Finite Group Databasebeta

← #41014636

Subgroups — #41014636

Order 27008.

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

normal subgroups only

This group has 318 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×#20269966C211yesmaximal, characteristic
2C2×#20269901C211yesmaximal, characteristic, complemented
3C2×#20269899C211yesmaximal, characteristic
4C2×#20269897C211yesmaximal, characteristic
5#27213045C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7C2×#103×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10#20269966—22no—
11#20269966—22no—
12C2×C4⋊Dic211—21no—
13C2×C4⋊Dic211C2211yescharacteristic
14C2×(C2×D211)⋊C4—21no—
15C2×(C2×D211)⋊C4—21no—
16C2×C4×Dic211C2211yescharacteristic
17#20269901—22no—
18#20269899—22no—
19#20269897—22no—
20C211⋊#95—21no—
21C22×Q8×C211C2211yescharacteristic
22C22×C8×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24#103×C211—22no—
25C23×D211—21no—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844D412yes—
32C22×Dic211—21no—
33C22×Dic211D411yescharacteristic
34C22×Dic211D411yescharacteristic
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C4⋊Dic211—21no—
39C4⋊Dic211—21no—
40C4⋊Dic211—42no—
41(C2×D211)⋊C4—24no—
42(C2×D211)⋊C4—24no—
43C4×Dic211—21no—
44C4×Dic211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—21no—
50C2×Q8×C211D412yes—
51C22×C4×C211—21no—
52C22×C4×C211—21no—
53C22×C4×C211C2311yescharacteristic
54C2×C8×C211—21no—
55C2×C8×C211—21no—
56C2×C8×C211—21no—
57C4⋊C4×C211—21no—
58C4⋊C4×C211D412yes—
59C22×D211—22no—
60C22×D211—41no—
61C22×D211—41no—
62C22×D211—41no—
63C22×D211—41no—
64C22×D211—41no—
65C22×D211—41no—
66D844—22no—
67D844—22no—
68D844—41no—
69D844—41no—
70C2×Dic211—21no—
71C2×Dic211—21no—
72C2×Dic211—21no—
73C2×Dic211—21no—
74C2×Dic211—21no—
75C2×Dic211—21no—
76C2×Dic211—41no—
77C2×Dic211—41no—
78C2×Dic211—41no—
79C211⋊C8—41no—
80C23×C211C2×D411yescharacteristic
81Q8×C211—22no—
82Q8×C211—22no—
83Q8×C211—41no—
84Q8×C211—41no—
85C2×C4×C211—22no—
86C2×C4×C211—22no—
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#12411yescharacteristic
101C22×C211#13511yescharacteristic
102C22×C211#13511yescharacteristic
103C22×C211AGL(1,ℤ/8)11yescharacteristic
104C22×C211#13711yescharacteristic
105C2×D211—42no—
106C2×D211—42no—
107C2×D211—42no—
108C2×D211—81no—
109C2×D211—81no—
110C2×D211—81no—
111C2×D211—81no—
112C4×C211—42no—
113C4×C211—42no—
114C4×C211#12111yescharacteristic
115C4×C211#12111yescharacteristic
116C4×C211#12311yescharacteristic
117C4×C211#12311yescharacteristic
118Dic211—41no—
119Dic211—41no—
120Dic211—81no—
121C2×C211#39411yescharacteristic
122C2×C211#44911yescharacteristic
123C2×C211#45211yescharacteristic
124C2×C211#46311yescharacteristic
125C2×C211#46411yescharacteristic
126C2×C211#48811yescharacteristic
127C2×C211#48911yescharacteristic
128D211—82no—
129C211#204211yescharacteristic, complemented, Sylow
130#2042—2111nomaximal, Sylow
131C2×#124—2111no—
132C2×#103—2111no—
133C2×#103D21111yescharacteristic, complemented
134C2×#102—2111no—
135C2×#102—2111no—
136#389—2111no—
137#336—2111no—
138C22×Q8C2×D21111yescharacteristic
139C22×D4—2111no—
140C22×C8—2111no—
141C22×C8C2×D21111yescharacteristic
142#124—4222no—
143#124—4222no—
144C2×C4⋊C4—4221no—
145C2×C4⋊C4—2111no—
146C2×C4⋊C4C2×D21111yescharacteristic
147C2×C22⋊C4—4221no—
148C2×C22⋊C4—4221no—
149C2×C42—2111no—
150#103—22no—
151#103—4222no—
152#102—4222no—
153#102—4222no—
154#95—4221no—
155C24—4221no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8D84412yes—
162C2×D4—4221no—
163C2×D4—4221no—
164C2×D4—4221no—
165C2×D4—4221no—
166C2×D4—4221no—
167C2×D4—2112no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—4221no—
171C22×C4—2111no—
172C22×C4—2111no—
173C22×C4C22×D21111yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—4221no—
178C2×C8—4221no—
179C2×C8—4221no—
180C4⋊C4—21no—
181C4⋊C4—8442no—
182C4⋊C4—4221no—
183C4⋊C4—4221no—
184C4⋊C4C22⋊D21112yes—
185C22⋊C4—4224no—
186C22⋊C4—4224no—
187C42—4221no—
188C42—4221no—
189C23—8441no—
190C23—8441no—
191C23—8441no—
192C23—8441no—
193C23—8441no—
194C23—8441no—
195C23—4222no—
196C23D4×D21111yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201D4—8441no—
202D4—8441no—
203D4—4222no—
204D4—4222no—
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—4221no—
215C2×C4—4221no—
216C2×C4—4221no—
217C2×C4—4221no—
218C2×C4—4221no—
219C2×C4—4221no—
220C2×C4D4×D21111yescharacteristic
221C2×C4C2×C22⋊D21111yescharacteristic
222C2×C4C2×D84411yescharacteristic
223C2×C4#1680996111yescharacteristic
224C2×C4#1680996111yescharacteristic
225C2×C4#1680996211yescharacteristic
226C8—41no—
227C8—8441no—
228C22—8442no—
229C22—8442no—
230C22—8442no—
231C22—16881no—
232C22—16881no—
233C22—16881no—
234C22—16881no—
235C22#2026994711yescharacteristic
236C22#2026994911yescharacteristic
237C22#2026996311yescharacteristic
238C22#2026999811yescharacteristic
239C22#2027000011yescharacteristic
240C22#2027000411yescharacteristic
241C22#2027000911yescharacteristic
242C4—42no—
243C4—42no—
244C4—8441no—
245C4—8441no—
246C4—16881no—
247C4#2026994811yescharacteristic
248C4#2026995111yescharacteristic
249C4#2026995911yescharacteristic
250C4#2026996111yescharacteristic
251C2—16882no—
252C2#2721306211yescharacteristic
253C2#2721329411yescharacteristic
254C2#2721331111yescharacteristic
255C2#2721336311yescharacteristic
256C2#2721337011yescharacteristic
257C2#2721350211yescharacteristic
258C2#2721350711yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#103×C211C2
socleC23×C211C2×D4
prime radicalC22×D844C22
hypercenterC2×#103D211
hypocenterC211#2042
LCSG ▷ C22×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
2. C2×#20269901C2semidirect
5. #27213045C2semidirect
6. #27212900C2semidirect
7. C2×#103×C211C2semidirect
129. C211#2042semidirect
133. C2×#103D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699662
C2×#202699012
C2×#202698992
C2×#202698972
#272130452
#272129002
C2×#103×C2112
#2042211