Finite Group Databasebeta

← #41014658

Subgroups — #41014658

Order 27008.

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

normal subgroups only

This group has 276 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×#16809949C211yesmaximal, characteristic
2#27213173C211yesmaximal, characteristic, complemented
3#27213150C211yesmaximal, characteristic
4C2×#20269896C211yesmaximal, characteristic, complemented
5C2×#20269893C211yesmaximal, characteristic
6#27212910C211yesmaximal, characteristic, complemented
7#419×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809949—22no—
10C2×#16809949—22no—
11C2×#16809949—22no—
12C2×#16809946C2211yescharacteristic
13C2×#16809946C2211yescharacteristic
14C22×C211⋊C8C2211yescharacteristic
15C4⋊(C211⋊C8)—22no—
16#20269897—22no—
17#20269896—22no—
18#20269893—22no—
19C22×Q8×C211C2211yescharacteristic
20C2×M16×C211C2211yescharacteristic
21#117×C211C2211yescharacteristic
22#103×C211—22no—
23C23×D211—21no—
24C2×D844—21no—
25C2×D844—21no—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844D412yes—
30#16809949—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—
40C2×Q8×C211—21no—
41C2×Q8×C211—21no—
42C2×Q8×C211—21no—
43C2×Q8×C211—21no—
44C2×Q8×C211—21no—
45C2×Q8×C211D412yes—
46C22×C4×C211—21no—
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#4111yescharacteristic
66Q8×C211—22no—
67Q8×C211—22no—
68Q8×C211—41no—
69Q8×C211—41no—
70C2×C4×C211—22no—
71C2×C4×C211—22no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C8×C211—41no—
81C22×C211—22no—
82C22×C211#12111yescharacteristic
83C22×C211#12311yescharacteristic
84C22×C211#12311yescharacteristic
85C2×D211—42no—
86C2×D211—82no—
87C2×D211—82no—
88C2×D211—82no—
89C4×C211—42no—
90C4×C211—42no—
91C4×C211#12011yescharacteristic
92C4×C211#12111yescharacteristic
93C4×C211#12111yescharacteristic
94C4×C211#12411yescharacteristic
95C2×C211—21no—
96C2×C211—21no—
97C2×C211#39411yescharacteristic
98C2×C211#47112yes—
99D211—82no—
100C211#204611yescharacteristic, complemented, Sylow
101#2046—2111nomaximal, Sylow
102C22×SD16—2111no—
103#424—2111no—
104#419D21111yescharacteristic, complemented
105#418—2111no—
106C2×#101—2111no—
107C2×#100—2111no—
108#341—2111no—
109C22×Q8C2×D21111yescharacteristic
110C22×D4—2111no—
111C2×SD16—4222no—
112C2×SD16—4222no—
113C2×SD16—4222no—
114C2×M16—2111no—
115C2×M16—2111no—
116C2×M16C2×D21111yescharacteristic
117C22×C8—2111no—
118#117C2×D21111yescharacteristic
119C4⋊C8—4222no—
120#103—22no—
121#102—4222no—
122#101—4222no—
123#100—4222no—
124C24—4221no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8D84412yes—
131C2×D4—4221no—
132C2×D4—4221no—
133C2×D4—4221no—
134C2×D4—4221no—
135C2×D4—4221no—
136C2×D4—2112no—
137C22×C4—21no—
138C22×C4C22×D21111yescharacteristic
139SD16—4228no—
140M16—21no—
141M16—21no—
142M16—4221no—
143M16—4221no—
144M16—4221no—
145M16—4221no—
146C2×C8—21no—
147C2×C8—4221no—
148C2×C8—4221no—
149C2×C8—4221no—
150C2×C8—4221no—
151C2×C8—2112no—
152C4⋊C4C22⋊D21112yes—
153C22⋊C4—21no—
154C42—21no—
155C23—8441no—
156C23—8441no—
157C23—16882no—
158C23—4222no—
159C23#1680996111yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164D4—8441no—
165D4—8441no—
166D4—4222no—
167D4—4222no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4D4×D21111yescharacteristic
173C2×C4D4×D21111yescharacteristic
174C2×C4C2×C22⋊D21111yescharacteristic
175C2×C4C2×D84411yescharacteristic
176C2×C4#1680996111yescharacteristic
177C2×C4#1680996211yescharacteristic
178C8—41no—
179C8—8441no—
180C8—8441no—
181C8—4222no—
182C22—22no—
183C22—8442no—
184C22—16882no—
185C22—16882no—
186C22—16882no—
187C22#2026994911yescharacteristic
188C22#2026995911yescharacteristic
189C22#2026996111yescharacteristic
190C4—42no—
191C4—42no—
192C4#2026994711yescharacteristic
193C4#2026994811yescharacteristic
194C4#2026995111yescharacteristic
195C4#2026996311yescharacteristic
196C2—21no—
197C2—21no—
198C2—16882no—
199C2#2721306211yescharacteristic
200C2#2721341112yes—
subgroupG/H
centerC22#20269949
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#419×C211C2
socleC22×C211#121
prime radicalC22×D844C22
hypercenter#419D211
hypocenterC211#2046
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. #27213173C2semidirect
4. C2×#20269896C2semidirect
6. #27212910C2semidirect
7. #419×C211C2semidirect
100. C211#2046semidirect
104. #419D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099492
#272131732
#272131502
C2×#202698962
C2×#202698932
#272129102
#419×C2112
#2046211