Finite Group Databasebeta

← #13144786

Subgroups — #13144786

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
2H(3,𝔽3)×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#11721850C211yesmaximal, characteristic, complemented
4#11721842C211yesmaximal, characteristic, complemented
5#11720586C211yesmaximal, characteristic, complemented
6#11720562C211yesmaximal, characteristic, complemented
7#11720560C211yesmaximal, characteristic, complemented
8C3×#5198—34nomaximal
9#41×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
11C2×#6235—22no—
12#88748—21no—
13D4×#272—22no—
14D4×#272—21no—
15#88740—21no—
16#88740C2211yescharacteristic
17C2×C4⋊#272—21no—
18C2×C4⋊#272C2211yescharacteristic
19SD16×H(3,𝔽3)—21no—
20D8×H(3,𝔽3)—21no—
21M16×H(3,𝔽3)C2211yescharacteristic, complemented
22#88546—21no—
23#88545—21no—
24C4×#1084C2211yescharacteristic, complemented
25#88538C2211yescharacteristic, complemented
26C3×#1009—34no—
27C32×AGL(1,ℤ/8)D314yescomplemented
28C3×#940—34no—
29C3×#932—34no—
30C3×#837—34no—
31C3×#825—34no—
32C3×#823—34no—
33C23×H(3,𝔽3)—21no—
34C22×#272—22no—
35C22×#272D412yes—
36Q8×H(3,𝔽3)D411yescharacteristic, complemented
37D4×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)—21no—
40D4×H(3,𝔽3)D411yescharacteristic, complemented
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43#6235—22no—
44#6235—21no—
45#6235—44no—
46C2×#1084—21no—
47C4⋊#272—22no—
48C4⋊#272—21no—
49C4⋊#272—21no—
50C4⋊#272D411yescharacteristic, complemented
51C4×#272—22no—
52C4×#272—21no—
53#6229D411yescharacteristic, complemented
54C8×H(3,𝔽3)—21no—
55#6180—21no—
56C3×#453—91no—
57#5198—34no—
58C32×#41C2×D314yes—
59C2×C32×D4C2×D314yescomplemented
60C2×C3×#67—68no—
61C3×#239—64no—
62C3×D3×D4—68no—
63C3×D3×D4—64no—
64C3×#235—34no—
65C3×#235—64no—
66C2×C3×D12—34no—
67C2×C3×D12—64no—
68C32×SD16—24no—
69C32×D8—24no—
70C32×M16C2×D314yescomplemented
71C3×#214—64no—
72C3×#213—64no—
73C3×C4×Dic3—34no—
74C3×#208—34no—
75C22×H(3,𝔽3)—21no—
76C22×H(3,𝔽3)—42no—
77C22×H(3,𝔽3)—41no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×#272—22no—
80C2×#272—21no—
81C2×#272—42no—
82C2×#272—82no—
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)C2×D411yescharacteristic
85C4×H(3,𝔽3)C2×D411yescharacteristic
86#1084—22no—
87#1084—21no—
88C3×H(5,𝔽2)—91no—
89#1009—34no—
90C3×AGL(1,ℤ/8)—34no—
91C3×AGL(1,ℤ/8)—91no—
92C3×AGL(1,ℤ/8)C3⋊D311yescharacteristic
93C3×#127—91no—
94#940—34no—
95#932—34no—
96C3×#104—91no—
97C3×#104—91no—
98C3×#100—91no—
99#837—34no—
100#825—34no—
101#823—34no—
102C23×C32—24no—
103C22×C3×D3—38no—
104C22×C3×D3—68no—
105C32×Q8#6714yescomplemented
106C32×D4—24no—
107C32×D4—24no—
108C32×D4—24no—
109C32×D4#6714yescomplemented
110C2×C32×C4—24no—
111C2×C32×C4C22×D314yes—
112C3×#67—68no—
113C3×#67—64no—
114C3×#67—1216no—
115C2×C3×Dic3—64no—
116C3×D12—34no—
117C3×D12—68no—
118C3×D12—64no—
119C3×D12—64no—
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)—81no—
129C2×H(3,𝔽3)#12011yescharacteristic
130#272—42no—
131#272—41no—
132C3×#41—34no—
133C3×#41—91no—
134C3×#41—181no—
135C3×#41—181no—
136C3×#41C2×C3⋊D311yescharacteristic
137C2×C3×D4—34no—
138C2×C3×D4—91no—
139C2×C3×D4—181no—
140C2×C3×D4—182no—
141C2×C3×D4—181no—
142C2×C3×D4—182no—
143C2×C3×D4C2×C3⋊D311yescharacteristic
144C2×#67—68no—
145#239—64no—
146D3×D4—64no—
147D3×D4—68no—
148#235—34no—
149#235—64no—
150C2×D12—34no—
151C2×D12—64no—
152C3×SD16—21no—
153C3×SD16—64no—
154C3×SD16—181no—
155C3×D8—21no—
156C3×D8—64no—
157C3×D8—181no—
158C3×M16—34no—
159C3×M16—91no—
160C3×M16C2×C3⋊D311yescharacteristic
161C3×C42—91no—
162#214—64no—
163#213—64no—
164C4×Dic3—34no—
165#208—34no—
166C22×C32—24no—
167C22×C32—48no—
168C22×C32—44no—
169C22×C32D3×D414yes—
170C2×C3×D3—64no—
171C2×C3×D3—68no—
172C2×C3×D3—128no—
173C2×C3×D3—248no—
174C32×C4—24no—
175C32×C4D3×D414yes—
176C32×C4C2×#6714yes—
177C3×Dic3—68no—
178C3×Dic3—64no—
179H(5,𝔽2)—91no—
180AGL(1,ℤ/8)—91no—
181AGL(1,ℤ/8)#27211yescharacteristic, complemented
182#127—91no—
183#104—91no—
184#104—91no—
185#100—91no—
186H(3,𝔽3)#45311yescharacteristic, complemented, Sylow
187C23×C3—21no—
188C23×C3—64no—
189C23×C3—92no—
190C23×C3—182no—
191C22×D3—38no—
192C22×D3—68no—
193C3×Q8—34no—
194C3×Q8—91no—
195C3×Q8#63811yescharacteristic
196C3×D4—21no—
197C3×D4—21no—
198C3×D4—21no—
199C3×D4—34no—
200C3×D4—64no—
201C3×D4—64no—
202C3×D4—64no—
203C3×D4—91no—
204C3×D4—182no—
205C3×D4—182no—
206C3×D4—181no—
207C3×D4—181no—
208C3×D4—181no—
209C3×D4—364no—
210C3×D4#63811yescharacteristic
211C2×C3×C4—21no—
212C2×C3×C4—34no—
213C2×C3×C4—64no—
214C2×C3×C4—182no—
215C2×C3×C4—181no—
216C2×C3×C4—181no—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218#67—64no—
219#67—68no—
220#67—1216no—
221C2×Dic3—64no—
222D12—34no—
223D12—64no—
224D12—64no—
225D12—68no—
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—84no—
236C2×C32#93714yes—
237C3×D3—128no—
238C3×D3—124no—
239#41—91no—
240#41—181no—
241#41—181no—
242#41C2×#27211yescharacteristic
243C2×D4—91no—
244C2×D4—181no—
245C2×D4—182no—
246C2×D4—181no—
247C2×D4—182no—
248C2×D4C2×#27211yescharacteristic, complemented
249SD16—21no—
250SD16—181no—
251D8—21no—
252D8—181no—
253M16—91no—
254M16C2×#27211yescharacteristic, complemented
255C42—91no—
256C22×C3—21no—
257C22×C3—34no—
258C22×C3—42no—
259C22×C3—41no—
260C22×C3—64no—
261C22×C3—124no—
262C22×C3—128no—
263C22×C3—181no—
264C22×C3—182no—
265C22×C3—362no—
266C22×C3—722no—
267C22×C3D4×C3⋊D311yescharacteristic
268C2×D3—64no—
269C2×D3—68no—
270C2×D3—128no—
271C2×D3—248no—
272C3×C4—21no—
273C3×C4—34no—
274C3×C4—34no—
275C3×C4—64no—
276C3×C4—182no—
277C3×C4—181no—
278C3×C4D4×C3⋊D311yescharacteristic
279C3×C4C2×#63811yescharacteristic
280Dic3—64no—
281Dic3—68no—
282C32#519814yescomplemented
283C23—21no—
284C23—92no—
285C23—182no—
286Q8—91no—
287Q8#623511yescharacteristic, complemented
288D4—21no—
289D4—21no—
290D4—21no—
291D4—91no—
292D4—182no—
293D4—182no—
294D4—181no—
295D4—181no—
296D4—181no—
297D4—364no—
298D4#623511yescharacteristic, complemented
299C2×C4—21no—
300C2×C4—182no—
301C2×C4—181no—
302C2×C4—181no—
303C2×C4C22×#27211yescharacteristic
304C8—21no—
305C8—181no—
306C2×C3—21no—
307C2×C3—34no—
308C2×C3—41no—
309C2×C3—64no—
310C2×C3—81no—
311C2×C3—124no—
312C2×C3—244no—
313C2×C3—362no—
314C2×C3—361no—
315C2×C3#6414611yescharacteristic
316D3—128no—
317D3—124no—
318C22—21no—
319C22—42no—
320C22—41no—
321C22—181no—
322C22—182no—
323C22—362no—
324C22—722no—
325C22D4×#27211yescharacteristic
326C4—21no—
327C4—182no—
328C4—181no—
329C4D4×#27211yescharacteristic
330C4C2×#623511yescharacteristic
331C3—34no—
332C3#1059289711yescharacteristic
333C2—21no—
334C2—41no—
335C2—81no—
336C2—362no—
337C2—361no—
338C2#1172184711yescharacteristic
subgroupG/H
center, socleC2×C3#64146
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×AGL(1,ℤ/8)C2
hypercenterC3×AGL(1,ℤ/8)C3⋊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×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #11722675C2semidirect
2. H(3,𝔽3)×AGL(1,ℤ/8)C2semidirect
3. #11721850C2semidirect
4. #11721842C2semidirect
5. #11720586C2semidirect
6. #11720562C2semidirect
7. #11720560C2semidirect
10. C2×D4×H(3,𝔽3)C22semidirect
21. M16×H(3,𝔽3)C22semidirect
24. C4×#1084C22semidirect
25. #88538C22semidirect
27. C32×AGL(1,ℤ/8)D3semidirect
36. Q8×H(3,𝔽3)D4semidirect
40. D4×H(3,𝔽3)D4semidirect
50. C4⋊#272D4semidirect
53. #6229D4semidirect
59. C2×C32×D4C2×D3semidirect
70. C32×M16C2×D3semidirect
105. C32×Q8#67semidirect
109. C32×D4#67semidirect
181. AGL(1,ℤ/8)#272semidirect
186. H(3,𝔽3)#453semidirect
248. C2×D4C2×#272semidirect
254. M16C2×#272semidirect
282. C32#5198semidirect
287. Q8#6235semidirect
298. D4#6235semidirect

Maximal subgroups

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

subgroupindex
#117226752
H(3,𝔽3)×AGL(1,ℤ/8)2
#117218502
#117218422
#117205862
#117205622
#117205602
C3×#51983