Finite Group Databasebeta

← #41013967

Subgroups — #41013967

Order 27008.

230 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 91 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#27213042C211yesmaximal, characteristic
3C4⋊C4×Dic211C211yesmaximal, characteristic
4#27212909C211yesmaximal, 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×Dic211—21no—
13C2×C4×Dic211C2211yescharacteristic
14#20269910C414yescomplemented
15C211⋊#95—21no—
16C211⋊#95—21no—
17C22×C8×C211C2211yescharacteristic
18C2×C4⋊C4×C211C2211yescharacteristic
19C2×C4⋊C4×C211C2211yescharacteristic
20C22×Dic211—21no—
21C22×Dic211—21no—
22C22×Dic211D411yescharacteristic
23C22×Dic211D411yescharacteristic
24C2×C211⋊C8—21no—
25C2×C211⋊C8—21no—
26C2×C211⋊C8C2×C412yes—
27C4⋊Dic211—21no—
28C4⋊Dic211—21no—
29C4⋊Dic211C2×C412yes—
30C4⋊Dic211Q812yes—
31Dic211⋊C4—42no—
32C4×Dic211—21no—
33C4×Dic211—21no—
34C4×Dic211—42no—
35C22×C4×C211—21no—
36C22×C4×C211—21no—
37C22×C4×C211C2311yescharacteristic
38C2×C8×C211—21no—
39C2×C8×C211—21no—
40C2×C8×C211—21no—
41C4⋊C4×C211—21no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211C2×C412yes—
45C2×Dic211—21no—
46C2×Dic211—21no—
47C2×Dic211—22no—
48C2×Dic211—22no—
49C2×Dic211—21no—
50C2×Dic211—21no—
51C2×Dic211—21no—
52C2×Dic211—41no—
53C2×Dic211—41no—
54C2×Dic211—41no—
55C2×Dic211—41no—
56C2×Dic211D812yes—
57C211⋊C8—22no—
58C23×C211C2×D411yescharacteristic
59C2×C4×C211—22no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211C22×C411yescharacteristic
66C2×C4×C211C2×Q811yescharacteristic
67C2×C4×C211#4111yescharacteristic
68C2×C4×C211#4111yescharacteristic
69C2×C4×C211#4111yescharacteristic
70C2×C4×C211#4111yescharacteristic
71C8×C211—41no—
72C22×C211C4×D411yescharacteristic
73C22×C211C22⋊Q811yescharacteristic
74C22×C211#12311yescharacteristic
75C22×C211C2×D811yescharacteristic
76C22×C211#13511yescharacteristic
77C22×C211#13711yescharacteristic
78C22×C211#13711yescharacteristic
79C4×C211—42no—
80C4×C211—81no—
81C4×C211#11711yescharacteristic
82C4×C211C4×Q811yescharacteristic
83C4×C211#12511yescharacteristic
84C4×C211#12611yescharacteristic
85Dic211—22no—
86Dic211—42no—
87Dic211—42no—
88Dic211—41no—
89C2×C211#38711yescharacteristic
90C2×C211C4×D811yescharacteristic
91C2×C211#44111yescharacteristic
92C2×C211#47411yescharacteristic
93C2×C211#47911yescharacteristic
94C2×C211#48011yescharacteristic
95C2×C211#48511yescharacteristic
96C211#191811yescharacteristic, complemented, Sylow
97#1918—2111nomaximal, Sylow
98C2×#107—2111no—
99#389—2111no—
100C4×C4⋊C4—2111no—
101#340—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—4221no—
113C2×C42—2111no—
114#107—2114no—
115#95—4221no—
116#95—4221no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—4221no—
120C22×C4—4221no—
121C22×C4—2111no—
122C22×C4—2111no—
123C22×C4C22×D21111yescharacteristic
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—4221no—
128C2×C8—4221no—
129C2×C8—2112no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—8442no—
134C4⋊C4—4221no—
135C4⋊C4—4221no—
136C4⋊C4—2112no—
137C4⋊C4—2112no—
138C4⋊C4C4×D21112yes—
139C42—8442no—
140C42—4221no—
141C42—4221no—
142C23D4×D21111yescharacteristic
143C2×C4—22no—
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—4222no—
154C2×C4—4222no—
155C2×C4—4221no—
156C2×C4—4221no—
157C2×C4—4221no—
158C2×C4—4221no—
159C2×C4—4221no—
160C2×C4—2112no—
161C2×C4C2×C4×D21111yescharacteristic
162C2×C4Q8×D21111yescharacteristic
163C2×C4#1680996111yescharacteristic
164C2×C4#1680996111yescharacteristic
165C2×C4#1680996211yescharacteristic
166C2×C4#1680996211yescharacteristic
167C8—41no—
168C8—4222no—
169C22(C22⋊D211)⋊C411yescharacteristic
170C22#2026995511yescharacteristic
171C22#2026995911yescharacteristic
172C22D8×D21111yescharacteristic
173C22#2026999811yescharacteristic
174C22#2027000611yescharacteristic
175C22#2027000911yescharacteristic
176C4—42no—
177C4—81no—
178C4—8442no—
179C4—8442no—
180C4—8441no—
181C4—4222no—
182C4#2026993411yescharacteristic
183C4#2026994211yescharacteristic
184C4#2026996711yescharacteristic
185C4#2026997011yescharacteristic
186C2#2721302811yescharacteristic
187C2#2721322611yescharacteristic
188C2#2721325011yescharacteristic
189C2#2721342511yescharacteristic
190C2#2721345611yescharacteristic
191C2#2721345811yescharacteristic
192C2#2721348511yescharacteristic
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#1918
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
14. #20269910C4semidirect
96. C211#1918semidirect

Maximal subgroups

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

subgroupindex
C2×#202699102
#272130422
C4⋊C4×Dic2112
#272129092
#272129082
#272129002
#340×C2112
#1918211