Finite Group Databasebeta

← #46837505

Subgroups — #46837505

Order 34688.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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×#22576160C211yesmaximal, characteristic
2C2×#22576084C212yesmaximal
3#31773659C212yesmaximal
4#31773652C211yesmaximal, characteristic
5#391×C271C211yesmaximal, characteristic
6C22×Dic542—21no—
7C22×C271⋊C8C2212yes—
8#22576160—24no—
9C2×C4⋊Dic271—21no—
10C2×C4⋊Dic271C2212yes—
11#22576084—24no—
12C22×Q8×C271C2211yescharacteristic
13C2×C4⋊C4×C271C2211yescharacteristic
14C2×C42×C271C2211yescharacteristic
15#95×C271—21no—
16#95×C271—21no—
17C2×Dic542—24no—
18C2×Dic542—42no—
19C2×Dic542—42no—
20C22×Dic271—22no—
21C2×C271⋊C8—22no—
22C2×C271⋊C8—22no—
23C2×C271⋊C8—22no—
24C4⋊Dic271—24no—
25C4⋊Dic271—22no—
26C4⋊Dic271D414yes—
27C2×Q8×C271—21no—
28C2×Q8×C271—22no—
29C2×Q8×C271—22no—
30C2×Q8×C271Q812yescomplemented
31C22×C4×C271—21no—
32C22×C4×C271—21no—
33C22×C4×C271D411yescharacteristic
34C22×C4×C271D411yescharacteristic
35C22×C4×C271C2311yescharacteristic
36C4⋊C4×C271—22no—
37C42×C271—22no—
38Dic542—48no—
39C2×Dic271—24no—
40C2×Dic271—42no—
41C2×Dic271—42no—
42C271⋊C8—42no—
43C23×C271C2×D411yescharacteristic
44Q8×C271—24no—
45Q8×C271—42no—
46C2×C4×C271—21no—
47C2×C4×C271—21no—
48C2×C4×C271—21no—
49C2×C4×C271—21no—
50C2×C4×C271—22no—
51C2×C4×C271—41no—
52C2×C4×C271—41no—
53C2×C4×C271—41no—
54C2×C4×C271—41no—
55C2×C4×C271—41no—
56C2×C4×C271SD1612yes—
57C2×C4×C271Q1612yes—
58C2×C4×C271C2×D412yes—
59C2×C4×C271C2×Q811yescharacteristic
60C2×C4×C271#4112yes—
61C2×C4×C271#4111yescharacteristic
62C22×C271#12011yescharacteristic
63C22×C271C22⋊Q811yescharacteristic
64C22×C271#12411yescharacteristic
65C22×C271C2×SD1611yescharacteristic
66C22×C271C2×Q1611yescharacteristic
67C22×C271AGL(1,ℤ/8)11yescharacteristic
68C22×C271#13711yescharacteristic
69C4×C271—22no—
70C4×C271—22no—
71C4×C271—42no—
72C4×C271—81no—
73C4×C271C22⋊Q812yes—
74C4×C271#12412yes—
75Dic271—44no—
76C2×C271#39311yescharacteristic
77C2×C271#45011yescharacteristic
78C2×C271#45111yescharacteristic
79C2×C271#47511yescharacteristic
80C2×C271#47711yescharacteristic
81C2×C271#48711yescharacteristic
82C2×C271#48911yescharacteristic
83C271#202411yescharacteristic, complemented, Sylow
84#2024—2711nomaximal, Sylow
85C2×C4⋊Q8—2711no—
86C2×#103—2712no—
87#391D27111yescharacteristic
88#340—2712no—
89#336—2711no—
90C22×Q8—5421no—
91C22×Q8C2×D27111yescharacteristic
92C22×C8—2712no—
93C4⋊Q8—5424no—
94C2×C4⋊C4—2712no—
95C2×C4⋊C4—5421no—
96C2×C4⋊C4C2×D27111yescharacteristic
97C2×C42C2×D27111yescharacteristic
98#103—5424no—
99#95—21no—
100#95—21no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—5424no—
105C2×Q8—10842no—
106C2×Q8—10842no—
107C2×Q8Dic54212yescomplemented
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—5422no—
111C22×C4C22⋊D27111yescharacteristic
112C22×C4C22⋊D27111yescharacteristic
113C22×C4C22×D27111yescharacteristic
114C2×C8—5422no—
115C2×C8—5422no—
116C2×C8—5422no—
117C4⋊C4—22no—
118C4⋊C4—2714no—
119C4⋊C4—5424no—
120C4⋊C4—5422no—
121C42—22no—
122C23C2×C22⋊D27111yescharacteristic
123Q8—24no—
124Q8—42no—
125Q8—10848no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—5424no—
137C2×C4—10842no—
138C2×C4—10842no—
139C2×C4#1686464112yes—
140C2×C4#1686464412yes—
141C2×C4D4×D27112yes—
142C2×C4C2×Dic54211yescharacteristic
143C2×C4#1686465312yes—
144C2×C4#1686465411yescharacteristic
145C8—10842no—
146C22#2257612811yescharacteristic
147C22#2257613911yescharacteristic
148C22#2257614711yescharacteristic
149C22C2×#1686464111yescharacteristic
150C22C2×#1686464411yescharacteristic
151C22#2257618511yescharacteristic
152C22#2257619011yescharacteristic
153C4—22no—
154C4—22no—
155C4—42no—
156C4—81no—
157C4—10844no—
158C4#2257613712yes—
159C4#2257614612yes—
160C2#3177380911yescharacteristic
161C2#3177404711yescharacteristic
162C2#3177405211yescharacteristic
163C2#3177418211yescharacteristic
164C2#3177419211yescharacteristic
165C2#3177424711yescharacteristic
166C2#3177425611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D271
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#391×C271C2
socle, prime radicalC23×C271C2×D4
hypercenter#391D271
hypocenterC271#2024
LCSG ▷ C22×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C271Q8semidirect
83. C271#2024semidirect
107. C2×Q8Dic542semidirect

Maximal subgroups

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

subgroupindex
C2×#225761602
C2×#225760842
#317736592
#317736522
#391×C2712
#2024271