Finite Group Databasebeta

← #41014900

Subgroups — #41014900

Order 27008.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20269908C211yesmaximal, characteristic
2C2×C4⋊(C211⋊C8)C211yesmaximal, characteristic
3#27213044C211yesmaximal, characteristic
4#27212908C211yesmaximal, 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#20269908—22no—
14C4⋊(C211⋊C8)—22no—
15C211⋊#95—21no—
16C2×C4⋊C4×C211—21no—
17C2×C4⋊C4×C211C2211yescharacteristic
18C2×C4⋊C4×C211C2211yescharacteristic
19C2×C42×C211C2211yescharacteristic
20#95×C211—21no—
21C22×Dic211—21no—
22C22×Dic211—21no—
23C2×C211⋊C8—21no—
24C2×C211⋊C8—21no—
25C2×C211⋊C8—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8D412yes—
29C4⋊Dic211—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211Q812yes—
33Dic211⋊C4—42no—
34C22×C4×C211—21no—
35C22×C4×C211—21no—
36C22×C4×C211D411yescharacteristic
37C22×C4×C211D411yescharacteristic
38C22×C4×C211C2311yescharacteristic
39C4⋊C4×C211—21no—
40C4⋊C4×C211—21no—
41C4⋊C4×C211—21no—
42C4⋊C4×C211—42no—
43C4⋊C4×C211Q812yes—
44C42×C211—21no—
45C42×C211—21no—
46C2×Dic211—22no—
47C2×Dic211—41no—
48C2×Dic211—41no—
49C2×Dic211—41no—
50C2×Dic211—41no—
51C2×Dic211—41no—
52C211⋊C8—22no—
53C211⋊C8—41no—
54C23×C211C2×D411yescharacteristic
55C2×C4×C211—21no—
56C2×C4×C211—21no—
57C2×C4×C211—21no—
58C2×C4×C211—22no—
59C2×C4×C211—21no—
60C2×C4×C211—21no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211—41no—
66C2×C4×C211SD1612yes—
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×Q811yescharacteristic
69C2×C4×C211C2×Q811yescharacteristic
70C2×C4×C211#4111yescharacteristic
71C2×C4×C211#4111yescharacteristic
72C2×C4×C211#4111yescharacteristic
73C22×C211#12111yescharacteristic
74C22×C211C22⋊Q811yescharacteristic
75C22×C211C22⋊Q811yescharacteristic
76C22×C211C2×SD1611yescharacteristic
77C22×C211#13511yescharacteristic
78C22×C211AGL(1,ℤ/8)11yescharacteristic
79C22×C211#13711yescharacteristic
80C4×C211—22no—
81C4×C211—41no—
82C4×C211—42no—
83C4×C211—81no—
84C4×C211#12311yescharacteristic
85C4×C211#12511yescharacteristic
86C4×C211#12511yescharacteristic
87C4×C211C4⋊Q811yescharacteristic
88Dic211—42no—
89Dic211—81no—
90C2×C211#39811yescharacteristic
91C2×C211#46511yescharacteristic
92C2×C211#46811yescharacteristic
93C2×C211#47511yescharacteristic
94C2×C211#47611yescharacteristic
95C2×C211#47811yescharacteristic
96C2×C211#47911yescharacteristic
97C211#208811yescharacteristic, complemented, Sylow
98#2088—2111nomaximal, Sylow
99C2×#106—2111no—
100C2×C4⋊C8—2111no—
101#389—2111no—
102#389D21111yescharacteristic
103#340—2111no—
104#340—2111no—
105#340—2111no—
106C22×C8—2111no—
107C22×C8—2111no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—4221no—
110C2×C4⋊C4—2111no—
111C2×C4⋊C4—2111no—
112C2×C4⋊C4C2×D21111yescharacteristic
113C2×C4⋊C4C2×D21111yescharacteristic
114C2×C42C2×D21111yescharacteristic
115#106—4222no—
116C4⋊C8—4222no—
117#95—21no—
118#95—4221no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—4221no—
122C22×C4—4221no—
123C22×C4C22⋊D21111yescharacteristic
124C22×C4C22⋊D21111yescharacteristic
125C22×C4C22×D21111yescharacteristic
126C2×C8—4221no—
127C2×C8—4221no—
128C2×C8—4221no—
129C2×C8—4221no—
130C2×C8—4221no—
131C2×C8—2112no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—42no—
136C4⋊C4—8442no—
137C4⋊C4—4221no—
138C4⋊C4—4221no—
139C4⋊C4—4221no—
140C4⋊C4—2112no—
141C4⋊C4Dic42212yes—
142C42—21no—
143C42—21no—
144C23C2×C22⋊D21111yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—8441no—
157C2×C4—8441no—
158C2×C4—8441no—
159C2×C4—8441no—
160C2×C4—8441no—
161C2×C4—4222no—
162C2×C4#1680994912yes—
163C2×C4D4×D21111yescharacteristic
164C2×C4Q8×D21111yescharacteristic
165C2×C4C2×Dic42211yescharacteristic
166C2×C4#1680996111yescharacteristic
167C2×C4#1680996211yescharacteristic
168C2×C4#1680996311yescharacteristic
169C8—8441no—
170C8—4222no—
171C22#2026994911yescharacteristic
172C22#2026995311yescharacteristic
173C22#2026995611yescharacteristic
174C22C2×#1680994911yescharacteristic
175C22#2026999711yescharacteristic
176C22#2027000311yescharacteristic
177C22#2027000711yescharacteristic
178C4—22no—
179C4—41no—
180C4—42no—
181C4—81no—
182C4—8442no—
183C4—16881no—
184C4#2026996211yescharacteristic
185C4#2026996711yescharacteristic
186C4#2026996811yescharacteristic
187C4#2026997611yescharacteristic
188C2#2721307911yescharacteristic
189C2#2721337711yescharacteristic
190C2#2721339011yescharacteristic
191C2#2721343111yescharacteristic
192C2#2721343311yescharacteristic
193C2#2721344611yescharacteristic
194C2#2721345011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#389×C211C2
socle, prime radicalC23×C211C2×D4
hypercenter#389D211
hypocenterC211#2088
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
97. C211#2088semidirect

Maximal subgroups

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

subgroupindex
C2×#202699082
C2×C4⋊(C211⋊C8)2
#272130442
#272129082
#389×C2112
#2088211