Finite Group Databasebeta

← #41014733

Subgroups — #41014733

Order 27008.

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

normal subgroups only

This group has 250 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
2C2×#20269908C211yesmaximal, characteristic
3#27213044C211yesmaximal, characteristic
4#27212908C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic
6#27212901C211yesmaximal, characteristic
7C2×#125×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12C2×Dic211⋊C4—21no—
13#20269910—22no—
14#20269908—22no—
15C211⋊#95—21no—
16#125×C211—22no—
17#125×C211—22no—
18C2×C4⋊C4×C211—21no—
19C2×C4⋊C4×C211—21no—
20C2×C4⋊C4×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C42×C211C2211yescharacteristic
23C22×Dic211—21no—
24C22×Dic211—21no—
25C2×C211⋊C8—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8Q812yes—
30C2×C211⋊C8Q812yes—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211Q812yes—
35Dic211⋊C4—42no—
36C22×C4×C211—21no—
37C22×C4×C211—21no—
38C22×C4×C211D411yescharacteristic
39C22×C4×C211D411yescharacteristic
40C22×C4×C211C2311yescharacteristic
41C4⋊C4×C211—24no—
42C4⋊C4×C211—24no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211D412yes—
46C4⋊C4×C211D412yes—
47C42×C211—21no—
48C42×C211—21no—
49C2×Dic211—22no—
50C2×Dic211—41no—
51C2×Dic211—41no—
52C2×Dic211—41no—
53C2×Dic211—41no—
54C2×Dic211—41no—
55C211⋊C8—22no—
56C211⋊C8—22no—
57C23×C211C2×D411yescharacteristic
58C2×C4×C211—21no—
59C2×C4×C211—21no—
60C2×C4×C211—22no—
61C2×C4×C211—22no—
62C2×C4×C211—21no—
63C2×C4×C211—21no—
64C2×C4×C211—21no—
65C2×C4×C211—21no—
66C2×C4×C211—41no—
67C2×C4×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211—41no—
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211C2×Q811yescharacteristic
73C2×C4×C211C2×Q811yescharacteristic
74C2×C4×C211C2×Q811yescharacteristic
75C2×C4×C211#4111yescharacteristic
76C22×C211#12011yescharacteristic
77C22×C211C22⋊Q811yescharacteristic
78C22×C211C4⋊Q811yescharacteristic
79C22×C211#13511yescharacteristic
80C22×C211#13511yescharacteristic
81C22×C211AGL(1,ℤ/8)11yescharacteristic
82C22×C211#13711yescharacteristic
83C4×C211—41no—
84C4×C211—42no—
85C4×C211—41no—
86C4×C211—42no—
87C4×C211C22⋊Q811yescharacteristic
88C4×C211C22⋊Q811yescharacteristic
89C4×C211C4⋊Q811yescharacteristic
90C4×C211C4⋊Q811yescharacteristic
91Dic211—42no—
92Dic211—81no—
93C2×C211#39511yescharacteristic
94C2×C211#44911yescharacteristic
95C2×C211#45211yescharacteristic
96C2×C211#47811yescharacteristic
97C2×C211#47911yescharacteristic
98C2×C211#49911yescharacteristic
99C2×C211#50111yescharacteristic
100C211#205911yescharacteristic, complemented, Sylow
101#2059—2111nomaximal, Sylow
102C2×#125D21111yescharacteristic
103C2×#107—2111no—
104C2×#106—2111no—
105#389—2111no—
106#340—2111no—
107#340—2111no—
108#336—2111no—
109C22×C8—2111no—
110C22×C8—2111no—
111#125—22no—
112#125—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—4221no—
116C2×C4⋊C4—2111no—
117C2×C4⋊C4—2111no—
118C2×C4⋊C4C2×D21111yescharacteristic
119C2×C4⋊C4C2×D21111yescharacteristic
120C2×C42C2×D21111yescharacteristic
121#107—4222no—
122#106—4222no—
123#95—4221no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—4221no—
127C22×C4—4221no—
128C22×C4C22⋊D21111yescharacteristic
129C22×C4C22⋊D21111yescharacteristic
130C22×C4C22×D21111yescharacteristic
131C2×C8—4221no—
132C2×C8—4221no—
133C2×C8—4221no—
134C2×C8—4221no—
135C2×C8—2112no—
136C2×C8—2112no—
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—8442no—
142C4⋊C4—4221no—
143C4⋊C4—4221no—
144C4⋊C4—4221no—
145C4⋊C4—2112no—
146C4⋊C4C22⋊D21112yes—
147C4⋊C4C22⋊D21112yes—
148C42—21no—
149C42—21no—
150C23C2×C22⋊D21111yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—8441no—
164C2×C4—8441no—
165C2×C4—8441no—
166C2×C4—8441no—
167C2×C4—8441no—
168C2×C4—4222no—
169C2×C4C2×C22⋊D21111yescharacteristic
170C2×C4C2×C22⋊D21111yescharacteristic
171C2×C4Q8×D21111yescharacteristic
172C2×C4Q8×D21111yescharacteristic
173C2×C4Q8×D21111yescharacteristic
174C2×C4#1680996311yescharacteristic
175C8—4222no—
176C8—4222no—
177C22#2026994611yescharacteristic
178C22#2026995311yescharacteristic
179C22#2026997511yescharacteristic
180C22#2026999711yescharacteristic
181C22#2026999711yescharacteristic
182C22#2027000211yescharacteristic
183C22#2027000711yescharacteristic
184C4—41no—
185C4—42no—
186C4—41no—
187C4—42no—
188C4—8442no—
189C4—16881no—
190C4#2026995311yescharacteristic
191C4#2026995311yescharacteristic
192C4#2026997511yescharacteristic
193C4#2026997511yescharacteristic
194C2#2721306511yescharacteristic
195C2#2721329111yescharacteristic
196C2#2721330811yescharacteristic
197C2#2721344311yescharacteristic
198C2#2721345011yescharacteristic
199C2#2721354511yescharacteristic
200C2#2721355611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#125×C211C2
socle, prime radicalC23×C211C2×D4
hypercenterC2×#125D211
hypocenterC211#2059
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
100. C211#2059semidirect

Maximal subgroups

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

subgroupindex
C2×#202699102
C2×#202699082
#272130442
#272129082
#272129012
C2×#125×C2112
#2059211