Finite Group Databasebeta

← #41013679

Subgroups — #41013679

Order 27008.

374 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20269892C211yesmaximal, characteristic
2#27213026C211yesmaximal, characteristic
3#27213018C212yesmaximal, complemented
4#27212913C212yesmaximal, complemented
5#518×C211C211yesmaximal, characteristic
6C23×Dic211C2211yescharacteristic
7C2×Dic211⋊C4—22no—
8C2×C22⋊Dic211—21no—
9C2×C22⋊Dic211C2211yescharacteristic
10C2×C22⋊Dic211C2212yes—
11#20269892C414yescomplemented
12C211⋊#95—22no—
13C211⋊#95—21no—
14C22×D4×C211C2211yescharacteristic
15C4×D4×C211—22no—
16C4×D4×C211—22no—
17#117×C211—21no—
18C2×C22⋊C4×C211—21no—
19C2×C22⋊C4×C211C2212yes—
20C22×Dic211—21no—
21C22×Dic211—22no—
22C22×Dic211—22no—
23C22×Dic211—21no—
24C22×Dic211—42no—
25Dic211⋊C4—44no—
26C22⋊Dic211—22no—
27C22⋊Dic211—22no—
28C22⋊Dic211—42no—
29C22⋊Dic211C2×C414yescomplemented
30C24×C211D411yescharacteristic
31C24×C211C2311yescharacteristic
32C2×D4×C211—22no—
33C2×D4×C211—22no—
34C2×D4×C211C2×C412yes—
35C2×D4×C211Q812yes—
36C22×C4×C211—21no—
37C22×C4×C211—21no—
38C22×C4×C211D411yescharacteristic
39C22×C4×C211D412yes—
40C22×C4×C211D412yes—
41C4⋊C4×C211—21no—
42C4⋊C4×C211—21no—
43C22⋊C4×C211—22no—
44C22⋊C4×C211—22no—
45C22⋊C4×C211—22no—
46C42×C211—22no—
47C2×Dic211—22no—
48C2×Dic211—24no—
49C2×Dic211—24no—
50C2×Dic211—42no—
51C2×Dic211—42no—
52C2×Dic211—42no—
53C2×Dic211—41no—
54C2×Dic211—41no—
55C2×Dic211—41no—
56C2×Dic211—41no—
57C2×Dic211—41no—
58C23×C211—21no—
59C23×C211—21no—
60C23×C211—41no—
61C23×C211—41no—
62C23×C211—41no—
63C23×C211—41no—
64C23×C211C4⋊C412yescomplemented
65C23×C211C22×C411yescharacteristic
66C23×C211C2×D411yescharacteristic
67C23×C211C2×D412yes—
68C23×C211C2×Q811yescharacteristic
69C23×C211#4112yes—
70D4×C211—24no—
71D4×C211—24no—
72C2×C4×C211—21no—
73C2×C4×C211—21no—
74C2×C4×C211—22no—
75C2×C4×C211—22no—
76C2×C4×C211—42no—
77C2×C4×C211—42no—
78C2×C4×C211—81no—
79C2×C4×C211—81no—
80C2×C4×C211C4⋊C412yescomplemented
81C22×C211—22no—
82C22×C211—21no—
83C22×C211—22no—
84C22×C211—21no—
85C22×C211—22no—
86C22×C211—41no—
87C22×C211—41no—
88C22×C211—42no—
89C22×C211—42no—
90C22×C211—41no—
91C22×C211—41no—
92C22×C211—81no—
93C22×C211C2×C4⋊C411yescharacteristic
94C22×C211C4×D412yes—
95C22×C211#12011yescharacteristic
96C22×C211C22⋊Q812yes—
97C22×C211#12311yescharacteristic
98C4×C211—22no—
99C4×C211—42no—
100C4×C211—42no—
101Dic211—44no—
102Dic211—42no—
103Dic211—81no—
104C2×C211—22no—
105C2×C211—21no—
106C2×C211—21no—
107C2×C211—22no—
108C2×C211—42no—
109C2×C211#38511yescharacteristic
110C2×C211B(4,𝔽2)11yescharacteristic
111C2×C211#45811yescharacteristic
112C211#186711yescharacteristic, complemented, Sylow
113#1867—2111nomaximal, Sylow
114#518D21111yescharacteristic
115C2×#99—2111no—
116#386—2111no—
117#385—2112no—
118#342—2112no—
119C22×D4C2×D21111yescharacteristic
120C23×C4—2111no—
121C4×D4—22no—
122C4×D4—22no—
123#117—21no—
124C2×C4⋊C4—4222no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—4221no—
127C2×C22⋊C4—2111no—
128C2×C22⋊C4—2112no—
129C2×C22⋊C4C2×D21112yes—
130#99—2114no—
131#95—4222no—
132#95—4221no—
133C24C22⋊D21111yescharacteristic
134C24C22×D21111yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4C4×D21112yes—
138C2×D4Dic42212yes—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—8442no—
142C22×C4—4221no—
143C22×C4—4221no—
144C22×C4—4222no—
145C22×C4—4222no—
146C22×C4C22⋊D21111yescharacteristic
147C22×C4C22⋊D21112yes—
148C22×C4C22⋊D21112yes—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—8444no—
152C22⋊C4—22no—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—8442no—
156C22⋊C4—4222no—
157C22⋊C4—4222no—
158C22⋊C4—2114no—
159C42—22no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23Dic211⋊C412yescomplemented
167C23C2×C4×D21111yescharacteristic
168C23C2×C22⋊D21111yescharacteristic
169C23C2×C22⋊D21112yes—
170C23C2×Dic42211yescharacteristic
171C23#1680996212yes—
172D4—24no—
173D4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—8442no—
183C2×C4—8442no—
184C2×C4—8441no—
185C2×C4—8442no—
186C2×C4—8441no—
187C2×C4—8441no—
188C2×C4—8441no—
189C2×C4—8441no—
190C2×C4—4222no—
191C2×C4—4224no—
192C2×C4—4224no—
193C2×C4Dic211⋊C412yescomplemented
194C22—22no—
195C22—21no—
196C22—22no—
197C22—21no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22C2×Dic211⋊C411yescharacteristic
207C22C4×C22⋊D21112yes—
208C22#2026994611yescharacteristic
209C22#2026995612yes—
210C22#2026996011yescharacteristic
211C4—22no—
212C4—42no—
213C4—42no—
214C4—8444no—
215C4—8442no—
216C4—16881no—
217C2—22no—
218C2—21no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2#2721301911yescharacteristic
223C2#2721333311yescharacteristic
224C2#2721333611yescharacteristic
subgroupG/H
centerC22#20269946
commutatorC23×C211C22×C4
FrattiniC24C22×D211
Fitting#518×C211C2
socleC22×C211#120
prime radicalC22×D4×C211C22
cosocleC24×C211C23
hypercenter#518D211
hypocenterC211#1867
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
3. #27213018C2semidirect
4. #27212913C2semidirect
11. #20269892C4semidirect
29. C22⋊Dic211C2×C4semidirect
64. C23×C211C4⋊C4semidirect
80. C2×C4×C211C4⋊C4semidirect
112. C211#1867semidirect
166. C23Dic211⋊C4semidirect
193. C2×C4Dic211⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#202698922
#272130262
#272130182
#272129132
#518×C2112
#1867211