Finite Group Databasebeta

← #41013685

Subgroups — #41013685

Order 27008.

260 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#27213912C211yesmaximal, characteristic
2#27213174C211yesmaximal, characteristic, complemented
3#27213173C211yesmaximal, characteristic, complemented
4#27213119C211yesmaximal, characteristic
5#27213114C211yesmaximal, characteristic
6#27212918C211yesmaximal, characteristic, complemented
7#344×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20269985—22no—
10#20269983—22no—
11C2×#16809946C2211yescharacteristic
12M16×D211—22no—
13C22×C211⋊C8C2211yescharacteristic
14#20269936C2211yescharacteristic
15C4⋊(C211⋊C8)—22no—
16Dic211⋊C8—22no—
17#20269891C414yes—
18(C2×D211)⋊C8—22no—
19C2×M16×C211C2211yescharacteristic
20C2×M16×C211C2211yescharacteristic
21#117×C211C2211yescharacteristic
22#16809962—21no—
23#16809962—21no—
24#16809962—21no—
25#16809962—21no—
26C2×Dic422Q811yescharacteristic
27C2×D844Q811yescharacteristic
28C2×C22⋊D211C2×C412yes—
29C2×C4×D211D412yes—
30#16809946—21no—
31#16809946—21no—
32#16809945—24no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8—21no—
36C2×C211⋊C8D412yes—
37C8×D211—24no—
38C4⋊Dic211—21no—
39C22⋊Dic211C2×C412yes—
40C4×Dic211—21no—
41C22×C4×C211C2311yescharacteristic
42M16×C211—21no—
43M16×C211—21no—
44M16×C211—21no—
45M16×C211—21no—
46C2×C8×C211—21no—
47C2×C8×C211D412yes—
48C4⋊C4×C211—21no—
49C22⋊C4×C211C2×C412yes—
50C42×C211—21no—
51C22×D211C4⋊C412yes—
52Dic422—21no—
53Dic422—21no—
54D844—21no—
55D844—21no—
56C22⋊D211—22no—
57C22⋊D211—22no—
58C2×Dic211—22no—
59C2×Dic211C4⋊C412yes—
60C4×D211—22no—
61C4×D211—22no—
62C211⋊C8—22no—
63C211⋊C8—41no—
64C23×C211C22×C411yescharacteristic
65C2×C4×C211—22no—
66C2×C4×C211C2×D411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×Q811yescharacteristic
70C2×C4×C211#4111yescharacteristic
71C2×C4×C211#4111yescharacteristic
72C8×C211—22no—
73C8×C211—41no—
74C22×C211—41no—
75C22×C211C2×C4⋊C411yescharacteristic
76C22×C211C4×D411yescharacteristic
77C22×C211C4×D411yescharacteristic
78C2×D211—22no—
79C2×D211—42no—
80C4×C211—42no—
81C4×C211#12011yescharacteristic
82C4×C211C22⋊Q811yescharacteristic
83C4×C211C22⋊Q811yescharacteristic
84C4×C211#12311yescharacteristic
85Dic211—22no—
86Dic211—42no—
87C2×C211—21no—
88C2×C211—21no—
89C2×C211—21no—
90C2×C211#38511yescharacteristic
91D211—42no—
92C211#186811yescharacteristic, complemented, Sylow
93#1868—2111nomaximal, Sylow
94#568—2111no—
95#424—2111no—
96#424—2111no—
97#410—2111no—
98#408—2111no—
99#344—2111no—
100#344D21111yescharacteristic, complemented
101C2×#41—2111no—
102#131—4222no—
103#131—4222no—
104C2×M16—4222no—
105C2×M16—2111no—
106C2×M16C2×D21111yescharacteristic
107C2×M16C2×D21111yescharacteristic
108C22×C8—2111no—
109#117—2111no—
110#117C2×D21111yescharacteristic
111C4⋊C8—4222no—
112C4⋊C8—4222no—
113#99—2114no—
114C22⋊C8—4222no—
115#41—4221no—
116#41—4221no—
117#41—4221no—
118#41—4221no—
119C2×Q8—2111no—
120C2×D4—2112no—
121C2×D4—2111no—
122C22×C4—2112no—
123C22×C4C22×D21111yescharacteristic
124M16—21no—
125M16—21no—
126M16—21no—
127M16—21no—
128M16—4224no—
129M16—4221no—
130M16—4221no—
131C2×C8—21no—
132C2×C8—4224no—
133C2×C8—4221no—
134C2×C8—4221no—
135C2×C8—4221no—
136C2×C8—2112no—
137C2×C8D84412yes—
138C4⋊C4—21no—
139C4⋊C4—4221no—
140C22⋊C4—2112no—
141C22⋊C4C4×D21112yes—
142C42—21no—
143C42—4221no—
144C23—2112no—
145C23C2×C4×D21111yescharacteristic
146Q8—4221no—
147Q8—4221no—
148D4—4222no—
149D4—4222no—
150D4—4221no—
151D4—4221no—
152C2×C4—22no—
153C2×C4—4222no—
154C2×C4—4222no—
155C2×C4—4222no—
156C2×C4—2112no—
157C2×C4D4×D21111yescharacteristic
158C2×C4D4×D21111yescharacteristic
159C2×C4C2×D84411yescharacteristic
160C2×C4Q8×D21111yescharacteristic
161C2×C4#1680996111yescharacteristic
162C2×C4#1680996211yescharacteristic
163C8—22no—
164C8—41no—
165C8—8441no—
166C8—4222no—
167C22—41no—
168C22—8442no—
169C22—4222no—
170C22C4⋊C4×D21111yescharacteristic
171C22(C22⋊D211)⋊C411yescharacteristic
172C22C4×D84411yescharacteristic
173C4—42no—
174C4—8442no—
175C4—4222no—
176C4#2026994711yescharacteristic
177C4#2026995511yescharacteristic
178C4#2026995711yescharacteristic
179C4#2026995911yescharacteristic
180C2—21no—
181C2—21no—
182C2—21no—
183C2—8442no—
184C2#2721302011yescharacteristic
subgroupG/H
centerC4#20269947
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#344×C211C2
socleC2×C211#385
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenter#344D211
hypocenterC211#1868
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #27213174C2semidirect
3. #27213173C2semidirect
6. #27212918C2semidirect
7. #344×C211C2semidirect
92. C211#1868semidirect
100. #344D211semidirect

Maximal subgroups

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

subgroupindex
#272139122
#272131742
#272131732
#272131192
#272131142
#272129182
#344×C2112
#1868211