Finite Group Databasebeta

← #41014810

Subgroups — #41014810

Order 27008.

238 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×#20269902C211yesmaximal, characteristic
2C2×#20269900C211yesmaximal, characteristic
3#27213051C211yesmaximal, characteristic
4#27213015C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic
6#27212900C211yesmaximal, characteristic
7#340×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×Dic211⋊C4—21no—
12C2×Dic211⋊C4—21no—
13C2×C4×Dic211C2211yescharacteristic
14#20269902—22no—
15#20269900—22no—
16C211⋊#95—21no—
17C211⋊#95—21no—
18C22×C8×C211C2211yescharacteristic
19C2×C4⋊C4×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21C2×Dic422—21no—
22C2×Dic422—21no—
23C2×Dic422—21no—
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422D412yes—
27C22×Dic211—21no—
28C22×Dic211—21no—
29C22×Dic211D411yescharacteristic
30C22×Dic211D411yescharacteristic
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36Dic211⋊C4—42no—
37Dic211⋊C4—42no—
38C4×Dic211—21no—
39C4×Dic211—21no—
40C22×C4×C211—21no—
41C22×C4×C211—21no—
42C22×C4×C211C2311yescharacteristic
43C2×C8×C211—21no—
44C2×C8×C211—21no—
45C2×C8×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211—21no—
49C4⋊C4×C211—21no—
50Dic422—22no—
51Dic422—22no—
52Dic422—41no—
53Dic422—41no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—22no—
59C2×Dic211—41no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211SD1612yes—
65C2×Dic211Q1612yes—
66C211⋊C8—41no—
67C23×C211C2×D411yescharacteristic
68C2×C4×C211—41no—
69C2×C4×C211—41no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211#4111yescharacteristic
76C2×C4×C211#4111yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C8×C211—41no—
81C22×C211#12111yescharacteristic
82C22×C211#12311yescharacteristic
83C22×C211#12411yescharacteristic
84C22×C211C2×SD1611yescharacteristic
85C22×C211C2×Q1611yescharacteristic
86C22×C211AGL(1,ℤ/8)11yescharacteristic
87C22×C211#13711yescharacteristic
88C4×C211—81no—
89C4×C211—81no—
90C4×C211#12311yescharacteristic
91C4×C211#12311yescharacteristic
92C4×C211#12611yescharacteristic
93C4×C211#12611yescharacteristic
94Dic211—22no—
95Dic211—22no—
96Dic211—42no—
97Dic211—81no—
98C2×C211#39711yescharacteristic
99C2×C211#46011yescharacteristic
100C2×C211#46211yescharacteristic
101C2×C211#48311yescharacteristic
102C2×C211#48411yescharacteristic
103C2×C211#48711yescharacteristic
104C2×C211#48911yescharacteristic
105C211#207211yescharacteristic, complemented, Sylow
106#2072—2111nomaximal, Sylow
107C2×#103—2111no—
108C2×#103—2111no—
109#391—2111no—
110#384—2111no—
111#340—2111no—
112#340D21111yescharacteristic
113#336—2111no—
114C22×Q8—2111no—
115C22×C8—2111no—
116C22×C8C2×D21111yescharacteristic
117C2×C4⋊C4—4221no—
118C2×C4⋊C4—4221no—
119C2×C4⋊C4—2111no—
120C2×C4⋊C4C2×D21111yescharacteristic
121C2×C4⋊C4C2×D21111yescharacteristic
122C2×C42—2111no—
123#103—4222no—
124#103—4222no—
125#95—4221no—
126#95—4221no—
127C2×Q8—4221no—
128C2×Q8—4221no—
129C2×Q8—4221no—
130C2×Q8—4221no—
131C2×Q8—4221no—
132C2×Q8—2112no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—4221no—
136C22×C4—4221no—
137C22×C4—2111no—
138C22×C4—2111no—
139C22×C4C22×D21111yescharacteristic
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—4221no—
144C2×C8—4221no—
145C2×C8—4221no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—8442no—
151C4⋊C4—8442no—
152C4⋊C4—4221no—
153C4⋊C4—4221no—
154C42—4221no—
155C42—4221no—
156C23D4×D21111yescharacteristic
157Q8—8441no—
158Q8—8441no—
159Q8—4222no—
160Q8—4222no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—8441no—
168C2×C4—8441no—
169C2×C4—8441no—
170C2×C4—8441no—
171C2×C4—8441no—
172C2×C4—4222no—
173C2×C4—4221no—
174C2×C4—4221no—
175C2×C4—4221no—
176C2×C4—4221no—
177C2×C4—2112no—
178C2×C4—2112no—
179C2×C4D4×D21111yescharacteristic
180C2×C4#1680996111yescharacteristic
181C2×C4#1680996211yescharacteristic
182C2×C4#1680996211yescharacteristic
183C2×C4#1680996211yescharacteristic
184C2×C4#1680996311yescharacteristic
185C8—41no—
186C8—8441no—
187C22#2026995011yescharacteristic
188C22#2026996211yescharacteristic
189C22#2026996311yescharacteristic
190C22SD16×D21111yescharacteristic
191C22Q16×D21111yescharacteristic
192C22#2027000111yescharacteristic
193C22#2027000611yescharacteristic
194C4—81no—
195C4—81no—
196C4—8442no—
197C4—16881no—
198C4—4222no—
199C4—4222no—
200C4#2026995911yescharacteristic
201C4#2026996211yescharacteristic
202C4#2026997111yescharacteristic
203C4#2026997211yescharacteristic
204C2#2721307511yescharacteristic
205C2#2721334511yescharacteristic
206C2#2721335511yescharacteristic
207C2#2721347811yescharacteristic
208C2#2721348311yescharacteristic
209C2#2721349511yescharacteristic
210C2#2721350411yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socle, prime radicalC23×C211C2×D4
hypercenter#340D211
hypocenterC211#2072
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
105. C211#2072semidirect

Maximal subgroups

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

subgroupindex
C2×#202699022
C2×#202699002
#272130512
#272130152
#272129082
#272129002
#340×C2112
#2072211