Finite Group Databasebeta

← #41014621

Subgroups — #41014621

Order 27008.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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
1#27213724C211yesmaximal, characteristic
2C2×#20269905C211yesmaximal, characteristic, complemented
3#27213148C211yesmaximal, characteristic, complemented
4#27213147C211yesmaximal, characteristic, complemented
5#27213016C211yesmaximal, characteristic, complemented
6#27212914C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269956—21no—
11#20269953—21no—
12#20269951—21no—
13#20269949—22no—
14#20269948—21no—
15C2×C4⋊Dic211—22no—
16C2×C4⋊Dic211C2211yescharacteristic
17C2×C4⋊Dic211C2211yescharacteristic
18#20269905—22no—
19#20269902—21no—
20#20269901—21no—
21#20269899—21no—
22#20269898—21no—
23C2×#41×C211C2211yescharacteristic
24C2×M16×C211C2211yescharacteristic
25C2×C42×C211C2211yescharacteristic
26#104×C211—22no—
27#16809962—21no—
28#16809962—21no—
29#16809962—21no—
30#16809962—21no—
31C2×Dic422D411yescharacteristic
32C2×D844D411yescharacteristic
33C2×C22⋊D211—22no—
34C2×C22⋊D211—21no—
35C22×Dic211—21no—
36C22×Dic211D412yes—
37C2×C4×D211—21no—
38#16809946—21no—
39#16809946—21no—
40C2×C211⋊C8—21no—
41C4⋊Dic211—21no—
42C4⋊Dic211—21no—
43C4⋊Dic211—21no—
44C4⋊Dic211—21no—
45C4⋊Dic211—44no—
46Dic211⋊C4—22no—
47C22⋊Dic211—22no—
48(C2×D211)⋊C4—22no—
49#41×C211—21no—
50#41×C211—21no—
51#41×C211—21no—
52#41×C211—21no—
53C2×Q8×C211D411yescharacteristic
54C2×D4×C211—21no—
55C2×D4×C211D411yescharacteristic
56C22×C4×C211—21no—
57C22×C4×C211—21no—
58C22×C4×C211C2311yescharacteristic
59M16×C211—21no—
60M16×C211—21no—
61C2×C8×C211—21no—
62C42×C211—21no—
63C42×C211D412yescomplemented
64C22×D211—21no—
65Dic422—21no—
66Dic422—21no—
67D844—21no—
68D844—21no—
69C22⋊D211—41no—
70C22⋊D211—41no—
71C22⋊D211—82no—
72C2×Dic211—22no—
73C2×Dic211—21no—
74C2×Dic211—42no—
75C2×Dic211—42no—
76C2×Dic211—41no—
77C4×D211—41no—
78C4×D211—41no—
79C211⋊C8—41no—
80C23×C211—21no—
81C23×C211C2×D411yescharacteristic
82Q8×C211—21no—
83Q8×C211—21no—
84D4×C211—21no—
85D4×C211—21no—
86D4×C211—41no—
87D4×C211—41no—
88C2×C4×C211—22no—
89C2×C4×C211—22no—
90C2×C4×C211—22no—
91C2×C4×C211—21no—
92C2×C4×C211—41no—
93C2×C4×C211—41no—
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211#4111yescharacteristic
98C2×C4×C211#4111yescharacteristic
99C2×C4×C211#4111yescharacteristic
100C8×C211—41no—
101C22×C211—21no—
102C22×C211—21no—
103C22×C211—41no—
104C22×C211—41no—
105C22×C211—41no—
106C22×C211#12011yescharacteristic
107C22×C211#12111yescharacteristic
108C22×C211#12411yescharacteristic
109C2×D211—41no—
110C2×D211—41no—
111C2×D211—41no—
112C4×C211—24no—
113C4×C211—41no—
114C4×C211#12111yescharacteristic
115C4×C211#12111yescharacteristic
116C4×C211#12311yescharacteristic
117C4×C211#12311yescharacteristic
118Dic211—41no—
119Dic211—42no—
120Dic211—81no—
121C2×C211—21no—
122C2×C211—21no—
123C2×C211—81no—
124C2×C211#39411yescharacteristic
125C2×C211#45311yescharacteristic
126C2×C211#45611yescharacteristic
127D211—81no—
128C211#203911yescharacteristic, complemented, Sylow
129#2039—2111nomaximal, Sylow
130#537—2111no—
131C2×#104—2111no—
132C2×#104D21111yescharacteristic, complemented
133#417—2111no—
134#417—2111no—
135#384—2111no—
136#343—2111no—
137C2×#41—2111no—
138C2×#41C2×D21111yescharacteristic
139C2×M16—2111no—
140C2×M16C2×D21111yescharacteristic
141C22⋊Q8—4221no—
142C22⋊Q8—4221no—
143#121—4221no—
144#121—4221no—
145#121—4222no—
146C2×C4⋊C4—4222no—
147C2×C4⋊C4—2111no—
148C2×C4⋊C4—2111no—
149C2×C42C2×D21111yescharacteristic
150#104—22no—
151#104—4222no—
152#103—4221no—
153#103—4221no—
154#102—4221no—
155#102—4221no—
156#41—21no—
157#41—21no—
158#41—21no—
159#41—21no—
160#41—4221no—
161#41—4221no—
162#41—4221no—
163#41—4221no—
164C2×Q8—2111no—
165C2×Q8D84411yescharacteristic
166C2×D4—21no—
167C2×D4—4221no—
168C2×D4—4222no—
169C2×D4—2111no—
170C2×D4D84411yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—4221no—
174C22×C4—4221no—
175C22×C4—2112no—
176C22×C4C22×D21111yescharacteristic
177M16—21no—
178M16—21no—
179M16—4221no—
180M16—4221no—
181C2×C8—21no—
182C2×C8—4221no—
183C4⋊C4—8444no—
184C4⋊C4—4222no—
185C4⋊C4—4221no—
186C4⋊C4—4221no—
187C4⋊C4—4221no—
188C4⋊C4—4221no—
189C22⋊C4—4222no—
190C22⋊C4—4222no—
191C42—21no—
192C42C22⋊D21112yescomplemented
193C23—21no—
194C23—4221no—
195C23D4×D21111yescharacteristic
196Q8—21no—
197Q8—21no—
198Q8—4221no—
199Q8—4221no—
200D4—21no—
201D4—21no—
202D4—41no—
203D4—41no—
204D4—8441no—
205D4—8441no—
206D4—16882no—
207D4—4221no—
208D4—4221no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—8441no—
216C2×C4—8441no—
217C2×C4—8442no—
218C2×C4—8442no—
219C2×C4—8441no—
220C2×C4—4222no—
221C2×C4—4221no—
222C2×C4D4×D21111yescharacteristic
223C2×C4C2×C22⋊D21111yescharacteristic
224C2×C4C2×D84411yescharacteristic
225C2×C4#1680996111yescharacteristic
226C2×C4#1680996111yescharacteristic
227C2×C4#1680996211yescharacteristic
228C8—41no—
229C8—8441no—
230C22—21no—
231C22—21no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—8441no—
236C22—8441no—
237C22—8441no—
238C22#2026994711yescharacteristic
239C22#2026994911yescharacteristic
240C22#2026996311yescharacteristic
241C4—24no—
242C4—41no—
243C4—8442no—
244C4—8441no—
245C4—16881no—
246C4#2026994811yescharacteristic
247C4#2026995111yescharacteristic
248C4#2026995911yescharacteristic
249C4#2026996111yescharacteristic
250C2—21no—
251C2—21no—
252C2—81no—
253C2—16881no—
254C2#2721306211yescharacteristic
255C2#2721331511yescharacteristic
256C2#2721333011yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC22×C211#120
prime radical#27213724C2
hypercenterC2×#104D211
hypocenterC211#2039
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#20269905C2semidirect
3. #27213148C2semidirect
4. #27213147C2semidirect
5. #27213016C2semidirect
6. #27212914C2semidirect
7. C2×#104×C211C2semidirect
63. C42×C211D4semidirect
128. C211#2039semidirect
132. C2×#104D211semidirect
192. C42C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272137242
C2×#202699052
#272131482
#272131472
#272130162
#272129142
C2×#104×C2112
#2039211