Finite Group Databasebeta

← #41013913

Subgroups — #41013913

Order 27008.

312 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20270007C211yesmaximal, characteristic
2#27213147C211yesmaximal, characteristic
3C2×#20269902C211yesmaximal, characteristic, complemented
4C2×#20269895C211yesmaximal, characteristic, complemented
5#27212914C211yesmaximal, characteristic, complemented
6#27212911C211yesmaximal, characteristic, complemented
7C2×#131×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9#20270007—22no—
10#20270007C414yescomplemented
11C2×#16809952—21no—
12C2×#16809950—21no—
13C2×#16809946C2211yescharacteristic
14C2×#16809946C2211yescharacteristic
15C2×C4⋊Dic211C2211yescharacteristic
16#20269902—21no—
17#20269902—21no—
18#20269901—21no—
19#20269898—21no—
20#20269895—22no—
21C2×#41×C211C2211yescharacteristic
22#131×C211—22no—
23#131×C211—22no—
24C2×M16×C211—21no—
25C2×M16×C211C2211yescharacteristic
26C22×C8×C211—21no—
27C22×C8×C211C2211yescharacteristic
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422C2×C412yes—
34C22×Dic211—21no—
35C22×Dic211—21no—
36#16809952—24no—
37#16809950—24no—
38#16809946—21no—
39#16809946—21no—
40#16809946—21no—
41#16809946C2×C412yes—
42C2×C211⋊C8—21no—
43C2×C211⋊C8—21no—
44C4⋊Dic211—21no—
45C4⋊Dic211—21no—
46#41×C211—21no—
47#41×C211—21no—
48#41×C211C2×C412yes—
49#41×C211D412yes—
50C2×Q8×C211D411yescharacteristic
51C2×D4×C211—21no—
52C2×D4×C211D411yescharacteristic
53C22×C4×C211—21no—
54C22×C4×C211C2311yescharacteristic
55M16×C211—24no—
56M16×C211—21no—
57M16×C211D412yes—
58C2×C8×C211—24no—
59C2×C8×C211—21no—
60C2×C8×C211—21no—
61C2×C8×C211—21no—
62C2×C8×C211D411yescharacteristic
63C2×C8×C211D411yescharacteristic
64Dic422—22no—
65Dic422—22no—
66Dic422—41no—
67Dic422—41no—
68C2×Dic211—22no—
69C2×Dic211—41no—
70C2×Dic211—41no—
71C2×Dic211—41no—
72C2×Dic211—81no—
73C211⋊C8—22no—
74C211⋊C8—41no—
75C23×C211—21no—
76C23×C211#4111yescharacteristic
77Q8×C211—21no—
78Q8×C211C22⋊C412yes—
79D4×C211—22no—
80D4×C211—22no—
81D4×C211—21no—
82D4×C211C22⋊C412yes—
83C2×C4×C211—21no—
84C2×C4×C211—22no—
85C2×C4×C211—22no—
86C2×C4×C211C22×C411yescharacteristic
87C2×C4×C211C2×D411yescharacteristic
88C2×C4×C211C2×D411yescharacteristic
89C2×C4×C211C2×D411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211#4111yescharacteristic
92C8×C211—21no—
93C8×C211—22no—
94C8×C211—21no—
95C22×C211—21no—
96C22×C211—21no—
97C22×C211—22no—
98C22×C211—41no—
99C22×C211—41no—
100C22×C211C4×D411yescharacteristic
101C22×C211#12111yescharacteristic
102C22×C211#12311yescharacteristic
103C4×C211—22no—
104C4×C211C2×C22⋊C411yescharacteristic
105C4×C211C4×D411yescharacteristic
106C4×C211#12011yescharacteristic
107C4×C211#12111yescharacteristic
108Dic211—42no—
109Dic211—81no—
110C2×C211—21no—
111C2×C211—21no—
112C2×C211—42no—
113C2×C211#38611yescharacteristic
114C2×C211#47111yescharacteristic
115C2×C211#47311yescharacteristic
116C211#190811yescharacteristic, complemented, Sylow
117#1908—2111nomaximal, Sylow
118C2×#137—2111no—
119C2×#131D21111yescharacteristic
120#417—2111no—
121C2×#103—2111no—
122C2×#101—2111no—
123#343—2111no—
124#341—2111no—
125C2×#41C2×D21111yescharacteristic
126C22×Q8—2111no—
127#137—4222no—
128#137—2114no—
129C2×Q16—4221no—
130C2×SD16—4221no—
131#131—22no—
132#131—22no—
133C2×M16—21no—
134C2×M16—2111no—
135C2×M16—2111no—
136C2×M16C2×D21111yescharacteristic
137C22×C8—21no—
138C22×C8C2×D21111yescharacteristic
139C2×C4⋊C4—2111no—
140#103—4221no—
141#103—4221no—
142#103—4221no—
143#102—4221no—
144#101—4222no—
145#41—21no—
146#41—21no—
147#41C4×D21112yes—
148#41C22⋊D21112yes—
149C2×Q8—4221no—
150C2×Q8—4221no—
151C2×Q8—4221no—
152C2×Q8—4221no—
153C2×Q8—4221no—
154C2×Q8—2112no—
155C2×Q8D84411yescharacteristic
156C2×D4—21no—
157C2×D4D84411yescharacteristic
158C22×C4—21no—
159C22×C4—4221no—
160C22×C4—4221no—
161C22×C4C22×D21111yescharacteristic
162Q16—4224no—
163SD16—4224no—
164M16—24no—
165M16—21no—
166M16—4221no—
167M16—4221no—
168M16—4221no—
169M16—2112no—
170M16C22⋊D21112yes—
171C2×C8—24no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—4221no—
176C2×C8—4221no—
177C2×C8C22⋊D21111yescharacteristic
178C2×C8C22⋊D21111yescharacteristic
179C4⋊C4—4221no—
180C4⋊C4—4221no—
181C23—21no—
182C23#1680996211yescharacteristic
183Q8—21no—
184Q8—8441no—
185Q8—8441no—
186Q8—4222no—
187Q8—4222no—
188Q8(C2×D211)⋊C412yes—
189D4—22no—
190D4—22no—
191D4—21no—
192D4(C2×D211)⋊C412yes—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—8441no—
197C2×C4—8441no—
198C2×C4—8441no—
199C2×C4—16881no—
200C2×C4—4222no—
201C2×C4C2×C4×D21111yescharacteristic
202C2×C4C2×C22⋊D21111yescharacteristic
203C2×C4C2×C22⋊D21111yescharacteristic
204C2×C4C2×C22⋊D21111yescharacteristic
205C2×C4C2×D84411yescharacteristic
206C2×C4#1680996211yescharacteristic
207C8—21no—
208C8—22no—
209C8—21no—
210C8—8441no—
211C8—4222no—
212C22—21no—
213C22—21no—
214C22—22no—
215C22—41no—
216C22—41no—
217C22C4×C22⋊D21111yescharacteristic
218C22#2026995111yescharacteristic
219C22#2026996011yescharacteristic
220C4—22no—
221C4—8442no—
222C4—16881no—
223C4C2×(C2×D211)⋊C411yescharacteristic
224C4C4×C22⋊D21111yescharacteristic
225C4#2026994611yescharacteristic
226C4#2026995111yescharacteristic
227C2—21no—
228C2—21no—
229C2—42no—
230C2#2721302511yescharacteristic
231C2#2721340911yescharacteristic
232C2#2721342111yescharacteristic
subgroupG/H
centerC22#20269951
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#131×C211C2
socleC22×C211#121
prime radicalC2×#41×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#131D211
hypocenterC211#1908
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
3. C2×#20269902C2semidirect
4. C2×#20269895C2semidirect
5. #27212914C2semidirect
6. #27212911C2semidirect
10. #20270007C4semidirect
116. C211#1908semidirect

Maximal subgroups

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

subgroupindex
C2×#202700072
#272131472
C2×#202699022
C2×#202698952
#272129142
#272129112
C2×#131×C2112
#1908211