Finite Group Databasebeta

← #41014369

Subgroups — #41014369

Order 27008.

276 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 99 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×#20269976C211yesmaximal, characteristic
2C2×#20269887C211yesmaximal, characteristic
3#27213011C211yesmaximal, characteristic
4#27212909C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic
6#27212908C211yesmaximal, characteristic
7#340×C211C211yesmaximal, characteristic
8C22×Dic422—21no—
9C22×C211⋊C8C2211yescharacteristic
10#20269976—22no—
11#20269976C414yescomplemented
12C2×C4⋊Dic211C2211yescharacteristic
13C2×C4⋊Dic211C2211yescharacteristic
14C2×Dic211⋊C4—21no—
15C2×C4×Dic211C2211yescharacteristic
16#20269887—22no—
17C211⋊#95—21no—
18C211⋊#95—21no—
19C22×C8×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×Dic422—24no—
23C2×Dic422—42no—
24C2×Dic422—42no—
25C22×Dic211—21no—
26C22×Dic211—21no—
27C22×Dic211D411yescharacteristic
28C22×Dic211D411yescharacteristic
29C2×C211⋊C8—21no—
30C2×C211⋊C8—21no—
31C2×C211⋊C8Q812yescomplemented
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211C2×C412yes—
36Dic211⋊C4—24no—
37C4×Dic211C2×C412yes—
38C4×Dic211D412yes—
39C22×C4×C211—21no—
40C22×C4×C211—21no—
41C22×C4×C211C2311yescharacteristic
42C2×C8×C211—21no—
43C2×C8×C211—21no—
44C2×C8×C211Q812yescomplemented
45C4⋊C4×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211C2×C412yes—
49Dic422—48no—
50C2×Dic211—21no—
51C2×Dic211—21no—
52C2×Dic211—21no—
53C2×Dic211—21no—
54C2×Dic211—22no—
55C2×Dic211—41no—
56C2×Dic211—41no—
57C2×Dic211—41no—
58C2×Dic211—41no—
59C2×Dic211—41no—
60C2×Dic211C22⋊C412yes—
61C2×Dic211C22⋊C412yes—
62C211⋊C8—22no—
63C23×C211C2×D411yescharacteristic
64C2×C4×C211—22no—
65C2×C4×C211—41no—
66C2×C4×C211—41no—
67C2×C4×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211—41no—
70C2×C4×C211C22×C411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211C2×Q811yescharacteristic
73C2×C4×C211C2×Q811yescharacteristic
74C2×C4×C211#4111yescharacteristic
75C2×C4×C211#4111yescharacteristic
76C8×C211—22no—
77C22×C211C2×C22⋊C411yescharacteristic
78C22×C211#12411yescharacteristic
79C22×C211C4⋊Q811yescharacteristic
80C22×C211AGL(1,ℤ/8)11yescharacteristic
81C22×C211AGL(1,ℤ/8)11yescharacteristic
82C22×C211#13711yescharacteristic
83C22×C211#13711yescharacteristic
84C4×C211—42no—
85C4×C211—81no—
86C4×C211C4×Q811yescharacteristic
87C4×C211C4×Q811yescharacteristic
88C4×C211C22⋊Q811yescharacteristic
89C4×C211C22⋊Q811yescharacteristic
90Dic211—22no—
91Dic211—22no—
92Dic211—42no—
93Dic211—81no—
94C2×C211#39111yescharacteristic
95C2×C211#41811yescharacteristic
96C2×C211#41911yescharacteristic
97C2×C211#48911yescharacteristic
98C2×C211#48911yescharacteristic
99C2×C211#50111yescharacteristic
100C2×C211#50111yescharacteristic
101C211#199311yescharacteristic, complemented, Sylow
102#1993—2111nomaximal, Sylow
103C2×C4⋊Q8—2111no—
104C2×#97—2111no—
105#382—2111no—
106#340—2111no—
107#340—2111no—
108#340—2111no—
109#340D21111yescharacteristic
110C22×Q8—4221no—
111C22×C8—2111no—
112C22×C8C2×D21111yescharacteristic
113C4⋊Q8—4222no—
114C4⋊Q8—2114no—
115C2×C4⋊C4—4221no—
116C2×C4⋊C4—2111no—
117C2×C4⋊C4—2111no—
118C2×C4⋊C4C2×D21111yescharacteristic
119C2×C4⋊C4C2×D21111yescharacteristic
120C2×C42—2111no—
121#97—4222no—
122#95—4221no—
123#95—4221no—
124C2×Q8—8442no—
125C2×Q8—8442no—
126C2×Q8—4224no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—4221no—
130C22×C4—4221no—
131C22×C4—2111no—
132C22×C4—2111no—
133C22×C4C22×D21111yescharacteristic
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—4221no—
137C2×C8—4221no—
138C2×C8—2112no—
139C2×C8Dic42212yescomplemented
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—4224no—
144C4⋊C4—4221no—
145C4⋊C4—4221no—
146C4⋊C4—4221no—
147C4⋊C4—2112no—
148C4⋊C4C4×D21112yes—
149C42—2112no—
150C42—2112no—
151C23D4×D21111yescharacteristic
152Q8—8448no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—8441no—
160C2×C4—8441no—
161C2×C4—8441no—
162C2×C4—8441no—
163C2×C4—8441no—
164C2×C4—4222no—
165C2×C4—4221no—
166C2×C4—4221no—
167C2×C4—4221no—
168C2×C4—4221no—
169C2×C4—2112no—
170C2×C4—2112no—
171C2×C4C2×C4×D21111yescharacteristic
172C2×C4D4×D21111yescharacteristic
173C2×C4Q8×D21111yescharacteristic
174C2×C4C2×Dic42211yescharacteristic
175C2×C4#1680996111yescharacteristic
176C2×C4#1680996211yescharacteristic
177C8—22no—
178C8—4222no—
179C22C22⋊C4×D21111yescharacteristic
180C22#2026996311yescharacteristic
181C22#2026997611yescharacteristic
182C22#2027000111yescharacteristic
183C22#2027000411yescharacteristic
184C22#2027000611yescharacteristic
185C22#2027000911yescharacteristic
186C4—42no—
187C4—81no—
188C4—8442no—
189C4—16881no—
190C4—4222no—
191C4—4222no—
192C4#2026994211yescharacteristic
193C4C4×Dic42211yescharacteristic
194C4#2026995411yescharacteristic
195C4#2026995511yescharacteristic
196C2#2721305011yescharacteristic
197C2#2721314911yescharacteristic
198C2#2721315311yescharacteristic
199C2#2721350411yescharacteristic
200C2#2721350711yescharacteristic
201C2#2721355311yescharacteristic
202C2#2721355411yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socle, prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenter#340D211
hypocenterC211#1993
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
11. #20269976C4semidirect
31. C2×C211⋊C8Q8semidirect
44. C2×C8×C211Q8semidirect
101. C211#1993semidirect
139. C2×C8Dic422semidirect

Maximal subgroups

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

subgroupindex
C2×#202699762
C2×#202698872
#272130112
#272129092
#272129082
#340×C2112
#1993211