Finite Group Databasebeta

← #41012467

Subgroups — #41012467

Order 27008.

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

normal subgroups only

This group has 236 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#27213857C211yesmaximal, characteristic, complemented
2#27213343C211yesmaximal, characteristic, complemented
3#27213338C211yesmaximal, characteristic
4#27212895C211yesmaximal, characteristic, complemented
5#27212878C211yesmaximal, characteristic, complemented
6#27212875C211yesmaximal, characteristic, complemented
7#492×C211C211yesmaximal, characteristic, complemented
8C2×#16809950—21no—
9C2×#16809947—21no—
10#20269968—21no—
11#20269957—21no—
12#20269953—22no—
13C4×Dic422C2211yescharacteristic
14C4×D844C2211yescharacteristic
15#20269939—21no—
16C4⋊C4×D211—21no—
17C4⋊(C211⋊C8)C2211yescharacteristic
18C4⋊(C211⋊C8)C2211yescharacteristic
19#20269898—21no—
20#20269898—21no—
21C2×SD16×C211—22no—
22C4⋊Q8×C211C2211yescharacteristic
23#127×C211C2211yescharacteristic
24C4×C8×C211C2211yescharacteristic
25C2×Dic422D411yescharacteristic
26C2×D844D411yescharacteristic
27C2×C4×D211—22no—
28C2×C4×D211—21no—
29#16809950—42no—
30#16809947—42no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C4⋊Dic211—22no—
34C4⋊Dic211D411yescharacteristic
35C4⋊Dic211D411yescharacteristic
36Dic211⋊C4—22no—
37Dic211⋊C4—21no—
38(C2×D211)⋊C4—22no—
39(C2×D211)⋊C4—21no—
40C4×Dic211—21no—
41C2×Q8×C211D412yes—
42C2×D4×C211—21no—
43C2×D4×C211—21no—
44C2×D4×C211—21no—
45SD16×C211—44no—
46C2×C8×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211—21no—
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51Dic422—21no—
52Dic422—21no—
53D844—21no—
54D844SD1612yescomplemented
55C2×Dic211—21no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C4×D211—41no—
59C4×D211—41no—
60C4×D211—82no—
61C211⋊C8—41no—
62C211⋊C8—41no—
63C23×C211—41no—
64Q8×C211—42no—
65D4×C211—42no—
66D4×C211—41no—
67D4×C211—41no—
68D4×C211—41no—
69D4×C211—41no—
70C2×C4×C211—22no—
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211C2×D411yescharacteristic
73C2×C4×C211C2×D411yescharacteristic
74C8×C211—22no—
75C22×C211—81no—
76C22×C211—81no—
77C22×C211—81no—
78C22×C211#12011yescharacteristic
79C2×D211—22no—
80C2×D211—41no—
81C2×D211—41no—
82C4×C211—21no—
83C4×C211—42no—
84C4×C211C2×SD1611yescharacteristic
85C4×C211#13511yescharacteristic
86C4×C211AGL(1,ℤ/8)11yescharacteristic
87C4×C211#13711yescharacteristic
88Dic211—41no—
89Dic211—41no—
90Dic211—41no—
91C2×C211—161no—
92C2×C211#44811yescharacteristic
93C2×C211#44911yescharacteristic
94C2×C211#45311yescharacteristic
95D211—42no—
96C211#162211yescharacteristic, complemented, Sylow
97#1622—2111nomaximal, Sylow
98#554—2111no—
99#492D21111yescharacteristic, complemented
100#460—2111no—
101#459—2111no—
102#333—2111no—
103Q8⋊C8—2111no—
104D4⋊C8—2111no—
105C2×SD16—22no—
106C2×SD16—4221no—
107C2×D8—4221no—
108C4⋊Q8C2×D21111yescharacteristic
109#127C2×D21111yescharacteristic
110#125—4221no—
111C22⋊Q8—4221no—
112C22⋊Q8—4222no—
113C4×Q8—2111no—
114C4×D4—4221no—
115C4×D4—2111no—
116C2×C4⋊C4—4221no—
117C4⋊C8—2111no—
118C4⋊C8—2111no—
119#102—4221no—
120#102—4221no—
121C4×C8C2×D21111yescharacteristic
122C2×Q8—2111no—
123C2×Q8C22⋊D21112yes—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—2111no—
128C22×C4—4222no—
129C22×C4—4221no—
130SD16—44no—
131SD16—8442no—
132D8—8442no—
133C2×C8—21no—
134C2×C8—4221no—
135C2×C8—4221no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—4222no—
139C4⋊C4—4222no—
140C4⋊C4—4221no—
141C4⋊C4—2111no—
142C4⋊C4—2111no—
143C22⋊C4—4222no—
144C22⋊C4—4221no—
145C42—4221no—
146C42C22×D21111yescharacteristic
147C23—41no—
148C23—4221no—
149Q8—42no—
150Q8—4221no—
151Q8—4221no—
152D4—42no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—4221no—
158D4—2112no—
159C2×C4—22no—
160C2×C4—8441no—
161C2×C4—8441no—
162C2×C4—16882no—
163C2×C4—4221no—
164C2×C4—4221no—
165C2×C4—4221no—
166C2×C4D4×D21111yescharacteristic
167C2×C4D4×D21111yescharacteristic
168C2×C4C2×C22⋊D21111yescharacteristic
169C8—22no—
170C8—8441no—
171C8—8441no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—8441no—
176C22—8441no—
177C22—4222no—
178C22#2026994511yescharacteristic
179C4—21no—
180C4—42no—
181C4—8441no—
182C4—8441no—
183C4—8441no—
184C4SD16×D21111yescharacteristic
185C4#2026999811yescharacteristic
186C4#2027000411yescharacteristic
187C4#2027000611yescharacteristic
188C2—161no—
189C2—8442no—
190C2#2721328311yescharacteristic
191C2#2721329011yescharacteristic
192C2#2721331311yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#492×C211C2
socleC22×C211#120
prime radical#27213338C2
hypercenter#492D211
hypocenterC211#1622
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #27213857C2semidirect
2. #27213343C2semidirect
4. #27212895C2semidirect
5. #27212878C2semidirect
6. #27212875C2semidirect
7. #492×C211C2semidirect
54. D844SD16semidirect
96. C211#1622semidirect
99. #492D211semidirect

Maximal subgroups

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

subgroupindex
#272138572
#272133432
#272133382
#272128952
#272128782
#272128752
#492×C2112
#1622211