Finite Group Databasebeta

← #41014429

Subgroups — #41014429

Order 27008.

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

normal subgroups only

This group has 394 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, complemented
2#27213713C211yesmaximal, characteristic
3C2×#20269964C211yesmaximal, characteristic, complemented
4C2×#20269903C211yesmaximal, characteristic, complemented
5#27213156C211yesmaximal, characteristic, complemented
6#27212907C211yesmaximal, characteristic, complemented
7C2×#100×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×Dic422C2211yescharacteristic
10#20270010—22no—
11#20270010—22no—
12C2×Dic844—21no—
13C2×#16809951—21no—
14#20269973—21no—
15#20269964—21no—
16#20269964—22no—
17#20269964—22no—
18#20269948—22no—
19#20269946—22no—
20#20269936C2211yescharacteristic
21C2×C22⋊Dic211—22no—
22C2×C4×Dic211C2211yescharacteristic
23#20269903—22no—
24#20269902—22no—
25C22×D4×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27C2×M16×C211C2211yescharacteristic
28#100×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×Dic422D412yescomplemented
40C2×D844D411yescharacteristic
41C2×C22⋊D211—24no—
42C2×C22⋊D211—21no—
43C22×Dic211—21no—
44C22×Dic211—21no—
45C2×C4×D211—21no—
46Dic844—24no—
47#16809951—24no—
48C4⋊Dic211D412yes—
49C22⋊Dic211—28no—
50C22⋊Dic211—24no—
51C22⋊Dic211—21no—
52C4×Dic211—21no—
53C4×Dic211—21no—
54C4×Dic211D412yescomplemented
55C24×C211D412yes—
56C2×D4×C211—21no—
57C2×D4×C211—21no—
58C2×D4×C211—21no—
59C2×D4×C211—21no—
60C2×D4×C211—21no—
61C2×D4×C211—21no—
62C22×C4×C211C2311yescharacteristic
63M16×C211—21no—
64M16×C211—21no—
65M16×C211—21no—
66M16×C211D412yescomplemented
67C2×C8×C211—21no—
68C2×C8×C211—21no—
69C22×D211—21no—
70Dic422—22no—
71Dic422—21no—
72Dic422—22no—
73Dic422—21no—
74Dic422—41no—
75Dic422—41no—
76D844—21no—
77D844—21no—
78C22⋊D211—41no—
79C22⋊D211—41no—
80C22⋊D211—84no—
81C2×Dic211—22no—
82C2×Dic211—22no—
83C2×Dic211—21no—
84C2×Dic211—22no—
85C2×Dic211—42no—
86C2×Dic211—41no—
87C2×Dic211—41no—
88C4×D211—41no—
89C4×D211—41no—
90C23×C211—22no—
91C23×C211—22no—
92C23×C211—22no—
93C23×C211—44no—
94C23×C211C2×D411yescharacteristic
95D4×C211—41no—
96D4×C211—41no—
97D4×C211—41no—
98D4×C211—41no—
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211C2×D411yescharacteristic
103C2×C4×C211C2×D411yescharacteristic
104C2×C4×C211#4111yescharacteristic
105C8×C211—22no—
106C8×C211—41no—
107C22×C211—22no—
108C22×C211—44no—
109C22×C211—42no—
110C22×C211—44no—
111C22×C211—82no—
112C22×C211#12011yescharacteristic
113C22×C211#12011yescharacteristic
114C22×C211#12111yescharacteristic
115C2×D211—41no—
116C2×D211—42no—
117C4×C211#12011yescharacteristic
118C4×C211#12111yescharacteristic
119C4×C211#12111yescharacteristic
120C4×C211#12711yescharacteristic
121Dic211—42no—
122Dic211—41no—
123Dic211—42no—
124Dic211—42no—
125C2×C211—21no—
126C2×C211—21no—
127C2×C211—82no—
128C2×C211#39211yescharacteristic
129C2×C211#45512yes—
130D211—81no—
131C211#200411yescharacteristic, complemented, Sylow
132#2004—2111nomaximal, Sylow
133C2×#137—2111no—
134#535—2111no—
135C2×#124—2111no—
136C2×#104—2111no—
137#419—2111no—
138C2×#100D21111yescharacteristic, complemented
139#339—2111no—
140C2×#41—2111no—
141C22×Q8—2111no—
142C22×D4C2×D21111yescharacteristic
143#137—4222no—
144#137—4222no—
145C2×Q16—4221no—
146C2×SD16—4221no—
147C2×M16C2×D21111yescharacteristic
148C2×M16C2×D21111yescharacteristic
149#127—4221no—
150#124—4222no—
151#124—4221no—
152#124—4222no—
153#121—4222no—
154#120—4222no—
155#117—2111no—
156C2×C22⋊C4—4222no—
157C2×C42—2111no—
158#104—4222no—
159#103—4222no—
160#100—22no—
161C24D84412yes—
162#41—4221no—
163#41—4221no—
164#41—4221no—
165#41—4221no—
166C2×Q8—4221no—
167C2×Q8—4221no—
168C2×Q8—4221no—
169C2×Q8—4221no—
170C2×Q8—4221no—
171C2×Q8—2111no—
172C2×Q8—2112no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—4221no—
180C2×D4—4224no—
181C2×D4—2111no—
182C22×C4—4221no—
183C22×C4—4221no—
184C22×C4—4221no—
185C22×C4C22×D21111yescharacteristic
186Q16—4224no—
187SD16—4224no—
188M16—21no—
189M16—21no—
190M16—21no—
191M16C22⋊D21112yescomplemented
192C2×C8—21no—
193C2×C8—21no—
194C4⋊C4—2112no—
195C22⋊C4—4228no—
196C22⋊C4—4224no—
197C22⋊C4—4221no—
198C42—4221no—
199C42—4221no—
200C42—2112no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—44no—
205C23—4221no—
206C23C2×D84411yescharacteristic
207Q8—8441no—
208Q8—8441no—
209Q8—4222no—
210Q8—4221no—
211Q8—4222no—
212Q8—4221no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—8441no—
218D4—8441no—
219D4—16884no—
220D4—4221no—
221D4—4221no—
222C2×C4—8442no—
223C2×C4—8441no—
224C2×C4—8441no—
225C2×C4—8441no—
226C2×C4—8441no—
227C2×C4—4222no—
228C2×C4—4222no—
229C2×C4—4222no—
230C2×C4—4221no—
231C2×C4D4×D21111yescharacteristic
232C2×C4D4×D21111yescharacteristic
233C2×C4D4×D21111yescharacteristic
234C2×C4D4×D21111yescharacteristic
235C2×C4C2×C22⋊D21111yescharacteristic
236C2×C4#1680996211yescharacteristic
237C8—22no—
238C8—41no—
239C22—22no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—82no—
244C22—8442no—
245C22—8441no—
246C22#2026994711yescharacteristic
247C22#2026994711yescharacteristic
248C22#2026995111yescharacteristic
249C4—8442no—
250C4—8442no—
251C4—8441no—
252C4—8442no—
253C4#2026994511yescharacteristic
254C4#2026995011yescharacteristic
255C4#2026995011yescharacteristic
256C4#2026997311yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—16881no—
261C2#2721305511yescharacteristic
262C2#2721332612yes—
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#100×C211C2
socleC22×C211#120
prime radical#27213713C2
hypercenterC2×#100D211
hypocenterC211#2004
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#20270010C2semidirect
3. C2×#20269964C2semidirect
4. C2×#20269903C2semidirect
5. #27213156C2semidirect
6. #27212907C2semidirect
7. C2×#100×C211C2semidirect
39. C2×Dic422D4semidirect
54. C4×Dic211D4semidirect
66. M16×C211D4semidirect
131. C211#2004semidirect
138. C2×#100D211semidirect
191. M16C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700102
#272137132
C2×#202699642
C2×#202699032
#272131562
#272129072
C2×#100×C2112
#2004211