Finite Group Databasebeta

← #41014573

Subgroups — #41014573

Order 27008.

244 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 63 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#27213699C211yesmaximal, characteristic, complemented
2#27213162C211yesmaximal, characteristic
3C2×#20269903C211yesmaximal, characteristic
4#27213119C211yesmaximal, characteristic
5#27212907C211yesmaximal, characteristic, complemented
6#27212903C211yesmaximal, characteristic, complemented
7#344×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20269976—21no—
10#20269968—21no—
11#20269954—22no—
12C4×Dic422—22no—
13#20269936C2211yescharacteristic
14#20269936C2211yescharacteristic
15C2×C4×Dic211C2211yescharacteristic
16#20269903—21no—
17#20269903—21no—
18#20269903—22no—
19#20269891—22no—
20C2×M16×C211C2211yescharacteristic
21C2×M16×C211C2211yescharacteristic
22#117×C211C2211yescharacteristic
23#16809962—21no—
24#16809962—21no—
25#16809962—21no—
26#16809962—21no—
27C2×Dic422—22no—
28C2×Dic422Q811yescharacteristic, complemented
29C2×D844Q811yescharacteristic, complemented
30C2×C22⋊D211—21no—
31C22×Dic211—21no—
32C2×C4×D211—21no—
33C4⋊Dic211—22no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36Dic211⋊C4—22no—
37Dic211⋊C4—22no—
38C22⋊Dic211—21no—
39C22⋊Dic211—21no—
40C4×Dic211—21no—
41C4×Dic211—21no—
42C4×Dic211D411yescharacteristic
43C4×Dic211D411yescharacteristic
44C4×Dic211D412yes—
45C22×C4×C211C2311yescharacteristic
46M16×C211—21no—
47M16×C211—21no—
48M16×C211—21no—
49M16×C211—21no—
50C2×C8×C211—21no—
51C2×C8×C211—21no—
52C4⋊C4×C211—21no—
53C22⋊C4×C211D412yes—
54C42×C211—21no—
55C22×D211—21no—
56Dic422—21no—
57Dic422—21no—
58Dic422—44no—
59D844—21no—
60D844—21no—
61C22⋊D211—41no—
62C22⋊D211—41no—
63C2×Dic211—21no—
64C2×Dic211—21no—
65C2×Dic211—22no—
66C2×Dic211—21no—
67C2×Dic211—42no—
68C2×Dic211—41no—
69C4×D211—41no—
70C4×D211—41no—
71C23×C211C2×D411yescharacteristic
72C2×C4×C211—22no—
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211C2×Q811yescharacteristic
76C2×C4×C211#4111yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C8×C211—41no—
80C8×C211—41no—
81C22×C211—41no—
82C22×C211#12011yescharacteristic
83C22×C211C22⋊Q811yescharacteristic
84C22×C211#12411yescharacteristic
85C2×D211—41no—
86C2×D211—81no—
87C4×C211—42no—
88C4×C211C22⋊Q811yescharacteristic
89C4×C211C22⋊Q811yescharacteristic
90C4×C211#12411yescharacteristic
91C4×C211#12411yescharacteristic
92Dic211—22no—
93Dic211—41no—
94Dic211—42no—
95Dic211—41no—
96Dic211—81no—
97C2×C211—21no—
98C2×C211—21no—
99C2×C211—21no—
100C2×C211#39311yescharacteristic
101D211—81no—
102C211#203111yescharacteristic, complemented, Sylow
103#2031—2111nomaximal, Sylow
104#533—2111no—
105#421—2111no—
106C2×#104—2111no—
107#410—2111no—
108#344D21111yescharacteristic, complemented
109#339—2111no—
110H(3,ℤ/4)—2111no—
111C2×#41—2111no—
112C2×M16C2×D21111yescharacteristic
113C2×M16C2×D21111yescharacteristic
114C4⋊Q8—4221no—
115#125—4221no—
116C22⋊Q8—4222no—
117C4×Q8—4222no—
118#117—2111no—
119#117—2111no—
120#117C2×D21111yescharacteristic
121C2×C42—2111no—
122#104—4221no—
123#104—4221no—
124#104—4222no—
125#99—4222no—
126#41—4221no—
127#41—4221no—
128#41—4221no—
129#41—4221no—
130C2×Q8—4222no—
131C2×Q8—2111no—
132C2×D4—4221no—
133C2×D4—2111no—
134C22×C4—4221no—
135C22×C4—4221no—
136C22×C4C22×D21111yescharacteristic
137M16—21no—
138M16—21no—
139M16—21no—
140M16—21no—
141C2×C8—21no—
142C2×C8—21no—
143C4⋊C4—21no—
144C4⋊C4—4222no—
145C4⋊C4—4222no—
146C4⋊C4—4222no—
147C4⋊C4—4221no—
148C4⋊C4—4221no—
149C22⋊C4—4221no—
150C22⋊C4—4221no—
151C22⋊C4D84412yes—
152C42—21no—
153C42—4221no—
154C42—4221no—
155C42—2111no—
156C42—2111no—
157C42—2112no—
158C23—4221no—
159C23C2×D84411yescharacteristic
160Q8—8444no—
161Q8—4221no—
162Q8—4221no—
163D4—8441no—
164D4—8441no—
165D4—4221no—
166D4—4221no—
167C2×C4—22no—
168C2×C4—8442no—
169C2×C4—8441no—
170C2×C4—8441no—
171C2×C4—8441no—
172C2×C4—4221no—
173C2×C4—4222no—
174C2×C4—4221no—
175C2×C4—4221no—
176C2×C4D4×D21111yescharacteristic
177C2×C4D4×D21111yescharacteristic
178C2×C4Q8×D21111yescharacteristic
179C2×C4#1680996111yescharacteristic
180C2×C4#1680996211yescharacteristic
181C2×C4#1680996211yescharacteristic
182C8—41no—
183C8—41no—
184C22—41no—
185C22—8441no—
186C22—16881no—
187C22#2026994711yescharacteristic
188C22#2026995711yescharacteristic
189C22#2026996511yescharacteristic
190C4—42no—
191C4—8441no—
192C4—8442no—
193C4—8441no—
194C4—16881no—
195C4—4222no—
196C4#2026995511yescharacteristic
197C4#2026995511yescharacteristic
198C4#2026996311yescharacteristic
199C4#2026996311yescharacteristic
200C2—21no—
201C2—21no—
202C2—21no—
203C2—16881no—
204C2#2721305911yescharacteristic
subgroupG/H
centerC4#20269963
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#344×C211C2
socleC2×C211#393
prime radicalC2×#16809962C22
hypercenter#344D211
hypocenterC211#2031
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
1. #27213699C2semidirect
5. #27212907C2semidirect
6. #27212903C2semidirect
7. #344×C211C2semidirect
28. C2×Dic422Q8semidirect
29. C2×D844Q8semidirect
102. C211#2031semidirect
108. #344D211semidirect

Maximal subgroups

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

subgroupindex
#272136992
#272131622
C2×#202699032
#272131192
#272129072
#272129032
#344×C2112
#2031211