Finite Group Databasebeta

← #46836554

Subgroups — #46836554

Order 34688.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C22×C4⋊Dic271C211yesmaximal, characteristic
2C2×#22576094C212yesmaximal, complemented
3#31773757C211yesmaximal, characteristic
4#31773660C212yesmaximal, complemented
5C2×C22⋊C8×C271C211yesmaximal, characteristic
6C23×Dic271—21no—
7C2×C4⋊Dic271—24no—
8C2×C4⋊Dic271—21no—
9C2×C4⋊Dic271C2211yescharacteristic
10C2×C4⋊Dic271C2211yescharacteristic
11C2×C4⋊Dic271C2212yes—
12C2×C22⋊Dic271—21no—
13#22576094—24no—
14C271⋊#95—21no—
15C23×C4×C271C2211yescharacteristic
16C22×C8×C271C2212yes—
17C22⋊C8×C271C414yescomplemented
18C22×Dic271—24no—
19C22×Dic271—22no—
20C22×Dic271—21no—
21C22×Dic271—21no—
22C22×Dic271—42no—
23C22×Dic271—42no—
24C4⋊Dic271—24no—
25C4⋊Dic271—22no—
26C4⋊Dic271—22no—
27C4⋊Dic271—22no—
28C4⋊Dic271D414yes—
29C22⋊Dic271—42no—
30C24×C271D411yescharacteristic
31C22×C4×C271—22no—
32C22×C4×C271—22no—
33C22×C4×C271C2×C412yes—
34C22×C4×C271D411yescharacteristic
35C22×C4×C271Q812yes—
36C22×C4×C271C2311yescharacteristic
37C2×C8×C271—22no—
38C2×C8×C271—22no—
39C2×C8×C271C2×C414yescomplemented
40C2×Dic271—24no—
41C2×Dic271—48no—
42C2×Dic271—42no—
43C2×Dic271—42no—
44C2×Dic271—41no—
45C2×Dic271—41no—
46C2×Dic271—41no—
47C2×Dic271—41no—
48C2×Dic271—41no—
49C2×Dic271—41no—
50C23×C271—21no—
51C23×C271—21no—
52C23×C271—21no—
53C23×C271—21no—
54C23×C271C4⋊C412yes—
55C23×C271D812yes—
56C23×C271Q1612yes—
57C23×C271C2×D411yescharacteristic
58C2×C4×C271—21no—
59C2×C4×C271—21no—
60C2×C4×C271—24no—
61C2×C4×C271—24no—
62C2×C4×C271C4⋊C412yes—
63C2×C4×C271C22×C411yescharacteristic
64C2×C4×C271C2×D412yes—
65C2×C4×C271C2×Q811yescharacteristic
66C2×C4×C271#4112yes—
67C8×C271—24no—
68C22×C271—22no—
69C22×C271—22no—
70C22×C271—22no—
71C22×C271—22no—
72C22×C271—22no—
73C22×C271—22no—
74C22×C271#10714yescomplemented
75C22×C271C2×C4⋊C411yescharacteristic
76C22×C271#12011yescharacteristic
77C22×C271#12311yescharacteristic
78C22×C271C2×D811yescharacteristic
79C22×C271C2×Q1611yescharacteristic
80C22×C271AGL(1,ℤ/8)11yescharacteristic
81C22×C271#13711yescharacteristic
82C4×C271—22no—
83C4×C271C4×D412yes—
84C4×C271C22⋊Q812yes—
85Dic271—44no—
86Dic271—81no—
87Dic271—81no—
88C2×C271—24no—
89C2×C271#38511yescharacteristic
90C2×C271C2×#10711yescharacteristic
91C2×C271#42811yescharacteristic
92C2×C271#44711yescharacteristic
93C2×C271#45111yescharacteristic
94C2×C271#48011yescharacteristic
95C2×C271#48411yescharacteristic
96C271#185211yescharacteristic, complemented, Sylow
97#1852—2711nomaximal, Sylow
98C22×C4⋊C4—2711no—
99C2×#107—2712no—
100C2×C22⋊C8D27111yescharacteristic
101#380—2711no—
102#340—2712no—
103C23×C4—5421no—
104C23×C4C2×D27111yescharacteristic
105C22×C8C2×D27112yes—
106C2×C4⋊C4—2712no—
107C2×C4⋊C4—2711no—
108C2×C4⋊C4—2711no—
109C2×C4⋊C4—5424no—
110C2×C4⋊C4—5421no—
111C2×C22⋊C4—5421no—
112#107—5424no—
113C22⋊C8Dic27114yescomplemented
114#95—5421no—
115C24D108411yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—5424no—
119C22×C4—5422no—
120C22×C4—5421no—
121C22×C4—5421no—
122C22×C4—10842no—
123C22×C4—10842no—
124C22×C4C2×Dic27112yes—
125C22×C4D108411yescharacteristic
126C22×C4Dic54212yes—
127C22×C4C22×D27111yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8C2×Dic27114yescomplemented
131C4⋊C4—2714no—
132C4⋊C4—5424no—
133C4⋊C4—5422no—
134C4⋊C4—5422no—
135C4⋊C4—5422no—
136C22⋊C4—10842no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23C4⋊Dic27112yes—
142C23D216812yes—
143C23Dic108412yes—
144C23C2×D108411yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—5424no—
150C2×C4—10848no—
151C2×C4—10842no—
152C2×C4—10842no—
153C2×C4—10841no—
154C2×C4—10841no—
155C2×C4—10841no—
156C2×C4—10841no—
157C2×C4—10841no—
158C2×C4—10841no—
159C2×C4C4⋊Dic27112yes—
160C2×C4C22×Dic27111yescharacteristic
161C2×C4D4×D27112yes—
162C2×C4C2×Dic54211yescharacteristic
163C2×C4#1686465312yes—
164C8—24no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#2257609414yescomplemented
172C22C2×C4⋊Dic27111yescharacteristic
173C22#2257613011yescharacteristic
174C22#2257614411yescharacteristic
175C22C2×D216811yescharacteristic
176C22C2×Dic108411yescharacteristic
177C22#2257618811yescharacteristic
178C22#2257619311yescharacteristic
179C4—22no—
180C4—10844no—
181C4—21681no—
182C4—21681no—
183C4D4×Dic27112yes—
184C4#2257613712yes—
185C2—24no—
186C2#3177377211yescharacteristic
187C2C2×#2257609411yescharacteristic
188C2#3177393911yescharacteristic
189C2#3177403311yescharacteristic
190C2#3177405511yescharacteristic
191C2#3177421111yescharacteristic
192C2#3177423311yescharacteristic
subgroupG/H
centerC23C2×D1084
commutatorC2×C4×C271C22×C4
FrattiniC22×C4C22×D271
FittingC2×C22⋊C8×C271C2
socleC23×C271C2×D4
prime radicalC24×C271D4
cosocleC22×C4×C271C23
hypercenterC2×C22⋊C8D271
hypocenterC271#1852
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#22576094C2semidirect
4. #31773660C2semidirect
17. C22⋊C8×C271C4semidirect
39. C2×C8×C271C2×C4semidirect
74. C22×C271#107semidirect
96. C271#1852semidirect
113. C22⋊C8Dic271semidirect
130. C2×C8C2×Dic271semidirect
171. C22#22576094semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic2712
C2×#225760942
#317737572
#317736602
C2×C22⋊C8×C2712
#1852271