Finite Group Databasebeta

← #41013323

Subgroups — #41013323

Order 27008.

584 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16809961C211yesmaximal, characteristic
2C2×#20269895C212yesmaximal, complemented
3C2×#20269894C211yesmaximal, characteristic, complemented
4#27213109C212yesmaximal, complemented
5C2×#100×C211C211yesmaximal, characteristic, complemented
6C2×#16809961—24no—
7C2×#16809961—24no—
8C2×#16809961—22no—
9C2×#16809961—22no—
10C22×Dic422C2211yescharacteristic
11C22×C22⋊D211C412yes—
12C23×Dic211C412yes—
13C22×C4×D211C2211yescharacteristic
14C2×#16809946C2212yes—
15#20269895—24no—
16#20269894—22no—
17(C2×D211)⋊C8—24no—
18C22×D4×C211C2211yescharacteristic
19C2×M16×C211C2212yes—
20#100×C211—22no—
21C23×D211C2×C411yescharacteristic
22#16809961—216no—
23#16809961—216no—
24C2×Dic422—22no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422D414yes—
28C2×C22⋊D211—24no—
29C2×C22⋊D211—24no—
30C2×C22⋊D211—24no—
31C22×Dic211—24no—
32C22×Dic211—24no—
33C22×Dic211—24no—
34C22×Dic211C2×C411yescharacteristic
35C22×Dic211C2×C412yes—
36C2×C4×D211—22no—
37C2×C4×D211—21no—
38C2×C4×D211—21no—
39C2×C4×D211D414yes—
40#16809946—22no—
41#16809946—22no—
42C2×C211⋊C8—22no—
43C24×C211C2×C412yes—
44C2×D4×C211—22no—
45C2×D4×C211—22no—
46C2×D4×C211D412yes—
47C2×D4×C211D412yes—
48C22×C4×C211C2311yescharacteristic
49M16×C211—22no—
50M16×C211—22no—
51C2×C8×C211—22no—
52C22×D211—21no—
53C22×D211—21no—
54C22×D211—42no—
55C22×D211C22⋊C412yes—
56Dic422—24no—
57Dic422—24no—
58C22⋊D211—216no—
59C2×Dic211—216no—
60C2×Dic211—21no—
61C2×Dic211—21no—
62C2×Dic211—22no—
63C2×Dic211—22no—
64C2×Dic211C22⋊C412yes—
65C2×Dic211C22⋊C414yes—
66C4×D211—24no—
67C4×D211—24no—
68C211⋊C8—42no—
69C23×C211—22no—
70C23×C211—22no—
71C23×C211—44no—
72C23×C211C22⋊C414yes—
73C23×C211C22×C411yescharacteristic
74D4×C211—24no—
75D4×C211—24no—
76C2×C4×C211C2×D412yes—
77C2×C4×C211C2×D412yes—
78C2×C4×C211C2×D411yescharacteristic
79C2×C4×C211C2×D411yescharacteristic
80C8×C211—42no—
81C22×C211—22no—
82C22×C211—24no—
83C22×C211—44no—
84C22×C211—44no—
85C22×C211—44no—
86C22×C211C2×C22⋊C411yescharacteristic
87C22×C211C2×C22⋊C411yescharacteristic
88C22×C211C2×C22⋊C411yescharacteristic
89C2×D211—22no—
90C2×D211—24no—
91C2×D211—42no—
92C2×D211—42no—
93C4×C211#12012yes—
94C4×C211#12012yes—
95Dic211—22no—
96Dic211—24no—
97C2×C211—21no—
98C2×C211—21no—
99C2×C211—44no—
100C2×C211#37911yescharacteristic
101C2×C211#41312yes—
102D211—42no—
103C211#178511yescharacteristic, complemented, Sylow
104#1785—2111nomaximal, Sylow
105C22×#41—2111no—
106C2×#101—2112no—
107C2×#100—2111no—
108C2×#100D21111yescharacteristic, complemented
109#407—2112no—
110C2×#41—4224no—
111C2×#41—4222no—
112C2×#41—4222no—
113C2×#41—4224no—
114C22×Q8—2111no—
115C22×D4—2112no—
116C22×D4C2×D21111yescharacteristic
117C23×C4—2112no—
118C23×C4—2111no—
119C2×M16—2112no—
120C2×M16C2×D21112yes—
121#101—4224no—
122#100—22no—
123#100—4222no—
124C22⋊C8—4224no—
125C24—2111no—
126C24C4×D21112yes—
127#41—42216no—
128#41—42216no—
129C2×Q8—4222no—
130C2×Q8—4221no—
131C2×Q8—4221no—
132C2×Q8—2114no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—4224no—
136C2×D4—4224no—
137C2×D4—4224no—
138C2×D4C22⋊D21112yes—
139C2×D4D84412yes—
140C22×C4—4224no—
141C22×C4—4224no—
142C22×C4—4224no—
143C22×C4—4222no—
144C22×C4—4221no—
145C22×C4—4221no—
146C22×C4—2111no—
147C22×C4—2112no—
148C22×C4—2114no—
149C22×C4C22×D21111yescharacteristic
150M16—22no—
151M16—22no—
152M16—4222no—
153M16—4222no—
154C2×C8—22no—
155C2×C8—4222no—
156C23—22no—
157C23—22no—
158C23—44no—
159C23—8442no—
160C23—4221no—
161C23—4221no—
162C23—2112no—
163C23(C2×D211)⋊C414yes—
164C23C2×C4×D21111yescharacteristic
165Q8—4224no—
166Q8—4224no—
167D4—24no—
168D4—24no—
169D4—42216no—
170C2×C4—42216no—
171C2×C4—4221no—
172C2×C4—4222no—
173C2×C4—4221no—
174C2×C4—4222no—
175C2×C4—4224no—
176C2×C4—4224no—
177C2×C4—2114no—
178C2×C4—2112no—
179C2×C4D4×D21112yes—
180C2×C4D4×D21112yes—
181C2×C4C2×C22⋊D21111yescharacteristic
182C2×C4C2×D84411yescharacteristic
183C8—42no—
184C8—8442no—
185C22—22no—
186C22—24no—
187C22—44no—
188C22—44no—
189C22—44no—
190C22—8442no—
191C22—8442no—
192C22—4224no—
193C22—4222no—
194C22C22⋊C4×D21111yescharacteristic
195C22C22⋊C4×D21111yescharacteristic
196C22C2×(C2×D211)⋊C411yescharacteristic
197C4—4222no—
198C4—4224no—
199C4#2026994512yes—
200C4#2026994712yes—
201C2—21no—
202C2—21no—
203C2—44no—
204C2—8442no—
205C2#2721300111yescharacteristic
206C2#2721312712yes—
subgroupG/H
centerC22C22⋊C4×D211
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#100×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC22×#16809961C2
cosocleC22×C4×C211C23
hypercenterC2×#100D211
hypocenterC211#1785
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×#20269895C2semidirect
3. C2×#20269894C2semidirect
4. #27213109C2semidirect
5. C2×#100×C211C2semidirect
103. C211#1785semidirect
108. C2×#100D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099612
C2×#202698952
C2×#202698942
#272131092
C2×#100×C2112
#1785211