Finite Group Databasebeta

← #41012417

Subgroups — #41012417

Order 27008.

288 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 53 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#27213708C211yesmaximal, characteristic
2#27213464C212yesmaximal, complemented
3#27212879C212yesmaximal, complemented
4#27212870C211yesmaximal, characteristic, complemented
5#399×C211C211yesmaximal, characteristic, complemented
6C22×C22⋊D211—21no—
7#20269961—22no—
8#20269951—21no—
9#20269951C2212yes—
10#20269947—22no—
11#20269908—22no—
12#20269897—22no—
13C22⋊(C211⋊C8)C2212yescomplemented
14C2×C4⋊C4×C211—21no—
15C2×C4⋊C4×C211C2211yescharacteristic, complemented
16C2×C22⋊C4×C211—21no—
17C2×C22⋊C4×C211C2211yescharacteristic
18#95×C211C2211yescharacteristic, complemented
19C23×D211—21no—
20C2×D844—21no—
21C2×D844D412yes—
22C2×C22⋊D211—22no—
23C2×C22⋊D211—22no—
24C2×C22⋊D211—42no—
25C2×C22⋊D211—42no—
26C22×Dic211—21no—
27C2×C211⋊C8—22no—
28C4⋊Dic211—21no—
29C4⋊Dic211D412yes—
30(C2×D211)⋊C4—22no—
31(C2×D211)⋊C4—22no—
32C24×C211D411yescharacteristic
33C22×C4×C211—21no—
34C22×C4×C211—21no—
35C22×C4×C211D411yescharacteristic
36C22×C4×C211C2311yescharacteristic
37C4⋊C4×C211—22no—
38C4⋊C4×C211—42no—
39C22⋊C4×C211—22no—
40C22⋊C4×C211—42no—
41C22×D211—22no—
42C22×D211—42no—
43C22×D211—81no—
44C22×D211—81no—
45D844—42no—
46D844—81no—
47C22⋊D211—44no—
48C22⋊D211—42no—
49C22⋊D211—42no—
50C2×Dic211—22no—
51C2×Dic211—42no—
52C211⋊C8—42no—
53C23×C211—21no—
54C23×C211—21no—
55C23×C211—41no—
56C23×C211—41no—
57C23×C211SD1612yes—
58C23×C211C2×D411yescharacteristic
59C2×C4×C211—22no—
60C2×C4×C211—21no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211—41no—
66C2×C4×C211—41no—
67C2×C4×C211—41no—
68C2×C4×C211C2×D412yes—
69C22×C211—21no—
70C22×C211—21no—
71C22×C211—22no—
72C22×C211—42no—
73C22×C211—41no—
74C22×C211—41no—
75C22×C211—81no—
76C22×C211#12011yescharacteristic
77C22×C211C2×SD1611yescharacteristic
78C22×C211AGL(1,ℤ/8)11yescharacteristic
79C2×D211—42no—
80C2×D211—42no—
81C2×D211—42no—
82C2×D211—84no—
83C4×C211—22no—
84C4×C211—42no—
85C4×C211—41no—
86C4×C211—81no—
87Dic211—42no—
88C2×C211—21no—
89C2×C211—21no—
90C2×C211—42no—
91C2×C211#45011yescharacteristic
92C2×C211#45311yescharacteristic
93C2×C211#45811yescharacteristic
94D211—82no—
95C211#161311yescharacteristic, complemented, Sylow
96#1613—2111nomaximal, Sylow
97#534—2111no—
98#481—2112no—
99#399D21111yescharacteristic, complemented
100#327—2112no—
101#323—2111no—
102C22×D4—4221no—
103#123—4222no—
104#121—4221no—
105#121—2112no—
106#120—4222no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C2×D21111yescharacteristic, complemented
109C2×C22⋊C4—21no—
110C2×C22⋊C4C2×D21111yescharacteristic
111#106—4222no—
112#102—4222no—
113C22⋊C8—2112no—
114#95C2×D21111yescharacteristic, complemented
115C24—4221no—
116C24C22⋊D21111yescharacteristic
117C2×D4—8442no—
118C2×D4—8442no—
119C2×D4—4221no—
120C2×D4—4222no—
121C2×D4—4222no—
122C2×D4—2112no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—4221no—
126C22×C4C22⋊D21111yescharacteristic
127C22×C4C22×D21111yescharacteristic
128C2×C8—4222no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—4221no—
132C4⋊C4—2112no—
133C22⋊C4—22no—
134C22⋊C4—42no—
135C22⋊C4—4222no—
136C22⋊C4—4222no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—8442no—
142C23—16881no—
143C23—16881no—
144C23—4222no—
145C23#1680994912yes—
146C23C2×C22⋊D21111yescharacteristic
147D4—8442no—
148D4—8444no—
149D4—8442no—
150D4—8442no—
151D4—16881no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—8442no—
162C2×C4—4222no—
163C2×C4D4×D21112yes—
164C8—8442no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—8442no—
173C22—8442no—
174C22—8442no—
175C22—16884no—
176C22#2026994511yescharacteristic
177C22C2×#1680994911yescharacteristic
178C22#2027000211yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—81no—
183C4—8442no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—16882no—
188C2#2721329611yescharacteristic
189C2#2721331311yescharacteristic
190C2#2721333511yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#399×C211C2
socleC22×C211#120
prime radical#27213708C2
hypercenter#399D211
hypocenterC211#1613
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #27213464C2semidirect
3. #27212879C2semidirect
4. #27212870C2semidirect
5. #399×C211C2semidirect
13. C22⋊(C211⋊C8)C22semidirect
15. C2×C4⋊C4×C211C22semidirect
18. #95×C211C22semidirect
95. C211#1613semidirect
99. #399D211semidirect
108. C2×C4⋊C4C2×D211semidirect
114. #95C2×D211semidirect

Maximal subgroups

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

subgroupindex
#272137082
#272134642
#272128792
#272128702
#399×C2112
#1613211