Finite Group Databasebeta

← #41012575

Subgroups — #41012575

Order 27008.

308 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D844C211yesmaximal, characteristic
2#27213494C211yesmaximal, characteristic, complemented
3#27213423C211yesmaximal, characteristic, complemented
4#27212889C211yesmaximal, characteristic, complemented
5D4⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6#27212873C211yesmaximal, characteristic, complemented
7#461×C211C211yesmaximal, characteristic, complemented
8C22×D844—21no—
9C2×D4×D211—21no—
10#20269974C2211yescharacteristic
11#20269952—21no—
12#20269951—21no—
13#20269947—22no—
14C4×D844C2211yescharacteristic
15#20269910—21no—
16C4⋊(C211⋊C8)C2211yescharacteristic, complemented
17#20269897—21no—
18#20269897—22no—
19C4×C211⋊C8C2211yescharacteristic, complemented
20C2×SD16×C211—21no—
21C4⋊Q8×C211C2211yescharacteristic, complemented
22C4×D4×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic, complemented
24#103×C211—21no—
25C23×D211—22no—
26C2×D844—22no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844—42no—
30C2×D844—42no—
31C2×D844D411yescharacteristic
32C2×D844D412yes—
33C2×C22⋊D211—22no—
34D4×D211—42no—
35D4×D211—42no—
36C2×C4×D211—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C4⋊Dic211D411yescharacteristic
40(C2×D211)⋊C4—22no—
41(C2×D211)⋊C4—21no—
42C2×Q8×C211—21no—
43C2×D4×C211D411yescharacteristic
44C22×C4×C211—21no—
45SD16×C211—42no—
46C2×C8×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211—21no—
49C4⋊C4×C211D411yescharacteristic
50C22⋊C4×C211—21no—
51C42×C211C2311yescharacteristic
52C22×D211—21no—
53C22×D211—22no—
54C22×D211—44no—
55C22×D211—44no—
56C22×D211—82no—
57D844—21no—
58D844—42no—
59D844—44no—
60D844—41no—
61D844—41no—
62D844—41no—
63D844—41no—
64D844SD1612yescomplemented
65C22⋊D211—44no—
66C2×Dic211—21no—
67C4×D211—41no—
68C4×D211—41no—
69C211⋊C8—41no—
70C211⋊C8—41no—
71C23×C211—21no—
72Q8×C211—41no—
73Q8×C211—41no—
74D4×C211—21no—
75D4×C211D812yes—
76C2×C4×C211—21no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C8×C211—41no—
84C22×C211—22no—
85C22×C211—41no—
86C22×C211—41no—
87C22×C211#12011yescharacteristic
88C2×D211—22no—
89C2×D211—41no—
90C2×D211—41no—
91C2×D211—42no—
92C2×D211—42no—
93C2×D211—42no—
94C2×D211—88no—
95C4×C211—21no—
96C4×C211—41no—
97C4×C211—81no—
98C4×C211C2×D811yescharacteristic
99C4×C211C2×SD1611yescharacteristic
100C4×C211AGL(1,ℤ/8)11yescharacteristic
101C4×C211AGL(1,ℤ/8)11yescharacteristic
102Dic211—41no—
103C2×C211—42no—
104C2×C211#44711yescharacteristic
105C2×C211#45011yescharacteristic
106C2×C211#45511yescharacteristic
107D211—42no—
108D211—82no—
109C211#163911yescharacteristic, complemented, Sylow
110#1639—2111nomaximal, Sylow
111D42—2111no—
112#486—2111no—
113#474—2111no—
114#461D21111yescharacteristic, complemented
115#331—2111no—
116D4⋊C8—2111no—
117D4⋊C8—2111no—
118C22×D4—4221no—
119C22×D4—4221no—
120C2×SD16—21no—
121C4⋊Q8C2×D21111yescharacteristic, complemented
122#127—2111no—
123#121—4221no—
124#121—4221no—
125#120—4222no—
126C4×D4—2111no—
127C4×D4C2×D21111yescharacteristic
128#107—4221no—
129C4⋊C8—2111no—
130C4⋊C8C2×D21111yescharacteristic, complemented
131#103—21no—
132#102—4221no—
133#102—4222no—
134C4×C8—2111no—
135C24—4222no—
136C2×Q8—21no—
137C2×D4—8442no—
138C2×D4—8442no—
139C2×D4—8442no—
140C2×D4—8442no—
141C2×D4—4222no—
142C2×D4—4222no—
143C2×D4—4221no—
144C2×D4—4221no—
145C2×D4—2112no—
146C2×D4—2111no—
147C2×D4C22⋊D21111yescharacteristic
148C22×C4—21no—
149C22×C4—4221no—
150SD16—42no—
151C2×C8—21no—
152C2×C8—4221no—
153C2×C8—4221no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—2111no—
157C4⋊C4C22⋊D21111yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—4222no—
160C22⋊C4—4221no—
161C42C22×D21111yescharacteristic
162C23—21no—
163C23—8444no—
164C23—8444no—
165C23—16882no—
166C23—4222no—
167C23—4221no—
168Q8—41no—
169Q8—41no—
170D4—21no—
171D4—8442no—
172D4—8444no—
173D4—8444no—
174D4—8441no—
175D4—8441no—
176D4—8441no—
177D4—8441no—
178D4—4221no—
179D4—2112no—
180D4#1680994712yes—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—8441no—
186C2×C4—8441no—
187C2×C4—4221no—
188C2×C4D4×D21111yescharacteristic
189C2×C4D4×D21111yescharacteristic
190C2×C4C2×C22⋊D21111yescharacteristic
191C8—41no—
192C8—8441no—
193C8—8441no—
194C22—22no—
195C22—41no—
196C22—41no—
197C22—8441no—
198C22—8441no—
199C22—8442no—
200C22—8442no—
201C22—8442no—
202C22—16888no—
203C22—4222no—
204C22#2026994511yescharacteristic
205C4—21no—
206C4—41no—
207C4—81no—
208C4—8441no—
209C4C2×#1680994711yescharacteristic
210C4SD16×D21111yescharacteristic
211C4#2027000211yescharacteristic
212C4#2027000411yescharacteristic
213C2—42no—
214C2—8442no—
215C2—16882no—
216C2#2721327911yescharacteristic
217C2#2721329511yescharacteristic
218C2#2721332211yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#461×C211C2
socleC22×C211#120
prime radicalD4×D844C2
hypercenter#461D211
hypocenterC211#1639
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #27213494C2semidirect
3. #27213423C2semidirect
4. #27212889C2semidirect
5. D4⋊(C211⋊C8)C2semidirect
6. #27212873C2semidirect
7. #461×C211C2semidirect
16. C4⋊(C211⋊C8)C22semidirect
19. C4×C211⋊C8C22semidirect
21. C4⋊Q8×C211C22semidirect
23. C4⋊C8×C211C22semidirect
64. D844SD16semidirect
109. C211#1639semidirect
114. #461D211semidirect
121. C4⋊Q8C2×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
D4×D8442
#272134942
#272134232
#272128892
D4⋊(C211⋊C8)2
#272128732
#461×C2112
#1639211