Finite Group Databasebeta

← #41013430

Subgroups — #41013430

Order 27008.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#16809945C211yesmaximal, characteristic
2C2×#20269910C211yesmaximal, characteristic, complemented
3C2×#20269908C211yesmaximal, characteristic, complemented
4#27213003C211yesmaximal, characteristic
5#27213003C211yesmaximal, characteristic
6#27212909C211yesmaximal, characteristic, complemented
7#340×C211C211yesmaximal, characteristic, complemented
8C22×C4×D211C2211yescharacteristic
9C2×#16809945—22no—
10C2×#16809945—22no—
11C2×#16809945C414yescomplemented
12C22×C211⋊C8C2211yescharacteristic
13C2×C4⋊Dic211C2211yescharacteristic
14C2×C4⋊Dic211C2211yescharacteristic
15C2×(C2×D211)⋊C4—21no—
16C2×(C2×D211)⋊C4—21no—
17#20269910—22no—
18#20269908—22no—
19C211⋊#95—21no—
20C211⋊#95—21no—
21C22×C8×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24C23×D211D411yescharacteristic
25C22×Dic211—21no—
26C22×Dic211—21no—
27C22×Dic211D411yescharacteristic
28C2×C4×D211—21no—
29C2×C4×D211—21no—
30C2×C4×D211—21no—
31C2×C4×D211—21no—
32C2×C4×D211C2×C412yes—
33C2×C4×D211Q812yes—
34#16809945—28no—
35C2×C211⋊C8C2×C412yes—
36C2×C211⋊C8D412yes—
37C2×C211⋊C8D412yes—
38C4⋊Dic211—21no—
39C4⋊Dic211—21no—
40C4⋊Dic211—21no—
41C4⋊Dic211—21no—
42(C2×D211)⋊C4—42no—
43(C2×D211)⋊C4—42no—
44C22×C4×C211—21no—
45C22×C4×C211—21no—
46C22×C4×C211C2311yescharacteristic
47C2×C8×C211—21no—
48C2×C8×C211—21no—
49C2×C8×C211C2×C412yes—
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211—21no—
54C22×D211—21no—
55C22×D211—21no—
56C22×D211—21no—
57C22×D211—21no—
58C22×D211—21no—
59C22×D211—21no—
60C22×D211C4⋊C412yes—
61C2×Dic211—21no—
62C2×Dic211—21no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C2×Dic211—41no—
67C2×Dic211—41no—
68C2×Dic211—41no—
69C2×Dic211C4⋊C412yes—
70C4×D211—22no—
71C4×D211—22no—
72C4×D211—22no—
73C4×D211—22no—
74C211⋊C8C22⋊C414yescomplemented
75C23×C211C2×D411yescharacteristic
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211C22×C411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211C2×Q811yescharacteristic
86C2×C4×C211#4111yescharacteristic
87C2×C4×C211#4111yescharacteristic
88C8×C211—22no—
89C22×C211C2×C4⋊C411yescharacteristic
90C22×C211#12111yescharacteristic
91C22×C211#12111yescharacteristic
92C22×C211AGL(1,ℤ/8)11yescharacteristic
93C22×C211AGL(1,ℤ/8)11yescharacteristic
94C22×C211#13711yescharacteristic
95C22×C211#13711yescharacteristic
96C2×D211—22no—
97C2×D211—22no—
98C2×D211—22no—
99C2×D211—41no—
100C2×D211—41no—
101C2×D211—41no—
102C2×D211—41no—
103C4×C211—81no—
104C4×C211—81no—
105C4×C211C2×C22⋊C411yescharacteristic
106C4×C211#11711yescharacteristic
107C4×C211C22⋊Q811yescharacteristic
108C4×C211C22⋊Q811yescharacteristic
109Dic211—22no—
110Dic211—81no—
111Dic211—81no—
112C2×C211#38011yescharacteristic
113C2×C211#42811yescharacteristic
114C2×C211#42811yescharacteristic
115C2×C211#46811yescharacteristic
116C2×C211#46811yescharacteristic
117C2×C211#46911yescharacteristic
118C2×C211#47011yescharacteristic
119D211—42no—
120C211#180911yescharacteristic, complemented, Sylow
121#1809—2111nomaximal, Sylow
122C22×M16—2111no—
123C2×#107—2111no—
124C2×#106—2111no—
125#380—2111no—
126#380—2111no—
127#340—2111no—
128#340D21111yescharacteristic, complemented
129C23×C4—2111no—
130C2×M16—4222no—
131C2×M16—4222no—
132C2×M16—2114no—
133C22×C8—2111no—
134C22×C8C2×D21111yescharacteristic
135C2×C4⋊C4—2111no—
136C2×C4⋊C4—2111no—
137C2×C4⋊C4C2×D21111yescharacteristic
138C2×C4⋊C4C2×D21111yescharacteristic
139C2×C22⋊C4—4221no—
140C2×C22⋊C4—4221no—
141#107—4222no—
142#106—4222no—
143#95—4221no—
144#95—4221no—
145C24—2111no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—4221no—
149C22×C4—4221no—
150C22×C4—4221no—
151C22×C4—4221no—
152C22×C4—4221no—
153C22×C4—4221no—
154C22×C4—2111no—
155C22×C4—2112no—
156C22×C4—2112no—
157C22×C4C22×D21111yescharacteristic
158M16—4228no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—2112no—
162C2×C8—2112no—
163C2×C8—2112no—
164C2×C8C4×D21112yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—4221no—
170C4⋊C4—4221no—
171C4⋊C4—4221no—
172C4⋊C4—4221no—
173C22⋊C4—8442no—
174C22⋊C4—8442no—
175C23—4221no—
176C23—4221no—
177C23—4221no—
178C23—4221no—
179C23—4221no—
180C23—4221no—
181C23—2112no—
182C23D4×D21111yescharacteristic
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—8441no—
190C2×C4—8441no—
191C2×C4—8441no—
192C2×C4—8441no—
193C2×C4—8441no—
194C2×C4—8441no—
195C2×C4—4221no—
196C2×C4—4221no—
197C2×C4—4222no—
198C2×C4—4222no—
199C2×C4—4222no—
200C2×C4—4222no—
201C2×C4—2112no—
202C2×C4C2×C4×D21111yescharacteristic
203C2×C4D4×D21111yescharacteristic
204C2×C4D4×D21111yescharacteristic
205C2×C4Q8×D21111yescharacteristic
206C2×C4#1680996211yescharacteristic
207C2×C4#1680996211yescharacteristic
208C8—22no—
209C8—2114no—
210C22—8441no—
211C22—8441no—
212C22—8441no—
213C22—8441no—
214C22—4222no—
215C22—4222no—
216C22—4222no—
217C22C4⋊C4×D21111yescharacteristic
218C22#2026995011yescharacteristic
219C22#2026995011yescharacteristic
220C22#2027000111yescharacteristic
221C22#2027000411yescharacteristic
222C22#2027000611yescharacteristic
223C22#2027000911yescharacteristic
224C4—81no—
225C4—81no—
226C4—16881no—
227C4—16881no—
228C4—4222no—
229C4C22⋊C4×D21111yescharacteristic
230C4#2026993411yescharacteristic
231C4#2026995511yescharacteristic
232C4#2026995511yescharacteristic
233C2—8442no—
234C2#2721300511yescharacteristic
235C2#2721318511yescharacteristic
236C2#2721318711yescharacteristic
237C2#2721339111yescharacteristic
238C2#2721339511yescharacteristic
239C2#2721339811yescharacteristic
240C2#2721340311yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socleC23×C211C2×D4
prime radicalC23×D211D4
cosocleC22×C4×C211C23
hypercenter#340D211
hypocenterC211#1809
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#20269910C2semidirect
3. C2×#20269908C2semidirect
6. #27212909C2semidirect
7. #340×C211C2semidirect
11. C2×#16809945C4semidirect
74. C211⋊C8C22⋊C4semidirect
120. C211#1809semidirect
128. #340D211semidirect

Maximal subgroups

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

subgroupindex
C22×#168099452
C2×#202699102
C2×#202699082
#272130032
#272129092
#340×C2112
#1809211