Finite Group Databasebeta

← #1647×C211 #41009267

Subgroups — #1647×C211 #41009267

Order 27008.

240 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 72 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
1#547×C211C211yesmaximal, characteristic
2#500×C211C211yesmaximal, characteristic, complemented
3#474×C211C211yesmaximal, characteristic, complemented
4#461×C211C211yesmaximal, characteristic, complemented
5#332×C211C211yesmaximal, characteristic, complemented
6D4⋊C8×C211C211yesmaximal, characteristic, complemented
7D4⋊C8×C211C211yesmaximal, characteristic, complemented
8C2×#41×C211—21no—
9C2×SD16×C211—21no—
10C4⋊Q8×C211C2211yescharacteristic
11C4⋊Q8×C211C2211yescharacteristic
12#123×C211—22no—
13C22⋊Q8×C211—21no—
14#121×C211—21no—
15C4×D4×C211C2211yescharacteristic
16C4×D4×C211C2211yescharacteristic
17C2×C4⋊C4×C211—21no—
18#107×C211—21no—
19#107×C211—22no—
20C4⋊C8×C211C2211yescharacteristic
21C4⋊C8×C211C2211yescharacteristic
22#103×C211—21no—
23#102×C211—21no—
24C4×C8×C211C2211yescharacteristic
25#41×C211—42no—
26#41×C211—42no—
27C2×Q8×C211—21no—
28C2×Q8×C211—21no—
29C2×D4×C211—22no—
30C2×D4×C211D411yescharacteristic
31C2×D4×C211D411yescharacteristic
32C22×C4×C211—22no—
33C22×C4×C211—21no—
34C22×C4×C211—21no—
35SD16×C211—42no—
36C2×C8×C211—21no—
37C2×C8×C211—21no—
38C2×C8×C211—21no—
39C4⋊C4×C211—22no—
40C4⋊C4×C211—21no—
41C4⋊C4×C211—21no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211D411yescharacteristic
44C4⋊C4×C211D411yescharacteristic
45C4⋊C4×C211D412yes—
46C22⋊C4×C211—22no—
47C22⋊C4×C211—21no—
48C22⋊C4×C211—21no—
49C42×C211C2311yescharacteristic
50C23×C211—21no—
51C23×C211—21no—
52Q8×C211—41no—
53Q8×C211—41no—
54Q8×C211—41no—
55Q8×C211—41no—
56D4×C211—21no—
57D4×C211—21no—
58D4×C211—21no—
59D4×C211—44no—
60D4×C211D812yes—
61C2×C4×C211—21no—
62C2×C4×C211—21no—
63C2×C4×C211—22no—
64C2×C4×C211—44no—
65C2×C4×C211—41no—
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×D411yescharacteristic
73C8×C211—22no—
74C8×C211—41no—
75C8×C211—41no—
76C22×C211—22no—
77C22×C211—41no—
78C22×C211—41no—
79C22×C211—41no—
80C22×C211—41no—
81C22×C211—41no—
82C22×C211#12011yescharacteristic
83C4×C211—21no—
84C4×C211—41no—
85C4×C211—42no—
86C4×C211—41no—
87C4×C211—81no—
88C4×C211C2×D811yescharacteristic
89C4×C211#13511yescharacteristic
90C4×C211AGL(1,ℤ/8)11yescharacteristic
91C4×C211#13711yescharacteristic
92C2×C211—42no—
93C2×C211—81no—
94C2×C211#44711yescharacteristic
95C2×C211#45211yescharacteristic
96C2×C211#45611yescharacteristic
97C211#164711yescharacteristic, direct product factor, Sylow
98#1647C21111yesmaximal, characteristic, direct product factor, Sylow
99#547C2×C21111yescharacteristic
100#500C2×C21111yescharacteristic, complemented
101#474C2×C21111yescharacteristic, complemented
102#461C2×C21111yescharacteristic, complemented
103#332C2×C21111yescharacteristic, complemented
104D4⋊C8C2×C21111yescharacteristic, complemented
105D4⋊C8C2×C21111yescharacteristic, complemented
106C2×#41—21no—
107C2×SD16—21no—
108C4⋊Q8C22×C21111yescharacteristic
109C4⋊Q8C22×C21111yescharacteristic
110#123—22no—
111C22⋊Q8—21no—
112#121—21no—
113C4×D4C22×C21111yescharacteristic
114C4×D4C22×C21111yescharacteristic
115C2×C4⋊C4—21no—
116#107—21no—
117#107—22no—
118C4⋊C8C22×C21111yescharacteristic
119C4⋊C8C22×C21111yescharacteristic
120#103—21no—
121#102—21no—
122C4×C8C22×C21111yescharacteristic
123#41—42no—
124#41—42no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×D4—22no—
128C2×D4D4×C21111yescharacteristic
129C2×D4D4×C21111yescharacteristic
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133SD16—42no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4D4×C21111yescharacteristic
142C4⋊C4D4×C21111yescharacteristic
143C4⋊C4D4×C21112yes—
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C42C23×C21111yescharacteristic
148C23—21no—
149C23—21no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—21no—
155D4—21no—
156D4—21no—
157D4—44no—
158D4D8×C21112yes—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4C2×D4×C21111yescharacteristic
169C2×C4C2×D4×C21111yescharacteristic
170C2×C4C2×D4×C21111yescharacteristic
171C8—22no—
172C8—41no—
173C8—41no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22#120×C21111yescharacteristic
181C4—21no—
182C4—41no—
183C4—42no—
184C4—41no—
185C4—81no—
186C4C2×D8×C21111yescharacteristic
187C4#135×C21111yescharacteristic
188C4AGL(1,ℤ/8)×C21111yescharacteristic
189C4#137×C21111yescharacteristic
190C2—42no—
191C2—81no—
192C2#447×C21111yescharacteristic
193C2#452×C21111yescharacteristic
194C2#456×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC42C23×C211
prime radical#547×C211C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #500×C211C2semidirect
3. #474×C211C2semidirect
4. #461×C211C2semidirect
5. #332×C211C2semidirect
6. D4⋊C8×C211C2semidirect
7. D4⋊C8×C211C2semidirect
97. C211#1647direct (N × H)
98. #1647C211direct (N × H)
100. #500C2×C211semidirect
101. #474C2×C211semidirect
102. #461C2×C211semidirect
103. #332C2×C211semidirect
104. D4⋊C8C2×C211semidirect
105. D4⋊C8C2×C211semidirect

Maximal subgroups

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

subgroupindex
#547×C2112
#500×C2112
#474×C2112
#461×C2112
#332×C2112
D4⋊C8×C2112
#1647211