Finite Group Databasebeta

← #13144482

Subgroups — #13144482

Order 1728.

528 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 109 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
1H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic
2#11721826C211yesmaximal, characteristic
3#11720583C211yesmaximal, characteristic, complemented
4#11720583C211yesmaximal, characteristic, complemented
5#11720573C211yesmaximal, characteristic
6#11720530C211yesmaximal, characteristic, complemented
7#11720529C211yesmaximal, characteristic, complemented
8C3×#5045—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)—21no—
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C22⋊C4×H(3,𝔽3)—21no—
14#88552—21no—
15C4⋊#1084C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17C4×#1084—21no—
18C2×#6180C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C32×C22⋊Q8D314yes—
21C3×#926—34no—
22C3×#835—34no—
23C3×#835—34no—
24C3×#830—34no—
25C3×#808—34no—
26C3×#807—34no—
27C23×H(3,𝔽3)D411yescharacteristic
28Q8×H(3,𝔽3)—21no—
29Q8×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)D411yescharacteristic
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×#1084—21no—
37C2×#1084—21no—
38#6180—21no—
39#6180—21no—
40C3×#485—91no—
41#5045—34no—
42C2×C32×Q8C2×D314yes—
43C22×C32×C4C2×D314yes—
44C32×C4⋊C4—24no—
45C32×C4⋊C4C2×D314yes—
46C32×C22⋊C4—24no—
47C3×#217—64no—
48C3×C4⋊Dic3—34no—
49C3×C4⋊Dic3—34no—
50C3×C4×Dic3—64no—
51C2×C3×C3⋊C8—34no—
52C2×C3×C3⋊C8—34no—
53C22×H(3,𝔽3)—21no—
54C22×H(3,𝔽3)—21no—
55C22×H(3,𝔽3)—21no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)—41no—
59C4×H(3,𝔽3)—41no—
60C4×H(3,𝔽3)#4111yescharacteristic
61C4×H(3,𝔽3)#4111yescharacteristic
62#1084—41no—
63#1084—41no—
64C3×C22⋊Q8—34no—
65C3×C22⋊Q8C3⋊D311yescharacteristic
66C3×#117—91no—
67#926—34no—
68C3×#107—91no—
69C3×#106—91no—
70C3×#103—91no—
71C3×#103—91no—
72C3×C22⋊C8—91no—
73#835—34no—
74#835—34no—
75#830—34no—
76#808—34no—
77#807—34no—
78C23×C32#6714yes—
79C32×Q8—24no—
80C32×Q8—24no—
81C2×C32×C4—24no—
82C2×C32×C4—24no—
83C2×C32×C4—24no—
84C2×C32×C4—24no—
85C2×C32×C4#6714yes—
86C2×C32×C4C22×D314yes—
87C2×C3×Dic3—64no—
88C2×C3×Dic3—64no—
89C3×C3⋊C8—64no—
90C3×C3⋊C8—64no—
91#485—91noSylow
92C2×H(3,𝔽3)—41no—
93C2×H(3,𝔽3)#12311yescharacteristic
94C2×H(3,𝔽3)#13511yescharacteristic
95C2×H(3,𝔽3)#13711yescharacteristic
96C2×C3×Q8—34no—
97C2×C3×Q8C2×C3⋊D311yescharacteristic
98C22×C3×C4—34no—
99C22×C3×C4C2×C3⋊D311yescharacteristic
100C2×C3×C8—91no—
101C2×C3×C8—91no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4—34no—
104C3×C4⋊C4—64no—
105C3×C4⋊C4—91no—
106C3×C4⋊C4—91no—
107C3×C4⋊C4C2×C3⋊D311yescharacteristic
108C3×C22⋊C4—21no—
109C3×C22⋊C4—64no—
110C3×C22⋊C4—181no—
111C3×C42—181no—
112#217—64no—
113C4⋊Dic3—34no—
114C4⋊Dic3—34no—
115C4×Dic3—64no—
116C2×C3⋊C8—34no—
117C2×C3⋊C8—34no—
118C22×C32—24no—
119C22×C32—24no—
120C22×C32—24no—
121C22×C32C2×#6714yes—
122C32×C4—24no—
123C32×C4—44no—
124C32×C4—44no—
125C32×C4#23714yes—
126C32×C4#23714yes—
127C3×Dic3—124no—
128C3×Dic3—124no—
129C22⋊Q8#27211yescharacteristic
130#117—91no—
131#107—91no—
132#106—91no—
133#103—91no—
134#103—91no—
135C22⋊C8—91no—
136H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
137C23×C3—34no—
138C23×C3#63811yescharacteristic
139C3×Q8—21no—
140C3×Q8—21no—
141C3×Q8—64no—
142C3×Q8—64no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—34no—
148C2×C3×C4—34no—
149C2×C3×C4—64no—
150C2×C3×C4—64no—
151C2×C3×C4—64no—
152C2×C3×C4—64no—
153C2×C3×C4—181no—
154C2×C3×C4—181no—
155C2×C3×C4#63811yescharacteristic
156C2×C3×C4C22×C3⋊D311yescharacteristic
157C2×Dic3—64no—
158C2×Dic3—64no—
159C3×C8—181no—
160C3×C8—181no—
161C3⋊C8—64no—
162C3⋊C8—64no—
163C2×C32—44no—
164C2×C32#93514yes—
165C2×C32#94214yes—
166C2×C32#95014yes—
167C2×Q8C2×#27211yescharacteristic
168C22×C4C2×#27211yescharacteristic
169C2×C8—91no—
170C2×C8—91no—
171C4⋊C4—21no—
172C4⋊C4—91no—
173C4⋊C4—91no—
174C4⋊C4C2×#27211yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—181no—
177C42—181no—
178C22×C3—21no—
179C22×C3—21no—
180C22×C3—21no—
181C22×C3—34no—
182C22×C3—64no—
183C22×C3—64no—
184C22×C3—64no—
185C22×C3C2×#63811yescharacteristic
186C3×C4—21no—
187C3×C4—34no—
188C3×C4—34no—
189C3×C4—41no—
190C3×C4—41no—
191C3×C4—64no—
192C3×C4—124no—
193C3×C4—124no—
194C3×C4—361no—
195C3×C4—361no—
196C3×C4#383011yescharacteristic
197C3×C4#383011yescharacteristic
198Dic3—124no—
199Dic3—124no—
200C32#504514yescomplemented
201C23#623511yescharacteristic
202Q8—21no—
203Q8—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—181no—
209C2×C4—181no—
210C2×C4#623511yescharacteristic
211C2×C4C22×#27211yescharacteristic
212C8—181no—
213C8—181no—
214C2×C3—34no—
215C2×C3—34no—
216C2×C3—34no—
217C2×C3—41no—
218C2×C3—124no—
219C2×C3#6414411yescharacteristic
220C2×C3#6415111yescharacteristic
221C2×C3#6415911yescharacteristic
222C22—21no—
223C22—21no—
224C22—21no—
225C22C2×#623511yescharacteristic
226C4—21no—
227C4—41no—
228C4—41no—
229C4—361no—
230C4—361no—
231C4#8874411yescharacteristic
232C4#8874411yescharacteristic
233C3—34no—
234C3#1059274411yescharacteristic
235C2—41no—
236C2#1172184511yescharacteristic
237C2#1172185911yescharacteristic
238C2#1172187111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×C22⋊Q8C2
prime radicalC23×H(3,𝔽3)D4
hypercenterC3×C22⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#485
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #11720583C2semidirect
4. #11720583C2semidirect
6. #11720530C2semidirect
7. #11720529C2semidirect
136. H(3,𝔽3)#485semidirect
200. C32#5045semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
#117218262
#117205832
#117205732
#117205302
#117205292
C3×#50453