Finite Group Databasebeta

← #41011956

Subgroups — #41011956

Order 27008.

264 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213691C211yesmaximal, characteristic
2#27213196C211yesmaximal, characteristic, complemented
3#27213010C211yesmaximal, characteristic, complemented
4#27212881C211yesmaximal, characteristic, complemented
5#27212880C211yesmaximal, characteristic, complemented
6#27212879C211yesmaximal, characteristic, complemented
7#327×C211C211yesmaximal, characteristic, complemented
8C2×#16809961—21no—
9#20269976C411yescharacteristic, complemented
10#20269973C411yescharacteristic, complemented
11#20269963—21no—
12#20269951C2211yescharacteristic
13#20269949—21no—
14D4×Dic211C411yescharacteristic, complemented
15#20269939C411yescharacteristic, complemented
16(C22⋊D211)⋊C4—21no—
17C2×Dic211⋊C4—21no—
18C2×C4×Dic211C2211yescharacteristic
19Dic211⋊C8—21no—
20C22⋊(C211⋊C8)C2211yescharacteristic
21C8×Dic211—21no—
22C211⋊#95—21no—
23C211⋊#95C2211yescharacteristic
24#121×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27#16809961—42no—
28#16809961—42no—
29C2×Dic422—21no—
30C2×D844C2×C411yescharacteristic
31C2×C22⋊D211—21no—
32C2×C22⋊D211—21no—
33C2×C22⋊D211—21no—
34C22×Dic211—21no—
35C22×Dic211—21no—
36C22×Dic211D411yescharacteristic
37C22×Dic211D411yescharacteristic
38C2×C4×D211—21no—
39C2×C211⋊C8—21no—
40C4⋊Dic211C2×C411yescharacteristic
41Dic211⋊C4—21no—
42Dic211⋊C4—42no—
43C22⋊Dic211—21no—
44(C2×D211)⋊C4—21no—
45(C2×D211)⋊C4—21no—
46C4×Dic211C2×C411yescharacteristic
47C4×Dic211C2×C411yescharacteristic
48C4×Dic211D411yescharacteristic
49C4×Dic211D411yescharacteristic
50C2×D4×C211—21no—
51C2×D4×C211C2×C411yescharacteristic
52C22×C4×C211—21no—
53C22×C4×C211C2311yescharacteristic
54C2×C8×C211—21no—
55C4⋊C4×C211C2×C411yescharacteristic
56C22⋊C4×C211—21no—
57C22×D211—21no—
58Dic422—42no—
59D844—41no—
60C22⋊D211—42no—
61C22⋊D211—41no—
62C22⋊D211—41no—
63C22⋊D211—42no—
64C2×Dic211—21no—
65C2×Dic211—41no—
66C2×Dic211—42no—
67C2×Dic211—41no—
68C2×Dic211—41no—
69C2×Dic211—41no—
70C2×Dic211C22⋊C411yescharacteristic
71C2×Dic211C22⋊C411yescharacteristic
72C2×Dic211C22⋊C411yescharacteristic
73C2×Dic211C22⋊C411yescharacteristic
74C2×Dic211D812yes—
75C2×Dic211SD1612yes—
76C4×D211—42no—
77C211⋊C8—41no—
78C23×C211—21no—
79C23×C211C2×D411yescharacteristic
80D4×C211—41no—
81D4×C211—41no—
82D4×C211—41no—
83C2×C4×C211—21no—
84C2×C4×C211—41no—
85C2×C4×C211—41no—
86C2×C4×C211—41no—
87C2×C4×C211—41no—
88C2×C4×C211C22×C411yescharacteristic
89C2×C4×C211C2×D411yescharacteristic
90C8×C211—41no—
91C22×C211—21no—
92C22×C211—21no—
93C22×C211—41no—
94C22×C211—41no—
95C22×C211—41no—
96C22×C211C2×C22⋊C411yescharacteristic
97C22×C211C2×D811yescharacteristic
98C22×C211C2×SD1611yescharacteristic
99C2×D211—41no—
100C2×D211—41no—
101C2×D211—41no—
102C4×C211—21no—
103C4×C211—21no—
104C4×C211—41no—
105C4×C211—81no—
106Dic211—21no—
107Dic211—21no—
108Dic211—41no—
109Dic211—81no—
110Dic211#10212yes—
111Dic211#10212yes—
112C2×C211—21no—
113C2×C211—21no—
114C2×C211—81no—
115C2×C211#41011yescharacteristic
116C2×C211C2×#10211yescharacteristic
117C2×C211#42111yescharacteristic
118D211—81no—
119C211#151711yescharacteristic, complemented, Sylow
120#1517—2111nomaximal, Sylow
121#532—2111no—
122#431—2111no—
123#382—2111no—
124#327—2111no—
125#327—2111no—
126#327—2111no—
127#327D21111yescharacteristic, complemented
128C2×#41—4221no—
129C4⋊Q8—2111no—
130#127—2111no—
131#124—4221no—
132#121—4221no—
133#121—2111no—
134#121C2×D21111yescharacteristic
135C4×D4—4221no—
136C4×D4—2111no—
137C4×D4—2111no—
138C2×C4⋊C4—4221no—
139C2×C42—2111no—
140C4⋊C8—4221no—
141C22⋊C8—2111no—
142C22⋊C8C2×D21111yescharacteristic
143C4×C8—4221no—
144#95—4221no—
145#95—2111no—
146#95C2×D21111yescharacteristic
147#41—8442no—
148#41—8442no—
149C2×Q8—4221no—
150C2×D4—21no—
151C2×D4—4221no—
152C2×D4—4221no—
153C2×D4—4221no—
154C2×D4—2111no—
155C2×D4C4×D21111yescharacteristic
156C22×C4—21no—
157C22×C4—4221no—
158C22×C4—4221no—
159C22×C4—4221no—
160C22×C4—2111no—
161C22×C4—2111no—
162C22×C4C22×D21111yescharacteristic
163C2×C8—21no—
164C2×C8—4221no—
165C4⋊C4—8442no—
166C4⋊C4—4221no—
167C4⋊C4—2111no—
168C4⋊C4C4×D21111yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—4221no—
171C22⋊C4—4221no—
172C22⋊C4—4221no—
173C42—2111no—
174C42—2111no—
175C42—2111no—
176C42—2111no—
177C23—21no—
178C23—4221no—
179C23D4×D21111yescharacteristic
180Q8—8442no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—8441no—
185D4—8441no—
186D4—8441no—
187D4—8442no—
188D4—8442no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—8442no—
195C2×C4—8442no—
196C2×C4—8441no—
197C2×C4—8441no—
198C2×C4—8441no—
199C2×C4—8441no—
200C2×C4—4221no—
201C2×C4—2111no—
202C2×C4—2111no—
203C2×C4—2111no—
204C2×C4—2111no—
205C2×C4—2112no—
206C2×C4—2112no—
207C2×C4C2×C4×D21111yescharacteristic
208C2×C4D4×D21111yescharacteristic
209C8—41no—
210C8—8441no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—8441no—
217C22—8441no—
218C22—8441no—
219C22C22⋊C4×D21111yescharacteristic
220C22D8×D21111yescharacteristic
221C22SD16×D21111yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—81no—
226C4—8441no—
227C4—16881no—
228C4—4221no—
229C4—4221no—
230C4—2112no—
231C4—2112no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—16881no—
236C2#2721311811yescharacteristic
237C2#102×D21111yescharacteristic
238C2#2721316111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213691C2
cosocleC22×C4×C211C23
hypercenter#327D211
hypocenterC211#1517
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. #27213196C2semidirect
3. #27213010C2semidirect
4. #27212881C2semidirect
5. #27212880C2semidirect
6. #27212879C2semidirect
7. #327×C211C2semidirect
9. #20269976C4semidirect
10. #20269973C4semidirect
14. D4×Dic211C4semidirect
15. #20269939C4semidirect
119. C211#1517semidirect
127. #327D211semidirect

Maximal subgroups

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

subgroupindex
#272136912
#272131962
#272130102
#272128812
#272128802
#272128792
#327×C2112
#1517211