Finite Group Databasebeta

← #41014884

Subgroups — #41014884

Order 27008.

214 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 83 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×#20269909C211yesmaximal, characteristic
3#27213042C211yesmaximal, characteristic
4#27213011C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic
6#27212900C211yesmaximal, characteristic
7#340×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C2×C4⋊Dic211C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×Dic211⋊C4—21no—
12C2×C4×Dic211C2211yescharacteristic
13#20269910—22no—
14#20269909—22no—
15C211⋊#95—21no—
16C211⋊#95—21no—
17C211⋊#95—21no—
18C22×C8×C211C2211yescharacteristic
19C2×C4⋊C4×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
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⋊Dic211D412yes—
32Dic211⋊C4—42no—
33C4×Dic211—21no—
34C4×Dic211—21no—
35C22×C4×C211—21no—
36C22×C4×C211—21no—
37C22×C4×C211C2311yescharacteristic
38C2×C8×C211—21no—
39C2×C8×C211—21no—
40C2×C8×C211Q812yes—
41C4⋊C4×C211—21no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211—21no—
45C2×Dic211—22no—
46C2×Dic211—21no—
47C2×Dic211—21no—
48C2×Dic211—21no—
49C2×Dic211—21no—
50C2×Dic211—21no—
51C2×Dic211—21no—
52C2×Dic211—41no—
53C2×Dic211—41no—
54C2×Dic211—41no—
55C2×Dic211—41no—
56C2×Dic211—41no—
57C211⋊C8—22no—
58C23×C211C2×D411yescharacteristic
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×Q811yescharacteristic
67C2×C4×C211C2×Q811yescharacteristic
68C2×C4×C211#4111yescharacteristic
69C2×C4×C211#4111yescharacteristic
70C2×C4×C211#4111yescharacteristic
71C8×C211—22no—
72C22×C211#12111yescharacteristic
73C22×C211#12311yescharacteristic
74C22×C211C4⋊Q811yescharacteristic
75C22×C211#13511yescharacteristic
76C22×C211#13511yescharacteristic
77C22×C211AGL(1,ℤ/8)11yescharacteristic
78C22×C211#13711yescharacteristic
79C4×C211—81no—
80C4×C211—81no—
81C4×C211C22⋊Q811yescharacteristic
82C4×C211C22⋊Q811yescharacteristic
83C4×C211#12511yescharacteristic
84C4×C211#12511yescharacteristic
85Dic211—42no—
86Dic211—41no—
87Dic211—41no—
88Dic211—81no—
89C2×C211#39811yescharacteristic
90C2×C211#46311yescharacteristic
91C2×C211#46411yescharacteristic
92C2×C211#48211yescharacteristic
93C2×C211#48511yescharacteristic
94C2×C211#49911yescharacteristic
95C2×C211#50111yescharacteristic
96C211#208511yescharacteristic, complemented, Sylow
97#2085—2111nomaximal, Sylow
98C2×#107—2111no—
99C2×#106—2111no—
100#389—2111no—
101#382—2111no—
102#340—2111no—
103#340D21111yescharacteristic
104#336—2111no—
105C22×C8—2111no—
106C22×C8C2×D21111yescharacteristic
107C2×C4⋊C4—4221no—
108C2×C4⋊C4—2111no—
109C2×C4⋊C4—2111no—
110C2×C4⋊C4C2×D21111yescharacteristic
111C2×C4⋊C4C2×D21111yescharacteristic
112C2×C42—2111no—
113#107—4222no—
114#106—4222no—
115#95—4221no—
116#95—4221no—
117#95—4221no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—4221no—
121C22×C4—4221no—
122C22×C4—2111no—
123C22×C4—2111no—
124C22×C4C22×D21111yescharacteristic
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—4221no—
128C2×C8—4221no—
129C2×C8—2112no—
130C2×C8Dic42212yes—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—8442no—
136C4⋊C4—4221no—
137C4⋊C4—4221no—
138C4⋊C4—4221no—
139C4⋊C4—2112no—
140C42—4221no—
141C42—4221no—
142C23D4×D21111yescharacteristic
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—8441no—
150C2×C4—8441no—
151C2×C4—8441no—
152C2×C4—8441no—
153C2×C4—8441no—
154C2×C4—4222no—
155C2×C4—4221no—
156C2×C4—4221no—
157C2×C4—4221no—
158C2×C4—4221no—
159C2×C4—4221no—
160C2×C4—4221no—
161C2×C4D4×D21111yescharacteristic
162C2×C4Q8×D21111yescharacteristic
163C2×C4C2×Dic42211yescharacteristic
164C2×C4#1680996111yescharacteristic
165C2×C4#1680996211yescharacteristic
166C2×C4#1680996211yescharacteristic
167C8—22no—
168C8—4222no—
169C22#2026995011yescharacteristic
170C22#2026995911yescharacteristic
171C22#2026997611yescharacteristic
172C22#2026999811yescharacteristic
173C22#2026999811yescharacteristic
174C22#2027000111yescharacteristic
175C22#2027000911yescharacteristic
176C4—81no—
177C4—81no—
178C4—8442no—
179C4—8441no—
180C4—8441no—
181C4—16881no—
182C4#2026995411yescharacteristic
183C4#2026995511yescharacteristic
184C4#2026996711yescharacteristic
185C4#2026996911yescharacteristic
186C2#2721308011yescharacteristic
187C2#2721336011yescharacteristic
188C2#2721337111yescharacteristic
189C2#2721346911yescharacteristic
190C2#2721348511yescharacteristic
191C2#2721354611yescharacteristic
192C2#2721355411yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socle, prime radicalC23×C211C2×D4
hypercenter#340D211
hypocenterC211#2085
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
96. C211#2085semidirect

Maximal subgroups

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

subgroupindex
C2×#202699102
C2×#202699092
#272130422
#272130112
#272129082
#272129002
#340×C2112
#2085211