Finite Group Databasebeta

← #41014435

Subgroups — #41014435

Order 27008.

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

normal subgroups only

This group has 362 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×#20270008C211yesmaximal, characteristic
2#27213722C211yesmaximal, characteristic
3C2×#20269975C211yesmaximal, characteristic, complemented
4C2×#20269903C211yesmaximal, characteristic
5C2×#20269895C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#419×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×Dic422C2211yescharacteristic
10#20270008—22no—
11#20270008—22no—
12C2×#16809952—21no—
13C2×#16809949—21no—
14C2×#16809946C2211yescharacteristic
15#20269977—21no—
16#20269975—22no—
17#20269975—22no—
18#20269965—21no—
19#20269961—22no—
20#20269957—22no—
21C2×Dic211⋊C4—22no—
22C2×C4×Dic211C2211yescharacteristic
23#20269903—22no—
24#20269895—22no—
25C22×Q8×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27#117×C211C2211yescharacteristic
28#103×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×Dic422—21no—
39C2×Dic422D411yescharacteristic
40C2×D844D411yescharacteristic
41C2×C22⋊D211—21no—
42C22×Dic211—21no—
43C22×Dic211D412yes—
44C2×C4×D211—21no—
45#16809952—24no—
46#16809949—24no—
47#16809946—21no—
48#16809946D412yes—
49C2×C211⋊C8—21no—
50C4⋊Dic211—24no—
51Dic211⋊C4—28no—
52(C2×D211)⋊C4—24no—
53C4×Dic211—21no—
54C4×Dic211D412yes—
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—
65C2×C8×C211—21no—
66C4⋊C4×C211D412yes—
67C22⋊C4×C211—21no—
68C42×C211—21no—
69C22×D211—21no—
70Dic422—21no—
71Dic422—21no—
72Dic422—41no—
73Dic422—41no—
74Dic422—41no—
75Dic422—41no—
76D844—21no—
77D844—21no—
78C22⋊D211—41no—
79C22⋊D211—41no—
80C2×Dic211—24no—
81C2×Dic211—22no—
82C2×Dic211—22no—
83C2×Dic211—22no—
84C2×Dic211—21no—
85C2×Dic211—22no—
86C4×D211—41no—
87C4×D211—41no—
88C211⋊C8—22no—
89C23×C211C2×D411yescharacteristic
90Q8×C211—22no—
91Q8×C211—22no—
92Q8×C211—41no—
93Q8×C211—41no—
94C2×C4×C211—22no—
95C2×C4×C211—22no—
96C2×C4×C211—41no—
97C2×C4×C211—41no—
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211C2×D411yescharacteristic
103C2×C4×C211#4111yescharacteristic
104C8×C211—41no—
105C22×C211—22no—
106C22×C211#12011yescharacteristic
107C22×C211#12011yescharacteristic
108C22×C211#12111yescharacteristic
109C2×D211—41no—
110C2×D211—42no—
111C4×C211—42no—
112C4×C211—42no—
113C4×C211#12011yescharacteristic
114C4×C211#12111yescharacteristic
115C4×C211#12111yescharacteristic
116C4×C211#12711yescharacteristic
117Dic211—24no—
118Dic211—41no—
119Dic211—42no—
120C2×C211—21no—
121C2×C211—21no—
122C2×C211#39211yescharacteristic
123C2×C211#45612yes—
124D211—81no—
125C211#200511yescharacteristic, complemented, Sylow
126#2005—2111nomaximal, Sylow
127C2×#137—2111no—
128#536—2111no—
129C2×C4⋊Q8—2111no—
130C2×#104—2111no—
131#419D21111yescharacteristic, complemented
132C2×#101—2111no—
133#339—2111no—
134C2×#41—2111no—
135C22×Q8—2111no—
136C22×Q8C2×D21111yescharacteristic
137#137—4222no—
138#137—4222no—
139C2×Q16—4221no—
140C2×SD16—4221no—
141C2×M16—2111no—
142C2×M16C2×D21111yescharacteristic
143C4⋊Q8—4221no—
144C4⋊Q8—4222no—
145C4⋊Q8—4222no—
146#124—4221no—
147#123—4222no—
148C22⋊Q8—4222no—
149#117C2×D21111yescharacteristic
150C2×C4⋊C4—4222no—
151C2×C42—2111no—
152#104—4222no—
153#103—22no—
154#101—4222no—
155#41—4221no—
156#41—4221no—
157#41—4221no—
158#41—4221no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—4221no—
165C2×Q8—4221no—
166C2×Q8—4221no—
167C2×Q8—4221no—
168C2×Q8—4221no—
169C2×Q8—4221no—
170C2×Q8—2111no—
171C2×Q8D84412yes—
172C2×D4—4221no—
173C2×D4—2111no—
174C22×C4—21no—
175C22×C4—4221no—
176C22×C4—4221no—
177C22×C4—2112no—
178C22×C4C22×D21111yescharacteristic
179Q16—4224no—
180SD16—4224no—
181M16—21no—
182M16—21no—
183M16—4221no—
184M16—2112no—
185C2×C8—21no—
186C2×C8—4221no—
187C4⋊C4—4228no—
188C4⋊C4—4224no—
189C4⋊C4C22⋊D21112yes—
190C22⋊C4—21no—
191C22⋊C4—4224no—
192C42—21no—
193C42—4221no—
194C42—2112no—
195C23—4221no—
196C23D4×D21111yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—8441no—
202Q8—8441no—
203Q8—8441no—
204Q8—8441no—
205Q8—4221no—
206Q8—4221no—
207D4—8441no—
208D4—8441no—
209D4—4221no—
210D4—4221no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—8441no—
216C2×C4—8441no—
217C2×C4—4222no—
218C2×C4—4221no—
219C2×C4—4222no—
220C2×C4—4222no—
221C2×C4—4224no—
222C2×C4—4222no—
223C2×C4D4×D21111yescharacteristic
224C2×C4D4×D21111yescharacteristic
225C2×C4D4×D21111yescharacteristic
226C2×C4C2×C22⋊D21111yescharacteristic
227C2×C4C2×D84411yescharacteristic
228C2×C4#1680996211yescharacteristic
229C8—41no—
230C8—4222no—
231C22—22no—
232C22—8442no—
233C22—8441no—
234C22#2026994511yescharacteristic
235C22#2026994711yescharacteristic
236C22#2026995011yescharacteristic
237C4—42no—
238C4—42no—
239C4—8441no—
240C4—8442no—
241C4—4224no—
242C4#2026994711yescharacteristic
243C4#2026995011yescharacteristic
244C4#2026995111yescharacteristic
245C4#2026997311yescharacteristic
246C2—21no—
247C2—21no—
248C2—16881no—
249C2#2721305511yescharacteristic
250C2#2721332712yes—
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#419×C211C2
socleC22×C211#120
prime radicalC2×#16809962C22
hypercenter#419D211
hypocenterC211#2005
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. C2×#20269975C2semidirect
5. C2×#20269895C2semidirect
6. #27212905C2semidirect
7. #419×C211C2semidirect
125. C211#2005semidirect
131. #419D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700082
#272137222
C2×#202699752
C2×#202699032
C2×#202698952
#272129052
#419×C2112
#2005211