Finite Group Databasebeta

← #13140752

Subgroups — #13140752

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#11721853C211yesmaximal, characteristic, complemented
2C2×C9×#215C211yesmaximal, characteristic, complemented
3#11720890C211yesmaximal, characteristic
4#11720862C211yesmaximal, characteristic, complemented
5C3×#63394C211yesmaximal, characteristic, complemented
6#11720296C211yesmaximal, characteristic, complemented
7#11720280C211yesmaximal, characteristic, complemented
8#10591542—31nomaximal
9#10588857—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11#88736C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13#88548—21no—
14#88547—21no—
15#88532C2211yescharacteristic
16C9×#215—21no—
17C9×#215—21no—
18C2×C9×C3⋊C8C2211yescharacteristic
19C3×#3667C2211yescharacteristic
20C3×#3667C2211yescharacteristic
21#88424—21no—
22#88422—21no—
23#88414—21no—
24(C3⋊D9)⋊C8—21no—
25#64151—31no—
26C2×C3×#215D311yescharacteristic
27#63830—31no—
28#63817—31no—
29C3×#836—31no—
30#63577—31no—
31#63558—31no—
32C2×C9×SD16—31no—
33#63505—31no—
34#63505—61no—
35#63495—31no—
36#63464—31no—
37#63394D311yescharacteristic, complemented
38#63391—31no—
39#63372—31no—
40C22×D3×C9—21no—
41C3×Q8×C9—21no—
42C3×Q8×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45#6233—21no—
46C4⋊(C3⋊D9)D411yescharacteristic
47C4×C3⋊D9—21no—
48#6226D411yescharacteristic
49C9×D12—21no—
50C9×D12—21no—
51C3⋊(C9⋊C8)—21no—
52C9×C3⋊C8D411yescharacteristic
53C9×C3⋊C8D411yescharacteristic
54C3×C9⋊C8—21no—
55C3×C9⋊C8—21no—
56#5177—91no—
57#4754—91no—
58C2×C32×Q8C2×D311yescharacteristic
59#3828—31no—
60C2×C3×D12C2×D311yescharacteristic
61#3756—61no—
62#3755—61no—
63#3748—31no—
64C3×#215—21no—
65C3×#215—21no—
66C3×#208—31no—
67C3×#208—31no—
68C2×C3×C3⋊C8C2×D311yescharacteristic
69#3716—61no—
70#3714—61no—
71#3712—61no—
72#3710—61no—
73C2×Q8×C9—21no—
74C2×Q8×C9C2×D311yescharacteristic
75C2×D4×C9—31no—
76#3697—31no—
77#3697—61no—
78C9×SD16—61no—
79C9×SD16—61no—
80C2×C8×C9—31no—
81#3675—61no—
82#3675—62no—
83#3674—61no—
84#3674—62no—
85#3673—61no—
86#3672—61no—
87#3667—31no—
88#3667—61no—
89#3667C2×D311yescharacteristic
90#3667C2×D311yescharacteristic
91#3663—61no—
92C8×D9—61no—
93C22×C3×C9#4111yescharacteristic
94C2×C3⋊D9—21no—
95C2×D3×C9—41no—
96C2×D3×C9—81no—
97C3×C4×C9—41no—
98C3×C4×C9C2×D411yescharacteristic
99C3×C4×C9C2×D411yescharacteristic
100C3⋊Dic9—21no—
101C2×C3×SD16—31no—
102#942—91no—
103#942—91no—
104#942—181no—
105C2×#215D911yescharacteristic, complemented
106#932—91no—
107#908—91no—
108#907—91no—
109#901—91no—
110C3×#101—91no—
111#836—31no—
112#833—91no—
113#823—91no—
114#822—91no—
115#819—91no—
116C22×C3×D3—21no—
117C32×Q8—21no—
118C32×Q8—21no—
119C2×C32×C4—21no—
120C2×C32×C4C22×D311yescharacteristic
121#638—61no—
122C4⋊(C3⋊D3)—31no—
123C4×C3⋊D3—61no—
124#634—31no—
125C3×D12—21no—
126C3×D12—21no—
127C23×C9—61no—
128C3⋊(C3⋊C8)—61no—
129C3×C3⋊C8—61no—
130C3×C3⋊C8—61no—
131C3×C3⋊C8#6711yescharacteristic
132C3×C3⋊C8#6711yescharacteristic
133Q8×C9—22no—
134Q8×C9—21no—
135Q8×C9—21no—
136Q8×C9—41no—
137D4×C9—61no—
138D4×C9—61no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—41no—
142C2×C4×C9C22×D311yescharacteristic
143#611—61no—
144#611—121no—
145D36—31no—
146D36—61no—
147C4×D9—61no—
148C4×D9—121no—
149Dic18—31no—
150Dic18—61no—
151C8×C9—31no—
152C8×C9—31no—
153C9⋊C8—21no—
154C9⋊C8—21no—
155C9⋊C8—62no—
156C9⋊C8—61no—
157#471—271noSylow
158C2×C3×C9—21no—
159C2×C3×C9#12111yescharacteristic
160C3⋊D9—41no—
161D3×C9—81no—
162C2×C3×Q8—21no—
163C2×C3×Q8C2×D911yescharacteristic
164C2×C3×Q8D3211yescharacteristic
165C2×C3×D4—31no—
166#235—91no—
167#235—91no—
168#235—181no—
169C2×D12C2×D911yescharacteristic
170C3×SD16—61no—
171C3×SD16—61no—
172C3×M16—91no—
173C3×M16—91no—
174C2×C3×C8—31no—
175#216—181no—
176#216—181no—
177#216—182no—
178#215—21no—
179#215—21no—
180#215—181no—
181#215—181no—
182#215—182no—
183#214—181no—
184#213—181no—
185#208—31no—
186#208—31no—
187#208—91no—
188#208—91no—
189#208—181no—
190C2×C3⋊C8C2×D911yescharacteristic
191D24—181no—
192#204—181no—
193ΓL(1,𝔽25)—181no—
194ΓL(1,𝔽25)—181no—
195D3×C8—181no—
196D3×C8—181no—
197C22×C32#23711yescharacteristic
198C2×C3⋊D3—61no—
199C2×C3×D3—41no—
200C2×C3×D3—81no—
201C32×C4—41no—
202C32×C4D3×D411yescharacteristic
203C32×C4C2×#6711yescharacteristic
204C3⋊Dic3—61no—
205C22×C9—21no—
206C22×C9—121no—
207C22×C9—241no—
208C22×C9#23511yescharacteristic
209C2×D9—61no—
210C2×D9—121no—
211C4×C9—21no—
212C4×C9—21no—
213C4×C9—42no—
214C4×C9—41no—
215C4×C9D3×D411yescharacteristic
216C4×C9D3×D411yescharacteristic
217Dic9—61no—
218Dic9—121no—
219#137—271no—
220AGL(1,ℤ/8)—271no—
221C2×SD16—31no—
222#131—271no—
223#108—271no—
224#101—91no—
225#100—271no—
226C3×C9#47111yescharacteristic, complemented, Sylow
227C23×C3—61no—
228C22×D3—21no—
229C3×Q8—22no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—41no—
235C3×D4—61no—
236C3×D4—61no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—41no—
241C2×C3×C4C22×D911yescharacteristic
242C2×C3×C4C2×D3211yescharacteristic
243#67—181no—
244#67—181no—
245#67—361no—
246D12—21no—
247D12—21no—
248D12—91no—
249D12—91no—
250D12—181no—
251C4×D3—181no—
252C4×D3—181no—
253C4×D3—361no—
254Dic6—91no—
255Dic6—91no—
256Dic6—181no—
257C3×C8—31no—
258C3×C8—31no—
259C3×C8—181no—
260C3×C8—181no—
261C3⋊C8—61no—
262C3⋊C8—61no—
263C3⋊C8—182no—
264C3⋊C8—181no—
265C3⋊C8—181no—
266C3⋊C8#61111yescharacteristic
267C3⋊C8#61111yescharacteristic
268C2×C32—21no—
269C2×C32#93811yescharacteristic
270C3⋊D3—121no—
271C3×D3—81no—
272C2×C9—21no—
273C2×C9—21no—
274C2×C9—41no—
275C2×C9—241no—
276C2×C9#88411yescharacteristic
277D9—121no—
278D9—241no—
279#41—271no—
280C2×Q8D3×D911yescharacteristic
281C2×D4—31no—
282Q16—541no—
283SD16—61no—
284SD16—61no—
285SD16—541no—
286SD16—541no—
287D8—541no—
288M16—91no—
289M16—91no—
290M16—271no—
291M16—541no—
292C2×C8—31no—
293C2×C8—541no—
294C22×C3—21no—
295C22×C3—121no—
296C22×C3—241no—
297C22×C3#369911yescharacteristic
298C22×C3#380411yescharacteristic
299C2×D3—41no—
300C2×D3—81no—
301C2×D3—181no—
302C2×D3—181no—
303C2×D3—361no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—42no—
307C3×C4—41no—
308C3×C4—41no—
309C3×C4D4×D911yescharacteristic
310C3×C4C2×#61111yescharacteristic
311C3×C4D3×#6711yescharacteristic
312C3×C4#381111yescharacteristic
313Dic3—181no—
314Dic3—181no—
315Dic3—361no—
316C32#517711yescharacteristic
317C9—21no—
318C9#475411yescharacteristic, complemented
319C23—61no—
320Q8—21no—
321Q8—21no—
322Q8—271no—
323D4—61no—
324D4—61no—
325D4—271no—
326D4—541no—
327C2×C4—21no—
328C2×C4—541no—
329C2×C4C2×D3×D911yescharacteristic
330C8—31no—
331C8—31no—
332C8—181no—
333C8—181no—
334C8—541no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—41no—
339C2×C3—241no—
340C2×C3#6350011yescharacteristic
341C2×C3#6395811yescharacteristic
342D3—81no—
343D3—361no—
344D3—361no—
345D3—721no—
346C22—121no—
347C22—241no—
348C22—541no—
349C22#8866211yescharacteristic
350C4—41no—
351C4—541no—
352C4D3×#61111yescharacteristic
353C4#8866811yescharacteristic
354C3—21no—
355C3#1058885711yescharacteristic, complemented
356C3#1059154211yescharacteristic
357C2—21no—
358C2—241no—
359C2—1081no—
360C2#1172126211yescharacteristic
subgroupG/H
centerC2#11721262
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC2×C32#938
prime radical#11720890C2
hypercenterC2×Q8D3×D9
hypocenterC3×C9#471
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11721853C2semidirect
2. C2×C9×#215C2semidirect
4. #11720862C2semidirect
5. C3×#63394C2semidirect
6. #11720296C2semidirect
7. #11720280C2semidirect
37. #63394D3semidirect
105. C2×#215D9semidirect
226. C3×C9#471semidirect
318. C9#4754semidirect
355. C3#10588857semidirect

Maximal subgroups

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

subgroupindex
#117218532
C2×C9×#2152
#117208902
#117208622
C3×#633942
#117202962
#117202802
#105915423
#105888573