Finite Group Databasebeta

← #41012627

Subgroups — #41012627

Order 27008.

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

normal subgroups only

This group has 212 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#27213862C211yesmaximal, characteristic
2#27213496C211yesmaximal, characteristic, complemented
3#27213427C211yesmaximal, characteristic, complemented
4#27213352C211yesmaximal, characteristic, complemented
5#27212873C211yesmaximal, characteristic, complemented
6#27212873C211yesmaximal, characteristic, complemented
7#333×C211C211yesmaximal, characteristic, complemented
8C2×#16809951—21no—
9#20269977C2211yescharacteristic
10#20269966—22no—
11#20269952—21no—
12#20269952—21no—
13C4×D844C2211yescharacteristic
14C4×D844C2211yescharacteristic
15#20269935—21no—
16#20269935—21no—
17#20269911—21no—
18#20269902—21no—
19#20269901—22no—
20#20269899—21no—
21C4×C211⋊C8C2211yescharacteristic
22C4⋊Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4⋊C8×C211C2211yescharacteristic
25C2×Dic422—21no—
26C2×D844—22no—
27C2×D844D411yescharacteristic
28C2×D844D411yescharacteristic
29C2×C4×D211—21no—
30C2×C4×D211—21no—
31#16809951—42no—
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211D411yescharacteristic
36C4⋊Dic211D411yescharacteristic
37(C2×D211)⋊C4—22no—
38(C2×D211)⋊C4—22no—
39(C2×D211)⋊C4—21no—
40(C2×D211)⋊C4—21no—
41C4×Dic211—22no—
42C2×Q8×C211D412yes—
43C2×C8×C211—21no—
44C2×C8×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211—21no—
47C42×C211C2311yescharacteristic
48C22×D211—21no—
49C22×D211—21no—
50Dic422—41no—
51Dic422—41no—
52D844—21no—
53D844—21no—
54D844—21no—
55D844—21no—
56D844—82no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—41no—
60C4×D211—41no—
61C4×D211—41no—
62C4×D211—41no—
63C4×D211—41no—
64C211⋊C8—41no—
65Q8×C211—42no—
66C2×C4×C211—22no—
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C8×C211—41no—
71C8×C211—41no—
72C22×C211#12011yescharacteristic
73C2×D211—41no—
74C2×D211—41no—
75C2×D211—41no—
76C2×D211—41no—
77C2×D211—41no—
78C2×D211—41no—
79C4×C211—21no—
80C4×C211—42no—
81C4×C211#13511yescharacteristic
82C4×C211#13511yescharacteristic
83C4×C211AGL(1,ℤ/8)11yescharacteristic
84C4×C211#13711yescharacteristic
85Dic211—41no—
86Dic211—41no—
87Dic211—81no—
88C2×C211#44911yescharacteristic
89C2×C211#45211yescharacteristic
90C2×C211#45511yescharacteristic
91D211—81no—
92D211—81no—
93C211#164611yescharacteristic, complemented, Sylow
94#1646—2111nomaximal, Sylow
95#555—2111no—
96#487—2111no—
97#474—2111no—
98#461—2111no—
99#333D21111yescharacteristic, complemented
100D4⋊C8—2111no—
101D4⋊C8—2111no—
102C2×SD16—4221no—
103C4⋊Q8—2111no—
104C4⋊Q8C2×D21111yescharacteristic
105#124—4222no—
106#121—4221no—
107#121—4221no—
108C4×D4—2111no—
109C4×D4—2111no—
110#117—4221no—
111#117—4221no—
112#107—4221no—
113C4⋊C8C2×D21111yescharacteristic
114C4⋊C8C2×D21111yescharacteristic
115#103—4222no—
116#103—4221no—
117#102—4221no—
118C4×C8—2111no—
119C2×Q8—4221no—
120C2×Q8D84412yes—
121C2×D4—4222no—
122C2×D4—2111no—
123C2×D4—2111no—
124C22×C4—4221no—
125C22×C4—4221no—
126SD16—8442no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—4221no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—4221no—
133C4⋊C4—4221no—
134C4⋊C4—2111no—
135C4⋊C4—2111no—
136C22⋊C4—4222no—
137C22⋊C4—4222no—
138C22⋊C4—4221no—
139C22⋊C4—4221no—
140C42—4222no—
141C42C22×D21111yescharacteristic
142C23—4221no—
143C23—4221no—
144Q8—42no—
145Q8—8441no—
146Q8—8441no—
147D4—16882no—
148D4—4221no—
149D4—4221no—
150D4—4221no—
151D4—4221no—
152C2×C4—22no—
153C2×C4—8441no—
154C2×C4—8441no—
155C2×C4—8441no—
156C2×C4—8441no—
157C2×C4—8441no—
158C2×C4—4221no—
159C2×C4—4221no—
160C2×C4D4×D21111yescharacteristic
161C2×C4D4×D21111yescharacteristic
162C2×C4C2×D84411yescharacteristic
163C8—41no—
164C8—41no—
165C8—8441no—
166C22—8441no—
167C22—8441no—
168C22—8441no—
169C22—8441no—
170C22—8441no—
171C22—8441no—
172C22#2026994711yescharacteristic
173C4—21no—
174C4—42no—
175C4—8441no—
176C4—8441no—
177C4—16881no—
178C4#2026999811yescharacteristic
179C4#2027000011yescharacteristic
180C4#2027000411yescharacteristic
181C4#2027000911yescharacteristic
182C2—16881no—
183C2—16881no—
184C2#2721329411yescharacteristic
185C2#2721331111yescharacteristic
186C2#2721332611yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#333×C211C2
socleC22×C211#120
prime radical#27213862C2
hypercenter#333D211
hypocenterC211#1646
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #27213496C2semidirect
3. #27213427C2semidirect
4. #27213352C2semidirect
5. #27212873C2semidirect
6. #27212873C2semidirect
7. #333×C211C2semidirect
93. C211#1646semidirect
99. #333D211semidirect

Maximal subgroups

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

subgroupindex
#272138622
#272134962
#272134272
#272133522
#272128732
#333×C2112
#1646211