Finite Group Databasebeta

← #41012713

Subgroups — #41012713

Order 27008.

208 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 45 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#27213894C211yesmaximal, characteristic
2#27213538C211yesmaximal, characteristic, complemented
3#27213508C211yesmaximal, characteristic, complemented
4#27212888C211yesmaximal, characteristic, complemented
5#27212885C211yesmaximal, characteristic, complemented
6#27212885C211yesmaximal, characteristic, complemented
7#501×C211C211yesmaximal, characteristic, complemented
8C2×#16809952—21no—
9C2×#16809949—21no—
10#20269965C2211yescharacteristic
11#20269965C2211yescharacteristic
12#20269962—21no—
13#20269962—21no—
14#20269955—21no—
15#20269955—21no—
16#20269952—21no—
17#20269952—21no—
18#20269900—21no—
19#20269897—21no—
20#20269888C2211yescharacteristic
21#20269888C2211yescharacteristic
22C4⋊Q8×C211C2211yescharacteristic
23#125×C211C2211yescharacteristic
24#107×C211—21no—
25#106×C211—21no—
26#97×C211C2211yescharacteristic
27C2×Dic422D411yescharacteristic
28C2×Dic422D411yescharacteristic
29C2×D844—21no—
30C2×D844—21no—
31C2×D844D411yescharacteristic
32C2×D844D411yescharacteristic
33C2×C4×D211—21no—
34C2×C4×D211—21no—
35#16809952—42no—
36#16809949—42no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39Dic211⋊C4—21no—
40Dic211⋊C4—21no—
41(C2×D211)⋊C4—21no—
42(C2×D211)⋊C4—21no—
43(C2×D211)⋊C4—21no—
44(C2×D211)⋊C4—21no—
45(C2×D211)⋊C4—21no—
46(C2×D211)⋊C4—21no—
47C2×Q8×C211—21no—
48C2×C8×C211—21no—
49C4⋊C4×C211—21no—
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211D411yescharacteristic
54C4⋊C4×C211D411yescharacteristic
55C42×C211C2311yescharacteristic
56C22×D211—21no—
57C22×D211—21no—
58Dic422—41no—
59Dic422—41no—
60D844—41no—
61D844—41no—
62D844—81no—
63D844—81no—
64C2×Dic211—21no—
65C2×Dic211—21no—
66C4×D211—81no—
67C4×D211—81no—
68C211⋊C8—22no—
69C211⋊C8—41no—
70Q8×C211—41no—
71Q8×C211—41no—
72C2×C4×C211—21no—
73C2×C4×C211—21no—
74C2×C4×C211—41no—
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C8×C211—22no—
79C22×C211#12011yescharacteristic
80C2×D211—41no—
81C2×D211—41no—
82C2×D211—41no—
83C2×D211—41no—
84C2×D211—41no—
85C2×D211—41no—
86C4×C211—21no—
87C4×C211—21no—
88C4×C211—21no—
89C4×C211—41no—
90C4×C211—41no—
91C4×C211—81no—
92Dic211—41no—
93Dic211—41no—
94C2×C211#45311yescharacteristic
95C2×C211#45511yescharacteristic
96C2×C211#45611yescharacteristic
97D211—81no—
98D211—81no—
99C211#166111yescharacteristic, complemented, Sylow
100#1661—2111nomaximal, Sylow
101#562—2111no—
102#501D21111yescharacteristic, complemented
103#497—2111no—
104#489—2111no—
105#330—2111no—
106#329—2111no—
107#329—2111no—
108C2×Q16—4221no—
109C2×SD16—4221no—
110C4⋊Q8C2×D21111yescharacteristic
111#125C2×D21111yescharacteristic
112#124—2111no—
113#124—2111no—
114#123—4221no—
115#123—4221no—
116C22⋊Q8—4221no—
117C22⋊Q8—4221no—
118#121—4221no—
119#121—4221no—
120#107—21no—
121#106—21no—
122#103—4221no—
123#102—4221no—
124#97—2111no—
125#97—2111no—
126#97C2×D21111yescharacteristic
127C2×Q8—21no—
128C2×Q8—2111no—
129C2×Q8—2111no—
130C2×D4—4221no—
131C2×D4—4221no—
132C2×D4—2111no—
133C2×D4—2111no—
134C22×C4—4221no—
135C22×C4—4221no—
136Q16—8442no—
137SD16—8442no—
138C2×C8—21no—
139C2×C8—4221no—
140C2×C8—4221no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—4221no—
146C4⋊C4—4221no—
147C4⋊C4C22⋊D21111yescharacteristic
148C4⋊C4C22⋊D21111yescharacteristic
149C22⋊C4—4221no—
150C22⋊C4—4221no—
151C22⋊C4—4221no—
152C22⋊C4—4221no—
153C22⋊C4—4221no—
154C22⋊C4—4221no—
155C42C22×D21111yescharacteristic
156C23—4221no—
157C23—4221no—
158Q8—41no—
159Q8—41no—
160Q8—8441no—
161Q8—8441no—
162D4—8441no—
163D4—8441no—
164D4—16881no—
165D4—16881no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—16881no—
170C2×C4—16881no—
171C2×C4—4221no—
172C2×C4—4221no—
173C2×C4D4×D21111yescharacteristic
174C2×C4D4×D21111yescharacteristic
175C2×C4C2×C22⋊D21111yescharacteristic
176C8—22no—
177C8—8441no—
178C8—4222no—
179C22—8441no—
180C22—8441no—
181C22—8441no—
182C22—8441no—
183C22—8441no—
184C22—8441no—
185C22#2026994511yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—41no—
191C4—81no—
192C4—8441no—
193C4—8441no—
194C2—16881no—
195C2—16881no—
196C2#2721331211yescharacteristic
197C2#2721332211yescharacteristic
198C2#2721332811yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#501×C211C2
socleC22×C211#120
prime radical#27213894C2
hypercenter#501D211
hypocenterC211#1661
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #27213538C2semidirect
3. #27213508C2semidirect
4. #27212888C2semidirect
5. #27212885C2semidirect
6. #27212885C2semidirect
7. #501×C211C2semidirect
99. C211#1661semidirect
102. #501D211semidirect

Maximal subgroups

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

subgroupindex
#272138942
#272135382
#272135082
#272128882
#272128852
#501×C2112
#1661211