Finite Group Databasebeta

← #41011920

Subgroups — #41011920

Order 27008.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#27213677C211yesmaximal, characteristic
2#27213200C211yesmaximal, characteristic, complemented
3#27212997C211yesmaximal, characteristic, complemented
4#27212884C211yesmaximal, characteristic, complemented
5#27212882C211yesmaximal, characteristic, complemented
6#27212880C211yesmaximal, characteristic
7#327×C211C211yesmaximal, characteristic
8#20269970—21no—
9#20269968C411yescharacteristic, complemented
10#20269964C411yescharacteristic, complemented
11#20269958—21no—
12#20269956C2211yescharacteristic
13#20269942C411yescharacteristic, complemented
14D4×Dic211—21no—
15D4×Dic211C411yescharacteristic, complemented
16#20269933—21no—
17C2×Dic211⋊C4—21no—
18C2×C4×Dic211—21no—
19C2×C4×Dic211C2211yescharacteristic
20Dic211⋊C8—21no—
21C22⋊(C211⋊C8)C2211yescharacteristic
22#20269887—21no—
23C211⋊#95C2211yescharacteristic
24#121×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C2×Dic422C2×C411yescharacteristic
28C22×Dic211—21no—
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C22×Dic211D411yescharacteristic
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211C2×C411yescharacteristic
35Dic211⋊C4—24no—
36Dic211⋊C4—21no—
37Dic211⋊C4—21no—
38Dic211⋊C4—21no—
39C22⋊Dic211—21no—
40C22⋊Dic211—21no—
41C22⋊Dic211—21no—
42C22⋊Dic211—21no—
43C4×Dic211—21no—
44C4×Dic211—42no—
45C4×Dic211C2×C411yescharacteristic
46C4×Dic211C2×C411yescharacteristic
47C4×Dic211D411yescharacteristic
48C4×Dic211D411yescharacteristic
49C2×D4×C211—21no—
50C2×D4×C211C2×C411yescharacteristic
51C22×C4×C211—21no—
52C22×C4×C211C2311yescharacteristic
53C2×C8×C211—21no—
54C4⋊C4×C211C2×C411yescharacteristic
55C22⋊C4×C211—21no—
56Dic422—41no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—22no—
60C2×Dic211—22no—
61C2×Dic211—21no—
62C2×Dic211—21no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—42no—
66C2×Dic211C22⋊C411yescharacteristic
67C2×Dic211C22⋊C411yescharacteristic
68C2×Dic211C22⋊C411yescharacteristic
69C2×Dic211C22⋊C411yescharacteristic
70C211⋊C8—41no—
71C23×C211—21no—
72C23×C211C2×D411yescharacteristic
73D4×C211—41no—
74D4×C211—41no—
75D4×C211—41no—
76C2×C4×C211—22no—
77C2×C4×C211—21no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211C22×C411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C8×C211—41no—
84C22×C211—21no—
85C22×C211—21no—
86C22×C211—41no—
87C22×C211—41no—
88C22×C211—41no—
89C22×C211C2×C22⋊C411yescharacteristic
90C22×C211AGL(1,ℤ/8)11yescharacteristic
91C22×C211#13711yescharacteristic
92C4×C211—21no—
93C4×C211—21no—
94C4×C211—42no—
95C4×C211—41no—
96Dic211—21no—
97Dic211—21no—
98Dic211—21no—
99Dic211—42no—
100Dic211—41no—
101Dic211—41no—
102Dic211#10412yes—
103C2×C211—21no—
104C2×C211—21no—
105C2×C211—81no—
106C2×C211#41011yescharacteristic
107C2×C211#41711yescharacteristic
108C2×C211C2×#10411yescharacteristic
109C211#150911yescharacteristic, complemented, Sylow
110#1509—2111nomaximal, Sylow
111#529—2111no—
112#432—2111no—
113C4×C4⋊C4—2111no—
114#328—2111no—
115#328—2111no—
116#327—2111no—
117#327D21111yescharacteristic
118#126—4221no—
119#125—2111no—
120#124—2111no—
121#123—4221no—
122C22⋊Q8—2111no—
123#121C2×D21111yescharacteristic
124C4×Q8—2111no—
125C4×D4—4221no—
126C4×D4—2111no—
127#117—4221no—
128C2×C4⋊C4—4221no—
129C2×C42—4221no—
130C2×C42—2111no—
131C4⋊C8—4221no—
132C22⋊C8—2111no—
133C22⋊C8C2×D21111yescharacteristic
134#97—4221no—
135#95—2111no—
136#95C2×D21111yescharacteristic
137C2×Q8—2111no—
138C2×D4—21no—
139C2×D4C4×D21111yescharacteristic
140C22×C4—21no—
141C22×C4—4221no—
142C22×C4—4221no—
143C22×C4—2111no—
144C22×C4—2111no—
145C22×C4C22×D21111yescharacteristic
146C2×C8—21no—
147C2×C8—4221no—
148C4⋊C4—4224no—
149C4⋊C4—4221no—
150C4⋊C4—4221no—
151C4⋊C4—4221no—
152C4⋊C4—4221no—
153C4⋊C4—2111no—
154C4⋊C4C4×D21111yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—4221no—
157C22⋊C4—4221no—
158C22⋊C4—4221no—
159C22⋊C4—4221no—
160C42—8442no—
161C42—4221no—
162C42—2111no—
163C42—2111no—
164C42—2111no—
165C42—2111no—
166C23—21no—
167C23D4×D21111yescharacteristic
168Q8—8441no—
169D4—41no—
170D4—41no—
171D4—41no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—8441no—
178C2×C4—8441no—
179C2×C4—8442no—
180C2×C4—4222no—
181C2×C4—4221no—
182C2×C4—4221no—
183C2×C4—4221no—
184C2×C4—4221no—
185C2×C4—4222no—
186C2×C4—2111no—
187C2×C4—2111no—
188C2×C4—2111no—
189C2×C4—2111no—
190C2×C4C2×C4×D21111yescharacteristic
191C2×C4D4×D21111yescharacteristic
192C8—41no—
193C8—8441no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22C22⋊C4×D21111yescharacteristic
200C22#2027000111yescharacteristic
201C22#2027000611yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—8442no—
207C4—8441no—
208C4—8441no—
209C4—4221no—
210C4—4221no—
211C4—4221no—
212C4—2112no—
213C2—21no—
214C2—21no—
215C2—81no—
216C2#2721311811yescharacteristic
217C2#2721314411yescharacteristic
218C2#104×D21111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211C2×C22⋊C4
prime radical#121×C211C22
cosocleC22×C4×C211C23
hypercenter#327D211
hypocenterC211#1509
LCSG ▷ C2×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. #27213200C2semidirect
3. #27212997C2semidirect
4. #27212884C2semidirect
5. #27212882C2semidirect
9. #20269968C4semidirect
10. #20269964C4semidirect
13. #20269942C4semidirect
15. D4×Dic211C4semidirect
109. C211#1509semidirect

Maximal subgroups

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

subgroupindex
#272136772
#272132002
#272129972
#272128842
#272128822
#272128802
#327×C2112
#1509211