Finite Group Databasebeta

← #41013707

Subgroups — #41013707

Order 27008.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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×#20269953C211yesmaximal, characteristic
2C2×#20269935C211yesmaximal, characteristic
3C2×#20269901C211yesmaximal, characteristic, complemented
4C2×(C2×D211)⋊C8C211yesmaximal, characteristic
5#27212909C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7C2×#103×C211C211yesmaximal, characteristic, complemented
8C22×C4×D211C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10#20269953—22no—
11#20269953C414yescomplemented
12#20269935—22no—
13#20269935—22no—
14C2×C4⋊Dic211C2211yescharacteristic
15C2×C4⋊Dic211C2211yescharacteristic
16C2×Dic211⋊C4—21no—
17C2×(C2×D211)⋊C4—21no—
18C2×(C2×D211)⋊C4—21no—
19C2×C4×Dic211—21no—
20#20269901—22no—
21(C2×D211)⋊C8—22no—
22C22×Q8×C211C2211yescharacteristic
23C22×C8×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25#103×C211—22no—
26C23×D211D411yescharacteristic
27C22×Dic211—21no—
28C22×Dic211—21no—
29C22×Dic211D411yescharacteristic
30C2×C4×D211—21no—
31C2×C4×D211—21no—
32C2×C4×D211—21no—
33C2×C4×D211—21no—
34C2×C4×D211C2×C412yes—
35C2×C4×D211D412yes—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C4⋊Dic211—21no—
40C4⋊Dic211—21no—
41C4⋊Dic211C2×C412yes—
42C4⋊Dic211D412yes—
43Dic211⋊C4—24no—
44(C2×D211)⋊C4—24no—
45(C2×D211)⋊C4—24no—
46C4×Dic211—24no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—21no—
50C2×Q8×C211—21no—
51C2×Q8×C211—21no—
52C2×Q8×C211C2×C412yes—
53C22×C4×C211—21no—
54C22×C4×C211—21no—
55C22×C4×C211C2311yescharacteristic
56C2×C8×C211—21no—
57C2×C8×C211—21no—
58C2×C8×C211—21no—
59C4⋊C4×C211—21no—
60C4⋊C4×C211D412yes—
61C22×D211—21no—
62C22×D211—21no—
63C22×D211—21no—
64C22×D211—21no—
65C22×D211—21no—
66C22×D211—21no—
67C22×D211C22⋊C412yes—
68C2×Dic211—21no—
69C2×Dic211—21no—
70C2×Dic211—22no—
71C2×Dic211—22no—
72C2×Dic211—41no—
73C2×Dic211—41no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C2×Dic211C22⋊C412yes—
77C4×D211—22no—
78C4×D211—22no—
79C4×D211—22no—
80C4×D211—22no—
81C211⋊C8—41no—
82C23×C211C2×D411yescharacteristic
83Q8×C211—22no—
84Q8×C211—22no—
85Q8×C211—41no—
86Q8×C211—41no—
87C2×C4×C211—22no—
88C2×C4×C211—22no—
89C2×C4×C211—41no—
90C2×C4×C211—41no—
91C2×C4×C211—41no—
92C2×C4×C211—41no—
93C2×C4×C211C22×C411yescharacteristic
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211#4111yescharacteristic
98C2×C4×C211#4111yescharacteristic
99C8×C211—41no—
100C22×C211C2×C22⋊C411yescharacteristic
101C22×C211#12011yescharacteristic
102C22×C211#12311yescharacteristic
103C22×C211#13511yescharacteristic
104C22×C211#13511yescharacteristic
105C22×C211#13711yescharacteristic
106C22×C211#13711yescharacteristic
107C2×D211—22no—
108C2×D211—22no—
109C2×D211—22no—
110C2×D211—41no—
111C2×D211—41no—
112C2×D211—41no—
113C2×D211—41no—
114C4×C211—42no—
115C4×C211—42no—
116C4×C211C4×D411yescharacteristic
117C4×C211C4×D411yescharacteristic
118C4×C211#12111yescharacteristic
119C4×C211#12111yescharacteristic
120Dic211—22no—
121Dic211—42no—
122Dic211—42no—
123C2×C211#38611yescharacteristic
124C2×C211#41611yescharacteristic
125C2×C211#41911yescharacteristic
126C2×C211#45211yescharacteristic
127C2×C211#45211yescharacteristic
128C2×C211#48511yescharacteristic
129C2×C211#48511yescharacteristic
130D211—42no—
131C211#187311yescharacteristic, complemented, Sylow
132#1873—2111nomaximal, Sylow
133C2×C22⋊Q8—2111no—
134C2×#117—2111no—
135C2×#103—2111no—
136C2×#103D21111yescharacteristic, complemented
137C2×C22⋊C8—2111no—
138#340—2111no—
139#340—2111no—
140C22×Q8C2×D21111yescharacteristic
141C23×C4—2111no—
142C22×C8—2111no—
143C22×C8C2×D21111yescharacteristic
144C22⋊Q8—4222no—
145C22⋊Q8—2114no—
146#117—4222no—
147#117—4222no—
148C2×C4⋊C4—4221no—
149C2×C4⋊C4—2111no—
150C2×C4⋊C4—2111no—
151C2×C4⋊C4C2×D21111yescharacteristic
152C2×C22⋊C4—4221no—
153C2×C22⋊C4—4221no—
154C2×C42—4221no—
155#103—22no—
156#103—4222no—
157C22⋊C8—4222no—
158C24—2111no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8C4×D21112yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—4221no—
168C22×C4—4221no—
169C22×C4—4221no—
170C22×C4—4221no—
171C22×C4—4221no—
172C22×C4—4221no—
173C22×C4—2111no—
174C22×C4—2112no—
175C22×C4—2112no—
176C22×C4C22×D21111yescharacteristic
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—4221no—
181C2×C8—4221no—
182C2×C8—4221no—
183C4⋊C4—21no—
184C4⋊C4—4224no—
185C4⋊C4—4221no—
186C4⋊C4—4221no—
187C4⋊C4—2112no—
188C4⋊C4—2112no—
189C4⋊C4C22⋊D21112yes—
190C22⋊C4—4224no—
191C22⋊C4—4224no—
192C42—4224no—
193C23—4221no—
194C23—4221no—
195C23—4221no—
196C23—4221no—
197C23—4221no—
198C23—4221no—
199C23—2112no—
200C23D4×D21111yescharacteristic
201Q8—22no—
202Q8—22no—
203Q8—41no—
204Q8—41no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—8441no—
212C2×C4—8441no—
213C2×C4—8441no—
214C2×C4—8441no—
215C2×C4—4222no—
216C2×C4—4222no—
217C2×C4—4221no—
218C2×C4—4221no—
219C2×C4—4222no—
220C2×C4—4222no—
221C2×C4—4222no—
222C2×C4—4222no—
223C2×C4—2112no—
224C2×C4C2×C4×D21111yescharacteristic
225C2×C4D4×D21111yescharacteristic
226C2×C4D4×D21111yescharacteristic
227C2×C4C2×C22⋊D21111yescharacteristic
228C2×C4#1680996111yescharacteristic
229C2×C4#1680996211yescharacteristic
230C8—41no—
231C8—8441no—
232C22—8441no—
233C22—8441no—
234C22—8441no—
235C22—8441no—
236C22—4222no—
237C22—4222no—
238C22—4222no—
239C22C22⋊C4×D21111yescharacteristic
240C22#2026994511yescharacteristic
241C22#2026995911yescharacteristic
242C22#2026999811yescharacteristic
243C22#2027000011yescharacteristic
244C22#2027000611yescharacteristic
245C22#2027000911yescharacteristic
246C4—42no—
247C4—42no—
248C4—8442no—
249C4—8442no—
250C4—4222no—
251C4(C22⋊D211)⋊C411yescharacteristic
252C4C4×C22⋊D21111yescharacteristic
253C4#2026994811yescharacteristic
254C4#2026995011yescharacteristic
255C2—8442no—
256C2#2721302211yescharacteristic
257C2#2721314111yescharacteristic
258C2#2721315311yescharacteristic
259C2#2721330511yescharacteristic
260C2#2721330711yescharacteristic
261C2#2721348511yescharacteristic
262C2#2721348611yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#103×C211C2
socleC23×C211C2×D4
prime radicalC23×D211D4
cosocleC22×C4×C211C23
hypercenterC2×#103D211
hypocenterC211#1873
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#20269901C2semidirect
5. #27212909C2semidirect
6. #27212908C2semidirect
7. C2×#103×C211C2semidirect
11. #20269953C4semidirect
131. C211#1873semidirect
136. C2×#103D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699532
C2×#202699352
C2×#202699012
C2×(C2×D211)⋊C82
#272129092
#272129082
C2×#103×C2112
#1873211