Finite Group Databasebeta

← #13143995

Subgroups — #13143995

Order 1728.

682 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 99 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#11721864C211yesmaximal, characteristic, complemented
2C2×#88549C211yesmaximal, characteristic, complemented
3C2×#88529C211yesmaximal, characteristic
4#11721761C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#11720547C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#4819—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#88549—22no—
16#88542—21no—
17#88542—21no—
18#88542—21no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21#88529—22no—
22C3×#947—34no—
23C2×C3×#215—34no—
24C2×C3×D24—34no—
25C3×#893—34no—
26C32×#103D314yescomplemented
27C3×Dic3⋊C8—34no—
28C3×#809—34no—
29C22×#272—21no—
30C22×#272—21no—
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×#1084D411yescharacteristic
37C2×#1084D411yescharacteristic
38C4⋊#272—21no—
39C4⋊#272—21no—
40C4⋊#272—21no—
41C4⋊#272D412yes—
42C4×#272—42no—
43C8×H(3,𝔽3)—21no—
44#6180—21no—
45C3×#463—91no—
46#4819—34no—
47C2×C32×Q8C2×D314yes—
48C2×C3×D12—34no—
49C2×C3×D12—34no—
50C2×C3×C4×D3—64no—
51C2×C32×C8C2×D314yes—
52C32×C4⋊C4C2×D314yes—
53C3×#215—68no—
54C3×(C2×D3)⋊C4—64no—
55C3×(C2×D3)⋊C4—64no—
56C3×(C2×D3)⋊C4—64no—
57C3×C4×Dic3—34no—
58C2×C3×C3⋊C8—34no—
59C3×D24—68no—
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C2×#272—22no—
62C2×#272—41no—
63C2×#272—41no—
64C2×#272—41no—
65C2×#272—41no—
66C2×#272—41no—
67C4×H(3,𝔽3)—41no—
68C4×H(3,𝔽3)—41no—
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)#4111yescharacteristic
71#1084—21no—
72#1084—21no—
73C2×C3×SD16—91no—
74C2×C3×D8—91no—
75C3×#124—91no—
76C3×C4×D4—91no—
77#947—34no—
78C2×#215—34no—
79C2×D24—34no—
80#893—34no—
81C3×C4⋊C8—91no—
82C3×#103—34no—
83C3×#103C3⋊D311yescharacteristic
84C3×#102—91no—
85Dic3⋊C8—34no—
86#809—34no—
87C22×C3×D3—64no—
88C22×C3×D3—64no—
89C32×Q8—24no—
90C32×Q8—24no—
91C2×C32×C4—24no—
92C2×C32×C4—24no—
93C2×C32×C4C22×D314yes—
94C2×C3×Dic3—34no—
95C2×C3×Dic3—34no—
96C3×D12—38no—
97C3×D12—64no—
98C3×D12—64no—
99C3×D12—64no—
100C3×C4×D3—128no—
101C32×C8—24no—
102C3×C3⋊C8—64no—
103#463—91noSylow
104C2×H(3,𝔽3)#12111yescharacteristic
105C2×H(3,𝔽3)#13511yescharacteristic
106C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
107#272—42no—
108#272—81no—
109C2×C3×Q8—34no—
110C2×C3×Q8C2×C3⋊D311yescharacteristic
111C2×C3×D4—91no—
112C2×C3×D4—91no—
113C22×C3×C4—181no—
114C2×D12—34no—
115C2×D12—34no—
116C2×C4×D3—64no—
117C3×SD16—182no—
118C3×D8—182no—
119C2×C3×C8—34no—
120C2×C3×C8—91no—
121C2×C3×C8C2×C3⋊D311yescharacteristic
122C3×C4⋊C4—34no—
123C3×C4⋊C4C2×C3⋊D311yescharacteristic
124C3×C22⋊C4—181no—
125C3×C22⋊C4—181no—
126C3×C22⋊C4—181no—
127C3×C42—91no—
128#215—68no—
129(C2×D3)⋊C4—64no—
130(C2×D3)⋊C4—64no—
131(C2×D3)⋊C4—64no—
132C4×Dic3—34no—
133C2×C3⋊C8—34no—
134D24—68no—
135C22×C32D3×D414yes—
136C2×C3×D3—68no—
137C2×C3×D3—124no—
138C2×C3×D3—124no—
139C2×C3×D3—124no—
140C2×C3×D3—124no—
141C2×C3×D3—124no—
142C32×C4—44no—
143C32×C4—44no—
144C32×C4#23514yes—
145C32×C4D3×D414yes—
146C3×Dic3—64no—
147C3×Dic3—64no—
148C2×SD16—91no—
149C2×D8—91no—
150#124—91no—
151C4×D4—91no—
152C4⋊C8—91no—
153#103#27211yescharacteristic, complemented
154#102—91no—
155H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
156C23×C3—181no—
157C23×C3—181no—
158C22×D3—64no—
159C22×D3—64no—
160C3×Q8—21no—
161C3×Q8—21no—
162C3×Q8—64no—
163C3×Q8—64no—
164C3×D4—92no—
165C3×D4—181no—
166C3×D4—181no—
167C3×D4—181no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—34no—
171C2×C3×C4—64no—
172C2×C3×C4—64no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4—362no—
176C2×C3×C4C22×C3⋊D311yescharacteristic
177C2×Dic3—34no—
178C2×Dic3—34no—
179D12—38no—
180D12—64no—
181D12—64no—
182D12—64no—
183C4×D3—128no—
184C3×C8—21no—
185C3×C8—64no—
186C3×C8—181no—
187C3⋊C8—64no—
188C2×C32#88414yes—
189C2×C32#91414yes—
190C2×C32#91914yes—
191C3×D3—128no—
192C3×D3—244no—
193C2×Q8C2×#27211yescharacteristic
194C2×D4—91no—
195C2×D4—91no—
196C22×C4—181no—
197SD16—182no—
198D8—182no—
199C2×C8—91no—
200C2×C8C2×#27211yescharacteristic
201C4⋊C4C2×#27211yescharacteristic
202C22⋊C4—181no—
203C22⋊C4—181no—
204C22⋊C4—181no—
205C42—91no—
206C22×C3—34no—
207C22×C3—182no—
208C22×C3—361no—
209C22×C3—361no—
210C22×C3—361no—
211C22×C3—361no—
212C22×C3—361no—
213C22×C3D4×C3⋊D311yescharacteristic
214C2×D3—68no—
215C2×D3—124no—
216C2×D3—124no—
217C2×D3—124no—
218C2×D3—124no—
219C2×D3—124no—
220C3×C4—34no—
221C3×C4—34no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4—124no—
225C3×C4—124no—
226C3×C4—181no—
227C3×C4—181no—
228C3×C4#382811yescharacteristic
229C3×C4D4×C3⋊D311yescharacteristic
230Dic3—64no—
231Dic3—64no—
232C32#481914yescomplemented
233C23—181no—
234C23—181no—
235Q8—21no—
236Q8—21no—
237D4—92no—
238D4—181no—
239D4—181no—
240D4—181no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—91no—
244C2×C4—91no—
245C2×C4—362no—
246C2×C4C22×#27211yescharacteristic
247C8—21no—
248C8—181no—
249C2×C3—34no—
250C2×C3—34no—
251C2×C3—34no—
252C2×C3—362no—
253C2×C3—721no—
254C2×C3#6409311yescharacteristic
255C2×C3#6412311yescharacteristic
256C2×C3#6412811yescharacteristic
257D3—128no—
258D3—244no—
259C22—182no—
260C22—361no—
261C22—361no—
262C22—361no—
263C22—361no—
264C22—361no—
265C22D4×#27211yescharacteristic
266C4—41no—
267C4—41no—
268C4—181no—
269C4—181no—
270C4#8874011yescharacteristic
271C4D4×#27211yescharacteristic
272C3—34no—
273C3#1059251811yescharacteristic
274C2—362no—
275C2—721no—
276C2#1172174011yescharacteristic
277C2#1172180011yescharacteristic
278C2#1172180811yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC2×#88529C2
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#463
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
1. #11721864C2semidirect
2. C2×#88549C2semidirect
4. #11721761C2semidirect
5. H(3,𝔽3)×#103C2semidirect
6. #11720547C2semidirect
7. #11720531C2semidirect
26. C32×#103D3semidirect
153. #103#272semidirect
155. H(3,𝔽3)#463semidirect
232. C32#4819semidirect

Maximal subgroups

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

subgroupindex
#117218642
C2×#885492
C2×#885292
#117217612
H(3,𝔽3)×#1032
#117205472
#117205312
C3×#48193