21no—
159C23—22no—
160C23—31no—
161C23—41no—
162C23—41no—
163C23C22×D311yescharacteristic
164C23C22×D311yescharacteristic
165C23C22×D311yescharacteristic
166C23C22×D311yescharacteristic
167Q8—121no—
168D4—22no—
169D4—41no—
170D4—41no—
171D4—62no—
172D4—121no—
173D4—121no—
174D4—121no—
175C2×C4—31no—
176C2×C4—31no—
177C2×C4—31no—
178C2×C4—31no—
179C2×C4—31no—
180C2×C4—31no—
181C2×C4—31no—
182C2×C4—121no—
183C2×C4—121no—
184C2×C4—121no—
185C2×C4—121no—
186C2×C4C22×D311yescharacteristic
187C2×C4C22×D311yescharacteristic
188C2×C4C22×D311yescharacteristic
189C2×C3—22no—
190C2×C3—41no—
191C2×C3—41no—
192C2×C3—41no—
193C2×C3C2×#4111yescharacteristic
194C2×C3H(5,𝔽2)11yescharacteristic
195C2×C3H(5,𝔽2)11yescharacteristic
196D3—41no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—44no—
209C22—61no—
210C22—61no—
211C22—61no—
212C22#23712yes—
213C22C23×D311yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—61no—
218C4—61no—
219C4—61no—
220C4—61no—
221C4—61no—
222C4—61no—
223C4—61no—
224C3#53811yescharacteristic, complemented, Sylow
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—121no—
230C2C2×#23711yescharacteristic
231C2#100411yescharacteristic
232C2#100411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3
commutator, socle, cosocleC22×C3C24
FittingC3×#120C2
hypercenter#120D3
hypocenterC3#538
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C3×#120C2semidirect
2. #953C2semidirect
3. C2×#217C2semidirect
4. #939C2semidirect
5. #939C2semidirect
6. #938C2semidirect
7. #936C2semidirect
8. #935C2semidirect
9. D4×Dic3C2semidirect
10. #885C2semidirect
11. #883C2semidirect
12. #67⋊C4C2semidirect
13. #879C2semidirect
14. #879C2semidirect
15. #878C2semidirect
24. C2×#67C22semidirect
27. C22×Dic3C22semidirect
28. C2×C4×D3C22semidirect
29. C2×Dic6C22semidirect
40. #217C22semidirect
41. #217C22semidirect
42. (C2×D3)⋊C4C22semidirect
43. (C2×D3)⋊C4C22semidirect
44. C4⋊Dic3C22semidirect
45. C4⋊Dic3C22semidirect
46. Dic3⋊C4C22semidirect
47. Dic3⋊C4C22semidirect
48. Dic3⋊C4C22semidirect
49. Dic3⋊C4C22semidirect
50. C4×Dic3C22semidirect
51. C4×Dic3C22semidirect
63. #120D3semidirect
224. C3#538semidirect

Maximal subgroups

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

subgroupindex
C3×#1202
#9532
C2×#2172
#9392
#9382
#9362
#9352
D4×Dic32
#8852
#8832
#67⋊C42
#8792
#8782
#5383