Finite Group Databasebeta

← #41014133

Subgroups — #41014133

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×#20269908C211yesmaximal, characteristic
3C2×#20269888C211yesmaximal, characteristic
4#27213044C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic
6#27212908C211yesmaximal, characteristic
7#389×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12C2×Dic211⋊C4—21no—
13#20269910C414yescomplemented
14#20269908—22no—
15#20269888—22no—
16C211⋊#95—21no—
17C2×C4⋊C4×C211—21no—
18C2×C4⋊C4×C211C2211yescharacteristic
19C2×C4⋊C4×C211C2211yescharacteristic
20C2×C42×C211C2211yescharacteristic
21#95×C211—21no—
22C22×Dic211—21no—
23C22×Dic211—21no—
24C2×C211⋊C8—21no—
25C2×C211⋊C8—21no—
26C2×C211⋊C8C2×C412yes—
27C2×C211⋊C8D412yes—
28C2×C211⋊C8Q812yes—
29C2×C211⋊C8Q812yes—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211C2×C412yes—
34Dic211⋊C4—42no—
35C22×C4×C211—21no—
36C22×C4×C211—21no—
37C22×C4×C211D411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211C2311yescharacteristic
40C4⋊C4×C211—21no—
41C4⋊C4×C211—21no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211—42no—
44C4⋊C4×C211C2×C412yes—
45C42×C211—21no—
46C42×C211—21no—
47C2×Dic211—22no—
48C2×Dic211—41no—
49C2×Dic211—41no—
50C2×Dic211—41no—
51C2×Dic211—41no—
52C2×Dic211—41no—
53C211⋊C8—22no—
54C211⋊C8C4⋊C414yescomplemented
55C23×C211C2×D411yescharacteristic
56C2×C4×C211—22no—
57C2×C4×C211—21no—
58C2×C4×C211—21no—
59C2×C4×C211—21no—
60C2×C4×C211—21no—
61C2×C4×C211—21no—
62C2×C4×C211—21no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211—41no—
66C2×C4×C211—41no—
67C2×C4×C211—41no—
68C2×C4×C211C22×C411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211C2×Q811yescharacteristic
71C2×C4×C211C2×Q811yescharacteristic
72C2×C4×C211#4111yescharacteristic
73C2×C4×C211#4111yescharacteristic
74C22×C211C4×D411yescharacteristic
75C22×C211#12111yescharacteristic
76C22×C211C4⋊Q811yescharacteristic
77C22×C211AGL(1,ℤ/8)11yescharacteristic
78C22×C211AGL(1,ℤ/8)11yescharacteristic
79C22×C211#13711yescharacteristic
80C22×C211#13711yescharacteristic
81C4×C211—42no—
82C4×C211—41no—
83C4×C211—41no—
84C4×C211—81no—
85C4×C211C2×C4⋊C411yescharacteristic
86C4×C211C4×Q811yescharacteristic
87C4×C211C22⋊Q811yescharacteristic
88C4×C211#12511yescharacteristic
89Dic211—42no—
90Dic211—81no—
91C2×C211#38911yescharacteristic
92C2×C211#44111yescharacteristic
93C2×C211#44211yescharacteristic
94C2×C211#46811yescharacteristic
95C2×C211#46811yescharacteristic
96C2×C211#50111yescharacteristic
97C2×C211#50111yescharacteristic
98C211#194311yescharacteristic, complemented, Sylow
99#1943—2111nomaximal, Sylow
100C2×#107—2111no—
101C2×#106—2111no—
102C2×#97—2111no—
103#389—2111no—
104#389D21111yescharacteristic
105#340—2111no—
106#340—2111no—
107C22×C8—2111no—
108C22×C8—2111no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—4221no—
111C2×C4⋊C4—2111no—
112C2×C4⋊C4—2111no—
113C2×C4⋊C4C2×D21111yescharacteristic
114C2×C4⋊C4C2×D21111yescharacteristic
115C2×C42C2×D21111yescharacteristic
116#107—2114no—
117#106—4222no—
118#97—4222no—
119#95—21no—
120#95—4221no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—4221no—
124C22×C4—4221no—
125C22×C4C22⋊D21111yescharacteristic
126C22×C4C22⋊D21111yescharacteristic
127C22×C4C22×D21111yescharacteristic
128C2×C8—4221no—
129C2×C8—4221no—
130C2×C8—2112no—
131C2×C8—2112no—
132C2×C8—2112no—
133C2×C8—2112no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—42no—
138C4⋊C4—8442no—
139C4⋊C4—4221no—
140C4⋊C4—4221no—
141C4⋊C4—4221no—
142C4⋊C4—2112no—
143C4⋊C4C4×D21112yes—
144C42—21no—
145C42—21no—
146C23C2×C22⋊D21111yescharacteristic
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—8441no—
160C2×C4—8441no—
161C2×C4—8441no—
162C2×C4—8441no—
163C2×C4—8441no—
164C2×C4—4222no—
165C2×C4C2×C4×D21111yescharacteristic
166C2×C4D4×D21111yescharacteristic
167C2×C4Q8×D21111yescharacteristic
168C2×C4Q8×D21111yescharacteristic
169C2×C4#1680996111yescharacteristic
170C2×C4#1680996211yescharacteristic
171C8—4222no—
172C8—2114no—
173C22C4×C22⋊D21111yescharacteristic
174C22#2026994911yescharacteristic
175C22#2026997511yescharacteristic
176C22#2027000211yescharacteristic
177C22#2027000311yescharacteristic
178C22#2027000711yescharacteristic
179C22#2027000811yescharacteristic
180C4—41no—
181C4—41no—
182C4—42no—
183C4—81no—
184C4—8442no—
185C4—16881no—
186C4C4⋊C4×D21111yescharacteristic
187C4#2026994211yescharacteristic
188C4#2026995511yescharacteristic
189C4#2026996711yescharacteristic
190C2#2721304111yescharacteristic
191C2#2721325111yescharacteristic
192C2#2721325611yescharacteristic
193C2#2721339011yescharacteristic
194C2#2721339411yescharacteristic
195C2#2721355211yescharacteristic
196C2#2721355611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#389×C211C2
socle, prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenter#389D211
hypocenterC211#1943
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
13. #20269910C4semidirect
54. C211⋊C8C4⋊C4semidirect
98. C211#1943semidirect

Maximal subgroups

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

subgroupindex
C2×#202699102
C2×#202699082
C2×#202698882
#272130442
#272129082
#389×C2112
#1943211