Finite Group Databasebeta

← #13140766

Subgroups — #13140766

Order 1728.

380 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#88548C211yesmaximal, characteristic, complemented
2C3×#63505C211yesmaximal, characteristic, complemented
3#11720873C211yesmaximal, characteristic
4#11720859C211yesmaximal, characteristic, complemented
5C9×#836C211yesmaximal, characteristic, complemented
6#11720288C211yesmaximal, characteristic, complemented
7#11720286C211yesmaximal, characteristic, complemented
8#10591643—31nomaximal
9#10588434—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C3×#3697C2211yescharacteristic
13#88548—21no—
14#88548—21no—
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×#3675—21no—
17C3×#3674—21no—
18C9×#208C2211yescharacteristic
19C9×#208C2211yescharacteristic
20C3×#3667C2211yescharacteristic
21#88417—21no—
22#88416—21no—
23#88413—21no—
24D9×C3⋊C8—21no—
25C2×#3755—31no—
26C3×#942—31no—
27#63825—31no—
28#63815—31no—
29C3×#836D311yescharacteristic
30#63578—31no—
31#63559—31no—
32#63505D311yescharacteristic, complemented
33C2×#3675—31no—
34C2×#3675—61no—
35#63470—31no—
36#63469—31no—
37C9×#101—31no—
38#63383—31no—
39#63381—31no—
40C22×C3⋊D9—21no—
41C3×Q8×C9—21no—
42C3×Q8×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)—21no—
47C3×#611—21no—
48C3×D36D411yescharacteristic
49C3×C4×D9—21no—
50C3×Dic18D411yescharacteristic
51C3⋊(C9⋊C8)D411yescharacteristic
52C3⋊(C9⋊C8)D411yescharacteristic
53C9×C3⋊C8—21no—
54C9×C3⋊C8—21no—
55C3×C9⋊C8—21no—
56#5203—91no—
57#4754—91no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C4⋊(C3⋊D3)—31no—
60C3×#235—31no—
61#3755—61no—
62#3755—61no—
63C2×C3⋊(C3⋊C8)—31no—
64C3×#216—61no—
65C3×#215—61no—
66C3×#208—31no—
67C3×#208C2×D311yescharacteristic
68C3×#208C2×D311yescharacteristic
69#3717—61no—
70#3714—61no—
71#3711—61no—
72D3×C3⋊C8—61no—
73C2×Q8×C9—21no—
74C2×Q8×C9C2×D311yescharacteristic
75#3697C2×D311yescharacteristic
76C2×D36—31no—
77C2×D36—61no—
78C9×M16—31no—
79C9×M16—31no—
80#3675—21no—
81#3675—61no—
82#3675—61no—
83#3675—62no—
84#3675—62no—
85#3674—21no—
86#3667C2×D311yescharacteristic
87C2×C9⋊C8—31no—
88C2×C9⋊C8—61no—
89D72—61no—
90#3664—61no—
91#3663—61no—
92C8×D9—61no—
93C22×C3×C9#4111yescharacteristic
94C2×C3⋊D9—41no—
95C2×C3⋊D9—81no—
96C2×C3×D9—21no—
97C3×C4×C9—41no—
98C3×C4×C9C2×D411yescharacteristic
99C3×C4×C9C2×D411yescharacteristic
100C3×Dic9—21no—
101C3×#137—91no—
102#949—91no—
103#948—91no—
104#942—31no—
105C2×#215—91no—
106C2×#215—91no—
107C2×#215—181no—
108#908—91no—
109#907—91no—
110C3×#101—31no—
111#836D911yescharacteristic, complemented
112#823—91no—
113#823—91no—
114#822—91no—
115#822—91no—
116C22×C3⋊D3—61no—
117C32×Q8—21no—
118C32×Q8—21no—
119C2×C32×C4—21no—
120C2×C32×C4C22×D311yescharacteristic
121C4⋊(C3⋊D3)—61no—
122C4⋊(C3⋊D3)—61no—
123C3×#67—61no—
124C3×D12—31no—
125C3×C4×D3—61no—
126C3×Dic6—31no—
127C22×D9—61no—
128C22×D9—121no—
129C3⋊(C3⋊C8)—31no—
130C3⋊(C3⋊C8)—31no—
131C3×C3⋊C8—21no—
132C3×C3⋊C8—21no—
133C3×C3⋊C8—61no—
134Q8×C9—22no—
135Q8×C9—21no—
136Q8×C9—21no—
137Q8×C9—41no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—41no—
141C2×C4×C9C22×D311yescharacteristic
142#611—21no—
143D36—62no—
144D36—61no—
145D36—61no—
146D36—121no—
147D36#6711yescharacteristic
148C4×D9—21no—
149Dic18#6711yescharacteristic
150C8×C9—61no—
151C8×C9—61no—
152C9⋊C8—21no—
153C9⋊C8—31no—
154C9⋊C8—31no—
155C9⋊C8—61no—
156C9⋊C8—61no—
157#471—271noSylow
158C2×C3×C9—21no—
159C2×C3×C9#12111yescharacteristic
160C3⋊D9—81no—
161C3×D9—41no—
162C3×#41—91no—
163C2×C3×Q8—21no—
164C2×C3×Q8C2×D911yescharacteristic
165C2×C3×Q8D3211yescharacteristic
166#235—31no—
167C2×D12—91no—
168C2×D12—91no—
169C2×D12—181no—
170C3×Q16—181no—
171C3×SD16—181no—
172C3×M16—31no—
173C3×M16—31no—
174C3×M16—91no—
175#216—61no—
176#215—61no—
177#215—181no—
178#215—181no—
179#215—181no—
180#215—181no—
181#215—181no—
182#215—182no—
183#215—182no—
184#213—181no—
185#208—31no—
186#208—181no—
187#208C2×D911yescharacteristic
188#208C2×D911yescharacteristic
189C2×C3⋊C8—91no—
190C2×C3⋊C8—91no—
191C2×C3⋊C8—181no—
192C2×C3⋊C8—181no—
193D24—181no—
194#204—181no—
195ΓL(1,𝔽25)—181no—
196D3×C8—181no—
197C22×C32#23511yescharacteristic
198C2×C3⋊D3—121no—
199C2×C3⋊D3—241no—
200C2×C3×D3—61no—
201C32×C4—41no—
202C32×C4D3×D411yescharacteristic
203C32×C4D3×D411yescharacteristic
204C3×Dic3—61no—
205C22×C9—21no—
206C22×C9#23711yescharacteristic
207C2×D9—21no—
208C2×D9—122no—
209C2×D9—121no—
210C2×D9—242no—
211C2×D9—241no—
212C4×C9—21no—
213C4×C9—21no—
214C4×C9—42no—
215C4×C9—41no—
216C4×C9D3×D411yescharacteristic
217C4×C9C2×#6711yescharacteristic
218Dic9—21no—
219#137—91no—
220AGL(1,ℤ/8)—271no—
221C2×SD16—271no—
222#131—271no—
223#108—271no—
224#101—31no—
225#100—271no—
226C3×C9#47111yescharacteristic, complemented, Sylow
227C22×D3—181no—
228C22×D3—181no—
229C22×D3—361no—
230C3×Q8—22no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—21no—
235C3×Q8—41no—
236C3×Q8—91no—
237C3×D4—91no—
238C3×D4—181no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—41no—
243C2×C3×C4—181no—
244C2×C3×C4C22×D911yescharacteristic
245C2×C3×C4C2×D3211yescharacteristic
246#67—61no—
247D12—31no—
248D12—182no—
249D12—181no—
250D12—181no—
251D12—181no—
252D12—181no—
253D12—361no—
254C4×D3—61no—
255Dic6—31no—
256C3×C8—61no—
257C3×C8—61no—
258C3×C8—181no—
259C3⋊C8—21no—
260C3⋊C8—21no—
261C3⋊C8—61no—
262C3⋊C8—91no—
263C3⋊C8—91no—
264C3⋊C8—91no—
265C3⋊C8—91no—
266C3⋊C8—181no—
267C3⋊C8—181no—
268C2×C32—21no—
269C2×C32#88411yescharacteristic
270C3⋊D3—241no—
271C3×D3—121no—
272C2×C9—22no—
273C2×C9—21no—
274C2×C9—21no—
275C2×C9#93911yescharacteristic
276D9—41no—
277D9—242no—
278D9—241no—
279#41—91no—
280C2×Q8D3×D911yescharacteristic
281C2×D4—271no—
282Q16—181no—
283SD16—181no—
284SD16—541no—
285SD16—541no—
286SD16—541no—
287D8—541no—
288M16—31no—
289M16—31no—
290M16—91no—
291M16—541no—
292C2×C8—271no—
293C2×C8—541no—
294C22×C3—21no—
295C22×C3—181no—
296C22×C3#369711yescharacteristic
297C22×C3#380411yescharacteristic
298C2×D3—61no—
299C2×D3—362no—
300C2×D3—361no—
301C2×D3—361no—
302C2×D3—722no—
303C2×D3—721no—
304C2×D3—721no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—42no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—181no—
311C3×C4D4×D911yescharacteristic
312C3×C4D4×D911yescharacteristic
313C3×C4D3×#6711yescharacteristic
314C3×C4#381111yescharacteristic
315Dic3—61no—
316C32#475411yescharacteristic
317C9—21no—
318C9#520311yescharacteristic, complemented
319C23—541no—
320Q8—21no—
321Q8—21no—
322Q8—91no—
323D4—91no—
324D4—181no—
325D4—541no—
326D4—541no—
327C2×C4—21no—
328C2×C4—181no—
329C2×C4C2×D3×D911yescharacteristic
330C8—61no—
331C8—61no—
332C8—181no—
333C8—271no—
334C8—271no—
335C2×C3—22no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—361no—
340C2×C3#6344611yescharacteristic
341C2×C3#6397111yescharacteristic
342D3—121no—
343D3—722no—
344D3—721no—
345D3—721no—
346C22—181no—
347C22—1081no—
348C22—2161no—
349C22#8865811yescharacteristic
350C4—41no—
351C4—181no—
352C4D9×#6711yescharacteristic
353C4#8866811yescharacteristic
354C3—21no—
355C3#1058843411yescharacteristic, complemented
356C3#1059164311yescharacteristic
357C2—21no—
358C2—361no—
359C2—2161no—
360C2#1172127111yescharacteristic
subgroupG/H
centerC2#11721271
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC2×C32#884
prime radical#11720873C2
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. C2×#88548C2semidirect
2. C3×#63505C2semidirect
4. #11720859C2semidirect
5. C9×#836C2semidirect
6. #11720288C2semidirect
7. #11720286C2semidirect
32. #63505D3semidirect
111. #836D9semidirect
226. C3×C9#471semidirect
318. C9#5203semidirect
355. C3#10588434semidirect

Maximal subgroups

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

subgroupindex
C2×#885482
C3×#635052
#117208732
#117208592
C9×#8362
#117202882
#117202862
#105916433
#105884343