Finite Group Databasebeta

← #41012679

Subgroups — #41012679

Order 27008.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#27213799C211yesmaximal, characteristic
2#27213423C211yesmaximal, characteristic, complemented
3#27213348C211yesmaximal, characteristic, complemented
4#27212892C211yesmaximal, characteristic, complemented
5#27212875C211yesmaximal, characteristic, complemented
6#27212875C211yesmaximal, characteristic, complemented
7#498×C211C211yesmaximal, characteristic, complemented
8C2×#16809963—21no—
9C2×#16809949—21no—
10#20269962—22no—
11#20269953—21no—
12#20269952—21no—
13C4×D844C2211yescharacteristic
14C4×D844C2211yescharacteristic
15C4⋊C4×D211—21no—
16#20269910—21no—
17C4⋊(C211⋊C8)C2211yescharacteristic
18C4⋊(C211⋊C8)C2211yescharacteristic
19#20269901—21no—
20#20269897—21no—
21C4⋊Q8×C211C2211yescharacteristic
22C4⋊Q8×C211C2211yescharacteristic
23#106×C211—22no—
24C4×C8×C211C2211yescharacteristic
25#16809963—42no—
26#16809963—42no—
27C2×D844—22no—
28C2×D844D411yescharacteristic
29C2×D844D411yescharacteristic
30C2×C4×D211—22no—
31C2×C4×D211—21no—
32C2×C4×D211—21no—
33#16809949—42no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8—21no—
36C4⋊Dic211D411yescharacteristic
37C4⋊Dic211D411yescharacteristic
38Dic211⋊C4—22no—
39(C2×D211)⋊C4—22no—
40(C2×D211)⋊C4—21no—
41(C2×D211)⋊C4—21no—
42C2×Q8×C211—21no—
43C2×Q8×C211—21no—
44C2×C8×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211D412yes—
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51C22×D211—21no—
52D844—21no—
53D844—21no—
54D844—21no—
55D844—44no—
56D844SD1612yescomplemented
57C2×Dic211—21no—
58C2×Dic211—21no—
59C4×D211—44no—
60C4×D211—41no—
61C4×D211—41no—
62C4×D211—41no—
63C4×D211—41no—
64C211⋊C8—41no—
65C211⋊C8—41no—
66Q8×C211—41no—
67Q8×C211—41no—
68Q8×C211—41no—
69Q8×C211—41no—
70C2×C4×C211—22no—
71C2×C4×C211—41no—
72C2×C4×C211C2×D411yescharacteristic
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C8×C211—22no—
76C22×C211#12011yescharacteristic
77C2×D211—22no—
78C2×D211—41no—
79C2×D211—41no—
80C2×D211—41no—
81C2×D211—41no—
82C2×D211—41no—
83C4×C211—21no—
84C4×C211—42no—
85C4×C211—81no—
86C4×C211C2×SD1611yescharacteristic
87C4×C211#13511yescharacteristic
88C4×C211AGL(1,ℤ/8)11yescharacteristic
89C4×C211AGL(1,ℤ/8)11yescharacteristic
90Dic211—41no—
91Dic211—41no—
92C2×C211#44911yescharacteristic
93C2×C211#45011yescharacteristic
94C2×C211#45611yescharacteristic
95D211—42no—
96D211—81no—
97C211#165411yescharacteristic, complemented, Sylow
98#1654—2111nomaximal, Sylow
99#547—2111no—
100#498D21111yescharacteristic, complemented
101#474—2111no—
102#461—2111no—
103#332—2111no—
104D4⋊C8—2111no—
105D4⋊C8—2111no—
106C2×#41—4221no—
107C2×SD16—4221no—
108C4⋊Q8C2×D21111yescharacteristic
109C4⋊Q8C2×D21111yescharacteristic
110#123—4222no—
111C22⋊Q8—4221no—
112#121—4221no—
113C4×D4—2111no—
114C4×D4—2111no—
115C2×C4⋊C4—4221no—
116#107—4221no—
117#106—22no—
118C4⋊C8—2111no—
119C4⋊C8—2111no—
120#103—4221no—
121#102—4221no—
122C4×C8C2×D21111yescharacteristic
123#41—8442no—
124#41—8442no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×D4—4222no—
128C2×D4—2111no—
129C2×D4—2111no—
130C22×C4—4222no—
131C22×C4—4221no—
132C22×C4—4221no—
133SD16—8442no—
134C2×C8—21no—
135C2×C8—4221no—
136C2×C8—4221no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—4222no—
141C4⋊C4—2111no—
142C4⋊C4—2111no—
143C4⋊C4C22⋊D21112yes—
144C22⋊C4—4222no—
145C22⋊C4—4221no—
146C22⋊C4—4221no—
147C42C22×D21111yescharacteristic
148C23—4221no—
149C23—4221no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—8444no—
155D4—4221no—
156D4—4221no—
157D4—4221no—
158D4—2112no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—8444no—
162C2×C4—8441no—
163C2×C4—8441no—
164C2×C4—8441no—
165C2×C4—8441no—
166C2×C4—4221no—
167C2×C4—4221no—
168C2×C4D4×D21111yescharacteristic
169C2×C4D4×D21111yescharacteristic
170C2×C4C2×C22⋊D21111yescharacteristic
171C8—22no—
172C8—8441no—
173C8—8441no—
174C22—8441no—
175C22—8441no—
176C22—8441no—
177C22—8441no—
178C22—8441no—
179C22—4222no—
180C22#2026994511yescharacteristic
181C4—21no—
182C4—42no—
183C4—81no—
184C4—8441no—
185C4—8441no—
186C4SD16×D21111yescharacteristic
187C4#2026999811yescharacteristic
188C4#2027000411yescharacteristic
189C4#2027000411yescharacteristic
190C2—8442no—
191C2—16881no—
192C2#2721329011yescharacteristic
193C2#2721329511yescharacteristic
194C2#2721332811yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#498×C211C2
socleC22×C211#120
prime radical#27213799C2
hypercenter#498D211
hypocenterC211#1654
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #27213423C2semidirect
3. #27213348C2semidirect
4. #27212892C2semidirect
5. #27212875C2semidirect
6. #27212875C2semidirect
7. #498×C211C2semidirect
56. D844SD16semidirect
97. C211#1654semidirect
100. #498D211semidirect

Maximal subgroups

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

subgroupindex
#272137992
#272134232
#272133482
#272128922
#272128752
#498×C2112
#1654211