Finite Group Databasebeta

← #13144034

Subgroups — #13144034

Order 1728.

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

normal subgroups only

This group has 924 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#11722675C211yesmaximal, characteristic, complemented
2#11721870C211yesmaximal, characteristic, complemented
3#11721787C211yesmaximal, characteristic, complemented
4#11721720C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#104C211yesmaximal, characteristic, complemented
6#11720560C211yesmaximal, characteristic, complemented
7#11720522C211yesmaximal, characteristic, complemented
8C3×#4822—34nomaximal
9#41×H(3,𝔽3)C2211yescharacteristic
10#88748—21no—
11D4×#272—22no—
12D4×#272—21no—
13#88740—21no—
14#88740C2211yescharacteristic
15C2×C4⋊#272—21no—
16C2×C4⋊#272—22no—
17C2×C4⋊#272C2211yescharacteristic
18C2×C4⋊#272C2211yescharacteristic, complemented
19M16×H(3,𝔽3)C2211yescharacteristic, complemented
20C42×H(3,𝔽3)C2211yescharacteristic, complemented
21#88549—21no—
22#88545—21no—
23#88538C2211yescharacteristic, complemented
24#88529—21no—
25#88528—21no—
26C3×#1009—34no—
27C3×#949—34no—
28C3×#908—34no—
29C3×#874—34no—
30C32×#104D314yescomplemented
31C3×#823—34no—
32C3×#805—34no—
33C22×#272—22no—
34C22×#272—21no—
35C22×#272D412yes—
36Q8×H(3,𝔽3)D411yescharacteristic, complemented
37D4×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)D411yescharacteristic, complemented
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42#6235—22no—
43#6235—21no—
44C4⋊#272—22no—
45C4⋊#272—21no—
46C4⋊#272—21no—
47C4⋊#272—21no—
48C4⋊#272—21no—
49C4⋊#272—44no—
50C4⋊#272D411yescharacteristic, complemented
51C4×#272—22no—
52C4×#272—21no—
53#6229D411yescharacteristic, complemented
54C8×H(3,𝔽3)—21no—
55#6180—21no—
56C3×#453—91no—
57#4822—34no—
58C32×#41C2×D314yes—
59C3×#239—64no—
60C3×D3×D4—68no—
61C3×D3×D4—64no—
62C3×#235—34no—
63C3×#235—64no—
64C2×C3×D12—34no—
65C2×C3×D12—34no—
66C2×C3×D12—64no—
67C2×C3×D12—68no—
68C32×M16C2×D314yescomplemented
69C32×C42C2×D314yescomplemented
70C3×#215—64no—
71C3×#213—64no—
72C3×#208—34no—
73C3×D24—64no—
74C3×#204—64no—
75C22×H(3,𝔽3)—21no—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C2×#272—22no—
78C2×#272—21no—
79C2×#272—42no—
80C2×#272—42no—
81C2×#272—41no—
82C2×#272—82no—
83C4×H(3,𝔽3)—22no—
84C4×H(3,𝔽3)—21no—
85C4×H(3,𝔽3)C2×D411yescharacteristic
86C4×H(3,𝔽3)C2×D411yescharacteristic
87#1084—21no—
88C3×H(5,𝔽2)—91no—
89#1009—34no—
90C3×AGL(1,ℤ/8)—91no—
91C3×AGL(1,ℤ/8)—91no—
92C3×#127—91no—
93#949—34no—
94#908—34no—
95#874—34no—
96C3×#104—34no—
97C3×#104—91no—
98C3×#104C3⋊D311yescharacteristic
99C3×#100—91no—
100#823—34no—
101#805—34no—
102C22×C3×D3—38no—
103C22×C3×D3—68no—
104C22×C3×D3—64no—
105C32×Q8D1214yescomplemented
106C32×D4—24no—
107C32×D4D1214yescomplemented
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4C22×D314yes—
111C3×#67—68no—
112C3×#67—64no—
113C3×D12—34no—
114C3×D12—68no—
115C3×D12—64no—
116C3×D12—64no—
117C3×D12—64no—
118C3×D12—64no—
119C3×D12—1216no—
120C3×C4×D3—68no—
121C3×C4×D3—64no—
122C3×Dic6—34no—
123C32×C8—24no—
124C3×C3⋊C8—64no—
125#453—91noSylow
126C2×H(3,𝔽3)—21no—
127C2×H(3,𝔽3)—41no—
128C2×H(3,𝔽3)#12011yescharacteristic
129#272—42no—
130#272—41no—
131#272—81no—
132C3×#41—34no—
133C3×#41—91no—
134C3×#41—181no—
135C3×#41—181no—
136C3×#41C2×C3⋊D311yescharacteristic
137C2×C3×D4—91no—
138C2×C3×D4—91no—
139C2×C3×D4—181no—
140C2×C3×D4—182no—
141C2×C3×D4—182no—
142C2×C3×D4—181no—
143#239—64no—
144D3×D4—64no—
145D3×D4—68no—
146#235—34no—
147#235—64no—
148C2×D12—34no—
149C2×D12—34no—
150C2×D12—64no—
151C2×D12—68no—
152C3×SD16—181no—
153C3×SD16—181no—
154C3×D8—181no—
155C3×D8—181no—
156C3×M16—34no—
157C3×M16—91no—
158C3×M16C2×C3⋊D311yescharacteristic
159C3×C42—34no—
160C3×C42C2×C3⋊D311yescharacteristic
161#215—64no—
162#213—64no—
163#208—34no—
164D24—64no—
165#204—64no—
166C22×C32—24no—
167C22×C32D3×D414yes—
168C2×C3×D3—64no—
169C2×C3×D3—68no—
170C2×C3×D3—128no—
171C2×C3×D3—128no—
172C2×C3×D3—124no—
173C2×C3×D3—248no—
174C32×C4—28no—
175C32×C4—24no—
176C32×C4C2×D1214yes—
177C32×C4D3×D414yes—
178C3×Dic3—64no—
179H(5,𝔽2)—91no—
180AGL(1,ℤ/8)—91no—
181AGL(1,ℤ/8)—91no—
182#127—91no—
183#104—91no—
184#104#27211yescharacteristic, complemented
185#100—91no—
186H(3,𝔽3)#45311yescharacteristic, complemented, Sylow
187C23×C3—92no—
188C23×C3—182no—
189C23×C3—181no—
190C22×D3—38no—
191C22×D3—68no—
192C22×D3—64no—
193C3×Q8—34no—
194C3×Q8—91no—
195C3×Q8C4⋊(C3⋊D3)11yescharacteristic
196C3×D4—21no—
197C3×D4—34no—
198C3×D4—64no—
199C3×D4—91no—
200C3×D4—182no—
201C3×D4—182no—
202C3×D4—181no—
203C3×D4—181no—
204C3×D4—181no—
205C3×D4—181no—
206C3×D4—181no—
207C3×D4—364no—
208C3×D4C4⋊(C3⋊D3)11yescharacteristic
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—34no—
212C2×C3×C4—64no—
213C2×C3×C4—64no—
214C2×C3×C4—182no—
215C2×C3×C4—181no—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217#67—64no—
218#67—68no—
219D12—34no—
220D12—64no—
221D12—64no—
222D12—64no—
223D12—64no—
224D12—68no—
225D12—1216no—
226C4×D3—64no—
227C4×D3—68no—
228Dic6—34no—
229C3×C8—21no—
230C3×C8—64no—
231C3×C8—181no—
232C3⋊C8—64no—
233C2×C32—24no—
234C2×C32—44no—
235C2×C32#88214yes—
236C3×D3—128no—
237C3×D3—124no—
238C3×D3—244no—
239#41—91no—
240#41—181no—
241#41—181no—
242#41C2×#27211yescharacteristic
243C2×D4—91no—
244C2×D4—91no—
245C2×D4—181no—
246C2×D4—182no—
247C2×D4—182no—
248C2×D4—181no—
249SD16—181no—
250SD16—181no—
251D8—181no—
252D8—181no—
253M16—91no—
254M16C2×#27211yescharacteristic, complemented
255C42C2×#27211yescharacteristic, complemented
256C22×C3—21no—
257C22×C3—34no—
258C22×C3—64no—
259C22×C3—181no—
260C22×C3—182no—
261C22×C3—362no—
262C22×C3—362no—
263C22×C3—361no—
264C22×C3—722no—
265C22×C3D4×C3⋊D311yescharacteristic
266C2×D3—64no—
267C2×D3—68no—
268C2×D3—128no—
269C2×D3—124no—
270C2×D3—128no—
271C2×D3—248no—
272C3×C4—22no—
273C3×C4—21no—
274C3×C4—34no—
275C3×C4—34no—
276C3×C4—64no—
277C3×C4—68no—
278C3×C4—181no—
279C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
280C3×C4D4×C3⋊D311yescharacteristic
281Dic3—64no—
282C32#482214yescomplemented
283C23—92no—
284C23—182no—
285C23—181no—
286Q8—91no—
287Q8C4⋊#27211yescharacteristic, complemented
288D4—21no—
289D4—91no—
290D4—182no—
291D4—182no—
292D4—181no—
293D4—181no—
294D4—181no—
295D4—181no—
296D4—181no—
297D4—364no—
298D4C4⋊#27211yescharacteristic, complemented
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—182no—
302C2×C4—181no—
303C2×C4C22×#27211yescharacteristic
304C8—21no—
305C8—181no—
306C2×C3—21no—
307C2×C3—34no—
308C2×C3—41no—
309C2×C3—64no—
310C2×C3—124no—
311C2×C3—362no—
312C2×C3—361no—
313C2×C3—721no—
314C2×C3#6409111yescharacteristic
315D3—128no—
316D3—124no—
317D3—244no—
318C22—21no—
319C22—181no—
320C22—182no—
321C22—362no—
322C22—362no—
323C22—361no—
324C22—722no—
325C22D4×#27211yescharacteristic
326C4—22no—
327C4—21no—
328C4—181no—
329C4C2×C4⋊#27211yescharacteristic
330C4D4×#27211yescharacteristic
331C3—34no—
332C3#1059252111yescharacteristic
333C2—21no—
334C2—41no—
335C2—362no—
336C2—361no—
337C2—721no—
338C2#1172173811yescharacteristic
subgroupG/H
center, socleC2×C3#64091
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#104C2
hypercenterC3×#104C3⋊D3
hypocenterH(3,𝔽3)#453
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#104

Complements

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

NH ≅ G/Ntype
1. #11722675C2semidirect
2. #11721870C2semidirect
3. #11721787C2semidirect
4. #11721720C2semidirect
5. H(3,𝔽3)×#104C2semidirect
6. #11720560C2semidirect
7. #11720522C2semidirect
18. C2×C4⋊#272C22semidirect
19. M16×H(3,𝔽3)C22semidirect
20. C42×H(3,𝔽3)C22semidirect
23. #88538C22semidirect
30. C32×#104D3semidirect
36. Q8×H(3,𝔽3)D4semidirect
38. D4×H(3,𝔽3)D4semidirect
50. C4⋊#272D4semidirect
53. #6229D4semidirect
68. C32×M16C2×D3semidirect
69. C32×C42C2×D3semidirect
105. C32×Q8D12semidirect
107. C32×D4D12semidirect
184. #104#272semidirect
186. H(3,𝔽3)#453semidirect
254. M16C2×#272semidirect
255. C42C2×#272semidirect
282. C32#4822semidirect
287. Q8C4⋊#272semidirect
298. D4C4⋊#272semidirect

Maximal subgroups

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

subgroupindex
#117226752
#117218702
#117217872
#117217202
H(3,𝔽3)×#1042
#117205602
#117205222
C3×#48223