Finite Group Databasebeta

← #41011933

Subgroups — #41011933

Order 27008.

222 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 61 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#27213010C211yesmaximal, 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×Dic211C2211yescharacteristic
19Dic211⋊C8—21no—
20C22⋊(C211⋊C8)C2211yescharacteristic
21#20269887—21no—
22C211⋊#95—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—21no—
36Dic211⋊C4—21no—
37Dic211⋊C4—21no—
38Dic211⋊C4—42no—
39C22⋊Dic211—21no—
40C22⋊Dic211—21no—
41C22⋊Dic211—21no—
42C22⋊Dic211—21no—
43C4×Dic211—21no—
44C4×Dic211C2×C411yescharacteristic
45C4×Dic211C2×C411yescharacteristic
46C4×Dic211D411yescharacteristic
47C4×Dic211D411yescharacteristic
48C2×D4×C211—21no—
49C2×D4×C211C2×C411yescharacteristic
50C22×C4×C211—21no—
51C22×C4×C211C2311yescharacteristic
52C2×C8×C211—21no—
53C4⋊C4×C211C2×C411yescharacteristic
54C22⋊C4×C211—21no—
55Dic422—41no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—41no—
61C2×Dic211—42no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211C22⋊C411yescharacteristic
66C2×Dic211C22⋊C411yescharacteristic
67C2×Dic211C22⋊C411yescharacteristic
68C2×Dic211C22⋊C411yescharacteristic
69C211⋊C8—41no—
70C23×C211—21no—
71C23×C211C2×D411yescharacteristic
72D4×C211—41no—
73D4×C211—41no—
74D4×C211—41no—
75C2×C4×C211—21no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211C22×C411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C8×C211—41no—
83C22×C211—21no—
84C22×C211—21no—
85C22×C211—41no—
86C22×C211—41no—
87C22×C211—41no—
88C22×C211C2×C22⋊C411yescharacteristic
89C22×C211AGL(1,ℤ/8)11yescharacteristic
90C22×C211#13711yescharacteristic
91C4×C211—21no—
92C4×C211—21no—
93C4×C211—41no—
94C4×C211—81no—
95Dic211—21no—
96Dic211—21no—
97Dic211—21no—
98Dic211—21no—
99Dic211—41no—
100Dic211—41no—
101Dic211—81no—
102C2×C211—21no—
103C2×C211—21no—
104C2×C211—81no—
105C2×C211#41011yescharacteristic
106C2×C211#41711yescharacteristic
107C2×C211#42111yescharacteristic
108C211#151211yescharacteristic, complemented, Sylow
109#1512—2111nomaximal, Sylow
110#529—2111no—
111#432—2111no—
112#382—2111no—
113#328—2111no—
114#328—2111no—
115#327—2111no—
116#327D21111yescharacteristic
117#126—4221no—
118#125—2111no—
119#124—2111no—
120#123—4221no—
121C22⋊Q8—2111no—
122#121C2×D21111yescharacteristic
123C4×Q8—2111no—
124C4×D4—4221no—
125C4×D4—2111no—
126#117—4221no—
127C2×C4⋊C4—4221no—
128C2×C42—2111no—
129C4⋊C8—4221no—
130C22⋊C8—2111no—
131C22⋊C8C2×D21111yescharacteristic
132#97—4221no—
133#95—4221no—
134#95—2111no—
135#95C2×D21111yescharacteristic
136C2×Q8—2111no—
137C2×D4—21no—
138C2×D4C4×D21111yescharacteristic
139C22×C4—21no—
140C22×C4—4221no—
141C22×C4—4221no—
142C22×C4—2111no—
143C22×C4—2111no—
144C22×C4C22×D21111yescharacteristic
145C2×C8—21no—
146C2×C8—4221no—
147C4⋊C4—8442no—
148C4⋊C4—4221no—
149C4⋊C4—4221no—
150C4⋊C4—4221no—
151C4⋊C4—4221no—
152C4⋊C4—2111no—
153C4⋊C4C4×D21111yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—4221no—
156C22⋊C4—4221no—
157C22⋊C4—4221no—
158C22⋊C4—4221no—
159C42—4221no—
160C42—2111no—
161C42—2111no—
162C42—2111no—
163C42—2111no—
164C23—21no—
165C23D4×D21111yescharacteristic
166Q8—8441no—
167D4—41no—
168D4—41no—
169D4—41no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—8442no—
176C2×C4—8441no—
177C2×C4—8441no—
178C2×C4—8441no—
179C2×C4—8441no—
180C2×C4—4221no—
181C2×C4—4221no—
182C2×C4—4221no—
183C2×C4—4221no—
184C2×C4—2111no—
185C2×C4—2111no—
186C2×C4—2111no—
187C2×C4—2111no—
188C2×C4C2×C4×D21111yescharacteristic
189C2×C4D4×D21111yescharacteristic
190C8—41no—
191C8—8441no—
192C22—21no—
193C22—21no—
194C22—41no—
195C22—41no—
196C22—41no—
197C22C22⋊C4×D21111yescharacteristic
198C22#2027000111yescharacteristic
199C22#2027000611yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—81no—
204C4—8441no—
205C4—8441no—
206C4—16881no—
207C4—4221no—
208C4—4221no—
209C4—4221no—
210C4—4221no—
211C2—21no—
212C2—21no—
213C2—81no—
214C2#2721311811yescharacteristic
215C2#2721314411yescharacteristic
216C2#2721316111yescharacteristic
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#1512
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. #27213010C2semidirect
4. #27212884C2semidirect
5. #27212882C2semidirect
9. #20269968C4semidirect
10. #20269964C4semidirect
13. #20269942C4semidirect
15. D4×Dic211C4semidirect
108. C211#1512semidirect

Maximal subgroups

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

subgroupindex
#272136772
#272132002
#272130102
#272128842
#272128822
#272128802
#327×C2112
#1512211