Finite Group Databasebeta

← #396×D239 #44487767

Subgroups — #396×D239 #44487767

Order 30592.

908 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 203 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×C4⋊C4×D239C213yesmaximal, complemented
2C2×C22⋊C4×D239C213yesmaximal
3#29504696C211yesmaximal, characteristic, complemented
4#29504695C213yesmaximal, complemented
5#29504694C213yesmaximal, complemented
6#95×D239C211yesmaximal, characteristic, complemented
7#396×C239C211yesmaximal, characteristic, complemented
8C24×D239C2211yescharacteristic
9C22×C4×D239C2213yes—
10C22×C4×D239C2213yes—
11C2×C4⋊Dic239C2213yescomplemented
12C2×Dic239⋊C4C2213yescomplemented
13C2×Dic239⋊C4C2213yescomplemented
14C4⋊C4×D239—212no—
15C2×C22⋊Dic239C2213yes—
16C2×(C2×D239)⋊C4C2213yes—
17C2×(C2×D239)⋊C4C2213yes—
18C22⋊C4×D239—212no—
19C239⋊#95C2213yescomplemented
20C2×C4⋊C4×C239C2213yescomplemented
21C2×C22⋊C4×C239C2213yes—
22#95×C239C2211yescharacteristic, complemented
23C23×D239—26no—
24C23×D239—26no—
25C23×D239Q814yes—
26C23×D239C2311yescharacteristic
27C23×D239C2311yescharacteristic
28C22×Dic239C2313yes—
29C22×Dic239C2313yes—
30C2×C4×D239—26no—
31C2×C4×D239—26no—
32C2×C4×D239—26no—
33C2×C4×D239—26no—
34C2×C4×D239—26no—
35C2×C4×D239D4112yes—
36C4⋊Dic239—26no—
37Dic239⋊C4—26no—
38Dic239⋊C4—26no—
39C22⋊Dic239—26no—
40(C2×D239)⋊C4—26no—
41(C2×D239)⋊C4—26no—
42C24×C239C2311yescharacteristic
43C22×C4×C239C2313yes—
44C22×C4×C239C2313yes—
45C4⋊C4×C239—26no—
46C22⋊C4×C239—26no—
47C22×D239—23no—
48C22×D239—23no—
49C22×D239—224no—
50C22×D239—412no—
51C22×D239—44no—
52C22×D239C2×D416yes—
53C22×D239C2×Q812yes—
54C22×D239C2×Q812yes—
55C22×D239#4116yes—
56C2×Dic239—23no—
57C2×Dic239—23no—
58C2×Dic239—23no—
59C2×Dic239—23no—
60C2×Dic239—23no—
61C2×Dic239C2×D416yes—
62C4×D239—224no—
63C4×D239—412no—
64C23×C239—23no—
65C23×C239—23no—
66C23×C239C2×Q812yes—
67C23×C239C2411yescharacteristic
68C2×C4×C239—23no—
69C2×C4×C239—23no—
70C2×C4×C239—23no—
71C2×C4×C239—23no—
72C2×C4×C239—23no—
73C2×C4×C239C2×D416yes—
74C22×C239—26no—
75C22×C239—43no—
76C22×C239—41no—
77C22×C239C22×D413yes—
78C22×C239C22×Q811yescharacteristic
79C22×C239C2×#4113yes—
80C2×D239—26no—
81C2×D239—43no—
82C2×D239—41no—
83C2×D239—416no—
84C2×D239#121112yes—
85C2×D239C22⋊Q8112yes—
86C2×D239#12614yes—
87C4×C239—26no—
88C4×C239—43no—
89Dic239—26no—
90Dic239—43no—
91C2×C239—42no—
92C2×C239C2×#12113yes—
93C2×C239C2×C22⋊Q813yes—
94C2×C239C2×#12611yescharacteristic
95D239—42no—
96D239#39618yesdirect product factor
97C239C2×#39611yescharacteristic, complemented, Sylow
98C2×#396—2391nomaximal, Sylow
99C22×C4⋊C4—2393no—
100C22×C22⋊C4—2393no—
101#396—2393no—
102#396—2393no—
103#396—2391no—
104#396D23911yescharacteristic, direct product factor
105C2×#95—2391no—
106C25—2391no—
107C23×C4—2393no—
108C23×C4—2393no—
109C2×C4⋊C4—47812no—
110C2×C4⋊C4—2393no—
111C2×C4⋊C4—2393no—
112C2×C4⋊C4—2393no—
113C2×C4⋊C4C2×D23913yescomplemented
114C2×C22⋊C4—47812no—
115C2×C22⋊C4—2393no—
116C2×C22⋊C4—2393no—
117C2×C22⋊C4—2393no—
118C2×C22⋊C4C2×D23913yes—
119#95—2393no—
120#95C2×D23911yescharacteristic, complemented
121C24—4786no—
122C24—4786no—
123C24—2394no—
124C24—2391no—
125C24—2391no—
126C24C22×D23911yescharacteristic
127C22×C4—4786no—
128C22×C4—4786no—
129C22×C4—4786no—
130C22×C4—4786no—
131C22×C4—4786no—
132C22×C4—2393no—
133C22×C4—2393no—
134C22×C4—23912no—
135C22×C4C22×D23913yes—
136C22×C4C22×D23913yes—
137C4⋊C4—26no—
138C4⋊C4—4786no—
139C4⋊C4—4786no—
140C4⋊C4—4786no—
141C22⋊C4—26no—
142C22⋊C4—4786no—
143C22⋊C4—4786no—
144C22⋊C4—4786no—
145C23—23no—
146C23—23no—
147C23—9564no—
148C23—95612no—
149C23—47824no—
150C23—4783no—
151C23—4783no—
152C23—2392no—
153C23—2396no—
154C23—2396no—
155C23—2392no—
156C23Q8×D23912yes—
157C23C23×D23911yescharacteristic
158C2×C4—23no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—23no—
162C2×C4—23no—
163C2×C4—95612no—
164C2×C4—47824no—
165C2×C4—4783no—
166C2×C4—4783no—
167C2×C4—4783no—
168C2×C4—4783no—
169C2×C4—4783no—
170C2×C4—2396no—
171C2×C4D4×D23916yes—
172C22—26no—
173C22—41no—
174C22—43no—
175C22—95616no—
176C22—9561no—
177C22—9563no—
178C22—4786no—
179C22—2394no—
180C22—23912no—
181C22—23912no—
182C22C2×D4×D23913yes—
183C22C2×Q8×D23911yescharacteristic
184C22#41×D23913yes—
185C4—26no—
186C4—43no—
187C4—9563no—
188C4—4786no—
189C2—42no—
190C2—9562no—
191C2—2398no—
192C2#121×D23913yes—
193C2C22⋊Q8×D23913yes—
194C2#126×D23911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D239
commutator, socle, cosocleC23×C239C24
Fitting#396×C239C2
prime radicalC24×D239C22
hypercenter#396D239
hypocenterC239C2×#396
LCSG ▷ C23×C239 ▷ C239
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D239C2semidirect
3. #29504696C2semidirect
4. #29504695C2semidirect
5. #29504694C2semidirect
6. #95×D239C2semidirect
7. #396×C239C2semidirect
11. C2×C4⋊Dic239C22semidirect
12. C2×Dic239⋊C4C22semidirect
13. C2×Dic239⋊C4C22semidirect
19. C239⋊#95C22semidirect
20. C2×C4⋊C4×C239C22semidirect
22. #95×C239C22semidirect
96. D239#396direct (N × H)
97. C239C2×#396semidirect
104. #396D239direct (N × H)
113. C2×C4⋊C4C2×D239semidirect
120. #95C2×D239semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D2392
C2×C22⋊C4×D2392
#295046962
#295046952
#295046942
#95×D2392
#396×C2392
C2×#396239