Finite Group Databasebeta

← #41012449

Subgroups — #41012449

Order 27008.

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

normal subgroups only

This group has 240 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#27213736C211yesmaximal, characteristic
2#27213479C211yesmaximal, characteristic, complemented
3#27213465C211yesmaximal, characteristic, complemented
4#27213083C211yesmaximal, characteristic, complemented
5#27212879C211yesmaximal, characteristic, complemented
6#27212869C211yesmaximal, characteristic, complemented
7#328×C211C211yesmaximal, characteristic, complemented
8#20269972—21no—
9#20269966—21no—
10#20269962—21no—
11#20269952—21no—
12#20269951C2211yescharacteristic
13#20269947—21no—
14(C22⋊D211)⋊C4—21no—
15C2×C4⋊Dic211—21no—
16C2×C4⋊Dic211C2211yescharacteristic
17C2×(C2×D211)⋊C4—21no—
18C2×(C2×D211)⋊C4C2211yescharacteristic
19#20269910—21no—
20#20269909—21no—
21#20269901—21no—
22#20269899—21no—
23C22⋊(C211⋊C8)C2211yescharacteristic
24C22⋊Q8×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C23×D211D411yescharacteristic
28C2×D844—21no—
29C2×D844—21no—
30C2×D844D411yescharacteristic
31C2×C22⋊D211—21no—
32C22×Dic211—21no—
33C22×Dic211D411yescharacteristic
34C2×C4×D211—21no—
35C2×C211⋊C8—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211—21no—
38C4⋊Dic211—42no—
39C4⋊Dic211D411yescharacteristic
40Dic211⋊C4—21no—
41(C2×D211)⋊C4—21no—
42(C2×D211)⋊C4—21no—
43(C2×D211)⋊C4—21no—
44(C2×D211)⋊C4—21no—
45(C2×D211)⋊C4—21no—
46(C2×D211)⋊C4—21no—
47(C2×D211)⋊C4—42no—
48C4×Dic211—21no—
49C2×Q8×C211D411yescharacteristic
50C22×C4×C211—21no—
51C22×C4×C211C2311yescharacteristic
52C2×C8×C211—21no—
53C4⋊C4×C211—21no—
54C4⋊C4×C211D411yescharacteristic
55C22⋊C4×C211—21no—
56C22×D211—21no—
57C22×D211—21no—
58C22×D211—21no—
59C22×D211—21no—
60C22×D211—41no—
61C22×D211—41no—
62D844—41no—
63D844—81no—
64D844—81no—
65C22⋊D211—41no—
66C22⋊D211—41no—
67C2×Dic211—21no—
68C2×Dic211—21no—
69C2×Dic211—41no—
70C2×Dic211—41no—
71C2×Dic211—41no—
72C2×Dic211—41no—
73C4×D211—81no—
74C211⋊C8—41no—
75C23×C211C2×D411yescharacteristic
76Q8×C211—41no—
77C2×C4×C211—22no—
78C2×C4×C211—21no—
79C2×C4×C211—21no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211—41no—
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C8×C211—41no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211#12011yescharacteristic
89C22×C211#13511yescharacteristic
90C22×C211#13711yescharacteristic
91C2×D211—42no—
92C2×D211—41no—
93C2×D211—41no—
94C2×D211—41no—
95C2×D211—41no—
96C2×D211—41no—
97C2×D211—41no—
98C2×D211—81no—
99C4×C211—21no—
100C4×C211—21no—
101C4×C211—42no—
102C4×C211—41no—
103C4×C211—41no—
104Dic211—41no—
105Dic211—41no—
106Dic211—81no—
107C2×C211—21no—
108C2×C211—21no—
109C2×C211#45211yescharacteristic
110C2×C211#45311yescharacteristic
111C2×C211#45811yescharacteristic
112D211—81no—
113D211—81no—
114C211#161811yescharacteristic, complemented, Sylow
115#1618—2111nomaximal, Sylow
116#538—2111no—
117#484—2111no—
118#481—2111no—
119#399—2111no—
120#328D21111yescharacteristic, complemented
121#327—2111no—
122#323—2111no—
123#126—4221no—
124#124—4221no—
125#123—4221no—
126C22⋊Q8C2×D21111yescharacteristic
127#121—4221no—
128#121—2111no—
129#120—4221no—
130C4×D4—4221no—
131C2×C4⋊C4—4221no—
132C2×C4⋊C4—2111no—
133C2×C22⋊C4—4221no—
134C2×C22⋊C4—2111no—
135#107—4221no—
136#106—4221no—
137#103—4221no—
138#102—4221no—
139C22⋊C8—2111no—
140C22⋊C8C2×D21111yescharacteristic
141#95C2×D21111yescharacteristic
142C24—2111no—
143C2×Q8D84411yescharacteristic
144C2×D4—4221no—
145C2×D4—4221no—
146C2×D4—4221no—
147C2×D4—2111no—
148C22×C4—21no—
149C22×C4—4221no—
150C22×C4—4221no—
151C22×C4—2111no—
152C22×C4C22×D21111yescharacteristic
153C2×C8—21no—
154C2×C8—4221no—
155C4⋊C4—21no—
156C4⋊C4—8442no—
157C4⋊C4—4221no—
158C4⋊C4—4221no—
159C4⋊C4—4221no—
160C4⋊C4—2111no—
161C4⋊C4D84411yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—8442no—
164C22⋊C4—4221no—
165C22⋊C4—4221no—
166C22⋊C4—4221no—
167C22⋊C4—4221no—
168C22⋊C4—4221no—
169C22⋊C4—4221no—
170C42—4221no—
171C23—8441no—
172C23—8441no—
173C23—4221no—
174C23—4221no—
175C23—4221no—
176C23—4221no—
177C23D4×D21111yescharacteristic
178Q8—41no—
179D4—8441no—
180D4—8441no—
181D4—8441no—
182D4—16881no—
183D4—16881no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—8441no—
191C2×C4—8441no—
192C2×C4—8441no—
193C2×C4—8441no—
194C2×C4—16881no—
195C2×C4—4221no—
196C2×C4—4221no—
197C2×C4D4×D21111yescharacteristic
198C2×C4C2×D84411yescharacteristic
199C8—41no—
200C8—8441no—
201C22—21no—
202C22—21no—
203C22—8442no—
204C22—8441no—
205C22—8441no—
206C22—8441no—
207C22—8441no—
208C22—8441no—
209C22—8441no—
210C22—16881no—
211C22#2026994711yescharacteristic
212C22#2026999811yescharacteristic
213C22#2027000911yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—41no—
219C4—8441no—
220C4—8441no—
221C4—16881no—
222C2—21no—
223C2—21no—
224C2—16881no—
225C2—16881no—
226C2#2721331111yescharacteristic
227C2#2721331511yescharacteristic
228C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#328×C211C2
socleC22×C211#120
prime radical#27213736C2
hypercenter#328D211
hypocenterC211#1618
LCSG ▷ C22×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. #27213479C2semidirect
3. #27213465C2semidirect
4. #27213083C2semidirect
5. #27212879C2semidirect
6. #27212869C2semidirect
7. #328×C211C2semidirect
114. C211#1618semidirect
120. #328D211semidirect

Maximal subgroups

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

subgroupindex
#272137362
#272134792
#272134652
#272130832
#272128792
#272128692
#328×C2112
#1618211