Finite Group Databasebeta

← #41014534

Subgroups — #41014534

Order 27008.

326 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20269977C211yesmaximal, characteristic, complemented
2C2×#20269898C212yesmaximal
3#27212908C212yesmaximal, complemented
4#27212901C211yesmaximal, characteristic, complemented
5#390×C211C211yesmaximal, characteristic
6C22×Dic422—21no—
7C22×C211⋊C8C2212yes—
8#20269977—24no—
9C2×C4⋊Dic211—21no—
10C2×C4⋊Dic211C2212yes—
11#20269898—24no—
12C22×D4×C211C2211yescharacteristic
13C2×C4⋊C4×C211C2211yescharacteristic
14C2×C22⋊C4×C211—21no—
15C2×C22⋊C4×C211—21no—
16C2×C42×C211C2211yescharacteristic
17C2×Dic422—24no—
18C2×Dic422—42no—
19C2×Dic422—42no—
20C22×Dic211—22no—
21C2×C211⋊C8—22no—
22C2×C211⋊C8—22no—
23C2×C211⋊C8—22no—
24C4⋊Dic211—24no—
25C4⋊Dic211—22no—
26C4⋊Dic211D414yes—
27C24×C211—21no—
28C2×D4×C211—22no—
29C2×D4×C211—22no—
30C2×D4×C211—21no—
31C2×D4×C211Q812yescomplemented
32C22×C4×C211—21no—
33C22×C4×C211D411yescharacteristic
34C22×C4×C211D411yescharacteristic
35C22×C4×C211C2311yescharacteristic
36C4⋊C4×C211—22no—
37C22⋊C4×C211—42no—
38C22⋊C4×C211—42no—
39C42×C211—22no—
40Dic422—48no—
41C2×Dic211—24no—
42C2×Dic211—42no—
43C2×Dic211—42no—
44C211⋊C8—42no—
45C23×C211—22no—
46C23×C211—41no—
47C23×C211—41no—
48C23×C211—41no—
49C23×C211—41no—
50C23×C211—41no—
51C23×C211—41no—
52C23×C211C2×D411yescharacteristic
53D4×C211—24no—
54D4×C211—42no—
55C2×C4×C211—21no—
56C2×C4×C211—21no—
57C2×C4×C211—21no—
58C2×C4×C211—21no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211D812yes—
63C2×C4×C211SD1612yes—
64C2×C4×C211C2×D412yes—
65C2×C4×C211C2×Q811yescharacteristic
66C2×C4×C211#4112yes—
67C2×C4×C211#4111yescharacteristic
68C22×C211—42no—
69C22×C211—42no—
70C22×C211—42no—
71C22×C211—81no—
72C22×C211—81no—
73C22×C211—81no—
74C22×C211—81no—
75C22×C211#12011yescharacteristic
76C22×C211C22⋊Q811yescharacteristic
77C22×C211#12411yescharacteristic
78C22×C211C2×D811yescharacteristic
79C22×C211C2×SD1611yescharacteristic
80C22×C211AGL(1,ℤ/8)11yescharacteristic
81C22×C211#13711yescharacteristic
82C4×C211—22no—
83C4×C211—22no—
84C4×C211—81no—
85C4×C211C22⋊Q812yes—
86C4×C211#12412yes—
87Dic211—44no—
88C2×C211—82no—
89C2×C211#39311yescharacteristic
90C2×C211#44711yescharacteristic
91C2×C211#44811yescharacteristic
92C2×C211#47411yescharacteristic
93C2×C211#47611yescharacteristic
94C2×C211#48611yescharacteristic
95C2×C211#48911yescharacteristic
96C211#202311yescharacteristic, complemented, Sylow
97#2023—2111nomaximal, Sylow
98C2×C4⋊Q8—2111no—
99C2×#102—2112no—
100#390D21111yescharacteristic
101#340—2112no—
102#336—2111no—
103C22×Q8—4221no—
104C22×D4C2×D21111yescharacteristic
105C22×C8—2112no—
106C4⋊Q8—4224no—
107C2×C4⋊C4—4221no—
108C2×C4⋊C4—2112no—
109C2×C4⋊C4C2×D21111yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C42C2×D21111yescharacteristic
113#102—4224no—
114C24—21no—
115C2×Q8—8442no—
116C2×Q8—8442no—
117C2×Q8—4224no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—21no—
121C2×D4Dic42212yescomplemented
122C22×C4—21no—
123C22×C4—4222no—
124C22×C4C22⋊D21111yescharacteristic
125C22×C4C22⋊D21111yescharacteristic
126C22×C4C22×D21111yescharacteristic
127C2×C8—4222no—
128C2×C8—4222no—
129C2×C8—4222no—
130C4⋊C4—22no—
131C4⋊C4—4224no—
132C4⋊C4—4222no—
133C4⋊C4—2114no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C42—22no—
137C23—22no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23C2×C22⋊D21111yescharacteristic
145Q8—8448no—
146D4—24no—
147D4—42no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—8442no—
156C2×C4—8442no—
157C2×C4—4224no—
158C2×C4#1680994712yes—
159C2×C4#1680995012yes—
160C2×C4D4×D21112yes—
161C2×C4C2×Dic42211yescharacteristic
162C2×C4#1680996112yes—
163C2×C4#1680996211yescharacteristic
164C8—8442no—
165C22—42no—
166C22—42no—
167C22—42no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22#2026994511yescharacteristic
173C22#2026995611yescharacteristic
174C22#2026996411yescharacteristic
175C22C2×#1680994711yescharacteristic
176C22C2×#1680995011yescharacteristic
177C22#2027000211yescharacteristic
178C22#2027000711yescharacteristic
179C4—22no—
180C4—22no—
181C4—81no—
182C4—8444no—
183C4#2026995412yes—
184C4#2026996312yes—
185C2—82no—
186C2#2721305811yescharacteristic
187C2#2721327911yescharacteristic
188C2#2721328411yescharacteristic
189C2#2721342611yescharacteristic
190C2#2721343611yescharacteristic
191C2#2721349211yescharacteristic
192C2#2721350511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#390×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
hypercenter#390D211
hypocenterC211#2023
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#20269977C2semidirect
3. #27212908C2semidirect
4. #27212901C2semidirect
31. C2×D4×C211Q8semidirect
96. C211#2023semidirect
121. C2×D4Dic422semidirect

Maximal subgroups

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

subgroupindex
C2×#202699772
C2×#202698982
#272129082
#272129012
#390×C2112
#2023211