Finite Group Databasebeta

← #396×D271 #46839583

Subgroups — #396×D271 #46839583

Order 34688.

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×D271C213yesmaximal, complemented
2C2×C22⋊C4×D271C213yesmaximal
3#31773821C211yesmaximal, characteristic, complemented
4#31773820C213yesmaximal, complemented
5#31773819C213yesmaximal, complemented
6#95×D271C211yesmaximal, characteristic, complemented
7#396×C271C211yesmaximal, characteristic, complemented
8C24×D271C2211yescharacteristic
9C22×C4×D271C2213yes—
10C22×C4×D271C2213yes—
11C2×C4⋊Dic271C2213yescomplemented
12C2×Dic271⋊C4C2213yescomplemented
13C2×Dic271⋊C4C2213yescomplemented
14C4⋊C4×D271—212no—
15C2×C22⋊Dic271C2213yes—
16C2×(C2×D271)⋊C4C2213yes—
17C2×(C2×D271)⋊C4C2213yes—
18C22⋊C4×D271—212no—
19C271⋊#95C2213yescomplemented
20C2×C4⋊C4×C271C2213yescomplemented
21C2×C22⋊C4×C271C2213yes—
22#95×C271C2211yescharacteristic, complemented
23C23×D271—26no—
24C23×D271—26no—
25C23×D271Q814yes—
26C23×D271C2311yescharacteristic
27C23×D271C2311yescharacteristic
28C22×Dic271C2313yes—
29C22×Dic271C2313yes—
30C2×C4×D271—26no—
31C2×C4×D271—26no—
32C2×C4×D271—26no—
33C2×C4×D271—26no—
34C2×C4×D271—26no—
35C2×C4×D271D4112yes—
36C4⋊Dic271—26no—
37Dic271⋊C4—26no—
38Dic271⋊C4—26no—
39C22⋊Dic271—26no—
40(C2×D271)⋊C4—26no—
41(C2×D271)⋊C4—26no—
42C24×C271C2311yescharacteristic
43C22×C4×C271C2313yes—
44C22×C4×C271C2313yes—
45C4⋊C4×C271—26no—
46C22⋊C4×C271—26no—
47C22×D271—23no—
48C22×D271—23no—
49C22×D271—224no—
50C22×D271—412no—
51C22×D271—44no—
52C22×D271C2×D416yes—
53C22×D271C2×Q812yes—
54C22×D271C2×Q812yes—
55C22×D271#4116yes—
56C2×Dic271—23no—
57C2×Dic271—23no—
58C2×Dic271—23no—
59C2×Dic271—23no—
60C2×Dic271—23no—
61C2×Dic271C2×D416yes—
62C4×D271—224no—
63C4×D271—412no—
64C23×C271—23no—
65C23×C271—23no—
66C23×C271C2×Q812yes—
67C23×C271C2411yescharacteristic
68C2×C4×C271—23no—
69C2×C4×C271—23no—
70C2×C4×C271—23no—
71C2×C4×C271—23no—
72C2×C4×C271—23no—
73C2×C4×C271C2×D416yes—
74C22×C271—26no—
75C22×C271—43no—
76C22×C271—41no—
77C22×C271C22×D413yes—
78C22×C271C22×Q811yescharacteristic
79C22×C271C2×#4113yes—
80C2×D271—26no—
81C2×D271—43no—
82C2×D271—41no—
83C2×D271—416no—
84C2×D271#121112yes—
85C2×D271C22⋊Q8112yes—
86C2×D271#12614yes—
87C4×C271—26no—
88C4×C271—43no—
89Dic271—26no—
90Dic271—43no—
91C2×C271—42no—
92C2×C271C2×#12113yes—
93C2×C271C2×C22⋊Q813yes—
94C2×C271C2×#12611yescharacteristic
95D271—42no—
96D271#39618yesdirect product factor
97C271C2×#39611yescharacteristic, complemented, Sylow
98C2×#396—2711nomaximal, Sylow
99C22×C4⋊C4—2713no—
100C22×C22⋊C4—2713no—
101#396—2713no—
102#396—2713no—
103#396—2711no—
104#396D27111yescharacteristic, direct product factor
105C2×#95—2711no—
106C25—2711no—
107C23×C4—2713no—
108C23×C4—2713no—
109C2×C4⋊C4—2713no—
110C2×C4⋊C4—2713no—
111C2×C4⋊C4—2713no—
112C2×C4⋊C4—54212no—
113C2×C4⋊C4C2×D27113yescomplemented
114C2×C22⋊C4—2713no—
115C2×C22⋊C4—2713no—
116C2×C22⋊C4—2713no—
117C2×C22⋊C4—54212no—
118C2×C22⋊C4C2×D27113yes—
119#95—2713no—
120#95C2×D27111yescharacteristic, complemented
121C24—2711no—
122C24—2714no—
123C24—2711no—
124C24—5426no—
125C24—5426no—
126C24C22×D27111yescharacteristic
127C22×C4—2713no—
128C22×C4—27112no—
129C22×C4—2713no—
130C22×C4—5426no—
131C22×C4—5426no—
132C22×C4—5426no—
133C22×C4—5426no—
134C22×C4—5426no—
135C22×C4C22×D27113yes—
136C22×C4C22×D27113yes—
137C4⋊C4—26no—
138C4⋊C4—5426no—
139C4⋊C4—5426no—
140C4⋊C4—5426no—
141C22⋊C4—26no—
142C22⋊C4—5426no—
143C22⋊C4—5426no—
144C22⋊C4—5426no—
145C23—23no—
146C23—23no—
147C23—2716no—
148C23—2712no—
149C23—2716no—
150C23—2712no—
151C23—54224no—
152C23—5423no—
153C23—5423no—
154C23—108412no—
155C23—10844no—
156C23Q8×D27112yes—
157C23C23×D27111yescharacteristic
158C2×C4—23no—
159C2×C4—23no—
160C2×C4—23no—
161C2×C4—23no—
162C2×C4—23no—
163C2×C4—2716no—
164C2×C4—54224no—
165C2×C4—5423no—
166C2×C4—5423no—
167C2×C4—5423no—
168C2×C4—5423no—
169C2×C4—5423no—
170C2×C4—108412no—
171C2×C4D4×D27116yes—
172C22—26no—
173C22—43no—
174C22—41no—
175C22—27112no—
176C22—27112no—
177C22—2714no—
178C22—5426no—
179C22—108416no—
180C22—10843no—
181C22—10841no—
182C22C2×D4×D27113yes—
183C22C2×Q8×D27111yescharacteristic
184C22#41×D27113yes—
185C4—26no—
186C4—43no—
187C4—5426no—
188C4—10843no—
189C2—42no—
190C2—2718no—
191C2—10842no—
192C2#121×D27113yes—
193C2C22⋊Q8×D27113yes—
194C2#126×D27111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#396×C271C2
prime radicalC24×D271C22
hypercenter#396D271
hypocenterC271C2×#396
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D271C2semidirect
3. #31773821C2semidirect
4. #31773820C2semidirect
5. #31773819C2semidirect
6. #95×D271C2semidirect
7. #396×C271C2semidirect
11. C2×C4⋊Dic271C22semidirect
12. C2×Dic271⋊C4C22semidirect
13. C2×Dic271⋊C4C22semidirect
19. C271⋊#95C22semidirect
20. C2×C4⋊C4×C271C22semidirect
22. #95×C271C22semidirect
96. D271#396direct (N × H)
97. C271C2×#396semidirect
104. #396D271direct (N × H)
113. C2×C4⋊C4C2×D271semidirect
120. #95C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D2712
C2×C22⋊C4×D2712
#317738212
#317738202
#317738192
#95×D2712
#396×C2712
C2×#396271