Finite Group Databasebeta

← #41014729

Subgroups — #41014729

Order 27008.

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

normal subgroups only

This group has 282 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×#20269976C211yesmaximal, characteristic
2C2×#20269908C211yesmaximal, characteristic
3#27213042C211yesmaximal, characteristic
4#27212909C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic
6#27212900C211yesmaximal, characteristic
7C2×#106×C211C211yesmaximal, characteristic
8C22×Dic422—21no—
9C22×C211⋊C8C2211yescharacteristic
10#20269976—22no—
11#20269976—22no—
12C2×C4⋊Dic211C2211yescharacteristic
13C2×C4⋊Dic211C2211yescharacteristic
14C2×Dic211⋊C4—21no—
15C2×Dic211⋊C4—21no—
16C2×C4×Dic211C2211yescharacteristic
17#20269908—22no—
18C211⋊#95—21no—
19C22×C8×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22#106×C211—22no—
23C2×Dic422—24no—
24C2×Dic422—42no—
25C2×Dic422—42no—
26C22×Dic211—21no—
27C22×Dic211—21no—
28C22×Dic211D411yescharacteristic
29C22×Dic211D411yescharacteristic
30C2×C211⋊C8—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8Q812yescomplemented
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211D412yes—
36C4⋊Dic211Q812yescomplemented
37Dic211⋊C4—24no—
38Dic211⋊C4—42no—
39C4×Dic211—21no—
40C4×Dic211—21no—
41C22×C4×C211—21no—
42C22×C4×C211—21no—
43C22×C4×C211C2311yescharacteristic
44C2×C8×C211—21no—
45C2×C8×C211—21no—
46C2×C8×C211Q812yescomplemented
47C4⋊C4×C211—21no—
48C4⋊C4×C211—21no—
49C4⋊C4×C211—21no—
50C4⋊C4×C211D412yes—
51Dic422—48no—
52C2×Dic211—21no—
53C2×Dic211—21no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—22no—
57C2×Dic211—22no—
58C2×Dic211—41no—
59C2×Dic211—41no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C2×Dic211SD1612yes—
63C2×Dic211SD1612yes—
64C211⋊C8—22no—
65C23×C211C2×D411yescharacteristic
66C2×C4×C211—22no—
67C2×C4×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211—41no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C2×C4×C211C2×D411yescharacteristic
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×Q811yescharacteristic
75C2×C4×C211C2×Q811yescharacteristic
76C2×C4×C211C2×Q811yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C8×C211—22no—
79C22×C211#12011yescharacteristic
80C22×C211C22⋊Q811yescharacteristic
81C22×C211C4⋊Q811yescharacteristic
82C22×C211C2×SD1611yescharacteristic
83C22×C211C2×SD1611yescharacteristic
84C22×C211AGL(1,ℤ/8)11yescharacteristic
85C22×C211#13711yescharacteristic
86C4×C211—42no—
87C4×C211—81no—
88C4×C211C22⋊Q811yescharacteristic
89C4×C211C22⋊Q811yescharacteristic
90C4×C211C4⋊Q811yescharacteristic
91C4×C211C4⋊Q811yescharacteristic
92Dic211—22no—
93Dic211—22no—
94Dic211—42no—
95Dic211—42no—
96C2×C211#39511yescharacteristic
97C2×C211#44811yescharacteristic
98C2×C211#45011yescharacteristic
99C2×C211#47511yescharacteristic
100C2×C211#47611yescharacteristic
101C2×C211#49811yescharacteristic
102C2×C211#50111yescharacteristic
103C211#205811yescharacteristic, complemented, Sylow
104#2058—2111nomaximal, Sylow
105C2×C4⋊Q8—2111no—
106C2×#106—2111no—
107C2×#106D21111yescharacteristic
108#389—2111no—
109#340—2111no—
110#340—2111no—
111#336—2111no—
112C22×Q8—4221no—
113C22×C8—2111no—
114C22×C8C2×D21111yescharacteristic
115C4⋊Q8—4222no—
116C4⋊Q8—4222no—
117C2×C4⋊C4—4221no—
118C2×C4⋊C4—4221no—
119C2×C4⋊C4—2111no—
120C2×C4⋊C4—2111no—
121C2×C4⋊C4C2×D21111yescharacteristic
122C2×C4⋊C4C2×D21111yescharacteristic
123C2×C42—2111no—
124#106—22no—
125#106—4222no—
126#95—4221no—
127C2×Q8—8442no—
128C2×Q8—8442no—
129C2×Q8—4224no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—4221no—
133C22×C4—4221no—
134C22×C4—2111no—
135C22×C4—2111no—
136C22×C4C22×D21111yescharacteristic
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—4221no—
140C2×C8—4221no—
141C2×C8—2112no—
142C2×C8Dic42212yescomplemented
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—8442no—
147C4⋊C4—4224no—
148C4⋊C4—4221no—
149C4⋊C4—4221no—
150C4⋊C4—2112no—
151C4⋊C4—2112no—
152C4⋊C4C22⋊D21112yes—
153C42—4221no—
154C42—4221no—
155C23D4×D21111yescharacteristic
156Q8—8448no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—8441no—
164C2×C4—8441no—
165C2×C4—8441no—
166C2×C4—8441no—
167C2×C4—4222no—
168C2×C4—4222no—
169C2×C4—4221no—
170C2×C4—4221no—
171C2×C4—4221no—
172C2×C4—4221no—
173C2×C4—2112no—
174C2×C4—2112no—
175C2×C4D4×D21111yescharacteristic
176C2×C4C2×C22⋊D21111yescharacteristic
177C2×C4Q8×D21111yescharacteristic
178C2×C4Q8×D21111yescharacteristic
179C2×C4C2×Dic42211yescharacteristic
180C2×C4#1680996211yescharacteristic
181C8—22no—
182C8—4222no—
183C22#2026994511yescharacteristic
184C22#2026995511yescharacteristic
185C22#2026997611yescharacteristic
186C22SD16×D21111yescharacteristic
187C22SD16×D21111yescharacteristic
188C22#2027000411yescharacteristic
189C22#2027000611yescharacteristic
190C4—42no—
191C4—81no—
192C4—8442no—
193C4—8442no—
194C4—4222no—
195C4—4222no—
196C4#2026995511yescharacteristic
197C4#2026995611yescharacteristic
198C4#2026997511yescharacteristic
199C4#2026997611yescharacteristic
200C2#2721306611yescharacteristic
201C2#2721328311yescharacteristic
202C2#2721329511yescharacteristic
203C2#2721343011yescharacteristic
204C2#2721343511yescharacteristic
205C2#2721354311yescharacteristic
206C2#2721355311yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#106×C211C2
socle, prime radicalC23×C211C2×D4
hypercenterC2×#106D211
hypocenterC211#2058
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
32. C2×C211⋊C8Q8semidirect
36. C4⋊Dic211Q8semidirect
46. C2×C8×C211Q8semidirect
103. C211#2058semidirect
142. 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×#202699082
#272130422
#272129092
#272129082
#272129002
C2×#106×C2112
#2058211