Finite Group Databasebeta

← #13139104

Subgroups — #13139104

Order 1728.

362 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 69 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#11721782C211yesmaximal, characteristic
2C2×#88427C211yesmaximal, characteristic, complemented
3#11720895C211yesmaximal, characteristic
4#11720871C211yesmaximal, characteristic
5C9×#824C211yesmaximal, characteristic, complemented
6C3×#63382C211yesmaximal, characteristic, complemented
7#11720286C211yesmaximal, characteristic, complemented
8#10590993—31nomaximal
9#10588435—31nomaximal
10#88736C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×M16C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15#88523—21no—
16C8×C3⋊D9—21no—
17C9×#208C2211yescharacteristic
18C3×#3667C2211yescharacteristic
19#88428—21no—
20#88427—21no—
21#88427—21no—
22#88426—21no—
23#88418—21no—
24#88416—21no—
25#64116—31no—
26C2×#3718—31no—
27#63833—31no—
28#63833—31no—
29C3×#824—31no—
30C3×#824D311yescharacteristic
31#63578—31no—
32#63505—31no—
33C2×#3674—31no—
34#63471—31no—
35#63469—31no—
36#63469—61no—
37C9×#101—31no—
38#63382D311yescharacteristic, complemented
39#63381—31no—
40C2×C3×C4×C9C2311yescharacteristic
41#6233—21no—
42C4⋊(C3⋊D9)D411yescharacteristic
43C4×C3⋊D9—21no—
44#6226D411yescharacteristic
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C9×Dic6—21no—
48C9×Dic6—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51C3×C8×C9—21no—
52C3⋊(C9⋊C8)D411yescharacteristic
53C3⋊(C9⋊C8)D411yescharacteristic
54C9×C3⋊C8—21no—
55C3×C9⋊C8—21no—
56#4755—91no—
57#4755—91no—
58#3828—31no—
59C2×C3×Dic6—31no—
60C2×C3×Dic6C2×D311yescharacteristic
61C32×M16C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)—31no—
63#3743—61no—
64C8×C3⋊D3—61no—
65C3×#208—31no—
66C3×#208C2×D311yescharacteristic
67#3719—61no—
68#3719—61no—
69#3718—61no—
70#3718—61no—
71#3717—61no—
72#3717—61no—
73C2×Q8×C9—31no—
74#3697—31no—
75#3697—61no—
76C2×Dic18C2×D311yescharacteristic
77C9×M16—21no—
78C9×M16—31no—
79C9×M16C2×D311yescharacteristic
80#3675—61no—
81#3674—61no—
82#3674—61no—
83#3674—61no—
84#3667C2×D311yescharacteristic
85C2×C9⋊C8—31no—
86C2×C9⋊C8—61no—
87#3664—61no—
88#3663—61no—
89#3663—62no—
90C8×D9—61no—
91C8×D9—62no—
92Dic36—61no—
93C22×C3×C9#4111yescharacteristic
94C2×C3⋊D9—21no—
95C3×C4×C9C2×D411yescharacteristic
96C3×C4×C9C2×D411yescharacteristic
97C3⋊Dic9—21no—
98C9×Dic3—41no—
99C3×Dic9—41no—
100C2×#216—91no—
101C2×#216—91no—
102#942—91no—
103#942—91no—
104#909—91no—
105#909—91no—
106#907—91no—
107#907—91no—
108#907—181no—
109C3×#101—31no—
110C3×#101—91no—
111#824—31no—
112#824D911yescharacteristic, complemented
113#822—91no—
114#822—91no—
115C2×C32×C4C22×D311yescharacteristic
116#638—61no—
117C4⋊(C3⋊D3)—31no—
118C4×C3⋊D3—61no—
119#634—31no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—61no—
122C3×Dic6—21no—
123C3×Dic6—21no—
124C3×Dic6—61no—
125C3×Dic6—61no—
126C32×C8—21no—
127C3⋊(C3⋊C8)—31no—
128C3⋊(C3⋊C8)—31no—
129C3×C3⋊C8—21no—
130C3×C3⋊C8—61no—
131Q8×C9—61no—
132Q8×C9—61no—
133C2×C4×C9—21no—
134C2×C4×C9—61no—
135C2×C4×C9C22×D311yescharacteristic
136#611—62no—
137#611—61no—
138C2×Dic9—21no—
139D36—31no—
140D36—61no—
141C4×D9—62no—
142C4×D9—61no—
143Dic18—21no—
144Dic18—21no—
145Dic18—31no—
146Dic18—61no—
147C8×C9—22no—
148C8×C9—21no—
149C8×C9—61no—
150C9⋊C8—21no—
151C9⋊C8—31no—
152C9⋊C8—31no—
153C9⋊C8—61no—
154C9⋊C8—61no—
155#473—271noSylow
156C2×C3×C9—21no—
157C2×C3×C9#12111yescharacteristic
158C3⋊D9—41no—
159C2×C3×Q8—31no—
160C2×C3×Q8—91no—
161#235—91no—
162#235—91no—
163#235—181no—
164C2×Dic6—31no—
165C2×Dic6C2×D911yescharacteristic
166C3×M16—21no—
167C3×M16—31no—
168C3×M16—91no—
169C3×M16C2×D911yescharacteristic
170C3×M16D3211yescharacteristic
171#216—181no—
172#216—181no—
173#216—181no—
174#216—181no—
175#216—181no—
176#216—181no—
177#215—181no—
178#215—181no—
179#208—31no—
180#208C2×D911yescharacteristic
181C2×C3⋊C8—91no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—181no—
184Dic12—181no—
185Dic12—181no—
186#204—181no—
187#204—181no—
188ΓL(1,𝔽25)—181no—
189ΓL(1,𝔽25)—181no—
190ΓL(1,𝔽25)—182no—
191D3×C8—181no—
192D3×C8—181no—
193D3×C8—182no—
194C22×C32#23511yescharacteristic
195C2×C3⋊D3—61no—
196C32×C4D3×D411yescharacteristic
197C32×C4D3×D411yescharacteristic
198C3⋊Dic3—61no—
199C3×Dic3—41no—
200C3×Dic3—121no—
201C22×C9—21no—
202C22×C9#23511yescharacteristic
203C2×D9—62no—
204C2×D9—61no—
205C4×C9—21no—
206C4×C9—21no—
207C4×C9—121no—
208C4×C9D3×D411yescharacteristic
209C4×C9D3×D411yescharacteristic
210Dic9—41no—
211Dic9—62no—
212Dic9—61no—
213#137—271no—
214#137—271no—
215C2×Q16—271no—
216#131—271no—
217#108—271no—
218#101—31no—
219#101—91no—
220C3×C9#47311yescharacteristic, complemented, Sylow
221C3×Q8—61no—
222C3×Q8—61no—
223C3×Q8—181no—
224C3×Q8—181no—
225C2×C3×C4—21no—
226C2×C3×C4—61no—
227C2×C3×C4—181no—
228C2×C3×C4C22×D911yescharacteristic
229C2×C3×C4C2×D3211yescharacteristic
230#67—182no—
231#67—181no—
232#67—181no—
233C2×Dic3—21no—
234C2×Dic3—61no—
235D12—91no—
236D12—91no—
237D12—181no—
238C4×D3—182no—
239C4×D3—181no—
240C4×D3—181no—
241Dic6—21no—
242Dic6—21no—
243Dic6—61no—
244Dic6—61no—
245Dic6—91no—
246Dic6—91no—
247Dic6—181no—
248C3×C8—22no—
249C3×C8—21no—
250C3×C8—21no—
251C3×C8—61no—
252C3×C8—181no—
253C3⋊C8—21no—
254C3⋊C8—61no—
255C3⋊C8—91no—
256C3⋊C8—91no—
257C3⋊C8—91no—
258C3⋊C8—91no—
259C3⋊C8—181no—
260C3⋊C8—181no—
261C2×C32—21no—
262C2×C32#88411yescharacteristic
263C3⋊D3—121no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—41no—
267C2×C9#88411yescharacteristic
268D9—122no—
269D9—121no—
270#41—271no—
271C2×Q8—31no—
272C2×Q8—91no—
273Q16—541no—
274Q16—541no—
275Q16—541no—
276Q16—541no—
277SD16—541no—
278SD16—541no—
279M16—31no—
280M16—91no—
281M16—541no—
282M16D3×D911yescharacteristic
283C2×C8—271no—
284C2×C8—541no—
285C22×C3—21no—
286C22×C3#369711yescharacteristic
287C22×C3#379811yescharacteristic
288C2×D3—182no—
289C2×D3—181no—
290C2×D3—181no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—121no—
294C3×C4—361no—
295C3×C4D4×D911yescharacteristic
296C3×C4D4×D911yescharacteristic
297C3×C4#380211yescharacteristic
298C3×C4#381111yescharacteristic
299Dic3—41no—
300Dic3—121no—
301Dic3—182no—
302Dic3—181no—
303Dic3—181no—
304C32#475511yescharacteristic
305C9—21no—
306C9#475511yescharacteristic, complemented
307Q8—61no—
308Q8—61no—
309Q8—181no—
310Q8—181no—
311Q8—271no—
312D4—271no—
313D4—541no—
314C2×C4—61no—
315C2×C4—181no—
316C2×C4—541no—
317C2×C4C2×D3×D911yescharacteristic
318C8—21no—
319C8—61no—
320C8—181no—
321C8—271no—
322C8—271no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—41no—
327C2×C3#6344611yescharacteristic
328C2×C3#6391211yescharacteristic
329D3—362no—
330D3—361no—
331D3—361no—
332C22—541no—
333C22#8864011yescharacteristic
334C4—121no—
335C4—361no—
336C4—541no—
337C4#8864611yescharacteristic
338C4#8866811yescharacteristic
339C3—21no—
340C3#1058843511yescharacteristic, complemented
341C3#1059099311yescharacteristic
342C2—21no—
343C2—1081no—
344C2#1172104111yescharacteristic
subgroupG/H
centerC2#11721041
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×M16C22
socleC2×C32#884
prime radical#88736C22
hypercenterM16D3×D9
hypocenterC3×C9#473
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
2. C2×#88427C2semidirect
5. C9×#824C2semidirect
6. C3×#63382C2semidirect
7. #11720286C2semidirect
38. #63382D3semidirect
112. #824D9semidirect
220. C3×C9#473semidirect
306. C9#4755semidirect
340. C3#10588435semidirect

Maximal subgroups

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

subgroupindex
#117217822
C2×#884272
#117208952
#117208712
C9×#8242
C3×#633822
#117202862
#105909933
#105884353