Finite Group Databasebeta

← #13143937

Subgroups — #13143937

Order 1728.

561 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 111 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
1D4×#1084C211yesmaximal, characteristic
2#11721758C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic
4#11720578C211yesmaximal, characteristic
5#11720551C211yesmaximal, characteristic, complemented
6#11720547C211yesmaximal, characteristic, complemented
7#11720530C211yesmaximal, characteristic, complemented
8C3×#4766—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084—21no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13#88552—21no—
14C4⋊#1084—21no—
15C4⋊#1084C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17#88540—21no—
18C4×#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×D4×Dic3—34no—
21C3×#890—34no—
22C32×#102D314yes—
23C3×#832—34no—
24C3×#818—34no—
25C3×Dic3⋊C8—34no—
26C3×#808—34no—
27C23×H(3,𝔽3)—21no—
28D4×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)Q812yes—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C2×#1084—21no—
34C2×#1084—42no—
35C2×#1084D411yescharacteristic
36C2×#1084D411yescharacteristic
37C8×H(3,𝔽3)—21no—
38#6180—21no—
39C3×#478—91no—
40#4766—34no—
41C2×C32×D4C2×D314yes—
42C22×C3×Dic3—64no—
43C2×C32×C8C2×D314yes—
44C32×C4⋊C4C2×D314yes—
45C3×#217—64no—
46C3×C4⋊Dic3—34no—
47C3×C4⋊Dic3—34no—
48C3×C4⋊Dic3—64no—
49C3×Dic3⋊C4—64no—
50C3×C4×Dic3—34no—
51C2×C3×C3⋊C8—34no—
52C22×H(3,𝔽3)—22no—
53C22×H(3,𝔽3)—41no—
54C22×H(3,𝔽3)—41no—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C4×H(3,𝔽3)—41no—
57C4×H(3,𝔽3)C2×Q811yescharacteristic
58C4×H(3,𝔽3)#4111yescharacteristic
59#1084—21no—
60#1084—21no—
61#1084—41no—
62#1084—41no—
63C3×#125—91no—
64C3×C4×D4—91no—
65D4×Dic3—34no—
66#890—34no—
67C3×#107—91no—
68C3×#106—91no—
69C3×C4⋊C8—91no—
70C3×#102—34no—
71C3×#102—91no—
72C3×#102C3⋊D311yescharacteristic
73#832—34no—
74#818—34no—
75Dic3⋊C8—34no—
76#808—34no—
77C23×C32—24no—
78C32×D4—24no—
79C32×D4Dic618yes—
80C2×C32×C4—24no—
81C2×C32×C4C22×D314yes—
82C2×C3×Dic3—34no—
83C2×C3×Dic3—34no—
84C2×C3×Dic3—64no—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—128no—
87C32×C8—24no—
88C3×C3⋊C8—64no—
89#478—91noSylow
90C2×H(3,𝔽3)—42no—
91C2×H(3,𝔽3)C22⋊Q811yescharacteristic
92C2×H(3,𝔽3)#13511yescharacteristic
93C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
94C2×C3×D4—34no—
95C2×C3×D4C2×C3⋊D311yescharacteristic
96C22×C3×C4—181no—
97C22×Dic3—64no—
98C2×C3×C8—34no—
99C2×C3×C8—91no—
100C2×C3×C8C2×C3⋊D311yescharacteristic
101C3×C4⋊C4—34no—
102C3×C4⋊C4—91no—
103C3×C4⋊C4—91no—
104C3×C4⋊C4—181no—
105C3×C4⋊C4—181no—
106C3×C4⋊C4C2×C3⋊D311yescharacteristic
107C3×C22⋊C4—181no—
108C3×C42—91no—
109#217—64no—
110C4⋊Dic3—34no—
111C4⋊Dic3—34no—
112C4⋊Dic3—64no—
113Dic3⋊C4—64no—
114C4×Dic3—34no—
115C2×C3⋊C8—34no—
116C22×C32—28no—
117C22×C32—44no—
118C22×C32—44no—
119C22×C32D3×D414yes—
120C32×C4—44no—
121C32×C4C2×Dic614yes—
122C32×C4#23714yes—
123C3×Dic3—64no—
124C3×Dic3—64no—
125C3×Dic3—124no—
126C3×Dic3—124no—
127#125—91no—
128C4×D4—91no—
129#107—91no—
130#106—91no—
131C4⋊C8—91no—
132#102—91no—
133#102#27211yescharacteristic
134H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
135C23×C3—21no—
136C23×C3—64no—
137C3×D4—21no—
138C3×D4—38no—
139C3×D4—64no—
140C3×D4#63412yes—
141C2×C3×C4—21no—
142C2×C3×C4—34no—
143C2×C3×C4—64no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—181no—
147C2×C3×C4—181no—
148C2×C3×C4—362no—
149C2×C3×C4C22×C3⋊D311yescharacteristic
150C2×Dic3—34no—
151C2×Dic3—34no—
152C2×Dic3—64no—
153C2×Dic3—64no—
154C2×Dic3—128no—
155C3×C8—21no—
156C3×C8—64no—
157C3×C8—181no—
158C3⋊C8—64no—
159C2×C32—48no—
160C2×C32#87814yes—
161C2×C32#91214yes—
162C2×C32#91414yes—
163C2×D4C2×#27211yescharacteristic
164C22×C4—181no—
165C2×C8—91no—
166C2×C8C2×#27211yescharacteristic
167C4⋊C4—91no—
168C4⋊C4—91no—
169C4⋊C4—181no—
170C4⋊C4—181no—
171C4⋊C4C2×#27211yescharacteristic
172C22⋊C4—181no—
173C42—91no—
174C22×C3—22no—
175C22×C3—34no—
176C22×C3—41no—
177C22×C3—41no—
178C22×C3—68no—
179C22×C3—124no—
180C22×C3—124no—
181C22×C3D4×C3⋊D311yescharacteristic
182C3×C4—34no—
183C3×C4—34no—
184C3×C4—41no—
185C3×C4—124no—
186C3×C4—181no—
187C3×C4—181no—
188C3×C4—361no—
189C3×C4—361no—
190C3×C4C2×#63411yescharacteristic
191C3×C4#383011yescharacteristic
192Dic3—64no—
193Dic3—64no—
194Dic3—124no—
195Dic3—124no—
196C32#476614yescomplemented
197C23—21no—
198D4—21no—
199D4#622912yes—
200C2×C4—21no—
201C2×C4—91no—
202C2×C4—91no—
203C2×C4—181no—
204C2×C4—181no—
205C2×C4—362no—
206C2×C4C22×#27211yescharacteristic
207C8—21no—
208C8—181no—
209C2×C3—34no—
210C2×C3—34no—
211C2×C3—34no—
212C2×C3—42no—
213C2×C3—128no—
214C2×C3#6408711yescharacteristic
215C2×C3#6412111yescharacteristic
216C2×C3#6412311yescharacteristic
217C22—22no—
218C22—41no—
219C22—41no—
220C22D4×#27211yescharacteristic
221C4—41no—
222C4—181no—
223C4—181no—
224C4—361no—
225C4—361no—
226C4C2×#622911yescharacteristic
227C4#8874411yescharacteristic
228C3—34no—
229C3#1059246511yescharacteristic
230C2—42no—
231C2#1172173411yescharacteristic
232C2#1172179411yescharacteristic
233C2#1172180011yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#478
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. #11721758C2semidirect
5. #11720551C2semidirect
6. #11720547C2semidirect
7. #11720530C2semidirect
134. H(3,𝔽3)#478semidirect
196. C32#4766semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
#117217582
H(3,𝔽3)×#1022
#117205782
#117205512
#117205472
#117205302
C3×#47663