Finite Group Databasebeta

← #41012370

Subgroups — #41012370

Order 27008.

238 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 51 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#27213775C211yesmaximal, characteristic
2#27213302C211yesmaximal, characteristic
3#27213065C211yesmaximal, characteristic
4#27212884C211yesmaximal, characteristic
5#27212883C211yesmaximal, characteristic
6#27212871C211yesmaximal, characteristic
7#451×C211C211yesmaximal, characteristic
8C2×#16809952—21no—
9#20269976—21no—
10#20269975—21no—
11#20269967—21no—
12#20269967—21no—
13#20269956C2211yescharacteristic
14#20269954—21no—
15#20269933—21no—
16C2×Dic211⋊C4—21no—
17C2×Dic211⋊C4C2211yescharacteristic
18#20269901—21no—
19C22⋊(C211⋊C8)C2211yescharacteristic
20C211⋊#95—21no—
21C211⋊#95C2211yescharacteristic
22C22×Q8×C211C2211yescharacteristic
23C2×Q16×C211—21no—
24C22⋊Q8×C211C2211yescharacteristic
25#103×C211—21no—
26C22⋊C8×C211C2211yescharacteristic
27C2×Dic422—21no—
28C2×Dic422D411yescharacteristic
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C22×Dic211D411yescharacteristic
32#16809952—42no—
33C2×C211⋊C8—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211D411yescharacteristic
36Dic211⋊C4—21no—
37Dic211⋊C4—21no—
38Dic211⋊C4—21no—
39Dic211⋊C4—21no—
40Dic211⋊C4—21no—
41Dic211⋊C4—21no—
42Dic211⋊C4—21no—
43Dic211⋊C4—42no—
44C22⋊Dic211—21no—
45C4×Dic211—21no—
46C4×Dic211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—42no—
50C2×Q8×C211D411yescharacteristic
51C22×C4×C211—21no—
52C22×C4×C211C2311yescharacteristic
53Q16×C211—42no—
54C2×C8×C211—21no—
55C4⋊C4×C211—21no—
56C4⋊C4×C211D411yescharacteristic
57C22⋊C4×C211—21no—
58Dic422—41no—
59Dic422—81no—
60C2×Dic211—21no—
61C2×Dic211—21no—
62C2×Dic211—22no—
63C2×Dic211—21no—
64C2×Dic211—21no—
65C2×Dic211—21no—
66C2×Dic211—41no—
67C2×Dic211—41no—
68C2×Dic211—41no—
69C2×Dic211Q1612yescomplemented
70C211⋊C8—41no—
71C23×C211C2×D411yescharacteristic
72Q8×C211—22no—
73Q8×C211—22no—
74Q8×C211—41no—
75Q8×C211—41no—
76Q8×C211—41no—
77C2×C4×C211—22no—
78C2×C4×C211—21no—
79C2×C4×C211—21no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211—41no—
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C8×C211—41no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211#12011yescharacteristic
89C22×C211C2×Q1611yescharacteristic
90C22×C211#13711yescharacteristic
91C4×C211—21no—
92C4×C211—21no—
93C4×C211—41no—
94C4×C211—42no—
95C4×C211—41no—
96Dic211—22no—
97Dic211—42no—
98Dic211—41no—
99Dic211—41no—
100Dic211—41no—
101C2×C211—21no—
102C2×C211—21no—
103C2×C211#45111yescharacteristic
104C2×C211#45611yescharacteristic
105C2×C211B(4,𝔽2)11yescharacteristic
106C211#160511yescharacteristic, complemented, Sylow
107#1605—2111nomaximal, Sylow
108#544—2111no—
109#451—2111no—
110#451D21111yescharacteristic
111#395—2111no—
112#328—2111no—
113#328—2111no—
114#324—2111no—
115C22×Q8C2×D21111yescharacteristic
116C2×Q16—21no—
117C2×Q16—4221no—
118C4⋊Q8—4221no—
119C4⋊Q8—4221no—
120#125—4221no—
121#125—4221no—
122C22⋊Q8—4221no—
123C22⋊Q8—2111no—
124C22⋊Q8C2×D21111yescharacteristic
125#117—4221no—
126C2×C4⋊C4—4221no—
127C2×C4⋊C4—2111no—
128#103—21no—
129#103—4221no—
130C22⋊C8—2111no—
131C22⋊C8C2×D21111yescharacteristic
132#95—4221no—
133#95—2111no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—42no—
137C2×Q8—4221no—
138C2×Q8—2111no—
139C2×Q8C22⋊D21111yescharacteristic
140C22×C4—21no—
141C22×C4—4221no—
142C22×C4—2111no—
143C22×C4—2111no—
144C22×C4C22×D21111yescharacteristic
145Q16—42no—
146Q16—8442no—
147C2×C8—21no—
148C2×C8—4221no—
149C4⋊C4—21no—
150C4⋊C4—8442no—
151C4⋊C4—4221no—
152C4⋊C4—4221no—
153C4⋊C4—4221no—
154C4⋊C4—4221no—
155C4⋊C4—4221no—
156C4⋊C4—4221no—
157C4⋊C4—4221no—
158C4⋊C4—4221no—
159C4⋊C4—2111no—
160C4⋊C4C22⋊D21111yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—4221no—
163C42—4221no—
164C42—4221no—
165C23D4×D21111yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—41no—
171Q8—8441no—
172Q8—16881no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—8441no—
180C2×C4—8441no—
181C2×C4—8441no—
182C2×C4—4222no—
183C2×C4—4221no—
184C2×C4—4221no—
185C2×C4—4221no—
186C2×C4—4221no—
187C2×C4—4221no—
188C2×C4—2112no—
189C2×C4D4×D21111yescharacteristic
190C2×C4C2×C22⋊D21111yescharacteristic
191C8—41no—
192C8—8441no—
193C22—21no—
194C22—21no—
195C22#2026994511yescharacteristic
196C22Q16×D21111yescharacteristic
197C22#2027000911yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—42no—
202C4—41no—
203C4—8442no—
204C4—8441no—
205C4—8441no—
206C4—8441no—
207C4—4222no—
208C2—21no—
209C2—21no—
210C2#2721330011yescharacteristic
211C2#2721332711yescharacteristic
212C2#2721333211yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#451×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#451D211
hypocenterC211#1605
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
69. C2×Dic211Q16semidirect
106. C211#1605semidirect

Maximal subgroups

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

subgroupindex
#272137752
#272133022
#272130652
#272128842
#272128832
#272128712
#451×C2112
#1605211