Finite Group Databasebeta

← #41012396

Subgroups — #41012396

Order 27008.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#27213741C211yesmaximal, characteristic
2#27213477C211yesmaximal, characteristic, complemented
3#27213457C211yesmaximal, characteristic
4#27213078C211yesmaximal, characteristic, complemented
5#27212882C211yesmaximal, characteristic, complemented
6#27212871C211yesmaximal, characteristic, complemented
7#327×C211C211yesmaximal, characteristic
8#20269970—21no—
9#20269967—21no—
10#20269958—21no—
11#20269958—21no—
12#20269956C2211yescharacteristic
13#20269954—21no—
14D4×Dic211—21no—
15C2×C4⋊Dic211—21no—
16C2×C4⋊Dic211C2211yescharacteristic
17C2×Dic211⋊C4C2211yescharacteristic
18#20269910—21no—
19#20269909—21no—
20#20269902—21no—
21#20269898—21no—
22C22⋊(C211⋊C8)C2211yescharacteristic
23C211⋊#95—21no—
24#121×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C2×Dic422D411yescharacteristic
28C22×Dic211—21no—
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C22×Dic211D411yescharacteristic
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—42no—
37C4⋊Dic211D411yescharacteristic
38Dic211⋊C4—21no—
39Dic211⋊C4—21no—
40Dic211⋊C4—21no—
41Dic211⋊C4—21no—
42Dic211⋊C4—21no—
43C22⋊Dic211—21no—
44C22⋊Dic211—21no—
45C22⋊Dic211—21no—
46C22⋊Dic211—21no—
47C4×Dic211—21no—
48C2×D4×C211—21no—
49C2×D4×C211D411yescharacteristic
50C22×C4×C211—21no—
51C22×C4×C211C2311yescharacteristic
52C2×C8×C211—21no—
53C4⋊C4×C211D411yescharacteristic
54C22⋊C4×C211—21no—
55Dic422—41no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—21no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C2×Dic211—81no—
67C211⋊C8—41no—
68C23×C211—21no—
69C23×C211C2×D411yescharacteristic
70D4×C211—41no—
71D4×C211—41no—
72D4×C211—41no—
73C2×C4×C211—22no—
74C2×C4×C211—21no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211C2×D411yescharacteristic
79C2×C4×C211C2×D411yescharacteristic
80C8×C211—41no—
81C22×C211—21no—
82C22×C211—21no—
83C22×C211—41no—
84C22×C211—41no—
85C22×C211—41no—
86C22×C211#12011yescharacteristic
87C22×C211#13511yescharacteristic
88C22×C211AGL(1,ℤ/8)11yescharacteristic
89C4×C211—21no—
90C4×C211—21no—
91C4×C211—42no—
92C4×C211—41no—
93Dic211—41no—
94Dic211—41no—
95Dic211—41no—
96Dic211—41no—
97Dic211—81no—
98C2×C211—21no—
99C2×C211—21no—
100C2×C211—81no—
101C2×C211#44911yescharacteristic
102C2×C211#45611yescharacteristic
103C2×C211#45811yescharacteristic
104C211#160911yescharacteristic, complemented, Sylow
105#1609—2111nomaximal, Sylow
106#539—2111no—
107#483—2111no—
108#480—2111no—
109#398—2111no—
110#328—2111no—
111#327D21111yescharacteristic
112#324—2111no—
113#126—4221no—
114#125—4221no—
115#123—4221no—
116#123—4221no—
117C22⋊Q8—4221no—
118C22⋊Q8—2111no—
119#121C2×D21111yescharacteristic
120C4×D4—4221no—
121C2×C4⋊C4—4221no—
122C2×C4⋊C4—2111no—
123C2×C4⋊C4—2111no—
124#107—4221no—
125#106—4221no—
126#103—4221no—
127#102—4221no—
128C22⋊C8—2111no—
129C22⋊C8C2×D21111yescharacteristic
130#95—4221no—
131#95C2×D21111yescharacteristic
132C2×Q8—2111no—
133C2×D4—21no—
134C2×D4D84411yescharacteristic
135C22×C4—21no—
136C22×C4—4221no—
137C22×C4—4221no—
138C22×C4—2111no—
139C22×C4—2111no—
140C22×C4C22×D21111yescharacteristic
141C2×C8—21no—
142C2×C8—4221no—
143C4⋊C4—8442no—
144C4⋊C4—4221no—
145C4⋊C4—4221no—
146C4⋊C4—4221no—
147C4⋊C4—4221no—
148C4⋊C4—4221no—
149C4⋊C4—4221no—
150C4⋊C4—4221no—
151C4⋊C4—4221no—
152C4⋊C4—2111no—
153C4⋊C4D84411yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—4221no—
156C22⋊C4—4221no—
157C22⋊C4—4221no—
158C22⋊C4—4221no—
159C42—4221no—
160C23—21no—
161C23D4×D21111yescharacteristic
162Q8—8441no—
163D4—41no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—8441no—
172C2×C4—8441no—
173C2×C4—8441no—
174C2×C4—8441no—
175C2×C4—16881no—
176C2×C4—4221no—
177C2×C4—4221no—
178C2×C4—4221no—
179C2×C4—4221no—
180C2×C4—4221no—
181C2×C4—4221no—
182C2×C4D4×D21111yescharacteristic
183C2×C4C2×D84411yescharacteristic
184C8—41no—
185C8—8441no—
186C22—21no—
187C22—21no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22#2026994711yescharacteristic
192C22#2026999811yescharacteristic
193C22#2027000111yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—8441no—
199C4—8441no—
200C4—8441no—
201C4—8441no—
202C4—16881no—
203C2—21no—
204C2—21no—
205C2—81no—
206C2#2721329211yescharacteristic
207C2#2721333011yescharacteristic
208C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211#120
prime radical#121×C211C22
hypercenter#327D211
hypocenterC211#1609
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27213477C2semidirect
4. #27213078C2semidirect
5. #27212882C2semidirect
6. #27212871C2semidirect
104. C211#1609semidirect

Maximal subgroups

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

subgroupindex
#272137412
#272134772
#272134572
#272130782
#272128822
#272128712
#327×C2112
#1609211