Finite Group Databasebeta

← #41014518

Subgroups — #41014518

Order 27008.

342 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 73 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×#20270010C211yesmaximal, characteristic
2C2×#20270008C211yesmaximal, characteristic
3C22×#16809952C211yesmaximal, characteristic, complemented
4C2×#20269895C211yesmaximal, characteristic, complemented
5#27213114C211yesmaximal, characteristic
6#27212910C211yesmaximal, characteristic, complemented
7C2×#101×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×Dic422C2211yescharacteristic
10#20270010—22no—
11#20270010—22no—
12#20270008—22no—
13#20270008—22no—
14C2×Dic844—21no—
15C2×#16809952—22no—
16C2×#16809952—21no—
17C2×#16809952—22no—
18C2×#16809952—22no—
19C2×#16809951—21no—
20C2×#16809949—21no—
21C2×#16809946C2211yescharacteristic
22C22×C211⋊C8C2211yescharacteristic
23#20269895—22no—
24(C2×D211)⋊C8—22no—
25C22×Q8×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27C2×M16×C211C2211yescharacteristic
28#101×C211—22no—
29#16809962—21no—
30#16809962—21no—
31#16809962—21no—
32#16809962—21no—
33C2×Dic422—21no—
34C2×Dic422—21no—
35C2×Dic422—21no—
36C2×Dic422—21no—
37C2×Dic422—21no—
38C2×Dic422D411yescharacteristic
39C2×Dic422D412yes—
40C2×D844D411yescharacteristic
41C2×C22⋊D211—21no—
42C22×Dic211—21no—
43C2×C4×D211—21no—
44Dic844—24no—
45#16809952—28no—
46#16809952—24no—
47#16809951—24no—
48#16809949—24no—
49#16809946—21no—
50#16809946D412yes—
51C2×C211⋊C8—21no—
52C2×C211⋊C8—21no—
53C2×C211⋊C8—21no—
54C2×C211⋊C8D412yes—
55C2×Q8×C211—21no—
56C2×Q8×C211—21no—
57C2×Q8×C211—21no—
58C2×Q8×C211—21no—
59C2×Q8×C211—21no—
60C2×Q8×C211D412yes—
61C22×C4×C211—21no—
62C22×C4×C211C2311yescharacteristic
63M16×C211—21no—
64M16×C211—21no—
65M16×C211—21no—
66M16×C211D412yes—
67C2×C8×C211—21no—
68C2×C8×C211—21no—
69C22×D211—21no—
70Dic422—22no—
71Dic422—22no—
72Dic422—21no—
73Dic422—21no—
74Dic422—41no—
75Dic422—41no—
76D844—21no—
77D844—21no—
78C22⋊D211—41no—
79C22⋊D211—41no—
80C2×Dic211—21no—
81C2×Dic211—22no—
82C2×Dic211—41no—
83C2×Dic211—41no—
84C4×D211—41no—
85C4×D211—41no—
86C211⋊C8—22no—
87C211⋊C8—22no—
88C23×C211#4111yescharacteristic
89Q8×C211—22no—
90Q8×C211—22no—
91Q8×C211—41no—
92Q8×C211—41no—
93C2×C4×C211—22no—
94C2×C4×C211—41no—
95C2×C4×C211—41no—
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C8×C211—22no—
103C8×C211—41no—
104C22×C211—22no—
105C22×C211#12111yescharacteristic
106C22×C211#12111yescharacteristic
107C22×C211#12111yescharacteristic
108C2×D211—41no—
109C2×D211—42no—
110C4×C211—42no—
111C4×C211#12011yescharacteristic
112C4×C211#12011yescharacteristic
113C4×C211#12011yescharacteristic
114C4×C211#12711yescharacteristic
115Dic211—41no—
116Dic211—42no—
117C2×C211—21no—
118C2×C211—21no—
119C2×C211#39211yescharacteristic
120C2×C211#47312yes—
121D211—81no—
122C211#202011yescharacteristic, complemented, Sylow
123#2020—2111nomaximal, Sylow
124C2×#137—2111no—
125C2×#137—2111no—
126C22×Q16—2111no—
127C2×#101—2111no—
128C2×#101D21111yescharacteristic, complemented
129#408—2111no—
130#341—2111no—
131C2×#41—2111no—
132C22×Q8—2111no—
133C22×Q8C2×D21111yescharacteristic
134#137—4222no—
135#137—4222no—
136#137—4222no—
137#137—4222no—
138C2×Q16—4221no—
139C2×Q16—4222no—
140C2×Q16—4221no—
141C2×Q16—4222no—
142C2×Q16—4222no—
143C2×SD16—4221no—
144C2×SD16—4221no—
145C2×M16—2111no—
146C2×M16C2×D21111yescharacteristic
147C2×M16C2×D21111yescharacteristic
148C22×C8—2111no—
149#101—22no—
150#101—4222no—
151C22⋊C8—4222no—
152#41—4221no—
153#41—4221no—
154#41—4221no—
155#41—4221no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—4221no—
162C2×Q8—4221no—
163C2×Q8—4221no—
164C2×Q8—4221no—
165C2×Q8—4221no—
166C2×Q8—2111no—
167C2×Q8—2112no—
168C2×Q8D84412yes—
169C2×D4—4221no—
170C2×D4—2111no—
171C22×C4—21no—
172C22×C4—4221no—
173C22×C4—4221no—
174C22×C4C22×D21111yescharacteristic
175Q16—4224no—
176Q16—4228no—
177Q16—4224no—
178SD16—4224no—
179SD16—4224no—
180M16—21no—
181M16—21no—
182M16—21no—
183M16—4221no—
184M16—2112no—
185M16C22⋊D21112yes—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—4221no—
189C2×C8—4221no—
190C2×C8—4221no—
191C2×C8—2112no—
192C23—4221no—
193C23#1680996211yescharacteristic
194Q8—22no—
195Q8—22no—
196Q8—41no—
197Q8—41no—
198Q8—8441no—
199Q8—8441no—
200Q8—4222no—
201Q8—4221no—
202Q8—4222no—
203Q8—4221no—
204D4—8441no—
205D4—8441no—
206D4—4221no—
207D4—4221no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—8441no—
212C2×C4—8441no—
213C2×C4—8441no—
214C2×C4—8441no—
215C2×C4—4222no—
216C2×C4—4221no—
217C2×C4D4×D21111yescharacteristic
218C2×C4D4×D21111yescharacteristic
219C2×C4D4×D21111yescharacteristic
220C2×C4D4×D21111yescharacteristic
221C2×C4C2×C22⋊D21111yescharacteristic
222C2×C4C2×D84411yescharacteristic
223C8—22no—
224C8—41no—
225C8—4222no—
226C8—4222no—
227C22—22no—
228C22—8442no—
229C22—8441no—
230C22#2026995011yescharacteristic
231C22#2026995011yescharacteristic
232C22#2026995111yescharacteristic
233C4—42no—
234C4—8441no—
235C4—8442no—
236C4#2026994511yescharacteristic
237C4#2026994711yescharacteristic
238C4#2026994711yescharacteristic
239C4#2026997311yescharacteristic
240C2—21no—
241C2—21no—
242C2—16881no—
243C2#2721305511yescharacteristic
244C2#2721342012yes—
subgroupG/H
centerC22#20269950
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#101×C211C2
socleC22×C211#121
prime radicalC2×#16809962C22
hypercenterC2×#101D211
hypocenterC211#2020
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
3. C22×#16809952C2semidirect
4. C2×#20269895C2semidirect
6. #27212910C2semidirect
7. C2×#101×C211C2semidirect
122. C211#2020semidirect
128. C2×#101D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700102
C2×#202700082
C22×#168099522
C2×#202698952
#272131142
#272129102
C2×#101×C2112
#2020211