Finite Group Databasebeta

← #41014872

Subgroups — #41014872

Order 27008.

230 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 89 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
2#27213045C211yesmaximal, characteristic
3#27213015C211yesmaximal, characteristic
4#27212909C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic
6#27212900C211yesmaximal, characteristic
7C2×#107×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C2×C4⋊Dic211—21no—
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12C2×Dic211⋊C4—21no—
13C2×Dic211⋊C4—21no—
14C2×C4×Dic211C2211yescharacteristic
15#20269910—22no—
16C211⋊#95—21no—
17C22×C8×C211C2211yescharacteristic
18C2×C4⋊C4×C211C2211yescharacteristic
19C2×C4⋊C4×C211C2211yescharacteristic
20#107×C211—22no—
21C22×Dic211—21no—
22C22×Dic211—21no—
23C22×Dic211D411yescharacteristic
24C22×Dic211D411yescharacteristic
25C2×C211⋊C8—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8Q812yes—
28C4⋊Dic211—21no—
29C4⋊Dic211—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211—42no—
33Dic211⋊C4—42no—
34Dic211⋊C4—42no—
35C4×Dic211—21no—
36C4×Dic211—21no—
37C22×C4×C211—21no—
38C22×C4×C211—21no—
39C22×C4×C211C2311yescharacteristic
40C2×C8×C211—21no—
41C2×C8×C211—21no—
42C2×C8×C211Q812yes—
43C4⋊C4×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211D412yes—
47C2×Dic211—21no—
48C2×Dic211—21no—
49C2×Dic211—21no—
50C2×Dic211—21no—
51C2×Dic211—41no—
52C2×Dic211—41no—
53C2×Dic211—41no—
54C2×Dic211—41no—
55C2×Dic211—41no—
56C2×Dic211—41no—
57C2×Dic211D812yes—
58C2×Dic211Q1612yes—
59C211⋊C8—22no—
60C23×C211C2×D411yescharacteristic
61C2×C4×C211—22no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211—41no—
66C2×C4×C211—41no—
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×Q811yescharacteristic
69C2×C4×C211C2×Q811yescharacteristic
70C2×C4×C211#4111yescharacteristic
71C2×C4×C211#4111yescharacteristic
72C2×C4×C211#4111yescharacteristic
73C8×C211—22no—
74C22×C211#12111yescharacteristic
75C22×C211#12311yescharacteristic
76C22×C211C4⋊Q811yescharacteristic
77C22×C211C2×D811yescharacteristic
78C22×C211C2×Q1611yescharacteristic
79C22×C211AGL(1,ℤ/8)11yescharacteristic
80C22×C211#13711yescharacteristic
81C4×C211—42no—
82C4×C211—81no—
83C4×C211C22⋊Q811yescharacteristic
84C4×C211C22⋊Q811yescharacteristic
85C4×C211#12511yescharacteristic
86C4×C211#12511yescharacteristic
87Dic211—22no—
88Dic211—22no—
89Dic211—81no—
90Dic211—81no—
91C2×C211#39811yescharacteristic
92C2×C211#45911yescharacteristic
93C2×C211#46211yescharacteristic
94C2×C211#48011yescharacteristic
95C2×C211#48411yescharacteristic
96C2×C211#50011yescharacteristic
97C2×C211#50111yescharacteristic
98C211#208311yescharacteristic, complemented, Sylow
99#2083—2111nomaximal, Sylow
100C2×#107—2111no—
101C2×#107D21111yescharacteristic
102#389—2111no—
103#384—2111no—
104#340—2111no—
105#340—2111no—
106#336—2111no—
107C22×C8—2111no—
108C22×C8C2×D21111yescharacteristic
109C2×C4⋊C4—4221no—
110C2×C4⋊C4—4221no—
111C2×C4⋊C4—4221no—
112C2×C4⋊C4—2111no—
113C2×C4⋊C4—2111no—
114C2×C4⋊C4C2×D21111yescharacteristic
115C2×C4⋊C4C2×D21111yescharacteristic
116C2×C42—2111no—
117#107—22no—
118#107—4222no—
119#95—4221no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—4221no—
123C22×C4—4221no—
124C22×C4—2111no—
125C22×C4—2111no—
126C22×C4C22×D21111yescharacteristic
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—4221no—
130C2×C8—4221no—
131C2×C8—2112no—
132C2×C8Dic42212yes—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—8442no—
137C4⋊C4—8442no—
138C4⋊C4—8442no—
139C4⋊C4—4221no—
140C4⋊C4—4221no—
141C4⋊C4—4221no—
142C4⋊C4—4221no—
143C4⋊C4C22⋊D21112yes—
144C42—4221no—
145C42—4221no—
146C23D4×D21111yescharacteristic
147C2×C4—22no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—8441no—
154C2×C4—8441no—
155C2×C4—8441no—
156C2×C4—8441no—
157C2×C4—8441no—
158C2×C4—8441no—
159C2×C4—4221no—
160C2×C4—4221no—
161C2×C4—4221no—
162C2×C4—4221no—
163C2×C4—2112no—
164C2×C4—2112no—
165C2×C4C2×C22⋊D21111yescharacteristic
166C2×C4Q8×D21111yescharacteristic
167C2×C4C2×Dic42211yescharacteristic
168C2×C4#1680996111yescharacteristic
169C2×C4#1680996211yescharacteristic
170C2×C4#1680996311yescharacteristic
171C8—22no—
172C8—4222no—
173C22#2026994911yescharacteristic
174C22#2026996211yescharacteristic
175C22#2026997611yescharacteristic
176C22D8×D21111yescharacteristic
177C22Q16×D21111yescharacteristic
178C22#2027000111yescharacteristic
179C22#2027000911yescharacteristic
180C4—42no—
181C4—81no—
182C4—16881no—
183C4—16881no—
184C4—4222no—
185C4—4222no—
186C4#2026995311yescharacteristic
187C4#2026995611yescharacteristic
188C4#2026996711yescharacteristic
189C4#2026996811yescharacteristic
190C2#2721307911yescharacteristic
191C2#2721333911yescharacteristic
192C2#2721335411yescharacteristic
193C2#2721346111yescharacteristic
194C2#2721348311yescharacteristic
195C2#2721355011yescharacteristic
196C2#2721355411yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#107×C211C2
socle, prime radicalC23×C211C2×D4
hypercenterC2×#107D211
hypocenterC211#2083
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
98. C211#2083semidirect

Maximal subgroups

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

subgroupindex
C2×#202699102
#272130452
#272130152
#272129092
#272129082
#272129002
C2×#107×C2112
#2083211