Finite Group Databasebeta

← #41012578

Subgroups — #41012578

Order 27008.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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#27213781C211yesmaximal, characteristic
2#27213492C211yesmaximal, characteristic, complemented
3#27213427C211yesmaximal, characteristic, complemented
4#27213352C211yesmaximal, characteristic, complemented
5#27212873C211yesmaximal, characteristic, complemented
6#27212873C211yesmaximal, characteristic, complemented
7#331×C211C211yesmaximal, characteristic, complemented
8C2×D4×D211—21no—
9C2×D4×D211—21no—
10C2×#16809951—21no—
11#20269977C2211yescharacteristic, complemented
12#20269948—21no—
13#20269948—21no—
14#20269945—22no—
15C4×D844C2211yescharacteristic
16C4×D844C2211yescharacteristic
17#20269911—21no—
18#20269902—21no—
19#20269899—21no—
20#20269898—22no—
21C4×C211⋊C8C2211yescharacteristic, complemented
22#127×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic, complemented
24C4⋊C8×C211C2211yescharacteristic, complemented
25C23×D211—22no—
26C2×Dic422—21no—
27C2×D844D411yescharacteristic
28C2×D844D411yescharacteristic
29C2×C22⋊D211—22no—
30C2×C22⋊D211—22no—
31D4×D211—42no—
32D4×D211—42no—
33D4×D211—42no—
34D4×D211—42no—
35C2×C4×D211—21no—
36C2×C4×D211—21no—
37#16809951—42no—
38C2×C211⋊C8—21no—
39C4⋊Dic211—21no—
40C4⋊Dic211—21no—
41C4⋊Dic211D411yescharacteristic
42C4⋊Dic211D411yescharacteristic
43C22⋊Dic211—22no—
44(C2×D211)⋊C4—21no—
45(C2×D211)⋊C4—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211D412yes—
49C2×C8×C211—21no—
50C2×C8×C211—21no—
51C42×C211C2311yescharacteristic
52C22×D211—21no—
53C22×D211—21no—
54C22×D211—44no—
55C22×D211—44no—
56C22×D211—82no—
57Dic422—41no—
58Dic422—41no—
59D844—21no—
60D844—21no—
61D844D812yes—
62D844SD1612yescomplemented
63C22⋊D211—44no—
64C22⋊D211—44no—
65C2×Dic211—21no—
66C2×Dic211—21no—
67C2×Dic211—41no—
68C4×D211—41no—
69C4×D211—41no—
70C4×D211—41no—
71C4×D211—41no—
72C211⋊C8—41no—
73C23×C211—22no—
74D4×C211—42no—
75D4×C211—41no—
76D4×C211—41no—
77D4×C211—41no—
78D4×C211—41no—
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C8×C211—41no—
83C8×C211—41no—
84C22×C211—42no—
85C22×C211—42no—
86C22×C211—42no—
87C22×C211#12011yescharacteristic
88C2×D211—22no—
89C2×D211—22no—
90C2×D211—41no—
91C2×D211—41no—
92C2×D211—41no—
93C2×D211—41no—
94C2×D211—88no—
95C4×C211—21no—
96C4×C211C2×D811yescharacteristic
97C4×C211C2×SD1611yescharacteristic
98C4×C211AGL(1,ℤ/8)11yescharacteristic
99C4×C211AGL(1,ℤ/8)11yescharacteristic
100Dic211—41no—
101Dic211—41no—
102Dic211—81no—
103C2×C211—82no—
104C2×C211#44711yescharacteristic
105C2×C211#45011yescharacteristic
106C2×C211#45511yescharacteristic
107D211—42no—
108D211—42no—
109C211#163911yescharacteristic, complemented, Sylow
110#1639—2111nomaximal, Sylow
111D42—2111no—
112#486—2111no—
113#474—2111no—
114#461—2111no—
115#331D21111yescharacteristic, complemented
116D4⋊C8—2111no—
117D4⋊C8—2111no—
118C22×D4—4221no—
119C22×D4—4221no—
120C2×SD16—4221no—
121C4⋊Q8—2111no—
122#127C2×D21111yescharacteristic
123#121—4221no—
124#121—4221no—
125#120—4222no—
126C4×D4—2111no—
127C4×D4—2111no—
128#107—4221no—
129C4⋊C8C2×D21111yescharacteristic, complemented
130C4⋊C8C2×D21111yescharacteristic, complemented
131#103—4221no—
132#102—4221no—
133#102—4222no—
134C4×C8—2111no—
135C24—4222no—
136C2×Q8—4221no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—8442no—
140C2×D4—8442no—
141C2×D4—8442no—
142C2×D4—8442no—
143C2×D4—4222no—
144C2×D4—4222no—
145C2×D4—2111no—
146C2×D4—2111no—
147C2×D4D84412yes—
148C22×C4—4221no—
149C22×C4—4221no—
150SD16—8442no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—4221no—
154C4⋊C4—4221no—
155C4⋊C4—4221no—
156C4⋊C4—2111no—
157C4⋊C4—2111no—
158C22⋊C4—4222no—
159C22⋊C4—4221no—
160C22⋊C4—4221no—
161C42C22×D21111yescharacteristic
162C23—22no—
163C23—8444no—
164C23—8444no—
165C23—16882no—
166C23—4221no—
167C23—4221no—
168Q8—8441no—
169Q8—8441no—
170D4—42no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—8444no—
176D4—8444no—
177D4—4221no—
178D4—4221no—
179D4—2112no—
180D4—2112no—
181C2×C4—8441no—
182C2×C4—8441no—
183C2×C4—8441no—
184C2×C4—8441no—
185C2×C4—8441no—
186C2×C4—4221no—
187C2×C4—4221no—
188C2×C4D4×D21111yescharacteristic
189C2×C4D4×D21111yescharacteristic
190C2×C4C2×D84411yescharacteristic
191C8—41no—
192C8—41no—
193C8—8441no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—8441no—
198C22—8441no—
199C22—8441no—
200C22—8441no—
201C22—16888no—
202C22—4222no—
203C22—4222no—
204C22#2026994711yescharacteristic
205C4—21no—
206C4—8441no—
207C4—8441no—
208C4—16881no—
209C4D8×D21111yescharacteristic
210C4SD16×D21111yescharacteristic
211C4#2027000111yescharacteristic
212C4#2027000411yescharacteristic
213C2—82no—
214C2—8442no—
215C2—8442no—
216C2#2721328111yescharacteristic
217C2#2721329811yescharacteristic
218C2#2721332611yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#331×C211C2
socleC22×C211#120
prime radical#27213781C2
hypercenter#331D211
hypocenterC211#1639
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27213492C2semidirect
3. #27213427C2semidirect
4. #27213352C2semidirect
5. #27212873C2semidirect
6. #27212873C2semidirect
7. #331×C211C2semidirect
11. #20269977C22semidirect
21. C4×C211⋊C8C22semidirect
23. C4⋊C8×C211C22semidirect
24. C4⋊C8×C211C22semidirect
62. D844SD16semidirect
109. C211#1639semidirect
115. #331D211semidirect
129. C4⋊C8C2×D211semidirect
130. C4⋊C8C2×D211semidirect

Maximal subgroups

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

subgroupindex
#272137812
#272134922
#272134272
#272133522
#272128732
#331×C2112
#1639211