Finite Group Databasebeta

← #13140751

Subgroups — #13140751

Order 1728.

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

normal subgroups only

This group has 360 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
2C2×C9×#216C211yesmaximal, characteristic, complemented
3#11720895C211yesmaximal, characteristic
4#11720862C211yesmaximal, characteristic
5C3×#63394C211yesmaximal, characteristic, complemented
6#11720299C211yesmaximal, characteristic, complemented
7#11720280C211yesmaximal, characteristic, complemented
8#10591543—31nomaximal
9#10588872—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11#88736C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13#88548—21no—
14#88547—21no—
15#88532C2211yescharacteristic
16C9×#216—21no—
17C9×#216—21no—
18C2×C9×C3⋊C8C2211yescharacteristic
19C3×#3667C2211yescharacteristic
20C3×#3667C2211yescharacteristic
21#88428—21no—
22#88426—21no—
23#88414—21no—
24(C3⋊D9)⋊C8—21no—
25#64151—31no—
26C2×C3×#216D311yescharacteristic
27#63833—31no—
28#63817—31no—
29C3×#836—31no—
30#63577—31no—
31#63560—31no—
32C2×C9×Q16—31no—
33#63505—31no—
34#63505—31no—
35#63505—61no—
36#63464—31no—
37#63394—31no—
38#63394D311yescharacteristic, complemented
39#63372—31no—
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44#6233—21no—
45C4⋊(C3⋊D9)D411yescharacteristic
46C4×C3⋊D9—21no—
47#6226D411yescharacteristic
48C2×C9×Dic3—21no—
49C9×Dic6—21no—
50C9×Dic6—21no—
51C3⋊(C9⋊C8)—21no—
52C9×C3⋊C8D411yescharacteristic
53C9×C3⋊C8D411yescharacteristic
54C3×C9⋊C8—21no—
55C3×C9⋊C8—21no—
56#5192—91no—
57#4755—91no—
58C2×C32×Q8C2×D311yescharacteristic
59#3828—31no—
60C2×C3×Dic6C2×D311yescharacteristic
61#3756—61no—
62#3755—61no—
63#3748—31no—
64C3×#216—21no—
65C3×#216—21no—
66C3×#208—31no—
67C3×#208—31no—
68C2×C3×C3⋊C8C2×D311yescharacteristic
69#3719—61no—
70#3717—61no—
71#3712—61no—
72#3710—61no—
73C2×Q8×C9—21no—
74C2×Q8×C9—31no—
75C2×Q8×C9C2×D311yescharacteristic
76#3697—31no—
77#3697—61no—
78C9×Q16—61no—
79C9×Q16—61no—
80C2×C8×C9—31no—
81#3675—61no—
82#3675—61no—
83#3675—62no—
84#3674—61no—
85#3674—61no—
86#3674—62no—
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—
95C3×C4×C9—41no—
96C3×C4×C9C2×D411yescharacteristic
97C3×C4×C9C2×D411yescharacteristic
98C3⋊Dic9—21no—
99C9×Dic3—41no—
100C2×C3×Q16—31no—
101C2×#216D911yescharacteristic, complemented
102#942—91no—
103#942—91no—
104#942—91no—
105#942—181no—
106#909—91no—
107#907—91no—
108#901—91no—
109C3×#101—91no—
110#836—31no—
111#836—91no—
112#824—91no—
113#822—91no—
114#819—91no—
115C32×Q8—21no—
116C32×Q8—21no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119#638—61no—
120C4⋊(C3⋊D3)—31no—
121C4×C3⋊D3—61no—
122#634—31no—
123C2×C3×Dic3—21no—
124C3×Dic6—21no—
125C3×Dic6—21no—
126C3⋊(C3⋊C8)—61no—
127C3×C3⋊C8—61no—
128C3×C3⋊C8—61no—
129C3×C3⋊C8#6711yescharacteristic
130C3×C3⋊C8#6711yescharacteristic
131Q8×C9—22no—
132Q8×C9—21no—
133Q8×C9—21no—
134Q8×C9—41no—
135Q8×C9—61no—
136Q8×C9—61no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—41no—
140C2×C4×C9—61no—
141C2×C4×C9C22×D311yescharacteristic
142#611—61no—
143#611—121no—
144D36—31no—
145D36—61no—
146C4×D9—61no—
147C4×D9—121no—
148Dic18—31no—
149Dic18—61no—
150C8×C9—31no—
151C8×C9—31no—
152C9⋊C8—21no—
153C9⋊C8—21no—
154C9⋊C8—62no—
155C9⋊C8—61no—
156#473—271noSylow
157C2×C3×C9—21no—
158C2×C3×C9#12111yescharacteristic
159C3⋊D9—41no—
160C2×C3×Q8—21no—
161C2×C3×Q8—31no—
162C2×C3×Q8C2×D911yescharacteristic
163C2×C3×Q8D3211yescharacteristic
164#235—91no—
165#235—91no—
166#235—181no—
167C2×Dic6C2×D911yescharacteristic
168C3×Q16—61no—
169C3×Q16—61no—
170C3×M16—91no—
171C3×M16—91no—
172C2×C3×C8—31no—
173#216—21no—
174#216—21no—
175#216—181no—
176#216—181no—
177#216—181no—
178#216—182no—
179#215—181no—
180#215—181no—
181#215—181no—
182#215—182no—
183#208—31no—
184#208—31no—
185#208—91no—
186#208—91no—
187#208—181no—
188C2×C3⋊C8C2×D911yescharacteristic
189Dic12—181no—
190#204—181no—
191ΓL(1,𝔽25)—181no—
192ΓL(1,𝔽25)—181no—
193D3×C8—181no—
194D3×C8—181no—
195C22×C32#23711yescharacteristic
196C2×C3⋊D3—61no—
197C32×C4—41no—
198C32×C4D3×D411yescharacteristic
199C32×C4C2×#6711yescharacteristic
200C3⋊Dic3—61no—
201C3×Dic3—41no—
202C22×C9—21no—
203C22×C9#23511yescharacteristic
204C2×D9—61no—
205C2×D9—121no—
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—42no—
209C4×C9—41no—
210C4×C9—121no—
211C4×C9D3×D411yescharacteristic
212C4×C9D3×D411yescharacteristic
213Dic9—61no—
214Dic9—121no—
215#137—271no—
216#137—271no—
217C2×Q16—31no—
218#131—271no—
219#108—271no—
220#101—91no—
221#101—271no—
222C3×C9#47311yescharacteristic, complemented, Sylow
223C3×Q8—22no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8—21no—
227C3×Q8—21no—
228C3×Q8—41no—
229C3×Q8—61no—
230C3×Q8—61no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—61no—
236C2×C3×C4C22×D911yescharacteristic
237C2×C3×C4C2×D3211yescharacteristic
238#67—181no—
239#67—181no—
240#67—361no—
241C2×Dic3—21no—
242D12—91no—
243D12—91no—
244D12—181no—
245C4×D3—181no—
246C4×D3—181no—
247C4×D3—361no—
248Dic6—21no—
249Dic6—21no—
250Dic6—91no—
251Dic6—91no—
252Dic6—181no—
253C3×C8—31no—
254C3×C8—31no—
255C3×C8—181no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—61no—
259C3⋊C8—182no—
260C3⋊C8—181no—
261C3⋊C8—181no—
262C3⋊C8#61111yescharacteristic
263C3⋊C8#61111yescharacteristic
264C2×C32—21no—
265C2×C32#93811yescharacteristic
266C3⋊D3—121no—
267C2×C9—21no—
268C2×C9—21no—
269C2×C9—41no—
270C2×C9#88411yescharacteristic
271D9—121no—
272D9—241no—
273#41—271no—
274C2×Q8—31no—
275C2×Q8D3×D911yescharacteristic
276Q16—61no—
277Q16—61no—
278Q16—541no—
279Q16—541no—
280SD16—541no—
281SD16—541no—
282M16—91no—
283M16—91no—
284M16—271no—
285M16—541no—
286C2×C8—31no—
287C2×C8—541no—
288C22×C3—21no—
289C22×C3#369911yescharacteristic
290C22×C3#380411yescharacteristic
291C2×D3—181no—
292C2×D3—181no—
293C2×D3—361no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—42no—
297C3×C4—41no—
298C3×C4—41no—
299C3×C4—121no—
300C3×C4D4×D911yescharacteristic
301C3×C4C2×#61111yescharacteristic
302C3×C4D3×#6711yescharacteristic
303C3×C4#381111yescharacteristic
304Dic3—41no—
305Dic3—181no—
306Dic3—181no—
307Dic3—361no—
308C32#519211yescharacteristic
309C9—21no—
310C9#475511yescharacteristic, complemented
311Q8—21no—
312Q8—21no—
313Q8—61no—
314Q8—61no—
315Q8—271no—
316D4—271no—
317D4—541no—
318C2×C4—21no—
319C2×C4—61no—
320C2×C4—541no—
321C2×C4C2×D3×D911yescharacteristic
322C8—31no—
323C8—31no—
324C8—181no—
325C8—181no—
326C8—541no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—41no—
331C2×C3#6350011yescharacteristic
332C2×C3#6395811yescharacteristic
333D3—361no—
334D3—361no—
335D3—721no—
336C22—541no—
337C22#8866211yescharacteristic
338C4—41no—
339C4—121no—
340C4—541no—
341C4D3×#61111yescharacteristic
342C4#8866811yescharacteristic
343C3—21no—
344C3#1058887211yescharacteristic, complemented
345C3#1059154311yescharacteristic
346C2—21no—
347C2—1081no—
348C2#1172126211yescharacteristic
subgroupG/H
centerC2#11721262
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC2×C32#938
prime radical#88736C22
hypercenterC2×Q8D3×D9
hypocenterC3×C9#473
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
2. C2×C9×#216C2semidirect
5. C3×#63394C2semidirect
6. #11720299C2semidirect
7. #11720280C2semidirect
38. #63394D3semidirect
101. C2×#216D9semidirect
222. C3×C9#473semidirect
310. C9#4755semidirect
344. C3#10588872semidirect

Maximal subgroups

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

subgroupindex
#117218532
C2×C9×#2162
#117208952
#117208622
C3×#633942
#117202992
#117202802
#105915433
#105888723