Finite Group Databasebeta

← #13140378

Subgroups — #13140378

Order 1728.

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

normal subgroups only

This group has 380 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#11721833C211yesmaximal, characteristic
2C2×C9×#214C211yesmaximal, characteristic, complemented
3#11720895C211yesmaximal, characteristic, complemented
4#11720862C211yesmaximal, characteristic, complemented
5C3×#63391C211yesmaximal, characteristic, complemented
6#11720299C211yesmaximal, characteristic, complemented
7#11720280C211yesmaximal, characteristic, complemented
8#10591407—31nomaximal
9#10588857—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11#88736C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13#88544—21no—
14#88543—21no—
15#88532C2211yescharacteristic
16C9×#214—21no—
17C9×#214—21no—
18C2×C9×C3⋊C8C2211yescharacteristic
19C3×#3667C2211yescharacteristic
20C3×#3667C2211yescharacteristic
21#88428—21no—
22#88426—21no—
23#88414—21no—
24(C3⋊D9)⋊C8—21no—
25#64141—31no—
26C2×C3×#214D311yescharacteristic
27#63833—31no—
28#63817—31no—
29C3×#833—31no—
30#63577—31no—
31#63560—31no—
32C2×C9×SD16—31no—
33#63505—31no—
34#63495—31no—
35#63495—61no—
36#63464—31no—
37#63394—31no—
38#63391D311yescharacteristic, complemented
39#63372—31no—
40C23×C3×C9—21no—
41C3×D4×C9—21no—
42C3×D4×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#5177—91no—
57#4748—91no—
58C2×C32×D4C2×D311yescharacteristic
59#3828—31no—
60C2×C3×Dic6C2×D311yescharacteristic
61#3754—61no—
62#3753—61no—
63#3748—31no—
64C3×#214—21no—
65C3×#214—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—31no—
74C2×D4×C9—21no—
75C2×D4×C9C2×D311yescharacteristic
76#3697—31no—
77#3697—61no—
78C9×SD16—61no—
79C9×SD16—61no—
80C2×C8×C9—31no—
81#3675—61no—
82#3674—61no—
83#3673—61no—
84#3673—62no—
85#3672—61no—
86#3672—62no—
87#3667—31no—
88#3667—61no—
89#3667C2×D311yescharacteristic
90#3667C2×D311yescharacteristic
91#3663—61no—
92C8×D9—61no—
93C22×C3×C9—41no—
94C22×C3×C9—81no—
95C22×C3×C9#4111yescharacteristic
96C2×C3⋊D9—21no—
97C3×C4×C9C2×D411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C3⋊Dic9—21no—
100C9×Dic3—41no—
101C2×C3×SD16—31no—
102#942—91no—
103C2×#214D911yescharacteristic, complemented
104#932—91no—
105#932—91no—
106#932—181no—
107#909—91no—
108#907—91no—
109#901—91no—
110C3×#100—91no—
111#836—91no—
112#833—31no—
113#824—91no—
114#822—91no—
115#819—91no—
116C23×C32—21no—
117C32×D4—21no—
118C32×D4—21no—
119C2×C32×C4C22×D311yescharacteristic
120#638—61no—
121C4⋊(C3⋊D3)—31no—
122C4×C3⋊D3—61no—
123#634—31no—
124C2×C3×Dic3—21no—
125C3×Dic6—21no—
126C3×Dic6—21no—
127C23×C9—21no—
128C23×C9—41no—
129C3⋊(C3⋊C8)—61no—
130C3×C3⋊C8—61no—
131C3×C3⋊C8—61no—
132C3×C3⋊C8#6711yescharacteristic
133C3×C3⋊C8#6711yescharacteristic
134Q8×C9—61no—
135Q8×C9—61no—
136D4×C9—22no—
137D4×C9—21no—
138D4×C9—21no—
139D4×C9—41no—
140C2×C4×C9—21no—
141C2×C4×C9—61no—
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—81no—
160C2×C3×C9#12111yescharacteristic
161C3⋊D9—41no—
162C2×C3×Q8—31no—
163C2×C3×D4—21no—
164C2×C3×D4C2×D911yescharacteristic
165C2×C3×D4D3211yescharacteristic
166#235—91no—
167#235—91no—
168#235—181no—
169C2×Dic6C2×D911yescharacteristic
170C3×SD16—61no—
171C3×SD16—61no—
172C3×M16—91no—
173C3×M16—91no—
174C2×C3×C8—31no—
175#216—181no—
176#215—181no—
177#214—21no—
178#214—21no—
179#214—181no—
180#214—181no—
181#214—182no—
182#213—181no—
183#213—181no—
184#213—182no—
185#208—31no—
186#208—31no—
187#208—91no—
188#208—91no—
189#208—181no—
190C2×C3⋊C8C2×D911yescharacteristic
191Dic12—181no—
192#204—181no—
193ΓL(1,𝔽25)—181no—
194ΓL(1,𝔽25)—181no—
195D3×C8—181no—
196D3×C8—181no—
197C22×C32—41no—
198C22×C32—81no—
199C22×C32#23711yescharacteristic
200C2×C3⋊D3—61no—
201C32×C4D3×D411yescharacteristic
202C32×C4C2×#6711yescharacteristic
203C3⋊Dic3—61no—
204C3×Dic3—41no—
205C22×C9—21no—
206C22×C9—42no—
207C22×C9—41no—
208C22×C9—81no—
209C22×C9—82no—
210C22×C9#23511yescharacteristic
211C2×D9—61no—
212C2×D9—121no—
213C4×C9—21no—
214C4×C9—21no—
215C4×C9—121no—
216C4×C9D3×D411yescharacteristic
217C4×C9D3×D411yescharacteristic
218Dic9—61no—
219Dic9—121no—
220#137—271no—
221AGL(1,ℤ/8)—271no—
222C2×SD16—31no—
223#131—271no—
224#108—271no—
225#101—271no—
226#100—91no—
227C3×C9#47111yescharacteristic, complemented, Sylow
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—41no—
231C3×Q8—61no—
232C3×Q8—61no—
233C3×D4—22no—
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—21no—
237C3×D4—21no—
238C3×D4—41no—
239C2×C3×C4—21no—
240C2×C3×C4—61no—
241C2×C3×C4C22×D911yescharacteristic
242C2×C3×C4C2×D3211yescharacteristic
243#67—181no—
244#67—181no—
245#67—361no—
246C2×Dic3—21no—
247D12—91no—
248D12—91no—
249D12—181no—
250C4×D3—181no—
251C4×D3—181no—
252C4×D3—361no—
253Dic6—21no—
254Dic6—21no—
255Dic6—91no—
256Dic6—91no—
257Dic6—181no—
258C3×C8—31no—
259C3×C8—31no—
260C3×C8—181no—
261C3×C8—181no—
262C3⋊C8—61no—
263C3⋊C8—61no—
264C3⋊C8—182no—
265C3⋊C8—181no—
266C3⋊C8—181no—
267C3⋊C8#61111yescharacteristic
268C3⋊C8#61111yescharacteristic
269C2×C32—21no—
270C2×C32—81no—
271C2×C32#93811yescharacteristic
272C3⋊D3—121no—
273C2×C9—21no—
274C2×C9—21no—
275C2×C9—41no—
276C2×C9—82no—
277C2×C9—81no—
278C2×C9#88411yescharacteristic
279D9—121no—
280D9—241no—
281#41—271no—
282C2×Q8—31no—
283C2×D4D3×D911yescharacteristic
284Q16—541no—
285SD16—61no—
286SD16—61no—
287SD16—541no—
288SD16—541no—
289D8—541no—
290M16—91no—
291M16—91no—
292M16—271no—
293M16—541no—
294C2×C8—31no—
295C2×C8—541no—
296C22×C3—21no—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—81no—
301C22×C3—81no—
302C22×C3—82no—
303C22×C3#369911yescharacteristic
304C22×C3#380411yescharacteristic
305C2×D3—181no—
306C2×D3—181no—
307C2×D3—361no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—121no—
311C3×C4D4×D911yescharacteristic
312C3×C4C2×#61111yescharacteristic
313C3×C4D3×#6711yescharacteristic
314C3×C4#381111yescharacteristic
315Dic3—41no—
316Dic3—181no—
317Dic3—181no—
318Dic3—361no—
319C32#517711yescharacteristic
320C9—21no—
321C9#474811yescharacteristic, complemented
322C23—21no—
323Q8—61no—
324Q8—61no—
325Q8—271no—
326D4—21no—
327D4—21no—
328D4—271no—
329D4—541no—
330C2×C4—61no—
331C2×C4—541no—
332C2×C4C2×D3×D911yescharacteristic
333C8—31no—
334C8—31no—
335C8—181no—
336C8—181no—
337C8—541no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—41no—
342C2×C3—82no—
343C2×C3—81no—
344C2×C3—81no—
345C2×C3#6350011yescharacteristic
346C2×C3#6395811yescharacteristic
347D3—361no—
348D3—361no—
349D3—721no—
350C22—41no—
351C22—81no—
352C22—541no—
353C22#8866211yescharacteristic
354C4—121no—
355C4—541no—
356C4D3×#61111yescharacteristic
357C4#8866811yescharacteristic
358C3—21no—
359C3#1058885711yescharacteristic, complemented
360C3#1059140711yescharacteristic
361C2—21no—
362C2—81no—
363C2—1081no—
364C2#1172126211yescharacteristic
subgroupG/H
centerC2#11721262
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32#938
prime radical#11721833C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#471
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C9×#214C2semidirect
3. #11720895C2semidirect
4. #11720862C2semidirect
5. C3×#63391C2semidirect
6. #11720299C2semidirect
7. #11720280C2semidirect
38. #63391D3semidirect
103. C2×#214D9semidirect
227. C3×C9#471semidirect
321. C9#4748semidirect
359. C3#10588857semidirect

Maximal subgroups

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

subgroupindex
#117218332
C2×C9×#2142
#117208952
#117208622
C3×#633912
#117202992
#117202802
#105914073
#105888573