Finite Group Databasebeta

← #41014557

Subgroups — #41014557

Order 27008.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213774C211yesmaximal, characteristic
2#27213155C211yesmaximal, characteristic
3#27213053C211yesmaximal, characteristic
4#27212914C211yesmaximal, characteristic
5#27212907C211yesmaximal, characteristic
6#27212905C211yesmaximal, characteristic
7#419×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9#20269977—21no—
10#20269976—21no—
11#20269969—21no—
12#20269968—21no—
13#20269954—22no—
14#20269936C2211yescharacteristic
15C2×C4⋊Dic211C2211yescharacteristic
16C2×C4×Dic211C2211yescharacteristic
17#20269901—21no—
18#20269901—21no—
19C211⋊#95—22no—
20C22×Q8×C211C2211yescharacteristic
21C2×M16×C211C2211yescharacteristic
22#117×C211C2211yescharacteristic
23#103×C211—21no—
24#103×C211—21no—
25C2×Dic422—22no—
26C22×Dic211—21no—
27C22×Dic211D412yes—
28#16809946—21no—
29#16809946—21no—
30C2×C211⋊C8—21no—
31C4⋊Dic211—22no—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211D411yescharacteristic
35C4⋊Dic211D411yescharacteristic
36Dic211⋊C4—22no—
37Dic211⋊C4—22no—
38C22⋊Dic211—21no—
39C4×Dic211—21no—
40C4×Dic211—21no—
41C4×Dic211—21no—
42C2×Q8×C211—21no—
43C2×Q8×C211—21no—
44C2×Q8×C211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211Q812yescomplemented
48C22×C4×C211—21no—
49C22×C4×C211C2311yescharacteristic
50M16×C211—21no—
51M16×C211—21no—
52C2×C8×C211—21no—
53C4⋊C4×C211D411yescharacteristic
54C4⋊C4×C211D411yescharacteristic
55C22⋊C4×C211—21no—
56C42×C211—21no—
57Dic422—82no—
58C2×Dic211—22no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—22no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—42no—
65C211⋊C8—41no—
66C23×C211C2×D411yescharacteristic
67Q8×C211—22no—
68Q8×C211—22no—
69Q8×C211—41no—
70Q8×C211—41no—
71C2×C4×C211—21no—
72C2×C4×C211—21no—
73C2×C4×C211—22no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C2×C4×C211C2×Q811yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C2×C4×C211#4111yescharacteristic
81C2×C4×C211#4111yescharacteristic
82C8×C211—41no—
83C22×C211—21no—
84C22×C211—21no—
85C22×C211#12011yescharacteristic
86C22×C211C22⋊Q811yescharacteristic
87C22×C211#12411yescharacteristic
88C4×C211—42no—
89C4×C211—41no—
90C4×C211—41no—
91C4×C211C22⋊Q811yescharacteristic
92C4×C211C22⋊Q811yescharacteristic
93C4×C211#12411yescharacteristic
94C4×C211#12411yescharacteristic
95Dic211—42no—
96Dic211—42no—
97Dic211—41no—
98Dic211—41no—
99C2×C211—21no—
100C2×C211—21no—
101C2×C211#39311yescharacteristic
102C2×C211#45511yescharacteristic
103C2×C211#45611yescharacteristic
104C211#202811yescharacteristic, complemented, Sylow
105#2028—2111nomaximal, Sylow
106#544—2111no—
107#419—2111no—
108#419D21111yescharacteristic
109#391—2111no—
110#343—2111no—
111#339—2111no—
112#339—2111no—
113C22×Q8C2×D21111yescharacteristic
114C2×M16—2111no—
115C2×M16C2×D21111yescharacteristic
116C4⋊Q8—4221no—
117C4⋊Q8—4221no—
118#125—4221no—
119#125—4221no—
120C22⋊Q8—4222no—
121#117—2111no—
122#117C2×D21111yescharacteristic
123C2×C4⋊C4—2111no—
124C2×C42—2111no—
125#103—21no—
126#103—21no—
127#103—4221no—
128#103—4221no—
129#95—4222no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—4222no—
136C2×Q8Dic42212yescomplemented
137C22×C4—21no—
138C22×C4—4221no—
139C22×C4—2112no—
140C22×C4C22×D21111yescharacteristic
141M16—21no—
142M16—21no—
143M16—4221no—
144M16—4221no—
145C2×C8—21no—
146C2×C8—4221no—
147C4⋊C4—4222no—
148C4⋊C4—4222no—
149C4⋊C4—4222no—
150C4⋊C4—4221no—
151C4⋊C4—4221no—
152C4⋊C4—2111no—
153C4⋊C4—2111no—
154C4⋊C4C22⋊D21111yescharacteristic
155C4⋊C4C22⋊D21111yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—4221no—
158C42—21no—
159C42—4221no—
160C42—4221no—
161C42—4221no—
162C23D4×D21111yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—41no—
166Q8—41no—
167Q8—16882no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—8441no—
174C2×C4—8441no—
175C2×C4—8442no—
176C2×C4—4222no—
177C2×C4—4221no—
178C2×C4—4221no—
179C2×C4—4222no—
180C2×C4D4×D21111yescharacteristic
181C2×C4C2×C22⋊D21111yescharacteristic
182C2×C4C2×Dic42211yescharacteristic
183C2×C4#1680996111yescharacteristic
184C2×C4#1680996111yescharacteristic
185C2×C4#1680996211yescharacteristic
186C8—41no—
187C8—8441no—
188C22—21no—
189C22—21no—
190C22#2026994511yescharacteristic
191C22#2026995411yescharacteristic
192C22#2026996311yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—8442no—
197C4—8442no—
198C4—8441no—
199C4—8441no—
200C4#2026995411yescharacteristic
201C4#2026995611yescharacteristic
202C4#2026996311yescharacteristic
203C4#2026996411yescharacteristic
204C2—21no—
205C2—21no—
206C2#2721305811yescharacteristic
207C2#2721332211yescharacteristic
208C2#2721332711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#419×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#419D211
hypocenterC211#2028
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
47. C2×Q8×C211Q8semidirect
104. C211#2028semidirect
136. C2×Q8Dic422semidirect

Maximal subgroups

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

subgroupindex
#272137742
#272131552
#272130532
#272129142
#272129072
#272129052
#419×C2112
#2028211