Finite Group Databasebeta

← #1612×C223 #43250627

Subgroups — #1612×C223 #43250627

Order 28544.

232 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 64 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#538×C223C211yesmaximal, characteristic
2#483×C223C211yesmaximal, characteristic, complemented
3#480×C223C211yesmaximal, characteristic, complemented
4#397×C223C211yesmaximal, characteristic, complemented
5#328×C223C211yesmaximal, characteristic, complemented
6#327×C223C211yesmaximal, characteristic, complemented
7#323×C223C211yesmaximal, characteristic, complemented
8#126×C223—21no—
9#124×C223—21no—
10#123×C223—21no—
11C22⋊Q8×C223C2211yescharacteristic
12#121×C223—21no—
13#121×C223C2211yescharacteristic
14#120×C223—21no—
15C4×D4×C223—21no—
16C2×C4⋊C4×C223C2211yescharacteristic
17C2×C22⋊C4×C223—21no—
18C2×C22⋊C4×C223C2211yescharacteristic
19#107×C223—21no—
20#106×C223—21no—
21#103×C223—21no—
22#102×C223—21no—
23C22⋊C8×C223C2211yescharacteristic
24C22⋊C8×C223C2211yescharacteristic
25#95×C223—21no—
26#95×C223C2211yescharacteristic
27C24×C223D411yescharacteristic
28C2×Q8×C223D411yescharacteristic
29C2×D4×C223—21no—
30C2×D4×C223—21no—
31C2×D4×C223—21no—
32C2×D4×C223D411yescharacteristic
33C22×C4×C223—21no—
34C22×C4×C223—21no—
35C22×C4×C223—21no—
36C22×C4×C223D411yescharacteristic
37C22×C4×C223C2311yescharacteristic
38C2×C8×C223—21no—
39C2×C8×C223—21no—
40C4⋊C4×C223—21no—
41C4⋊C4×C223—21no—
42C4⋊C4×C223—21no—
43C4⋊C4×C223—21no—
44C4⋊C4×C223D411yescharacteristic
45C4⋊C4×C223D411yescharacteristic
46C22⋊C4×C223—21no—
47C22⋊C4×C223—21no—
48C22⋊C4×C223—21no—
49C22⋊C4×C223—21no—
50C22⋊C4×C223—21no—
51C22⋊C4×C223—21no—
52C22⋊C4×C223—21no—
53C22⋊C4×C223—42no—
54C42×C223—21no—
55C23×C223—21no—
56C23×C223—21no—
57C23×C223—21no—
58C23×C223—21no—
59C23×C223—41no—
60C23×C223—41no—
61C23×C223C2×D411yescharacteristic
62Q8×C223—41no—
63D4×C223—41no—
64D4×C223—41no—
65D4×C223—41no—
66D4×C223—81no—
67D4×C223—81no—
68C2×C4×C223—21no—
69C2×C4×C223—21no—
70C2×C4×C223—21no—
71C2×C4×C223—21no—
72C2×C4×C223—41no—
73C2×C4×C223—41no—
74C2×C4×C223—41no—
75C2×C4×C223—41no—
76C2×C4×C223—41no—
77C2×C4×C223—41no—
78C2×C4×C223—41no—
79C2×C4×C223—41no—
80C2×C4×C223—81no—
81C2×C4×C223C2×D411yescharacteristic
82C2×C4×C223C2×D411yescharacteristic
83C8×C223—41no—
84C8×C223—41no—
85C22×C223—21no—
86C22×C223—21no—
87C22×C223—42no—
88C22×C223—41no—
89C22×C223—41no—
90C22×C223—41no—
91C22×C223—41no—
92C22×C223—41no—
93C22×C223—41no—
94C22×C223—81no—
95C22×C223#12011yescharacteristic
96C22×C223#13511yescharacteristic
97C22×C223AGL(1,ℤ/8)11yescharacteristic
98C4×C223—21no—
99C4×C223—21no—
100C4×C223—41no—
101C4×C223—41no—
102C4×C223—41no—
103C4×C223—41no—
104C4×C223—81no—
105C4×C223—81no—
106C2×C223—21no—
107C2×C223—21no—
108C2×C223—81no—
109C2×C223—81no—
110C2×C223#44911yescharacteristic
111C2×C223#45511yescharacteristic
112C2×C223#45811yescharacteristic
113C223#161211yescharacteristic, direct product factor, Sylow
114#1612C22311yesmaximal, characteristic, direct product factor, Sylow
115#538C2×C22311yescharacteristic
116#483C2×C22311yescharacteristic, complemented
117#480C2×C22311yescharacteristic, complemented
118#397C2×C22311yescharacteristic, complemented
119#328C2×C22311yescharacteristic, complemented
120#327C2×C22311yescharacteristic, complemented
121#323C2×C22311yescharacteristic, complemented
122#126—21no—
123#124—21no—
124#123—21no—
125C22⋊Q8C22×C22311yescharacteristic
126#121—21no—
127#121C22×C22311yescharacteristic
128#120—21no—
129C4×D4—21no—
130C2×C4⋊C4C22×C22311yescharacteristic
131C2×C22⋊C4—21no—
132C2×C22⋊C4C22×C22311yescharacteristic
133#107—21no—
134#106—21no—
135#103—21no—
136#102—21no—
137C22⋊C8C22×C22311yescharacteristic
138C22⋊C8C22×C22311yescharacteristic
139#95—21no—
140#95C22×C22311yescharacteristic
141C24D4×C22311yescharacteristic
142C2×Q8D4×C22311yescharacteristic
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4D4×C22311yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4D4×C22311yescharacteristic
151C22×C4C23×C22311yescharacteristic
152C2×C8—21no—
153C2×C8—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4D4×C22311yescharacteristic
159C4⋊C4D4×C22311yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—42no—
168C42—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23C2×D4×C22311yescharacteristic
176Q8—41no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—81no—
181D4—81no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4C2×D4×C22311yescharacteristic
196C2×C4C2×D4×C22311yescharacteristic
197C8—41no—
198C8—41no—
199C22—21no—
200C22—21no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—81no—
209C22#120×C22311yescharacteristic
210C22#135×C22311yescharacteristic
211C22AGL(1,ℤ/8)×C22311yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—81no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—81no—
224C2#449×C22311yescharacteristic
225C2#455×C22311yescharacteristic
226C2#458×C22311yescharacteristic
subgroupG/H
center, socleC22×C223#120
commutator, Frattini, cosocleC22×C4C23×C223
prime radical#538×C223C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C223 ◁ C22×C4×C223 ◁ G

Complements

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

NH ≅ G/Ntype
2. #483×C223C2semidirect
3. #480×C223C2semidirect
4. #397×C223C2semidirect
5. #328×C223C2semidirect
6. #327×C223C2semidirect
7. #323×C223C2semidirect
113. C223#1612direct (N × H)
114. #1612C223direct (N × H)
116. #483C2×C223semidirect
117. #480C2×C223semidirect
118. #397C2×C223semidirect
119. #328C2×C223semidirect
120. #327C2×C223semidirect
121. #323C2×C223semidirect

Maximal subgroups

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

subgroupindex
#538×C2232
#483×C2232
#480×C2232
#397×C2232
#328×C2232
#327×C2232
#323×C2232
#1612223