Finite Group Databasebeta

← #41014604

Subgroups — #41014604

Order 27008.

316 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 85 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×#20269957C211yesmaximal, characteristic
2C2×#20269910C211yesmaximal, characteristic, complemented
3C2×#20269902C211yesmaximal, characteristic, complemented
4C2×#20269900C211yesmaximal, characteristic, complemented
5C2×(C2×D211)⋊C8C211yesmaximal, characteristic
6#27213003C211yesmaximal, characteristic
7#340×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C22×C4×D211C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11#20269957—22no—
12#20269957—22no—
13C2×C4⋊Dic211—21no—
14C2×C4⋊Dic211C2211yescharacteristic
15C2×(C2×D211)⋊C4—21no—
16C2×(C2×D211)⋊C4—21no—
17#20269910—22no—
18#20269902—22no—
19#20269900—22no—
20(C2×D211)⋊C8—22no—
21C211⋊#95—21no—
22C22×C8×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25C23×D211D411yescharacteristic
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422D412yes—
32C22×Dic211—21no—
33C22×Dic211—21no—
34C22×Dic211D411yescharacteristic
35C2×C4×D211—21no—
36C2×C4×D211—21no—
37C2×C4×D211—21no—
38C2×C4×D211—21no—
39C2×C4×D211—21no—
40C2×C4×D211—21no—
41C2×C211⋊C8—21no—
42C2×C211⋊C8—21no—
43C2×C211⋊C8D412yes—
44C4⋊Dic211—24no—
45C4⋊Dic211—21no—
46C4⋊Dic211—21no—
47(C2×D211)⋊C4—24no—
48(C2×D211)⋊C4—42no—
49C22×C4×C211—21no—
50C22×C4×C211—21no—
51C22×C4×C211C2311yescharacteristic
52C2×C8×C211—21no—
53C2×C8×C211—21no—
54C2×C8×C211—21no—
55C4⋊C4×C211—21no—
56C4⋊C4×C211—21no—
57C4⋊C4×C211—21no—
58C4⋊C4×C211D412yes—
59C22×D211—21no—
60C22×D211—21no—
61C22×D211—21no—
62C22×D211—21no—
63C22×D211—21no—
64C22×D211—21no—
65C22×D211Q1612yes—
66Dic422—22no—
67Dic422—22no—
68Dic422—41no—
69Dic422—41no—
70C2×Dic211—22no—
71C2×Dic211—21no—
72C2×Dic211—21no—
73C2×Dic211—21no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C2×Dic211—41no—
77C2×Dic211—41no—
78C2×Dic211—41no—
79C4×D211—41no—
80C4×D211—41no—
81C4×D211—41no—
82C4×D211—41no—
83C211⋊C8—22no—
84C23×C211C2×D411yescharacteristic
85C2×C4×C211—22no—
86C2×C4×C211—41no—
87C2×C4×C211—41no—
88C2×C4×C211—41no—
89C2×C4×C211—41no—
90C2×C4×C211—41no—
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211#4111yescharacteristic
95C2×C4×C211#4111yescharacteristic
96C2×C4×C211#4111yescharacteristic
97C8×C211—41no—
98C22×C211#12011yescharacteristic
99C22×C211#12111yescharacteristic
100C22×C211#12311yescharacteristic
101C22×C211C2×Q1611yescharacteristic
102C22×C211#13511yescharacteristic
103C22×C211AGL(1,ℤ/8)11yescharacteristic
104C22×C211#13711yescharacteristic
105C2×D211—22no—
106C2×D211—22no—
107C2×D211—22no—
108C2×D211—41no—
109C2×D211—41no—
110C2×D211—41no—
111C2×D211—41no—
112C4×C211—42no—
113C4×C211—81no—
114C4×C211#12111yescharacteristic
115C4×C211#12111yescharacteristic
116C4×C211#12311yescharacteristic
117C4×C211#12411yescharacteristic
118Dic211—41no—
119Dic211—42no—
120Dic211—81no—
121C2×C211#39411yescharacteristic
122C2×C211#44911yescharacteristic
123C2×C211#45111yescharacteristic
124C2×C211#46711yescharacteristic
125C2×C211#46811yescharacteristic
126C2×C211#48411yescharacteristic
127C2×C211#48511yescharacteristic
128D211—42no—
129C211#203611yescharacteristic, complemented, Sylow
130#2036—2111nomaximal, Sylow
131C2×C22⋊Q8—2111no—
132C2×#107—2111no—
133C2×#103—2111no—
134C2×#103—2111no—
135C2×C22⋊C8—2111no—
136#380—2111no—
137#340D21111yescharacteristic, complemented
138C22×Q8—2111no—
139C23×C4—2111no—
140C22×C8—2111no—
141C22×C8C2×D21111yescharacteristic
142C22⋊Q8—4222no—
143C22⋊Q8—4222no—
144C2×C4⋊C4—4221no—
145C2×C4⋊C4—2111no—
146C2×C4⋊C4C2×D21111yescharacteristic
147C2×C4⋊C4C2×D21111yescharacteristic
148C2×C22⋊C4—4221no—
149C2×C22⋊C4—4221no—
150#107—4222no—
151#103—4222no—
152#103—4222no—
153C22⋊C8—4222no—
154#95—4221no—
155C24—2111no—
156C2×Q8—4221no—
157C2×Q8—4221no—
158C2×Q8—4221no—
159C2×Q8—4221no—
160C2×Q8—4221no—
161C2×Q8—2112no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—4221no—
165C22×C4—4221no—
166C22×C4—4221no—
167C22×C4—4221no—
168C22×C4—4221no—
169C22×C4—4221no—
170C22×C4—4221no—
171C22×C4—4221no—
172C22×C4—2111no—
173C22×C4C22×D21111yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—4221no—
178C2×C8—4221no—
179C2×C8—2112no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—4224no—
184C4⋊C4—4221no—
185C4⋊C4—4221no—
186C4⋊C4D84412yes—
187C22⋊C4—8442no—
188C22⋊C4—4224no—
189C23—4221no—
190C23—4221no—
191C23—4221no—
192C23—4221no—
193C23—4221no—
194C23—4221no—
195C23—2112no—
196C23D4×D21111yescharacteristic
197Q8—8441no—
198Q8—8441no—
199Q8—4222no—
200Q8—4222no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—8441no—
208C2×C4—8441no—
209C2×C4—8441no—
210C2×C4—8441no—
211C2×C4—8441no—
212C2×C4—8441no—
213C2×C4—8441no—
214C2×C4—8441no—
215C2×C4—8441no—
216C2×C4—4222no—
217C2×C4—4221no—
218C2×C4—4221no—
219C2×C4—4221no—
220C2×C4D4×D21111yescharacteristic
221C2×C4D4×D21111yescharacteristic
222C2×C4C2×D84411yescharacteristic
223C2×C4#1680996211yescharacteristic
224C2×C4#1680996211yescharacteristic
225C2×C4#1680996311yescharacteristic
226C8—41no—
227C8—4222no—
228C22—8441no—
229C22—8441no—
230C22—8441no—
231C22—8441no—
232C22—4222no—
233C22—4222no—
234C22—4222no—
235C22#2026994711yescharacteristic
236C22#2026995011yescharacteristic
237C22#2026996211yescharacteristic
238C22Q16×D21111yescharacteristic
239C22#2026999811yescharacteristic
240C22#2027000111yescharacteristic
241C22#2027000611yescharacteristic
242C4—42no—
243C4—81no—
244C4—8441no—
245C4—8442no—
246C4—16881no—
247C4#2026995011yescharacteristic
248C4#2026995211yescharacteristic
249C4#2026996211yescharacteristic
250C4#2026996511yescharacteristic
251C2—8442no—
252C2#2721306411yescharacteristic
253C2#2721329211yescharacteristic
254C2#2721330311yescharacteristic
255C2#2721338611yescharacteristic
256C2#2721339111yescharacteristic
257C2#2721348311yescharacteristic
258C2#2721349011yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socleC23×C211C2×D4
prime radicalC23×D211D4
hypercenter#340D211
hypocenterC211#2036
LCSG ▷ C22×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×#20269902C2semidirect
4. C2×#20269900C2semidirect
7. #340×C211C2semidirect
129. C211#2036semidirect
137. #340D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699572
C2×#202699102
C2×#202699022
C2×#202699002
C2×(C2×D211)⋊C82
#272130032
#340×C2112
#2036211