Finite Group Databasebeta

← #13139074

Subgroups — #13139074

Order 1728.

372 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 73 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#11721784C211yesmaximal, characteristic, complemented
2#11720868C211yesmaximal, characteristic
3C2×#88415C211yesmaximal, characteristic, complemented
4#11720862C211yesmaximal, characteristic, complemented
5C3×#63383C211yesmaximal, characteristic, complemented
6C9×#822C211yesmaximal, characteristic, complemented
7#11720290C211yesmaximal, characteristic, complemented
8#10590989—31nomaximal
9#10588578—31nomaximal
10#88736C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C3×C9×M16C2211yescharacteristic
14#88524—21no—
15#88521—21no—
16C9×#208C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C3×#3667C2211yescharacteristic
19#88417—21no—
20#88415—21no—
21#88415—21no—
22#88415—21no—
23#88414—21no—
24(C3⋊D9)⋊C8—21no—
25#64118—31no—
26#63830—31no—
27C2×#3716—31no—
28#63817—31no—
29C3×#823—31no—
30C3×#822D311yescharacteristic
31#63561—31no—
32#63471—31no—
33#63471—61no—
34#63470—31no—
35C2×#3664—31no—
36#63464—31no—
37C9×#108—31no—
38#63383D311yescharacteristic, complemented
39#63382—31no—
40C22×C3×D9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42#6233—21no—
43C2×C3⋊Dic9—21no—
44C4⋊(C3⋊D9)D411yescharacteristic
45C4×C3⋊D9—21no—
46#6226—21no—
47#6226—21no—
48#6226D411yescharacteristic
49C3×D36—21no—
50C3×D36—21no—
51C3×C8×C9—21no—
52C9×C3⋊C8—21no—
53C9×C3⋊C8D411yescharacteristic
54C9×C3⋊C8D411yescharacteristic
55C3×C9⋊C8—21no—
56#4898—91no—
57#4748—91no—
58#3828—31no—
59C2×#634—31no—
60C2×C3×D12—31no—
61C32×M16C2×D311yescharacteristic
62#3746—61no—
63#3744—61no—
64C3×#208—31no—
65C3×#208C2×D311yescharacteristic
66C2×C3×C3⋊C8C2×D311yescharacteristic
67#3716—61no—
68#3716—61no—
69#3716—61no—
70#3714—61no—
71#3712—61no—
72#3710—61no—
73#3697—31no—
74#3697—61no—
75C2×D36C2×D311yescharacteristic
76C2×Dic18—31no—
77C2×Dic18—61no—
78C9×M16—21no—
79C9×M16—31no—
80C9×M16C2×D311yescharacteristic
81C2×C8×C9—31no—
82#3667C2×D311yescharacteristic
83D72—61no—
84#3664—61no—
85#3664—62no—
86#3664—61no—
87#3664—61no—
88#3664—61no—
89#3663—61no—
90C8×D9—61no—
91Dic36—61no—
92Dic36—62no—
93C22×C3×C9#4111yescharacteristic
94C2×C3⋊D9—21no—
95C2×C3×D9—41no—
96C2×C3×D9—81no—
97C3×C4×C9C2×D411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C3⋊Dic9—21no—
100C3⋊Dic9—41no—
101C2×#214—91no—
102#932—91no—
103#909—91no—
104#909—91no—
105#909—181no—
106#908—91no—
107#907—91no—
108C2×#204—91no—
109#901—91no—
110C3×#108—31no—
111C3×#100—91no—
112#824—91no—
113#824—91no—
114#823—31no—
115#822D911yescharacteristic, complemented
116C22×C3×D3—61no—
117C2×C32×C4C22×D311yescharacteristic
118#638—61no—
119C2×C3⋊Dic3—61no—
120C4⋊(C3⋊D3)—31no—
121C4×C3⋊D3—61no—
122#634—31no—
123#634—61no—
124#634—61no—
125C3×D12—61no—
126C3×D12—61no—
127C22×D9—21no—
128C32×C8—21no—
129C3×C3⋊C8—21no—
130C3×C3⋊C8—61no—
131C3×C3⋊C8D1211yescharacteristic
132C3×C3⋊C8D1211yescharacteristic
133C2×C4×C9—21no—
134C2×C4×C9C22×D311yescharacteristic
135#611—61no—
136#611—121no—
137C2×Dic9—61no—
138C2×Dic9—121no—
139D36—21no—
140D36—21no—
141D36—31no—
142D36—61no—
143C4×D9—61no—
144C4×D9—121no—
145Dic18—31no—
146Dic18—61no—
147Dic18—62no—
148Dic18—61no—
149Dic18—61no—
150Dic18—121no—
151C8×C9—22no—
152C8×C9—21no—
153C8×C9—31no—
154C8×C9—31no—
155C8×C9—61no—
156C9⋊C8—21no—
157#471—271noSylow
158C2×C3×C9—21no—
159C2×C3×C9#12111yescharacteristic
160C3⋊D9—41no—
161C3×D9—81no—
162C2×C3×D4—91no—
163#235—91no—
164#235—91no—
165#235—181no—
166C2×D12—31no—
167C2×Dic6—91no—
168C2×Dic6—91no—
169C2×Dic6—181no—
170C3×M16—21no—
171C3×M16—31no—
172C3×M16—91no—
173C3×M16C2×D911yescharacteristic
174C3×M16D3211yescharacteristic
175C2×C3×C8—31no—
176#214—181no—
177#214—181no—
178#214—181no—
179#213—181no—
180#208—31no—
181#208C2×D911yescharacteristic
182C2×C3⋊C8C2×D911yescharacteristic
183Dic12—181no—
184Dic12—181no—
185Dic12—182no—
186D24—181no—
187#204—181no—
188#204—181no—
189#204—182no—
190#204—181no—
191#204—181no—
192#204—181no—
193ΓL(1,𝔽25)—181no—
194ΓL(1,𝔽25)—181no—
195D3×C8—181no—
196D3×C8—181no—
197C22×C32#23911yescharacteristic
198C2×C3⋊D3—61no—
199C2×C3×D3—121no—
200C2×C3×D3—241no—
201C32×C4C2×D1211yescharacteristic
202C32×C4D3×D411yescharacteristic
203C3⋊Dic3—61no—
204C3⋊Dic3—121no—
205C22×C9—21no—
206C22×C9#23511yescharacteristic
207C2×D9—41no—
208C2×D9—61no—
209C2×D9—81no—
210C2×D9—121no—
211C4×C9—21no—
212C4×C9—21no—
213C4×C9D3×D411yescharacteristic
214C4×C9D3×D411yescharacteristic
215Dic9—61no—
216Dic9—121no—
217Dic9—122no—
218Dic9—121no—
219#137—271no—
220AGL(1,ℤ/8)—271no—
221C2×SD16—271no—
222#131—271no—
223#108—31no—
224#101—271no—
225#100—91no—
226C3×C9#47111yescharacteristic, complemented, Sylow
227C23×C3—181no—
228C22×D3—61no—
229C3×D4—181no—
230C3×D4—181no—
231C2×C3×C4—21no—
232C2×C3×C4C22×D911yescharacteristic
233C2×C3×C4C2×D3211yescharacteristic
234#67—181no—
235#67—181no—
236#67—361no—
237C2×Dic3—181no—
238C2×Dic3—181no—
239C2×Dic3—361no—
240D12—61no—
241D12—61no—
242D12—91no—
243D12—91no—
244D12—181no—
245C4×D3—181no—
246C4×D3—181no—
247C4×D3—361no—
248Dic6—91no—
249Dic6—91no—
250Dic6—181no—
251Dic6—182no—
252Dic6—181no—
253Dic6—181no—
254Dic6—181no—
255Dic6—181no—
256Dic6—361no—
257C3×C8—22no—
258C3×C8—21no—
259C3×C8—21no—
260C3×C8—31no—
261C3×C8—31no—
262C3×C8—61no—
263C3×C8—181no—
264C3⋊C8—21no—
265C3⋊C8—61no—
266C3⋊C8D3611yescharacteristic
267C3⋊C8D3611yescharacteristic
268C2×C32—21no—
269C2×C32#89511yescharacteristic
270C3⋊D3—121no—
271C3×D3—241no—
272C2×C9—21no—
273C2×C9—21no—
274C2×C9—41no—
275C2×C9#88411yescharacteristic
276D9—81no—
277D9—121no—
278D9—241no—
279#41—271no—
280C2×Q8—271no—
281C2×D4—91no—
282Q16—541no—
283SD16—541no—
284SD16—541no—
285SD16—541no—
286SD16—541no—
287D8—541no—
288M16—31no—
289M16—91no—
290M16—541no—
291M16D3×D911yescharacteristic
292C2×C8—31no—
293C2×C8—541no—
294C22×C3—21no—
295C22×C3—361no—
296C22×C3—721no—
297C22×C3#370111yescharacteristic
298C22×C3#379911yescharacteristic
299C2×D3—121no—
300C2×D3—181no—
301C2×D3—181no—
302C2×D3—241no—
303C2×D3—361no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4C2×D3611yescharacteristic
307C3×C4D4×D911yescharacteristic
308C3×C4D3×D1211yescharacteristic
309C3×C4#381111yescharacteristic
310Dic3—181no—
311Dic3—181no—
312Dic3—361no—
313Dic3—362no—
314Dic3—361no—
315Dic3—361no—
316C32#489811yescharacteristic
317C9—21no—
318C9#474811yescharacteristic, complemented
319C23—181no—
320Q8—271no—
321Q8—541no—
322Q8—541no—
323D4—181no—
324D4—181no—
325D4—271no—
326D4—541no—
327C2×C4—541no—
328C2×C4—541no—
329C2×C4C2×D3×D911yescharacteristic
330C8—21no—
331C8—31no—
332C8—31no—
333C8—61no—
334C8—181no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—41no—
339C2×C3—721no—
340C2×C3#6345711yescharacteristic
341C2×C3#6391011yescharacteristic
342D3—241no—
343D3—361no—
344D3—361no—
345D3—721no—
346C22—361no—
347C22—541no—
348C22—721no—
349C22#8863511yescharacteristic
350C4—541no—
351C4—1081no—
352C4D3×D3611yescharacteristic
353C4#8866811yescharacteristic
354C3—21no—
355C3#1058857811yescharacteristic, complemented
356C3#1059098911yescharacteristic
357C2—21no—
358C2—721no—
359C2—1081no—
360C2#1172100511yescharacteristic
subgroupG/H
centerC2#11721005
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×M16C22
socleC2×C32#895
prime radical#11720868C2
hypercenterM16D3×D9
hypocenterC3×C9#471
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #11721784C2semidirect
3. C2×#88415C2semidirect
4. #11720862C2semidirect
5. C3×#63383C2semidirect
6. C9×#822C2semidirect
7. #11720290C2semidirect
38. #63383D3semidirect
115. #822D9semidirect
226. C3×C9#471semidirect
318. C9#4748semidirect
355. C3#10588578semidirect

Maximal subgroups

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

subgroupindex
#117217842
#117208682
C2×#884152
#117208622
C3×#633832
C9×#8222
#117202902
#105909893
#105885783