Finite Group Databasebeta

← #41013505

Subgroups — #41013505

Order 27008.

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

normal subgroups only

This group has 296 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
1#27213619C211yesmaximal, characteristic
2C2×#20269891C211yesmaximal, characteristic
3C2×#20269891C211yesmaximal, characteristic
4#27212913C211yesmaximal, characteristic, complemented
5#27212912C211yesmaximal, characteristic, complemented
6#27212912C211yesmaximal, characteristic, complemented
7#342×C211C211yesmaximal, characteristic, complemented
8C22×C22⋊D211C2211yescharacteristic
9C4×C22⋊D211—22no—
10(C22⋊D211)⋊C4—22no—
11#20269933C412yescomplemented
12C2×C22⋊Dic211C2211yescharacteristic
13C2×C22⋊Dic211C2211yescharacteristic
14C2×C22⋊Dic211C2211yescharacteristic
15C22⋊C4×D211C412yescomplemented
16#20269891—22no—
17#20269891—22no—
18C2×C22⋊C4×C211C2211yescharacteristic
19C2×C22⋊C4×C211C2211yescharacteristic
20C2×C22⋊C4×C211C2211yescharacteristic
21C23×D211C2×C411yescharacteristic
22C2×C22⋊D211—21no—
23C2×C22⋊D211—21no—
24C2×C22⋊D211—21no—
25C2×C22⋊D211—21no—
26C2×C22⋊D211D412yes—
27C2×C22⋊D211D412yes—
28C22×Dic211—21no—
29C22×Dic211—21no—
30C22×Dic211—21no—
31C22×Dic211C2×C411yescharacteristic
32C2×C4×D211—22no—
33Dic211⋊C4—22no—
34C22⋊Dic211—21no—
35C22⋊Dic211—21no—
36C22⋊Dic211—21no—
37C22⋊Dic211—21no—
38C22⋊Dic211—21no—
39C22⋊Dic211C2×C412yes—
40(C2×D211)⋊C4—22no—
41C4×Dic211—22no—
42C24×C211C2311yescharacteristic
43C22×C4×C211—21no—
44C22×C4×C211—21no—
45C22×C4×C211—21no—
46C22⋊C4×C211—21no—
47C22⋊C4×C211—21no—
48C22⋊C4×C211—21no—
49C22⋊C4×C211—21no—
50C22⋊C4×C211—21no—
51C22⋊C4×C211C2×C412yes—
52C22×D211—21no—
53C22×D211—21no—
54C22×D211—81no—
55C22×D211C22⋊C412yes—
56C22⋊D211—22no—
57C22⋊D211—22no—
58C22⋊D211—22no—
59C22⋊D211—22no—
60C2×Dic211—21no—
61C2×Dic211—21no—
62C2×Dic211—22no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—81no—
66C2×Dic211—81no—
67C2×Dic211—81no—
68C2×Dic211C22⋊C412yes—
69C4×D211—82no—
70C23×C211—81no—
71C23×C211C22×C411yescharacteristic
72C23×C211C2×D411yescharacteristic
73C23×C211C2×D411yescharacteristic
74C23×C211#4111yescharacteristic
75C23×C211#4111yescharacteristic
76C23×C211#4111yescharacteristic
77C23×C211#4111yescharacteristic
78C2×C4×C211—22no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211—81no—
82C2×C4×C211—81no—
83C2×C4×C211—81no—
84C22×C211—41no—
85C22×C211—41no—
86C22×C211—41no—
87C22×C211—41no—
88C22×C211—41no—
89C22×C211—41no—
90C22×C211—41no—
91C22×C211C2×C22⋊C411yescharacteristic
92C22×C211#11711yescharacteristic
93C22×C211#11711yescharacteristic
94C22×C211#12311yescharacteristic
95C22×C211#12311yescharacteristic
96C22×C211#12311yescharacteristic
97C22×C211#12311yescharacteristic
98C2×D211—22no—
99C2×D211—42no—
100C2×D211—81no—
101C2×D211—81no—
102C4×C211—42no—
103C4×C211—81no—
104C4×C211—81no—
105Dic211—22no—
106Dic211—42no—
107Dic211—81no—
108Dic211—81no—
109C2×C211—21no—
110C2×C211—21no—
111C2×C211—21no—
112C2×C211—21no—
113C2×C211—21no—
114C2×C211—21no—
115C2×C211—21no—
116C2×C211#38111yescharacteristic
117D211—42no—
118C211#182811yescharacteristic, complemented, Sylow
119#1828—2111nomaximal, Sylow
120#518—2111no—
121C2×#99—2111no—
122C2×#99—2111no—
123#342—2111no—
124#342—2111no—
125#342—2111no—
126#342D21111yescharacteristic, complemented
127C22×D4—2111no—
128C4×D4—4222no—
129C4×D4—4222no—
130#117—2112no—
131C2×C22⋊C4—2111no—
132C2×C22⋊C4—2111no—
133C2×C22⋊C4—2111no—
134C2×C22⋊C4—2112no—
135C2×C22⋊C4C2×D21111yescharacteristic
136C2×C22⋊C4C2×D21111yescharacteristic
137C2×C22⋊C4C2×D21111yescharacteristic
138#99—4222no—
139#99—4222no—
140C24—2111no—
141C24C22×D21111yescharacteristic
142C2×D4—4221no—
143C2×D4—4221no—
144C2×D4—4221no—
145C2×D4—4221no—
146C2×D4—2112no—
147C2×D4—2112no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—4222no—
152C22×C4—4221no—
153C22×C4—4221no—
154C22×C4—4221no—
155C22×C4—2111no—
156C4⋊C4—4222no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—4222no—
163C22⋊C4—4221no—
164C22⋊C4—4221no—
165C22⋊C4—4221no—
166C22⋊C4—4221no—
167C22⋊C4—4221no—
168C22⋊C4—2112no—
169C22⋊C4C4×D21112yes—
170C42—4222no—
171C23—81no—
172C23—16881no—
173C23—4221no—
174C23—4221no—
175C23—2112no—
176C23C2×C4×D21111yescharacteristic
177C23D4×D21111yescharacteristic
178C23D4×D21111yescharacteristic
179C23#1680996111yescharacteristic
180C23#1680996211yescharacteristic
181C23#1680996211yescharacteristic
182C23#1680996311yescharacteristic
183D4—4222no—
184D4—4222no—
185D4—4222no—
186D4—4222no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—8441no—
194C2×C4—8441no—
195C2×C4—16882no—
196C2×C4—16881no—
197C2×C4—16881no—
198C2×C4—16881no—
199C2×C4—4222no—
200C2×C4—4221no—
201C2×C4—4221no—
202C2×C4—2112no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—8442no—
211C22—16881no—
212C22—16881no—
213C22—4222no—
214C22C22⋊C4×D21111yescharacteristic
215C22#2026993411yescharacteristic
216C22#2026993511yescharacteristic
217C22#2026995911yescharacteristic
218C22#2026995911yescharacteristic
219C22#2026996211yescharacteristic
220C22#2026996211yescharacteristic
221C4—42no—
222C4—81no—
223C4—81no—
224C4—8442no—
225C4—16881no—
226C4—16881no—
227C4—4222no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—21no—
234C2—21no—
235C2—8442no—
236C2#2721300811yescharacteristic
subgroupG/H
centerC2#27213008
commutatorC23×C211C22×C4
FrattiniC24C22×D211
Fitting#342×C211C2
socleC2×C211#381
prime radicalC22×C22⋊D211C22
cosocleC24×C211C23
hypercenter#342D211
hypocenterC211#1828
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
4. #27212913C2semidirect
5. #27212912C2semidirect
6. #27212912C2semidirect
7. #342×C211C2semidirect
11. #20269933C4semidirect
15. C22⋊C4×D211C4semidirect
118. C211#1828semidirect
126. #342D211semidirect

Maximal subgroups

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

subgroupindex
#272136192
C2×#202698912
#272129132
#272129122
#342×C2112
#1828211