Finite Group Databasebeta

← #41014165

Subgroups — #41014165

Order 27008.

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

normal subgroups only

This group has 240 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×#20269910C211yesmaximal, characteristic
2C2×#20269909C211yesmaximal, characteristic
3C2×Dic211⋊C8C211yesmaximal, characteristic
4#27213011C211yesmaximal, characteristic
5C4⋊C4×Dic211C211yesmaximal, characteristic
6#27212908C211yesmaximal, characteristic
7#340×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C2×C4⋊Dic211C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4×Dic211—21no—
12C2×C4×Dic211C2211yescharacteristic
13#20269910—22no—
14#20269909—22no—
15Dic211⋊C8C414yescomplemented
16C211⋊#95—21no—
17C211⋊#95—21no—
18C211⋊#95—21no—
19C22×C8×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C22×Dic211—21no—
23C22×Dic211—21no—
24C22×Dic211D411yescharacteristic
25C22×Dic211D411yescharacteristic
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8C2×C412yes—
29C4⋊Dic211—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211D412yes—
33C4×Dic211—42no—
34C4×Dic211C2×C412yes—
35C4×Dic211Q812yes—
36C22×C4×C211—21no—
37C22×C4×C211—21no—
38C22×C4×C211C2311yescharacteristic
39C2×C8×C211—21no—
40C2×C8×C211—21no—
41C2×C8×C211C2×C412yes—
42C4⋊C4×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211Q812yes—
46C2×Dic211—21no—
47C2×Dic211—21no—
48C2×Dic211—21no—
49C2×Dic211—21no—
50C2×Dic211—22no—
51C2×Dic211—41no—
52C2×Dic211—41no—
53C2×Dic211—41no—
54C2×Dic211—41no—
55C2×Dic211—41no—
56C2×Dic211C4⋊C412yes—
57C2×Dic211C4⋊C412yes—
58C211⋊C8—22no—
59C23×C211C2×D411yescharacteristic
60C2×C4×C211—22no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211—41no—
66C2×C4×C211C22×C411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×Q811yescharacteristic
69C2×C4×C211C2×Q811yescharacteristic
70C2×C4×C211#4111yescharacteristic
71C2×C4×C211#4111yescharacteristic
72C8×C211—22no—
73C22×C211C2×C4⋊C411yescharacteristic
74C22×C211#12111yescharacteristic
75C22×C211C22⋊Q811yescharacteristic
76C22×C211#13511yescharacteristic
77C22×C211#13511yescharacteristic
78C22×C211AGL(1,ℤ/8)11yescharacteristic
79C22×C211#13711yescharacteristic
80C4×C211—42no—
81C4×C211—81no—
82C4×C211C4×D411yescharacteristic
83C4×C211C4×Q811yescharacteristic
84C4×C211#12511yescharacteristic
85C4×C211C4⋊Q811yescharacteristic
86Dic211—22no—
87Dic211—22no—
88Dic211—42no—
89Dic211—81no—
90C2×C211#38911yescharacteristic
91C2×C211#42711yescharacteristic
92C2×C211#42811yescharacteristic
93C2×C211#46311yescharacteristic
94C2×C211#46411yescharacteristic
95C2×C211#47811yescharacteristic
96C2×C211#47911yescharacteristic
97C211#194911yescharacteristic, complemented, Sylow
98#1949—2111nomaximal, Sylow
99C2×#107—2111no—
100C2×#106—2111no—
101C2×C4⋊C8—2111no—
102#382—2111no—
103C4×C4⋊C4—2111no—
104#340—2111no—
105#340D21111yescharacteristic
106C22×C8—2111no—
107C22×C8C2×D21111yescharacteristic
108C2×C4⋊C4—2111no—
109C2×C4⋊C4—2111no—
110C2×C4⋊C4C2×D21111yescharacteristic
111C2×C4⋊C4C2×D21111yescharacteristic
112C2×C42—4221no—
113C2×C42—2111no—
114#107—4222no—
115#106—4222no—
116C4⋊C8—2114no—
117#95—4221no—
118#95—4221no—
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×C8C4×D21112yes—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—4221no—
137C4⋊C4—4221no—
138C4⋊C4—4221no—
139C4⋊C4—2112no—
140C4⋊C4Dic42212yes—
141C42—8442no—
142C42—2112no—
143C42—2112no—
144C23D4×D21111yescharacteristic
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—8441no—
152C2×C4—8441no—
153C2×C4—8441no—
154C2×C4—8441no—
155C2×C4—8441no—
156C2×C4—4222no—
157C2×C4—4221no—
158C2×C4—4221no—
159C2×C4—4221no—
160C2×C4—4221no—
161C2×C4—2112no—
162C2×C4—2112no—
163C2×C4C2×C4×D21111yescharacteristic
164C2×C4D4×D21111yescharacteristic
165C2×C4Q8×D21111yescharacteristic
166C2×C4C2×Dic42211yescharacteristic
167C2×C4#1680996111yescharacteristic
168C2×C4#1680996211yescharacteristic
169C8—22no—
170C8—4222no—
171C22C4⋊C4×D21111yescharacteristic
172C22#2026995011yescharacteristic
173C22#2026995411yescharacteristic
174C22#2026999811yescharacteristic
175C22#2026999811yescharacteristic
176C22#2027000111yescharacteristic
177C22#2027000911yescharacteristic
178C4—42no—
179C4—81no—
180C4—8442no—
181C4—16881no—
182C4—4222no—
183C4—4222no—
184C4(C22⋊D211)⋊C411yescharacteristic
185C4C4×Dic42211yescharacteristic
186C4#2026996711yescharacteristic
187C4#2026997611yescharacteristic
188C2#2721304311yescharacteristic
189C2#2721318111yescharacteristic
190C2#2721318711yescharacteristic
191C2#2721336011yescharacteristic
192C2#2721337111yescharacteristic
193C2#2721344811yescharacteristic
194C2#2721345111yescharacteristic
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#1949
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
15. Dic211⋊C8C4semidirect
97. C211#1949semidirect

Maximal subgroups

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

subgroupindex
C2×#202699102
C2×#202699092
C2×Dic211⋊C82
#272130112
C4⋊C4×Dic2112
#272129082
#340×C2112
#1949211