Finite Group Databasebeta

← #41014638

Subgroups — #41014638

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×#20269964C211yesmaximal, characteristic
2C2×#20269902C211yesmaximal, characteristic, complemented
3C2×#20269900C211yesmaximal, characteristic, complemented
4C2×#20269898C211yesmaximal, characteristic
5#27213045C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7C2×#102×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10#20269964—22no—
11#20269964—22no—
12C2×C4⋊Dic211—21no—
13C2×C4⋊Dic211C2211yescharacteristic
14C2×C22⋊Dic211—21no—
15C2×C22⋊Dic211—21no—
16C2×C4×Dic211C2211yescharacteristic
17#20269902—22no—
18#20269900—22no—
19#20269898—22no—
20C211⋊#95—21no—
21C22×D4×C211C2211yescharacteristic
22C22×C8×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24#102×C211—22no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422D412yes—
31C22×Dic211—21no—
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—
41C22⋊Dic211—24no—
42C22⋊Dic211—24no—
43C4×Dic211—21no—
44C4×Dic211—21no—
45C24×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211—21no—
49C2×D4×C211—21no—
50C2×D4×C211—21no—
51C2×D4×C211D412yes—
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—
59Dic422—22no—
60Dic422—22no—
61Dic422—41no—
62Dic422—41no—
63C2×Dic211—21no—
64C2×Dic211—21no—
65C2×Dic211—22no—
66C2×Dic211—21no—
67C2×Dic211—21no—
68C2×Dic211—21no—
69C2×Dic211—21no—
70C2×Dic211—41no—
71C2×Dic211—41no—
72C2×Dic211—41no—
73C2×Dic211—41no—
74C2×Dic211—41no—
75C211⋊C8—41no—
76C23×C211—22no—
77C23×C211—41no—
78C23×C211—41no—
79C23×C211—41no—
80C23×C211—41no—
81C23×C211—41no—
82C23×C211—41no—
83C23×C211C2×D411yescharacteristic
84D4×C211—22no—
85D4×C211—22no—
86D4×C211—41no—
87D4×C211—41no—
88C2×C4×C211—22no—
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—42no—
99C22×C211—42no—
100C22×C211—42no—
101C22×C211—81no—
102C22×C211—81no—
103C22×C211—81no—
104C22×C211—81no—
105C22×C211#12011yescharacteristic
106C22×C211#12111yescharacteristic
107C22×C211#12411yescharacteristic
108C22×C211#13511yescharacteristic
109C22×C211#13511yescharacteristic
110C22×C211AGL(1,ℤ/8)11yescharacteristic
111C22×C211#13711yescharacteristic
112C4×C211—42no—
113C4×C211#12111yescharacteristic
114C4×C211#12111yescharacteristic
115C4×C211#12311yescharacteristic
116C4×C211#12311yescharacteristic
117Dic211—42no—
118Dic211—41no—
119Dic211—41no—
120Dic211—81no—
121C2×C211—82no—
122C2×C211#39411yescharacteristic
123C2×C211#44911yescharacteristic
124C2×C211#45211yescharacteristic
125C2×C211#46311yescharacteristic
126C2×C211#46411yescharacteristic
127C2×C211#48811yescharacteristic
128C2×C211#48911yescharacteristic
129C211#204211yescharacteristic, complemented, Sylow
130#2042—2111nomaximal, Sylow
131C2×#124—2111no—
132C2×#103—2111no—
133C2×#103—2111no—
134C2×#102—2111no—
135C2×#102D21111yescharacteristic
136#389—2111no—
137#336—2111no—
138C22×Q8—2111no—
139C22×D4C2×D21111yescharacteristic
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—4222no—
151#103—4222no—
152#102—22no—
153#102—4222no—
154#95—4221no—
155C24—21no—
156C2×Q8—4221no—
157C2×Q8—4221no—
158C2×Q8—4221no—
159C2×Q8—4221no—
160C2×Q8—4221no—
161C2×Q8—2112no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4D84412yes—
168C22×C4—21no—
169C22×C4—4221no—
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—22no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23D4×D21111yescharacteristic
197Q8—8441no—
198Q8—8441no—
199Q8—4222no—
200Q8—4222no—
201D4—22no—
202D4—22no—
203D4—41no—
204D4—41no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—8441no—
209C2×C4—8441no—
210C2×C4—8441no—
211C2×C4—8441no—
212C2×C4—8441no—
213C2×C4—4221no—
214C2×C4—4221no—
215C2×C4—4222no—
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—42no—
229C22—42no—
230C22—42no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22#2026994711yescharacteristic
236C22#2026994911yescharacteristic
237C22#2026996311yescharacteristic
238C22#2026999611yescharacteristic
239C22#2026999811yescharacteristic
240C22#2027000111yescharacteristic
241C22#2027000611yescharacteristic
242C4—42no—
243C4—8442no—
244C4—8441no—
245C4—8441no—
246C4—16881no—
247C4#2026994811yescharacteristic
248C4#2026995111yescharacteristic
249C4#2026995911yescharacteristic
250C4#2026996111yescharacteristic
251C2—82no—
252C2#2721306211yescharacteristic
253C2#2721329211yescharacteristic
254C2#2721330911yescharacteristic
255C2#2721335911yescharacteristic
256C2#2721336611yescharacteristic
257C2#2721349911yescharacteristic
258C2#2721350411yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
hypercenterC2×#102D211
hypocenterC211#2042
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#20269902C2semidirect
3. C2×#20269900C2semidirect
5. #27213045C2semidirect
6. #27212900C2semidirect
129. C211#2042semidirect

Maximal subgroups

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

subgroupindex
C2×#202699642
C2×#202699022
C2×#202699002
C2×#202698982
#272130452
#272129002
C2×#102×C2112
#2042211