Finite Group Databasebeta

← #41012436

Subgroups — #41012436

Order 27008.

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

normal subgroups only

This group has 246 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#27213465C211yesmaximal, characteristic, complemented
3#27213462C211yesmaximal, characteristic, complemented
4#27213086C211yesmaximal, 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⋊Dic211C2211yescharacteristic
16C2×Dic211⋊C4C2211yescharacteristic
17#20269909—21no—
18#20269908—21no—
19#20269899—21no—
20#20269898—21no—
21C22⋊(C211⋊C8)C2211yescharacteristic
22C211⋊#95—21no—
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⋊Dic211D411yescharacteristic
43Dic211⋊C4—21no—
44Dic211⋊C4—21no—
45Dic211⋊C4—21no—
46C22⋊Dic211—21no—
47(C2×D211)⋊C4—21no—
48(C2×D211)⋊C4—21no—
49C2×D4×C211—21no—
50C2×D4×C211D411yescharacteristic
51C22×C4×C211—21no—
52C22×C4×C211C2311yescharacteristic
53C2×C8×C211—21no—
54C4⋊C4×C211D411yescharacteristic
55C22⋊C4×C211—21no—
56C22×D211—21no—
57Dic422—42no—
58D844—41no—
59C22⋊D211—42no—
60C22⋊D211—41no—
61C22⋊D211—41no—
62C22⋊D211—81no—
63C2×Dic211—21no—
64C2×Dic211—21no—
65C2×Dic211—21no—
66C2×Dic211—21no—
67C2×Dic211—42no—
68C2×Dic211—41no—
69C2×Dic211—42no—
70C2×Dic211—41no—
71C2×Dic211SD1612yes—
72C4×D211—42no—
73C211⋊C8—41no—
74C23×C211—21no—
75C23×C211C2×D411yescharacteristic
76D4×C211—41no—
77D4×C211—41no—
78D4×C211—41no—
79C2×C4×C211—21no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211—41no—
83C2×C4×C211—41no—
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211C2×D411yescharacteristic
86C8×C211—41no—
87C22×C211—21no—
88C22×C211—21no—
89C22×C211—41no—
90C22×C211—41no—
91C22×C211—41no—
92C22×C211#12011yescharacteristic
93C22×C211C2×SD1611yescharacteristic
94C22×C211AGL(1,ℤ/8)11yescharacteristic
95C2×D211—41no—
96C2×D211—41no—
97C2×D211—41no—
98C4×C211—21no—
99C4×C211—21no—
100C4×C211—41no—
101C4×C211—81no—
102Dic211—22no—
103Dic211—41no—
104Dic211—41no—
105Dic211—81no—
106C2×C211—21no—
107C2×C211—21no—
108C2×C211—81no—
109C2×C211#45011yescharacteristic
110C2×C211#45411yescharacteristic
111C2×C211#45811yescharacteristic
112D211—81no—
113C211#161611yescharacteristic, complemented, Sylow
114#1616—2111nomaximal, Sylow
115#537—2111no—
116#481—2111no—
117#481—2111no—
118#400—2111no—
119#327—2111no—
120#327D21111yescharacteristic, complemented
121#324—2111no—
122C2×#41—4221no—
123C22⋊Q8—4221no—
124C22⋊Q8—4221no—
125#121—4221no—
126#121—4221no—
127#121—4221no—
128#121—2111no—
129#121C2×D21111yescharacteristic
130C2×C4⋊C4—2111no—
131C2×C4⋊C4—2111no—
132#106—4221no—
133#106—4221no—
134#102—4221no—
135#102—4221no—
136C22⋊C8—2111no—
137C22⋊C8C2×D21111yescharacteristic
138#95—4221no—
139#95—4221no—
140#95C2×D21111yescharacteristic
141#41—8442no—
142#41—8442no—
143C2×Q8—4221no—
144C2×D4—21no—
145C2×D4—4221no—
146C2×D4—4221no—
147C2×D4—4221no—
148C2×D4—2111no—
149C2×D4D84411yescharacteristic
150C22×C4—21no—
151C22×C4—4221no—
152C22×C4—4221no—
153C22×C4—4221no—
154C22×C4—2111no—
155C22×C4—2111no—
156C22×C4C22×D21111yescharacteristic
157C2×C8—21no—
158C2×C8—4221no—
159C4⋊C4—4221no—
160C4⋊C4—4221no—
161C4⋊C4—4221no—
162C4⋊C4—4221no—
163C4⋊C4—4221no—
164C4⋊C4—2111no—
165C4⋊C4D84411yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—4221no—
168C22⋊C4—4221no—
169C22⋊C4—4221no—
170C23—21no—
171C23—4221no—
172C23D4×D21111yescharacteristic
173Q8—8442no—
174D4—41no—
175D4—41no—
176D4—41no—
177D4—8441no—
178D4—8441no—
179D4—8441no—
180D4—8442no—
181D4—16881no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—8442no—
188C2×C4—8442no—
189C2×C4—8441no—
190C2×C4—8442no—
191C2×C4—8441no—
192C2×C4—4221no—
193C2×C4—4221no—
194C2×C4—4221no—
195C2×C4—4221no—
196C2×C4—2112no—
197C2×C4D4×D21111yescharacteristic
198C2×C4C2×D84411yescharacteristic
199C8—41no—
200C8—8441no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—8441no—
207C22—8441no—
208C22—8441no—
209C22#2026994711yescharacteristic
210C22SD16×D21111yescharacteristic
211C22#2027000111yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—8441no—
217C4—8441no—
218C4—16881no—
219C4—4222no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—16881no—
224C2#2721329811yescharacteristic
225C2#2721332011yescharacteristic
226C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211#120
prime radical#27213723C2
hypercenter#327D211
hypocenterC211#1616
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. #27213465C2semidirect
3. #27213462C2semidirect
4. #27213086C2semidirect
5. #27212879C2semidirect
6. #27212871C2semidirect
7. #327×C211C2semidirect
113. C211#1616semidirect
120. #327D211semidirect

Maximal subgroups

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

subgroupindex
#272137232
#272134652
#272134622
#272130862
#272128792
#272128712
#327×C2112
#1616211