Finite Group Databasebeta

← #41014258

Subgroups — #41014258

Order 27008.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16809952C211yesmaximal, characteristic
2C2×#20269910C211yesmaximal, characteristic
3C2×#20269901C211yesmaximal, characteristic
4C2×#20269900C211yesmaximal, characteristic
5C2×C4×C211⋊C8C211yesmaximal, characteristic
6#27213052C211yesmaximal, characteristic
7#391×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809952—22no—
10C2×#16809952—22no—
11C2×#16809952C414yescomplemented
12C22×C211⋊C8C2211yescharacteristic
13C22×C211⋊C8C2211yescharacteristic
14C2×C4⋊Dic211C2211yescharacteristic
15#20269910—22no—
16#20269901—22no—
17#20269900—22no—
18C4×C211⋊C8—22no—
19C211⋊#95—22no—
20C22×Q8×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C42×C211C2211yescharacteristic
23#95×C211—22no—
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422C2×C412yes—
30C22×Dic211—21no—
31C22×Dic211—21no—
32#16809952—28no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8C2×C412yes—
36C2×C211⋊C8D412yes—
37C2×C211⋊C8D412yes—
38C2×C211⋊C8D412yes—
39C4⋊Dic211—21no—
40C4⋊Dic211—21no—
41C2×Q8×C211—21no—
42C2×Q8×C211—21no—
43C2×Q8×C211—21no—
44C2×Q8×C211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211C2×C412yes—
47C22×C4×C211—21no—
48C22×C4×C211—21no—
49C22×C4×C211D412yes—
50C22×C4×C211C2311yescharacteristic
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C42×C211—21no—
54C42×C211—21no—
55Dic422—22no—
56Dic422—22no—
57Dic422—41no—
58Dic422—41no—
59C2×Dic211—22no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C2×Dic211—41no—
63C2×Dic211—42no—
64C211⋊C8—22no—
65C211⋊C8C22⋊C414yes—
66C23×C211C2×D411yescharacteristic
67Q8×C211—22no—
68Q8×C211—22no—
69Q8×C211—41no—
70Q8×C211—41no—
71C2×C4×C211—22no—
72C2×C4×C211—22no—
73C2×C4×C211—22no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211—42no—
78C2×C4×C211Q1614yescomplemented
79C2×C4×C211C22×C411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C22×C211C4×D411yescharacteristic
86C22×C211#12111yescharacteristic
87C22×C211#12711yescharacteristic
88C22×C211C2×Q1612yes—
89C22×C211#13512yes—
90C4×C211—24no—
91C4×C211—42no—
92C4×C211—81no—
93C4×C211C2×C22⋊C411yescharacteristic
94C4×C211C4×D411yescharacteristic
95C4×C211#12111yescharacteristic
96C4×C211#12411yescharacteristic
97Dic211—42no—
98Dic211—81no—
99C2×C211#39011yescharacteristic
100C2×C211C4×Q1612yes—
101C2×C211#46712yes—
102C2×C211#49411yescharacteristic
103C2×C211#49511yescharacteristic
104C211#196911yescharacteristic, complemented, Sylow
105#1969—2111nomaximal, Sylow
106C22×Q16—2111no—
107C2×#107—2111no—
108C2×#103—2111no—
109C2×#103—2111no—
110C2×C4×C8—2111no—
111#391—2111no—
112#391D21111yescharacteristic
113C22×Q8—2111no—
114C22×Q8C2×D21111yescharacteristic
115C2×Q16—4222no—
116C2×Q16—4222no—
117C2×Q16—2114no—
118C22×C8—2111no—
119C22×C8—2111no—
120C2×C4⋊C4—2111no—
121C2×C4⋊C4C2×D21111yescharacteristic
122C2×C42C2×D21111yescharacteristic
123#107—4222no—
124#103—4222no—
125#103—4222no—
126C4×C8—4222no—
127#95—22no—
128#95—4222no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—4221no—
135C2×Q8—4221no—
136C2×Q8—4221no—
137C2×Q8—4221no—
138C2×Q8—4221no—
139C2×Q8—2112no—
140C2×Q8C4×D21112yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—4221no—
144C22×C4—4221no—
145C22×C4C22⋊D21112yes—
146C22×C4C22×D21111yescharacteristic
147Q16—4228no—
148C2×C8—4221no—
149C2×C8—4221no—
150C2×C8—2112no—
151C2×C8—2112no—
152C2×C8—2112no—
153C2×C8—2112no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—4221no—
157C4⋊C4—4221no—
158C42—21no—
159C42—21no—
160C23C2×C22⋊D21111yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165Q8—8441no—
166Q8—8441no—
167Q8—4222no—
168Q8—4222no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—8441no—
177C2×C4—8441no—
178C2×C4—8441no—
179C2×C4—8442no—
180C2×C4—4222no—
181C2×C4#1680995214yescomplemented
182C2×C4C2×C4×D21111yescharacteristic
183C2×C4D4×D21111yescharacteristic
184C2×C4D4×D21111yescharacteristic
185C2×C4D4×D21111yescharacteristic
186C2×C4#1680996111yescharacteristic
187C2×C4#1680996211yescharacteristic
188C8—4222no—
189C8—2114no—
190C22C4×C22⋊D21111yescharacteristic
191C22#2026994911yescharacteristic
192C22#2026997311yescharacteristic
193C22C2×#1680995212yes—
194C22#2026999712yes—
195C4—24no—
196C4—42no—
197C4—81no—
198C4—8442no—
199C4—16881no—
200C4C22⋊C4×D21111yescharacteristic
201C4(C22⋊D211)⋊C411yescharacteristic
202C4#2026995011yescharacteristic
203C4#2026996311yescharacteristic
204C2#2721304611yescharacteristic
205C2C4×#1680995212yes—
206C2#2721338512yes—
207C2#2721352511yescharacteristic
208C2#2721352811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#391×C211C2
socle, prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenter#391D211
hypocenterC211#1969
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
11. C2×#16809952C4semidirect
78. C2×C4×C211Q16semidirect
104. C211#1969semidirect
181. C2×C4#16809952semidirect

Maximal subgroups

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

subgroupindex
C22×#168099522
C2×#202699102
C2×#202699012
C2×#202699002
C2×C4×C211⋊C82
#272130522
#391×C2112
#1969211