Finite Group Databasebeta

← #41011935

Subgroups — #41011935

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#27213682C211yesmaximal, characteristic
2#27213200C211yesmaximal, characteristic, complemented
3#27213010C211yesmaximal, characteristic, complemented
4#27212883C211yesmaximal, characteristic, complemented
5#27212881C211yesmaximal, characteristic
6#27212879C211yesmaximal, characteristic
7#328×C211C211yesmaximal, characteristic, complemented
8#20269972—21no—
9#20269968C411yescharacteristic, complemented
10#20269966C411yescharacteristic, complemented
11#20269961—21no—
12#20269951C2211yescharacteristic
13Q8×Dic211C411yescharacteristic, complemented
14#20269939C411yescharacteristic, complemented
15(C22⋊D211)⋊C4—21no—
16#20269935—21no—
17C2×Dic211⋊C4—21no—
18C2×C4×Dic211C2211yescharacteristic
19Dic211⋊C8—21no—
20C22⋊(C211⋊C8)C2211yescharacteristic
21#20269887—21no—
22C211⋊#95—21no—
23C211⋊#95C2211yescharacteristic
24C22⋊Q8×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C2×D844C2×C411yescharacteristic
28C2×C22⋊D211—21no—
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C22×Dic211D411yescharacteristic
32C2×C4×D211—21no—
33C2×C211⋊C8—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211C2×C411yescharacteristic
37Dic211⋊C4—21no—
38Dic211⋊C4—42no—
39(C2×D211)⋊C4—21no—
40(C2×D211)⋊C4—21no—
41(C2×D211)⋊C4—21no—
42(C2×D211)⋊C4—21no—
43C4×Dic211—21no—
44C4×Dic211C2×C411yescharacteristic
45C4×Dic211C2×C411yescharacteristic
46C4×Dic211D411yescharacteristic
47C4×Dic211D411yescharacteristic
48C2×Q8×C211C2×C411yescharacteristic
49C22×C4×C211—21no—
50C22×C4×C211C2311yescharacteristic
51C2×C8×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211C2×C411yescharacteristic
54C22⋊C4×C211—21no—
55C22×D211—21no—
56D844—41no—
57C22⋊D211—41no—
58C22⋊D211—41no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—21no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C2×Dic211C22⋊C411yescharacteristic
67C2×Dic211C22⋊C411yescharacteristic
68C2×Dic211C22⋊C411yescharacteristic
69C2×Dic211C22⋊C411yescharacteristic
70C4×D211—42no—
71C211⋊C8—41no—
72C23×C211C2×D411yescharacteristic
73Q8×C211—41no—
74C2×C4×C211—21no—
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×C211C2×C22⋊C411yescharacteristic
86C22×C211AGL(1,ℤ/8)11yescharacteristic
87C22×C211#13711yescharacteristic
88C2×D211—41no—
89C2×D211—41no—
90C2×D211—41no—
91C4×C211—21no—
92C4×C211—21no—
93C4×C211—41no—
94C4×C211—41no—
95C4×C211—81no—
96Dic211—21no—
97Dic211—21no—
98Dic211—21no—
99Dic211—21no—
100Dic211—41no—
101Dic211—81no—
102C2×C211—21no—
103C2×C211—21no—
104C2×C211#41011yescharacteristic
105C2×C211#41711yescharacteristic
106C2×C211#42111yescharacteristic
107D211—81no—
108C211#151211yescharacteristic, complemented, Sylow
109#1512—2111nomaximal, Sylow
110#529—2111no—
111#432—2111no—
112#382—2111no—
113#328—2111no—
114#328D21111yescharacteristic, complemented
115#327—2111no—
116#327—2111no—
117#126—4221no—
118#125—2111no—
119#124—2111no—
120#123—4221no—
121C22⋊Q8C2×D21111yescharacteristic
122#121—2111no—
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×Q8C4×D21111yescharacteristic
137C2×D4—4221no—
138C2×D4—2111no—
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—21no—
148C4⋊C4—8442no—
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—4221no—
165C23D4×D21111yescharacteristic
166Q8—41no—
167D4—8441no—
168D4—8441no—
169D4—8441no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—8442no—
177C2×C4—8441no—
178C2×C4—8441no—
179C2×C4—8441no—
180C2×C4—8441no—
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—8441no—
195C22—8441no—
196C22—8441no—
197C22C22⋊C4×D21111yescharacteristic
198C22#2027000411yescharacteristic
199C22#2027000911yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—81no—
205C4—8441no—
206C4—16881no—
207C4—4221no—
208C4—4221no—
209C4—4221no—
210C4—4221no—
211C2—21no—
212C2—21no—
213C2—16881no—
214C2#2721311811yescharacteristic
215C2#2721314611yescharacteristic
216C2#2721316111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#328×C211C2
socleC22×C211C2×C22⋊C4
prime radical#20269951C22
cosocleC22×C4×C211C23
hypercenter#328D211
hypocenterC211#1512
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #27213200C2semidirect
3. #27213010C2semidirect
4. #27212883C2semidirect
7. #328×C211C2semidirect
9. #20269968C4semidirect
10. #20269966C4semidirect
13. Q8×Dic211C4semidirect
14. #20269939C4semidirect
108. C211#1512semidirect
114. #328D211semidirect

Maximal subgroups

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

subgroupindex
#272136822
#272132002
#272130102
#272128832
#272128812
#272128792
#328×C2112
#1512211