Finite Group Databasebeta

← #41014516

Subgroups — #41014516

Order 27008.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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×#20270007C211yesmaximal, characteristic
2C22×#16809952C211yesmaximal, characteristic, complemented
3C2×#20269896C211yesmaximal, characteristic, complemented
4C2×#20269895C211yesmaximal, characteristic, complemented
5#27213113C211yesmaximal, characteristic
6#27212910C211yesmaximal, characteristic, complemented
7C2×#137×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9#20270007—22no—
10#20270007—22no—
11C2×#16809952—22no—
12C2×#16809952—22no—
13C2×#16809952—22no—
14C2×#16809952—21no—
15C2×#16809950—21no—
16C2×#16809946C2211yescharacteristic
17C2×#16809946C2211yescharacteristic
18C22×C211⋊C8C2211yescharacteristic
19#20269896—22no—
20#20269895—22no—
21C22⋊(C211⋊C8)—22no—
22C2×#41×C211C2211yescharacteristic
23C22×Q8×C211C2211yescharacteristic
24#137×C211—22no—
25#137×C211—22no—
26C2×Q16×C211—21no—
27C2×SD16×C211—21no—
28C2×M16×C211C2211yescharacteristic
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422D412yes—
35C22×Dic211—21no—
36#16809952—28no—
37#16809952—24no—
38#16809950—24no—
39#16809946—21no—
40#16809946—21no—
41#16809946—21no—
42#16809946D412yes—
43C2×C211⋊C8—21no—
44C2×C211⋊C8—21no—
45C2×C211⋊C8—21no—
46C2×C211⋊C8—21no—
47C2×C211⋊C8D412yes—
48#41×C211—21no—
49#41×C211—21no—
50#41×C211—21no—
51#41×C211—21no—
52C2×Q8×C211—21no—
53C2×Q8×C211—21no—
54C2×Q8×C211—21no—
55C2×Q8×C211—21no—
56C2×Q8×C211—21no—
57C2×Q8×C211D411yescharacteristic
58C2×Q8×C211D412yes—
59C2×D4×C211—21no—
60C2×D4×C211D411yescharacteristic
61C22×C4×C211—21no—
62C22×C4×C211—21no—
63C22×C4×C211C2311yescharacteristic
64Q16×C211—24no—
65SD16×C211—24no—
66M16×C211—21no—
67M16×C211D412yes—
68C2×C8×C211—21no—
69Dic422—22no—
70Dic422—22no—
71Dic422—41no—
72Dic422—41no—
73C2×Dic211—22no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C211⋊C8—22no—
77C211⋊C8—22no—
78C211⋊C8—41no—
79C23×C211—21no—
80C23×C211#4111yescharacteristic
81Q8×C211—22no—
82Q8×C211—21no—
83Q8×C211—22no—
84Q8×C211—21no—
85Q8×C211—41no—
86Q8×C211—41no—
87D4×C211—21no—
88D4×C211—21no—
89D4×C211—41no—
90D4×C211—41no—
91C2×C4×C211—21no—
92C2×C4×C211—22no—
93C2×C4×C211—41no—
94C2×C4×C211—41no—
95C2×C4×C211—41no—
96C2×C4×C211—41no—
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211C2×D411yescharacteristic
103C8×C211—22no—
104C22×C211—22no—
105C22×C211—41no—
106C22×C211—42no—
107C22×C211#12111yescharacteristic
108C22×C211#12111yescharacteristic
109C22×C211#12111yescharacteristic
110C4×C211—41no—
111C4×C211—42no—
112C4×C211#12011yescharacteristic
113C4×C211#12011yescharacteristic
114C4×C211#12011yescharacteristic
115C4×C211#12711yescharacteristic
116Dic211—42no—
117C2×C211—21no—
118C2×C211—21no—
119C2×C211—81no—
120C2×C211#39211yescharacteristic
121C2×C211#47312yes—
122C211#202011yescharacteristic, complemented, Sylow
123#2020—2111nomaximal, Sylow
124C2×#137—2111no—
125C2×#137D21111yescharacteristic
126C22×Q16—2111no—
127C2×#101—2111no—
128C2×#101—2111no—
129#408—2111no—
130#341—2111no—
131C2×#41C2×D21111yescharacteristic
132C22×Q8—2111no—
133C22×Q8C2×D21111yescharacteristic
134#137—22no—
135#137—22no—
136#137—4222no—
137#137—4222no—
138C2×Q16—21no—
139C2×Q16—4222no—
140C2×Q16—4222no—
141C2×Q16—4221no—
142C2×Q16—4222no—
143C2×SD16—21no—
144C2×SD16—4221no—
145C2×M16—2111no—
146C2×M16—2111no—
147C2×M16C2×D21111yescharacteristic
148C22×C8—2111no—
149#101—4222no—
150#101—4222no—
151C22⋊C8—4222no—
152#41—21no—
153#41—21no—
154#41—21no—
155#41—21no—
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—2112no—
167C2×Q8C22⋊D21111yescharacteristic
168C2×Q8C22⋊D21112yes—
169C2×D4—21no—
170C2×D4C22⋊D21111yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—4221no—
174C22×C4C22×D21111yescharacteristic
175Q16—24no—
176Q16—4228no—
177Q16—4224no—
178SD16—24no—
179SD16—4224no—
180M16—21no—
181M16—4221no—
182M16—4221no—
183M16—4221no—
184M16—2112no—
185M16C22⋊D21112yes—
186C2×C8—21no—
187C2×C8—4221no—
188C2×C8—4221no—
189C2×C8—4221no—
190C2×C8—4221no—
191C2×C8—2112no—
192C23—21no—
193C23#1680996111yescharacteristic
194Q8—22no—
195Q8—21no—
196Q8—22no—
197Q8—21no—
198Q8—41no—
199Q8—41no—
200Q8—8441no—
201Q8—8441no—
202Q8—4222no—
203Q8—4222no—
204D4—21no—
205D4—21no—
206D4—41no—
207D4—41no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—8441no—
215C2×C4—8441no—
216C2×C4—4222no—
217C2×C4D4×D21111yescharacteristic
218C2×C4D4×D21111yescharacteristic
219C2×C4D4×D21111yescharacteristic
220C2×C4C2×C22⋊D21111yescharacteristic
221C2×C4C2×C22⋊D21111yescharacteristic
222C2×C4C2×C22⋊D21111yescharacteristic
223C8—22no—
224C8—8441no—
225C8—4222no—
226C8—4222no—
227C22—22no—
228C22—42no—
229C22—41no—
230C22#2026994811yescharacteristic
231C22#2026994911yescharacteristic
232C22#2026994911yescharacteristic
233C4—41no—
234C4—42no—
235C4—8442no—
236C4#2026994511yescharacteristic
237C4#2026994511yescharacteristic
238C4#2026994611yescharacteristic
239C4#2026997311yescharacteristic
240C2—21no—
241C2—21no—
242C2—81no—
243C2#2721305411yescharacteristic
244C2#2721341912yes—
subgroupG/H
centerC22#20269949
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#137×C211C2
socleC22×C211#121
prime radicalC2×#41×C211C22
hypercenterC2×#137D211
hypocenterC211#2020
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#16809952C2semidirect
3. C2×#20269896C2semidirect
4. C2×#20269895C2semidirect
6. #27212910C2semidirect
122. C211#2020semidirect

Maximal subgroups

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

subgroupindex
C2×#202700072
C22×#168099522
C2×#202698962
C2×#202698952
#272131132
#272129102
C2×#137×C2112
#2020211