Finite Group Databasebeta

← #37565757

Subgroups — #37565757

Order 24256.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#20258607C211yesmaximal, characteristic, complemented
2C22×C4×D379C211yesmaximal, characteristic, complemented
3#27131316C212yesmaximal, complemented
4#27131307C211yesmaximal, characteristic, complemented
5#27131306C211yesmaximal, characteristic, complemented
6#27131305C211yesmaximal, characteristic, complemented
7#27131301C212yesmaximal, complemented
8#27131293C212yesmaximal, complemented
9(C22⋊D379)⋊C4C212yesmaximal, complemented
10#27131289C211yesmaximal, characteristic, complemented
11C22⋊Q8×C379C211yesmaximal, characteristic, complemented
12C23×D379C2211yescharacteristic
13#20258607—24no—
14C2×D1516C2212yes—
15C2×D1516C2212yes—
16C2×D1516C2211yescharacteristic, complemented
17C2×C22⋊D379C2212yes—
18C22×Dic379C2211yescharacteristic
19C2×C4×D379—24no—
20C2×C4×D379C2211yescharacteristic
21C2×C4×D379C2211yescharacteristic
22C2×C4×D379C2211yescharacteristic
23C2×C4×D379C2211yescharacteristic
24C2×C4×D379C2212yescomplemented
25C4⋊Dic379C2211yescharacteristic, complemented
26Dic379⋊C4C2212yescomplemented
27(C2×D379)⋊C4C2212yes—
28(C2×D379)⋊C4C2212yes—
29(C2×D379)⋊C4C2212yescomplemented
30(C2×D379)⋊C4C2212yescomplemented
31C4×Dic379C2212yescomplemented
32C2×Q8×C379C2211yescharacteristic, complemented
33C22×C4×C379C2211yescharacteristic
34C4⋊C4×C379C2211yescharacteristic, complemented
35C4⋊C4×C379C2212yescomplemented
36C22⋊C4×C379C2212yes—
37C22×D379—22no—
38C22×D379—22no—
39C22×D379—22no—
40C22×D379C2312yes—
41C22×D379C2311yescharacteristic
42C22×D379C2311yescharacteristic
43D1516—24no—
44D1516—24no—
45D1516—22no—
46C22⋊D379—24no—
47C2×Dic379—22no—
48C2×Dic379C2312yes—
49C2×Dic379C2311yescharacteristic
50C2×Dic379C2311yescharacteristic
51C4×D379—22no—
52C4×D379—22no—
53C4×D379—22no—
54C4×D379—24no—
55C4×D379D414yescomplemented
56C23×C379C2311yescharacteristic
57Q8×C379—22no—
58C2×C4×C379—22no—
59C2×C4×C379C2312yes—
60C2×C4×C379C2312yes—
61C2×C4×C379C2311yescharacteristic
62C2×C4×C379C2311yescharacteristic
63C22×C379—21no—
64C22×C379—21no—
65C22×C379#4112yescomplemented
66C22×C379C2411yescharacteristic
67C2×D379—28no—
68C2×D379—21no—
69C2×D379—21no—
70C2×D379—21no—
71C2×D379—21no—
72C2×D379—22no—
73C2×D379—22no—
74C2×D379—22no—
75C2×D379C2×D412yes—
76C2×D379#4112yescomplemented
77C4×C379—21no—
78C4×C379—22no—
79C4×C379—22no—
80C4×C379C2×D412yes—
81Dic379—21no—
82Dic379—22no—
83Dic379C2×D412yes—
84C2×C379—22no—
85C2×C379C22×D411yescharacteristic
86C2×C379C2×#4111yescharacteristic
87C2×C379C2×#4111yescharacteristic
88D379—22no—
89D379—22no—
90D379—42no—
91C379#52511yescharacteristic, complemented, Sylow
92#525—3791nomaximal, Sylow
93C2×#41—3791no—
94C23×C4—3791no—
95#123—3792no—
96C22⋊Q8—3791no—
97C22⋊Q8D37911yescharacteristic, complemented
98#121—3791no—
99#121—3791no—
100#120—3792no—
101C4×D4—3792no—
102C4×D4—3792no—
103#117—3791no—
104C24—3791no—
105#41—7584no—
106C2×Q8C2×D37911yescharacteristic, complemented
107C2×D4—3792no—
108C2×D4—3792no—
109C2×D4—3792no—
110C2×D4—3791no—
111C22×C4—3791no—
112C22×C4—3791no—
113C22×C4—3791no—
114C22×C4—3791no—
115C22×C4—3791no—
116C22×C4—3792no—
117C22×C4—7584no—
118C22×C4C2×D37911yescharacteristic
119C4⋊C4—3792no—
120C4⋊C4—3791no—
121C4⋊C4C2×D37911yescharacteristic, complemented
122C4⋊C4C2×D37912yescomplemented
123C22⋊C4—3792no—
124C22⋊C4—3792no—
125C22⋊C4—3792no—
126C22⋊C4—3792no—
127C22⋊C4C2×D37912yes—
128C42—3792no—
129C23—3791no—
130C23—3791no—
131C23—3792no—
132C23—7582no—
133C23—7582no—
134C23—7582no—
135C23C22×D37911yescharacteristic
136Q8—22no—
137D4—7584no—
138D4—7584no—
139D4—7584no—
140D4—7582no—
141C2×C4—22no—
142C2×C4—3791no—
143C2×C4—3794no—
144C2×C4—3792no—
145C2×C4—3791no—
146C2×C4—7582no—
147C2×C4—7584no—
148C2×C4—7582no—
149C2×C4—7582no—
150C2×C4—7582no—
151C2×C4C22×D37912yes—
152C2×C4C22×D37911yescharacteristic
153C2×C4C22×D37912yes—
154C2×C4C22×D37911yescharacteristic
155C22—21no—
156C22—21no—
157C22—3792no—
158C22—3792no—
159C22—7588no—
160C22—7582no—
161C22—7582no—
162C22—7582no—
163C22—7581no—
164C22—7581no—
165C22—7581no—
166C22—7581no—
167C22#2025860712yescomplemented
168C22C23×D37911yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—3792no—
173C4—7581no—
174C4—7582no—
175C4D4×D37912yes—
176C2—22no—
177C2—15162no—
178C2—7582no—
179C2—7582no—
180C2C2×D4×D37911yescharacteristic
181C2#41×D37911yescharacteristic
182C2C2×#2025860711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D379
commutator, socle, cosocleC22×C379C24
FittingC22⋊Q8×C379C2
hypercenterC22⋊Q8D379
hypocenterC379#525
LCSG ▷ C22×C379 ▷ C379
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#20258607C2semidirect
2. C22×C4×D379C2semidirect
3. #27131316C2semidirect
4. #27131307C2semidirect
5. #27131306C2semidirect
6. #27131305C2semidirect
7. #27131301C2semidirect
8. #27131293C2semidirect
9. (C22⋊D379)⋊C4C2semidirect
10. #27131289C2semidirect
11. C22⋊Q8×C379C2semidirect
16. C2×D1516C22semidirect
24. C2×C4×D379C22semidirect
25. C4⋊Dic379C22semidirect
26. Dic379⋊C4C22semidirect
29. (C2×D379)⋊C4C22semidirect
30. (C2×D379)⋊C4C22semidirect
31. C4×Dic379C22semidirect
32. C2×Q8×C379C22semidirect
34. C4⋊C4×C379C22semidirect
35. C4⋊C4×C379C22semidirect
55. C4×D379D4semidirect
65. C22×C379#41semidirect
76. C2×D379#41semidirect
91. C379#525semidirect
97. C22⋊Q8D379semidirect
106. C2×Q8C2×D379semidirect
121. C4⋊C4C2×D379semidirect
122. C4⋊C4C2×D379semidirect
167. C22#20258607semidirect

Maximal subgroups

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

subgroupindex
C2×#202586072
C22×C4×D3792
#271313162
#271313072
#271313062
#271313052
#271313012
#271312932
(C22⋊D379)⋊C42
#271312892
C22⋊Q8×C3792
#525379