Finite Group Databasebeta

← #41014039

Subgroups — #41014039

Order 27008.

254 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 97 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×#20269910C211yesmaximal, characteristic
2C2×#20269900C211yesmaximal, characteristic
3C2×#20269900C211yesmaximal, characteristic
4#27213052C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic
6#27212901C211yesmaximal, characteristic
7C4×C4⋊C4×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12#20269910—22no—
13#20269900—22no—
14#20269900C414yescomplemented
15C211⋊#95—21no—
16C211⋊#95—21no—
17C2×C4⋊C4×C211C2211yescharacteristic
18C2×C4⋊C4×C211C2211yescharacteristic
19C2×C42×C211—21no—
20C2×C42×C211C2211yescharacteristic
21#95×C211—21no—
22C2×Dic422—21no—
23C2×Dic422—21no—
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422C2×C412yes—
28C22×Dic211—21no—
29C22×Dic211—21no—
30C2×C211⋊C8—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8C2×C412yes—
36C4⋊Dic211—21no—
37C4⋊Dic211—21no—
38C22×C4×C211—21no—
39C22×C4×C211—21no—
40C22×C4×C211D411yescharacteristic
41C22×C4×C211D411yescharacteristic
42C22×C4×C211C2311yescharacteristic
43C4⋊C4×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211C2×C412yes—
46C4⋊C4×C211D412yes—
47C42×C211—21no—
48C42×C211—21no—
49C42×C211—42no—
50Dic422—22no—
51Dic422—22no—
52Dic422—41no—
53Dic422—41no—
54C2×Dic211—22no—
55C2×Dic211—41no—
56C2×Dic211—41no—
57C2×Dic211—41no—
58C2×Dic211—41no—
59C2×Dic211—41no—
60C211⋊C8—22no—
61C211⋊C8—41no—
62C23×C211C2×D411yescharacteristic
63C2×C4×C211—21no—
64C2×C4×C211—21no—
65C2×C4×C211—22no—
66C2×C4×C211—22no—
67C2×C4×C211—21no—
68C2×C4×C211—21no—
69C2×C4×C211—21no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211Q1612yes—
75C2×C4×C211C22×C411yescharacteristic
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C2×C4×C211#4111yescharacteristic
81C22×C211C4×D411yescharacteristic
82C22×C211#12111yescharacteristic
83C22×C211#12311yescharacteristic
84C22×C211C2×Q1611yescharacteristic
85C22×C211#13511yescharacteristic
86C22×C211AGL(1,ℤ/8)11yescharacteristic
87C22×C211#13711yescharacteristic
88C4×C211—22no—
89C4×C211—42no—
90C4×C211—42no—
91C4×C211—41no—
92C4×C211#11711yescharacteristic
93C4×C211C4×D411yescharacteristic
94C4×C211#12411yescharacteristic
95C4×C211#12611yescharacteristic
96Dic211—42no—
97Dic211—81no—
98C2×C211#38811yescharacteristic
99C2×C211C4×Q1611yescharacteristic
100C2×C211#44011yescharacteristic
101C2×C211#46211yescharacteristic
102C2×C211#46311yescharacteristic
103C2×C211#48411yescharacteristic
104C2×C211#48511yescharacteristic
105C211#192811yescharacteristic, complemented, Sylow
106#1928—2111nomaximal, Sylow
107C2×#107—2111no—
108C2×#103—2111no—
109C2×#103—2111no—
110#391—2111no—
111C4×C4⋊C4D21111yescharacteristic
112#340—2111no—
113#336—2111no—
114C22×Q8—2111no—
115C22×C8—2111no—
116C22×C8—2111no—
117C2×C4⋊C4—2111no—
118C2×C4⋊C4C2×D21111yescharacteristic
119C2×C4⋊C4C2×D21111yescharacteristic
120C2×C42—21no—
121C2×C42C2×D21111yescharacteristic
122#107—4222no—
123#103—4222no—
124#103—2114no—
125#95—21no—
126#95—4221no—
127#95—4221no—
128C2×Q8—4221no—
129C2×Q8—4221no—
130C2×Q8—4221no—
131C2×Q8—4221no—
132C2×Q8—4221no—
133C2×Q8—2112no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—4221no—
137C22×C4—4221no—
138C22×C4C22⋊D21111yescharacteristic
139C22×C4C22⋊D21111yescharacteristic
140C22×C4C22×D21111yescharacteristic
141C2×C8—4221no—
142C2×C8—4221no—
143C2×C8—4221no—
144C2×C8—4221no—
145C2×C8—4221no—
146C2×C8—2112no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—4221no—
150C4⋊C4—4221no—
151C4⋊C4C4×D21112yes—
152C4⋊C4D84412yes—
153C42—21no—
154C42—21no—
155C42—42no—
156C23C2×C22⋊D21111yescharacteristic
157Q8—8441no—
158Q8—8441no—
159Q8—4222no—
160Q8—4222no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—8441no—
173C2×C4—8441no—
174C2×C4—8441no—
175C2×C4—8441no—
176C2×C4—8441no—
177C2×C4—4222no—
178C2×C4#1680995212yes—
179C2×C4C2×C4×D21111yescharacteristic
180C2×C4C2×D84411yescharacteristic
181C2×C4#1680996211yescharacteristic
182C2×C4#1680996211yescharacteristic
183C2×C4#1680996211yescharacteristic
184C2×C4#1680996211yescharacteristic
185C8—8441no—
186C8—4222no—
187C22C4×C22⋊D21111yescharacteristic
188C22#2026995111yescharacteristic
189C22#2026996011yescharacteristic
190C22C2×#1680995211yescharacteristic
191C22#2026999711yescharacteristic
192C22#2027000311yescharacteristic
193C22#2027000711yescharacteristic
194C4—22no—
195C4—42no—
196C4—41no—
197C4—42no—
198C4—8442no—
199C4—16881no—
200C4#2026993411yescharacteristic
201C4C4×D84411yescharacteristic
202C4#2026996511yescharacteristic
203C4#2026997111yescharacteristic
204C2#2721303711yescharacteristic
205C2C4×#1680995211yescharacteristic
206C2#2721324611yescharacteristic
207C2#2721335611yescharacteristic
208C2#2721336111yescharacteristic
209C2#2721348111yescharacteristic
210C2#2721348711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC4×C4⋊C4×C211C2
socle, prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenterC4×C4⋊C4D211
hypocenterC211#1928
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
14. #20269900C4semidirect
105. C211#1928semidirect

Maximal subgroups

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

subgroupindex
C2×#202699102
C2×#202699002
#272130522
#272129082
#272129012
C4×C4⋊C4×C2112
#1928211