Finite Group Databasebeta

← #2085×C211 #41009705

Subgroups — #2085×C211 #41009705

Order 27008.

214 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 100 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×#107×C211C211yesmaximal, characteristic
2C2×#106×C211C211yesmaximal, characteristic
3#389×C211C211yesmaximal, characteristic
4#382×C211C211yesmaximal, characteristic
5#340×C211C211yesmaximal, characteristic
6#340×C211C211yesmaximal, characteristic
7#336×C211C211yesmaximal, characteristic
8C22×C8×C211C2211yescharacteristic
9C22×C8×C211C2211yescharacteristic
10C2×C4⋊C4×C211—21no—
11C2×C4⋊C4×C211C2211yescharacteristic
12C2×C4⋊C4×C211C2211yescharacteristic
13C2×C4⋊C4×C211C2211yescharacteristic
14C2×C4⋊C4×C211C2211yescharacteristic
15C2×C42×C211C2211yescharacteristic
16#107×C211—22no—
17#106×C211—22no—
18#95×C211—21no—
19#95×C211—21no—
20#95×C211—21no—
21C22×C4×C211—21no—
22C22×C4×C211—21no—
23C22×C4×C211—21no—
24C22×C4×C211—21no—
25C22×C4×C211D411yescharacteristic
26C22×C4×C211D411yescharacteristic
27C22×C4×C211C2311yescharacteristic
28C2×C8×C211—21no—
29C2×C8×C211—21no—
30C2×C8×C211—21no—
31C2×C8×C211—21no—
32C2×C8×C211Q812yes—
33C2×C8×C211Q812yes—
34C4⋊C4×C211—21no—
35C4⋊C4×C211—21no—
36C4⋊C4×C211—21no—
37C4⋊C4×C211—21no—
38C4⋊C4×C211—21no—
39C4⋊C4×C211—21no—
40C4⋊C4×C211—21no—
41C4⋊C4×C211—42no—
42C4⋊C4×C211D412yes—
43C42×C211—21no—
44C42×C211—21no—
45C23×C211C2×D411yescharacteristic
46C2×C4×C211—22no—
47C2×C4×C211—21no—
48C2×C4×C211—21no—
49C2×C4×C211—21no—
50C2×C4×C211—21no—
51C2×C4×C211—21no—
52C2×C4×C211—21no—
53C2×C4×C211—41no—
54C2×C4×C211—41no—
55C2×C4×C211—41no—
56C2×C4×C211—41no—
57C2×C4×C211—41no—
58C2×C4×C211—41no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211C2×D411yescharacteristic
65C2×C4×C211C2×Q811yescharacteristic
66C2×C4×C211C2×Q811yescharacteristic
67C2×C4×C211#4111yescharacteristic
68C2×C4×C211#4111yescharacteristic
69C2×C4×C211#4111yescharacteristic
70C8×C211—22no—
71C8×C211—22no—
72C22×C211#12111yescharacteristic
73C22×C211#12311yescharacteristic
74C22×C211C4⋊Q811yescharacteristic
75C22×C211#13511yescharacteristic
76C22×C211#13511yescharacteristic
77C22×C211AGL(1,ℤ/8)11yescharacteristic
78C22×C211#13711yescharacteristic
79C4×C211—42no—
80C4×C211—41no—
81C4×C211—41no—
82C4×C211—81no—
83C4×C211—81no—
84C4×C211—81no—
85C4×C211C22⋊Q811yescharacteristic
86C4×C211C22⋊Q811yescharacteristic
87C4×C211#12511yescharacteristic
88C4×C211#12511yescharacteristic
89C2×C211#39811yescharacteristic
90C2×C211#46311yescharacteristic
91C2×C211#46411yescharacteristic
92C2×C211#48211yescharacteristic
93C2×C211#48511yescharacteristic
94C2×C211#49911yescharacteristic
95C2×C211#50111yescharacteristic
96C211#208511yescharacteristic, direct product factor, Sylow
97#2085C21111yesmaximal, characteristic, direct product factor, Sylow
98C2×#107C2×C21111yescharacteristic
99C2×#106C2×C21111yescharacteristic
100#389C2×C21111yescharacteristic
101#382C2×C21111yescharacteristic
102#340C2×C21111yescharacteristic
103#340C2×C21111yescharacteristic
104#336C2×C21111yescharacteristic
105C22×C8C22×C21111yescharacteristic
106C22×C8C22×C21111yescharacteristic
107C2×C4⋊C4—21no—
108C2×C4⋊C4C22×C21111yescharacteristic
109C2×C4⋊C4C22×C21111yescharacteristic
110C2×C4⋊C4C22×C21111yescharacteristic
111C2×C4⋊C4C22×C21111yescharacteristic
112C2×C42C22×C21111yescharacteristic
113#107—22no—
114#106—22no—
115#95—21no—
116#95—21no—
117#95—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4D4×C21111yescharacteristic
123C22×C4D4×C21111yescharacteristic
124C22×C4C23×C21111yescharacteristic
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8Q8×C21112yes—
130C2×C8Q8×C21112yes—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—42no—
139C4⋊C4D4×C21112yes—
140C42—21no—
141C42—21no—
142C23C2×D4×C21111yescharacteristic
143C2×C4—22no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4C2×D4×C21111yescharacteristic
162C2×C4C2×Q8×C21111yescharacteristic
163C2×C4C2×Q8×C21111yescharacteristic
164C2×C4#41×C21111yescharacteristic
165C2×C4#41×C21111yescharacteristic
166C2×C4#41×C21111yescharacteristic
167C8—22no—
168C8—22no—
169C22#121×C21111yescharacteristic
170C22#123×C21111yescharacteristic
171C22C4⋊Q8×C21111yescharacteristic
172C22#135×C21111yescharacteristic
173C22#135×C21111yescharacteristic
174C22AGL(1,ℤ/8)×C21111yescharacteristic
175C22#137×C21111yescharacteristic
176C4—42no—
177C4—41no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—81no—
182C4C22⋊Q8×C21111yescharacteristic
183C4C22⋊Q8×C21111yescharacteristic
184C4#125×C21111yescharacteristic
185C4#125×C21111yescharacteristic
186C2#398×C21111yescharacteristic
187C2#463×C21111yescharacteristic
188C2#464×C21111yescharacteristic
189C2#482×C21111yescharacteristic
190C2#485×C21111yescharacteristic
191C2#499×C21111yescharacteristic
192C2#501×C21111yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C211C2×D4
commutator, Frattini, cosocleC22×C4C23×C211
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C211 ◁ C2×C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
96. C211#2085direct (N × H)
97. #2085C211direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#107×C2112
C2×#106×C2112
#389×C2112
#382×C2112
#340×C2112
#336×C2112
#2085211