Finite Group Databasebeta

← #41014653

Subgroups — #41014653

Order 27008.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C22×#16809947C211yesmaximal, characteristic
2#27213173C211yesmaximal, characteristic, complemented
3#27213150C211yesmaximal, characteristic, complemented
4C2×#20269894C211yesmaximal, characteristic, complemented
5C2×#20269893C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7#418×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809947—22no—
10C2×#16809947—22no—
11C2×#16809947—22no—
12C2×#16809946C2211yescharacteristic
13C2×#16809946C2211yescharacteristic
14C22×C211⋊C8C2211yescharacteristic
15C4⋊(C211⋊C8)—22no—
16#20269897—22no—
17#20269894—22no—
18#20269893—22no—
19C22×D4×C211C2211yescharacteristic
20C2×M16×C211C2211yescharacteristic
21#117×C211C2211yescharacteristic
22#102×C211—22no—
23C23×D211—21no—
24C2×D844—21no—
25C2×D844—21no—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844D412yes—
30#16809947—28no—
31#16809946—21no—
32#16809946—21no—
33#16809946—21no—
34#16809946—21no—
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8D412yes—
40C24×C211—21no—
41C2×D4×C211—21no—
42C2×D4×C211—21no—
43C2×D4×C211—21no—
44C2×D4×C211—21no—
45C2×D4×C211—21no—
46C2×D4×C211D412yes—
47C22×C4×C211C2311yescharacteristic
48M16×C211—21no—
49M16×C211—21no—
50C2×C8×C211—21no—
51C4⋊C4×C211D412yes—
52C22⋊C4×C211—21no—
53C42×C211—21no—
54C22×D211—22no—
55C22×D211—41no—
56C22×D211—41no—
57C22×D211—82no—
58D844—22no—
59D844—22no—
60D844—41no—
61D844—41no—
62C211⋊C8—22no—
63C211⋊C8—41no—
64C211⋊C8—41no—
65C23×C211—22no—
66C23×C211—41no—
67C23×C211—41no—
68C23×C211—82no—
69C23×C211#4111yescharacteristic
70D4×C211—22no—
71D4×C211—22no—
72D4×C211—41no—
73D4×C211—41no—
74C2×C4×C211—22no—
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C2×C4×C211C2×D411yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C2×C4×C211#4111yescharacteristic
81C8×C211—41no—
82C22×C211—22no—
83C22×C211—42no—
84C22×C211—82no—
85C22×C211—82no—
86C22×C211—82no—
87C22×C211#12111yescharacteristic
88C22×C211#12311yescharacteristic
89C22×C211#12311yescharacteristic
90C2×D211—42no—
91C2×D211—82no—
92C2×D211—82no—
93C2×D211—82no—
94C4×C211—42no—
95C4×C211#12011yescharacteristic
96C4×C211#12111yescharacteristic
97C4×C211#12111yescharacteristic
98C4×C211#12411yescharacteristic
99C2×C211—21no—
100C2×C211—21no—
101C2×C211—82no—
102C2×C211#39411yescharacteristic
103C2×C211#47212yes—
104D211—82no—
105C211#204511yescharacteristic, complemented, Sylow
106#2045—2111nomaximal, Sylow
107C22×D8—2111no—
108#424—2111no—
109#418—2111no—
110#418D21111yescharacteristic, complemented
111C2×#100—2111no—
112C2×#100—2111no—
113#341—2111no—
114C22×D4—2111no—
115C22×D4C2×D21111yescharacteristic
116C2×D8—4222no—
117C2×D8—4222no—
118C2×D8—4222no—
119C2×M16—2111no—
120C2×M16—2111no—
121C2×M16C2×D21111yescharacteristic
122C22×C8—2111no—
123#117C2×D21111yescharacteristic
124C4⋊C8—4222no—
125#102—22no—
126#102—4222no—
127#100—4222no—
128#100—4222no—
129C24—21no—
130C24—4221no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—4221no—
137C2×D4—4221no—
138C2×D4—4221no—
139C2×D4—4221no—
140C2×D4—4221no—
141C2×D4—2112no—
142C2×D4D84412yes—
143C22×C4C22×D21111yescharacteristic
144D8—4228no—
145M16—21no—
146M16—21no—
147M16—4221no—
148M16—4221no—
149M16—4221no—
150M16—4221no—
151C2×C8—21no—
152C2×C8—4221no—
153C2×C8—4221no—
154C2×C8—4221no—
155C2×C8—4221no—
156C2×C8—2112no—
157C4⋊C4C22⋊D21112yes—
158C22⋊C4—21no—
159C42—21no—
160C23—22no—
161C23—41no—
162C23—41no—
163C23—82no—
164C23—8441no—
165C23—8441no—
166C23—16882no—
167C23—4222no—
168C23#1680996111yescharacteristic
169D4—22no—
170D4—22no—
171D4—41no—
172D4—41no—
173D4—8441no—
174D4—8441no—
175D4—4222no—
176D4—4222no—
177C2×C4—22no—
178C2×C4D4×D21111yescharacteristic
179C2×C4D4×D21111yescharacteristic
180C2×C4C2×C22⋊D21111yescharacteristic
181C2×C4C2×D84411yescharacteristic
182C2×C4#1680996111yescharacteristic
183C2×C4#1680996211yescharacteristic
184C8—41no—
185C8—8441no—
186C8—8441no—
187C8—4222no—
188C22—22no—
189C22—42no—
190C22—82no—
191C22—82no—
192C22—82no—
193C22—8442no—
194C22—16882no—
195C22—16882no—
196C22—16882no—
197C22#2026994911yescharacteristic
198C22#2026995911yescharacteristic
199C22#2026996111yescharacteristic
200C4—42no—
201C4#2026994711yescharacteristic
202C4#2026994811yescharacteristic
203C4#2026995111yescharacteristic
204C4#2026996311yescharacteristic
205C2—21no—
206C2—21no—
207C2—82no—
208C2—16882no—
209C2#2721306211yescharacteristic
210C2#2721341412yes—
subgroupG/H
centerC22#20269949
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#418×C211C2
socleC22×C211#121
prime radicalC22×#16809947C2
hypercenter#418D211
hypocenterC211#2045
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #27213173C2semidirect
3. #27213150C2semidirect
4. C2×#20269894C2semidirect
5. C2×#20269893C2semidirect
6. #27212910C2semidirect
7. #418×C211C2semidirect
105. C211#2045semidirect
110. #418D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099472
#272131732
#272131502
C2×#202698942
C2×#202698932
#272129102
#418×C2112
#2045211