Finite Group Databasebeta

← #41012402

Subgroups — #41012402

Order 27008.

254 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 51 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#27213723C211yesmaximal, characteristic
2#27213460C211yesmaximal, characteristic, complemented
3#27213457C211yesmaximal, characteristic, complemented
4#27213078C211yesmaximal, characteristic, complemented
5#27212879C211yesmaximal, characteristic, complemented
6#27212871C211yesmaximal, characteristic, complemented
7#327×C211C211yesmaximal, characteristic, complemented
8C2×#16809961—21no—
9#20269955—21no—
10#20269954—21no—
11#20269951C2211yescharacteristic
12#20269950—21no—
13#20269949—21no—
14#20269949—21no—
15C2×C4⋊Dic211—21no—
16C2×C4⋊Dic211C2211yescharacteristic
17C2×Dic211⋊C4C2211yescharacteristic
18#20269911—21no—
19#20269910—21no—
20#20269899—21no—
21#20269898—21no—
22C22⋊(C211⋊C8)C2211yescharacteristic
23C211⋊#95—21no—
24#121×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27#16809961—42no—
28#16809961—42no—
29C2×Dic422—21no—
30C2×D844D411yescharacteristic
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⋊Dic211—21no—
41C4⋊Dic211—21no—
42C4⋊Dic211—42no—
43C4⋊Dic211D411yescharacteristic
44Dic211⋊C4—21no—
45Dic211⋊C4—21no—
46Dic211⋊C4—21no—
47C22⋊Dic211—21no—
48(C2×D211)⋊C4—21no—
49(C2×D211)⋊C4—21no—
50C2×D4×C211—21no—
51C2×D4×C211D411yescharacteristic
52C22×C4×C211—21no—
53C22×C4×C211C2311yescharacteristic
54C2×C8×C211—21no—
55C4⋊C4×C211D411yescharacteristic
56C22⋊C4×C211—21no—
57C22×D211—21no—
58Dic422—42no—
59D844—41no—
60C22⋊D211—42no—
61C22⋊D211—41no—
62C22⋊D211—41no—
63C22⋊D211—81no—
64C2×Dic211—21no—
65C2×Dic211—21no—
66C2×Dic211—21no—
67C2×Dic211—21no—
68C2×Dic211—42no—
69C2×Dic211—41no—
70C2×Dic211—41no—
71C2×Dic211—41no—
72C2×Dic211—41no—
73C2×Dic211D812yes—
74C4×D211—42no—
75C211⋊C8—41no—
76C23×C211—21no—
77C23×C211C2×D411yescharacteristic
78D4×C211—41no—
79D4×C211—41no—
80D4×C211—41no—
81C2×C4×C211—22no—
82C2×C4×C211—21no—
83C2×C4×C211—41no—
84C2×C4×C211—41no—
85C2×C4×C211—41no—
86C2×C4×C211C2×D411yescharacteristic
87C2×C4×C211C2×D411yescharacteristic
88C8×C211—41no—
89C22×C211—21no—
90C22×C211—21no—
91C22×C211—41no—
92C22×C211—41no—
93C22×C211—41no—
94C22×C211#12011yescharacteristic
95C22×C211C2×D811yescharacteristic
96C22×C211AGL(1,ℤ/8)11yescharacteristic
97C2×D211—41no—
98C2×D211—41no—
99C2×D211—41no—
100C4×C211—21no—
101C4×C211—21no—
102C4×C211—42no—
103C4×C211—41no—
104Dic211—22no—
105Dic211—41no—
106Dic211—41no—
107Dic211—81no—
108C2×C211—21no—
109C2×C211—21no—
110C2×C211—81no—
111C2×C211#44711yescharacteristic
112C2×C211#45611yescharacteristic
113C2×C211#45811yescharacteristic
114D211—81no—
115C211#161011yescharacteristic, complemented, Sylow
116#1610—2111nomaximal, Sylow
117#537—2111no—
118#480—2111no—
119#480—2111no—
120#398—2111no—
121#327—2111no—
122#327D21111yescharacteristic, complemented
123#324—2111no—
124C2×#41—4221no—
125C22⋊Q8—4221no—
126C22⋊Q8—4221no—
127#121—4221no—
128#121—4221no—
129#121—4221no—
130#121—2111no—
131#121C2×D21111yescharacteristic
132C2×C4⋊C4—4221no—
133C2×C4⋊C4—2111no—
134C2×C4⋊C4—2111no—
135#107—4221no—
136#107—4221no—
137#102—4221no—
138#102—4221no—
139C22⋊C8—2111no—
140C22⋊C8C2×D21111yescharacteristic
141#95—4221no—
142#95C2×D21111yescharacteristic
143#41—8442no—
144#41—8442no—
145C2×Q8—4221no—
146C2×D4—21no—
147C2×D4—4221no—
148C2×D4—4221no—
149C2×D4—4221no—
150C2×D4—2111no—
151C2×D4D84411yescharacteristic
152C22×C4—21no—
153C22×C4—4221no—
154C22×C4—4221no—
155C22×C4—4221no—
156C22×C4—2111no—
157C22×C4—2111no—
158C22×C4C22×D21111yescharacteristic
159C2×C8—21no—
160C2×C8—4221no—
161C4⋊C4—8442no—
162C4⋊C4—4221no—
163C4⋊C4—4221no—
164C4⋊C4—4221no—
165C4⋊C4—4221no—
166C4⋊C4—4221no—
167C4⋊C4—2111no—
168C4⋊C4D84411yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—4221no—
171C22⋊C4—4221no—
172C22⋊C4—4221no—
173C23—21no—
174C23—4221no—
175C23D4×D21111yescharacteristic
176Q8—8442no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—8441no—
181D4—8441no—
182D4—8441no—
183D4—8442no—
184D4—16881no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—8442no—
191C2×C4—8442no—
192C2×C4—8441no—
193C2×C4—8441no—
194C2×C4—8441no—
195C2×C4—8441no—
196C2×C4—4221no—
197C2×C4—4221no—
198C2×C4—4221no—
199C2×C4—4221no—
200C2×C4—2112no—
201C2×C4D4×D21111yescharacteristic
202C2×C4C2×D84411yescharacteristic
203C8—41no—
204C8—8441no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—8441no—
211C22—8441no—
212C22—8441no—
213C22#2026994711yescharacteristic
214C22D8×D21111yescharacteristic
215C22#2027000411yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—8441no—
221C4—8441no—
222C4—16881no—
223C4—4222no—
224C2—21no—
225C2—21no—
226C2—81no—
227C2—16881no—
228C2#2721328111yescharacteristic
229C2#2721333011yescharacteristic
230C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211#120
prime radical#27213723C2
hypercenter#327D211
hypocenterC211#1610
LCSG ▷ C22×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. #27213460C2semidirect
3. #27213457C2semidirect
4. #27213078C2semidirect
5. #27212879C2semidirect
6. #27212871C2semidirect
7. #327×C211C2semidirect
115. C211#1610semidirect
122. #327D211semidirect

Maximal subgroups

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

subgroupindex
#272137232
#272134602
#272134572
#272130782
#272128792
#272128712
#327×C2112
#1610211