Finite Group Databasebeta

← #13143929

Subgroups — #13143929

Order 1728.

770 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 101 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#11721850C211yesmaximal, characteristic, complemented
2C2×#88545C211yesmaximal, characteristic, complemented
3C2×#88529C211yesmaximal, characteristic, complemented
4#11721761C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#11720547C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#4786—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C2×#6235—21no—
12C2×C4⋊#272C2211yescharacteristic
13C2×C4⋊#272C2211yescharacteristic
14C2×C4×#272—21no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16C4⋊C4×H(3,𝔽3)C2211yescharacteristic
17#88545—22no—
18#88542—21no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21#88529—22no—
22C3×#940—34no—
23C2×C3×#213—34no—
24C2×C3×D24—34no—
25C3×#893—34no—
26C32×#102D314yescomplemented
27C3×Dic3⋊C8—34no—
28C3×#809—34no—
29C23×H(3,𝔽3)—21no—
30C22×#272—21no—
31C22×#272—21no—
32D4×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36#6235—42no—
37#6235—42no—
38C2×#1084D411yescharacteristic
39C2×#1084D411yescharacteristic
40C4⋊#272—21no—
41C4⋊#272—21no—
42C4⋊#272—21no—
43C4⋊#272D412yescomplemented
44C4×#272—42no—
45C8×H(3,𝔽3)—21no—
46#6180—21no—
47C3×#459—91no—
48#4786—34no—
49C2×C32×D4C2×D314yes—
50C2×C3×#67—64no—
51C2×C3×#67—64no—
52C2×C3×D12—34no—
53C2×C3×D12—34no—
54C2×C3×C4×D3—64no—
55C2×C32×C8C2×D314yes—
56C32×C4⋊C4C2×D314yes—
57C3×#213—68no—
58C3×(C2×D3)⋊C4—64no—
59C3×C4×Dic3—34no—
60C2×C3×C3⋊C8—34no—
61C3×D24—68no—
62C22×H(3,𝔽3)—41no—
63C22×H(3,𝔽3)—41no—
64C22×H(3,𝔽3)—41no—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C2×#272—22no—
67C2×#272—41no—
68C2×#272—41no—
69C2×#272—41no—
70C2×#272—41no—
71C2×#272—41no—
72C4×H(3,𝔽3)—41no—
73C4×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)#4111yescharacteristic
75#1084—21no—
76#1084D812yescomplemented
77C2×C3×D8—91no—
78C2×C3×D8—91no—
79C3×#127—91no—
80C3×C4×D4—91no—
81#940—34no—
82C2×#213—34no—
83C2×D24—34no—
84#893—34no—
85C3×C4⋊C8—91no—
86C3×#102—34no—
87C3×#102—91no—
88C3×#102C3⋊D311yescharacteristic
89Dic3⋊C8—34no—
90#809—34no—
91C23×C32—24no—
92C22×C3×D3—64no—
93C22×C3×D3—64no—
94C32×D4—24no—
95C32×D4—24no—
96C2×C32×C4—24no—
97C2×C32×C4C22×D314yes—
98C3×#67—128no—
99C3×#67—128no—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—34no—
102C3×D12—38no—
103C3×D12—64no—
104C3×D12—64no—
105C3×D12—64no—
106C3×C4×D3—128no—
107C32×C8—24no—
108C3×C3⋊C8—64no—
109#459—91noSylow
110C2×H(3,𝔽3)—81no—
111C2×H(3,𝔽3)#12111yescharacteristic
112C2×H(3,𝔽3)C2×D811yescharacteristic
113C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
114#272—42no—
115#272—81no—
116C2×C3×D4—34no—
117C2×C3×D4—91no—
118C2×C3×D4—91no—
119C2×C3×D4—181no—
120C2×C3×D4—181no—
121C2×C3×D4C2×C3⋊D311yescharacteristic
122C22×C3×C4—181no—
123C2×#67—64no—
124C2×#67—64no—
125C2×D12—34no—
126C2×D12—34no—
127C2×C4×D3—64no—
128C3×D8—182no—
129C3×D8—182no—
130C2×C3×C8—34no—
131C2×C3×C8—91no—
132C2×C3×C8C2×C3⋊D311yescharacteristic
133C3×C4⋊C4—34no—
134C3×C4⋊C4C2×C3⋊D311yescharacteristic
135C3×C22⋊C4—181no—
136C3×C42—91no—
137#213—68no—
138(C2×D3)⋊C4—64no—
139C4×Dic3—34no—
140C2×C3⋊C8—34no—
141D24—68no—
142C22×C32—44no—
143C22×C32—44no—
144C22×C32—44no—
145C22×C32D3×D414yes—
146C2×C3×D3—68no—
147C2×C3×D3—124no—
148C2×C3×D3—124no—
149C2×C3×D3—124no—
150C2×C3×D3—124no—
151C2×C3×D3—124no—
152C32×C4—44no—
153C32×C4#23514yes—
154C32×C4D3×D414yes—
155C3×Dic3—38no—
156C3×Dic3—64no—
157C2×D8—91no—
158C2×D8—91no—
159#127—91no—
160C4×D4—91no—
161C4⋊C8—91no—
162#102—91no—
163#102#27211yescharacteristic, complemented
164H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
165C23×C3—21no—
166C23×C3—64no—
167C23×C3—181no—
168C23×C3—181no—
169C22×D3—64no—
170C22×D3—64no—
171C3×D4—21no—
172C3×D4—21no—
173C3×D4—64no—
174C3×D4—64no—
175C3×D4—92no—
176C3×D4—181no—
177C3×D4—181no—
178C3×D4—181no—
179C3×D4—362no—
180C3×D4—362no—
181C2×C3×C4—21no—
182C2×C3×C4—34no—
183C2×C3×C4—64no—
184C2×C3×C4—91no—
185C2×C3×C4—91no—
186C2×C3×C4—362no—
187C2×C3×C4C22×C3⋊D311yescharacteristic
188#67—128no—
189#67—128no—
190C2×Dic3—34no—
191C2×Dic3—34no—
192D12—38no—
193D12—64no—
194D12—64no—
195D12—64no—
196C4×D3—128no—
197C3×C8—21no—
198C3×C8—64no—
199C3×C8—181no—
200C3⋊C8—64no—
201C2×C32—84no—
202C2×C32#88414yes—
203C2×C32D3×D814yes—
204C2×C32#91414yes—
205C3×D3—128no—
206C3×D3—244no—
207C2×D4—91no—
208C2×D4—91no—
209C2×D4—181no—
210C2×D4—181no—
211C2×D4C2×#27211yescharacteristic
212C22×C4—181no—
213D8—182no—
214D8—182no—
215C2×C8—91no—
216C2×C8C2×#27211yescharacteristic
217C4⋊C4C2×#27211yescharacteristic
218C22⋊C4—181no—
219C42—91no—
220C22×C3—34no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—124no—
225C22×C3—124no—
226C22×C3—124no—
227C22×C3—182no—
228C22×C3—361no—
229C22×C3—361no—
230C22×C3—361no—
231C22×C3—361no—
232C22×C3—361no—
233C22×C3D4×C3⋊D311yescharacteristic
234C2×D3—68no—
235C2×D3—124no—
236C2×D3—124no—
237C2×D3—124no—
238C2×D3—124no—
239C2×D3—124no—
240C3×C4—34no—
241C3×C4—34no—
242C3×C4—41no—
243C3×C4—92no—
244C3×C4—124no—
245C3×C4—181no—
246C3×C4#382811yescharacteristic
247C3×C4D4×C3⋊D311yescharacteristic
248Dic3—38no—
249Dic3—64no—
250C32#478614yescomplemented
251C23—21no—
252C23—181no—
253C23—181no—
254D4—21no—
255D4—21no—
256D4—92no—
257D4—181no—
258D4—181no—
259D4—181no—
260D4—362no—
261D4—362no—
262C2×C4—21no—
263C2×C4—91no—
264C2×C4—91no—
265C2×C4—362no—
266C2×C4C22×#27211yescharacteristic
267C8—21no—
268C8—181no—
269C2×C3—34no—
270C2×C3—34no—
271C2×C3—34no—
272C2×C3—81no—
273C2×C3—244no—
274C2×C3—362no—
275C2×C3—721no—
276C2×C3#6409311yescharacteristic
277C2×C3D8×C3⋊D311yescharacteristic
278C2×C3#6412311yescharacteristic
279D3—128no—
280D3—244no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—182no—
285C22—361no—
286C22—361no—
287C22—361no—
288C22—361no—
289C22—361no—
290C22D4×#27211yescharacteristic
291C4—41no—
292C4—92no—
293C4—181no—
294C4#8874011yescharacteristic
295C4D4×#27211yescharacteristic
296C3—34no—
297C3#1059248511yescharacteristic
298C2—81no—
299C2—362no—
300C2—721no—
301C2#1172174011yescharacteristic
302C2D8×#27211yescharacteristic
303C2#1172180011yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#459
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
1. #11721850C2semidirect
2. C2×#88545C2semidirect
3. C2×#88529C2semidirect
4. #11721761C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #11720547C2semidirect
7. #11720531C2semidirect
26. C32×#102D3semidirect
43. C4⋊#272D4semidirect
76. #1084D8semidirect
163. #102#272semidirect
164. H(3,𝔽3)#459semidirect
250. C32#4786semidirect

Maximal subgroups

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

subgroupindex
#117218502
C2×#885452
C2×#885292
#117217612
H(3,𝔽3)×#1022
#117205472
#117205312
C3×#47863