Finite Group Databasebeta

← #41013338

Subgroups — #41013338

Order 27008.

612 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#20269901C212yesmaximal, complemented
2C2×#20269898C212yesmaximal, complemented
3C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
4#27213006C211yesmaximal, characteristic, complemented
5C22×#41×C211C211yesmaximal, characteristic
6C22×C211⋊C8C2212yes—
7C2×C4⋊Dic211C2212yes—
8C2×C22⋊Dic211—21no—
9#20269901—24no—
10#20269898—24no—
11C22⋊(C211⋊C8)—22no—
12C211⋊#95—21no—
13C2×#41×C211—24no—
14C2×#41×C211—24no—
15C2×#41×C211—22no—
16C2×#41×C211C414yescomplemented
17C22×Q8×C211C2211yescharacteristic
18C22×D4×C211—21no—
19C22×D4×C211C2211yescharacteristic
20C23×C4×C211—21no—
21C23×C4×C211C2211yescharacteristic
22C22×Dic211—22no—
23C2×C211⋊C8—22no—
24C2×C211⋊C8—22no—
25C2×C211⋊C8—22no—
26C4⋊Dic211—22no—
27C4⋊Dic211—22no—
28C22⋊Dic211—42no—
29C24×C211—21no—
30C24×C211D411yescharacteristic
31#41×C211—216no—
32#41×C211—28no—
33#41×C211—28no—
34C2×Q8×C211—21no—
35C2×Q8×C211—22no—
36C2×Q8×C211C2×C412yes—
37C2×Q8×C211D414yes—
38C2×D4×C211—28no—
39C2×D4×C211—24no—
40C2×D4×C211—22no—
41C2×D4×C211—21no—
42C2×D4×C211C2×C412yes—
43C2×D4×C211D414yes—
44C22×C4×C211—28no—
45C22×C4×C211—21no—
46C22×C4×C211—24no—
47C22×C4×C211—22no—
48C22×C4×C211—22no—
49C22×C4×C211C2×C412yes—
50C22×C4×C211D411yescharacteristic
51C22×C4×C211D412yes—
52C22×C4×C211C2311yescharacteristic
53C2×Dic211—42no—
54C2×Dic211—42no—
55C2×Dic211—42no—
56C211⋊C8—42no—
57C23×C211—22no—
58C23×C211—22no—
59C23×C211—24no—
60C23×C211—22no—
61C23×C211—21no—
62C23×C211—21no—
63C23×C211—42no—
64C23×C211—41no—
65C23×C211—41no—
66C23×C211C22⋊C412yes—
67C23×C211C2×D411yescharacteristic
68Q8×C211—22no—
69Q8×C211—24no—
70Q8×C211C22⋊C414yescomplemented
71D4×C211—216no—
72D4×C211—22no—
73D4×C211—24no—
74D4×C211C22⋊C414yescomplemented
75C2×C4×C211—216no—
76C2×C4×C211—22no—
77C2×C4×C211—22no—
78C2×C4×C211—24no—
79C2×C4×C211—22no—
80C2×C4×C211—22no—
81C2×C4×C211—24no—
82C2×C4×C211C22⋊C412yes—
83C2×C4×C211C22×C411yescharacteristic
84C2×C4×C211C2×D412yes—
85C2×C4×C211C2×D412yes—
86C2×C4×C211C2×D411yescharacteristic
87C22×C211—22no—
88C22×C211—24no—
89C22×C211—22no—
90C22×C211—22no—
91C22×C211—42no—
92C22×C211—42no—
93C22×C211—42no—
94C22×C211—44no—
95C22×C211—42no—
96C22×C211—44no—
97C22×C211C2×C22⋊C411yescharacteristic
98C22×C211#12012yes—
99C22×C211#13512yes—
100C22×C211AGL(1,ℤ/8)11yescharacteristic
101C22×C211#13711yescharacteristic
102C4×C211—22no—
103C4×C211—24no—
104C4×C211C2×C22⋊C411yescharacteristic
105C4×C211C2×C22⋊C411yescharacteristic
106C4×C211#12012yes—
107Dic211—82no—
108C2×C211—42no—
109C2×C211—44no—
110C2×C211#37911yescharacteristic
111C2×C211#41611yescharacteristic
112C2×C211#41711yescharacteristic
113C2×C211#44912yes—
114C2×C211#45212yes—
115C211#178811yescharacteristic, complemented, Sylow
116#1788—2111nomaximal, Sylow
117C22×#41D21111yescharacteristic
118C2×#103—2112no—
119C2×#102—2112no—
120C2×C22⋊C8—2111no—
121#380—2111no—
122C2×#41—24no—
123C2×#41—24no—
124C2×#41—22no—
125C2×#41Dic21114yescomplemented
126C22×Q8C2×D21111yescharacteristic
127C22×D4—21no—
128C22×D4C2×D21111yescharacteristic
129C23×C4—21no—
130C23×C4C2×D21111yescharacteristic
131C22×C8—2112no—
132C2×C4⋊C4—2112no—
133C2×C22⋊C4—4221no—
134#103—4224no—
135#102—4224no—
136C22⋊C8—4222no—
137#95—4221no—
138C24—21no—
139C24C22⋊D21111yescharacteristic
140#41—216no—
141#41—28no—
142#41—28no—
143C2×Q8—21no—
144C2×Q8—22no—
145C2×Q8C2×Dic21112yes—
146C2×Q8C22⋊D21114yes—
147C2×D4—28no—
148C2×D4—24no—
149C2×D4—22no—
150C2×D4—21no—
151C2×D4C2×Dic21112yes—
152C2×D4C22⋊D21114yes—
153C22×C4—28no—
154C22×C4—21no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—4222no—
159C22×C4C2×Dic21112yes—
160C22×C4C22⋊D21111yescharacteristic
161C22×C4C22⋊D21112yes—
162C22×C4C22×D21111yescharacteristic
163C2×C8—4222no—
164C2×C8—4222no—
165C2×C8—4222no—
166C4⋊C4—4222no—
167C4⋊C4—4222no—
168C22⋊C4—8442no—
169C23—22no—
170C23—22no—
171C23—24no—
172C23—22no—
173C23—21no—
174C23—21no—
175C23—42no—
176C23—41no—
177C23—41no—
178C23C22⋊Dic21112yes—
179C23C2×C22⋊D21111yescharacteristic
180Q8—22no—
181Q8—24no—
182Q8C22⋊Dic21114yescomplemented
183D4—216no—
184D4—22no—
185D4—24no—
186D4C22⋊Dic21114yescomplemented
187C2×C4—216no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—8442no—
195C2×C4—8442no—
196C2×C4—8442no—
197C2×C4C22⋊Dic21112yes—
198C2×C4C22×Dic21111yescharacteristic
199C2×C4C2×C22⋊D21112yes—
200C2×C4C2×C22⋊D21112yes—
201C2×C4C2×C22⋊D21111yescharacteristic
202C8—8442no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—24no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—44no—
211C22—42no—
212C22—44no—
213C22C2×C22⋊Dic21111yescharacteristic
214C22#2026994612yes—
215C22#2026999712yes—
216C22#2027000211yescharacteristic
217C22#2027000711yescharacteristic
218C4—22no—
219C4—24no—
220C4—16882no—
221C4C2×C22⋊Dic21111yescharacteristic
222C4C2×C22⋊Dic21111yescharacteristic
223C4#2026994612yes—
224C2—42no—
225C2—44no—
226C2#2721300211yescharacteristic
227C2#2721314211yescharacteristic
228C2#2721314711yescharacteristic
229C2#2721329112yes—
230C2#2721330812yes—
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting, prime radicalC22×#41×C211C2
socleC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenterC22×#41D211
hypocenterC211#1788
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#20269901C2semidirect
2. C2×#20269898C2semidirect
3. C2×C22⋊(C211⋊C8)C2semidirect
4. #27213006C2semidirect
16. C2×#41×C211C4semidirect
70. Q8×C211C22⋊C4semidirect
74. D4×C211C22⋊C4semidirect
115. C211#1788semidirect
125. C2×#41Dic211semidirect
182. Q8C22⋊Dic211semidirect
186. D4C22⋊Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699012
C2×#202698982
C2×C22⋊(C211⋊C8)2
#272130062
C22×#41×C2112
#1788211