Finite Group Databasebeta

← #13144029

Subgroups — #13144029

Order 1728.

760 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 135 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#11722672C211yesmaximal, characteristic
2#11721866C211yesmaximal, characteristic, complemented
3#11721784C211yesmaximal, characteristic, complemented
4#11721712C211yesmaximal, characteristic, complemented
5C3×C9×#104C211yesmaximal, characteristic, complemented
6#11720557C211yesmaximal, characteristic, complemented
7#11720518C211yesmaximal, characteristic, complemented
8#10592524—31nomaximal
9#10588505—33nomaximal
10C3×C9×#41C2211yescharacteristic
11#88746—21no—
12D4×C3⋊D9—22no—
13D4×C3⋊D9—21no—
14#88736—21no—
15#88736C2211yescharacteristic
16C2×C4⋊(C3⋊D9)—21no—
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C2×#6226C2211yescharacteristic, complemented
19C3×C9×M16C2211yescharacteristic, complemented
20C3×C42×C9C2211yescharacteristic, complemented
21#88547—21no—
22#88543—21no—
23#88536—22no—
24#88532C2211yescharacteristic, complemented
25#88524—21no—
26#88521—21no—
27#64366—31no—
28#64160—31no—
29#64118—31no—
30#64084—31no—
31#63715—33no—
32C32×#104D311yescharacteristic
33#63650—31no—
34#63632—31no—
35#63511—33no—
36#63471—33no—
37#63437—33no—
38C9×#104D313yescomplemented
39#63383—33no—
40#63364—33no—
41C22×C3⋊D9—22no—
42C22×C3⋊D9D412yes—
43C3×Q8×C9D411yescharacteristic
44C3×D4×C9—21no—
45C3×D4×C9D411yescharacteristic
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9C2311yescharacteristic
49#6233—22no—
50#6233—21no—
51C2×C3⋊Dic9—21no—
52C4⋊(C3⋊D9)—22no—
53C4⋊(C3⋊D9)—21no—
54C4⋊(C3⋊D9)—21no—
55C4⋊(C3⋊D9)D411yescharacteristic
56C4×C3⋊D9—22no—
57C4×C3⋊D9—21no—
58#6226—21no—
59#6226—21no—
60#6226D411yescharacteristic
61C3×C8×C9—21no—
62C3⋊(C9⋊C8)—21no—
63#4825—93no—
64#4825—91no—
65C32×#41C2×D311yescharacteristic
66#3832—61no—
67D4×C3⋊D3—62no—
68D4×C3⋊D3—61no—
69#3828—31no—
70#3828—61no—
71C2×C4⋊(C3⋊D3)—31no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×#634—31no—
74C32×M16C2×D311yescharacteristic
75C32×C42C2×D311yescharacteristic
76#3756—61no—
77#3754—61no—
78#3752—62no—
79#3748—31no—
80#3746—61no—
81#3744—61no—
82C9×#41C2×D313yes—
83#3701—63no—
84D4×D9—66no—
85D4×D9—63no—
86#3697—33no—
87#3697—63no—
88C2×D36—33no—
89C2×D36—63no—
90C2×Dic18—33no—
91C9×M16C2×D313yescomplemented
92C42×C9C2×D313yescomplemented
93#3674—63no—
94#3672—63no—
95(C2×D9)⋊C4—66no—
96#3667—33no—
97#3664—63no—
98Dic36—63no—
99C22×C3×C9—21no—
100C22×C3×C9C2×D411yescharacteristic
101C2×C3⋊D9—22no—
102C2×C3⋊D9—21no—
103C2×C3⋊D9—42no—
104C2×C3⋊D9—82no—
105C3×C4×C9—21no—
106C3×C4×C9—41no—
107C3×C4×C9C2×D411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C3⋊Dic9—21no—
110C3⋊Dic9—41no—
111#1009—93no—
112#1009—91no—
113#951—93no—
114#951—91no—
115#909—93no—
116#909—91no—
117#875—93no—
118#875—91no—
119C3×#104D913yescomplemented
120C3×#104C3⋊D311yescharacteristic
121#823—93no—
122#823—91no—
123#805—93no—
124#805—91no—
125C22×C3⋊D3—32no—
126C22×C3⋊D3—62no—
127C32×Q8D1211yescharacteristic
128C32×D4—21no—
129C32×D4D1211yescharacteristic
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4C22×D311yescharacteristic
133#638—62no—
134#638—61no—
135C2×C3⋊Dic3—61no—
136C4⋊(C3⋊D3)—31no—
137C4⋊(C3⋊D3)—62no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—61no—
140C4×C3⋊D3—62no—
141C4×C3⋊D3—61no—
142#634—31no—
143#634—61no—
144#634—61no—
145C22×D9—36no—
146C22×D9—66no—
147C32×C8—21no—
148C3⋊(C3⋊C8)—61no—
149Q8×C9D1213yes—
150D4×C9—23no—
151D4×C9D1213yes—
152C2×C4×C9—23no—
153C2×C4×C9—23no—
154C2×C4×C9C22×D313yes—
155#611—66no—
156#611—63no—
157C2×Dic9—63no—
158D36—33no—
159D36—66no—
160D36—63no—
161D36—63no—
162C4×D9—66no—
163C4×D9—63no—
164Dic18—33no—
165Dic18—63no—
166Dic18—63no—
167C8×C9—23no—
168C9⋊C8—63no—
169#455—271noSylow
170C2×C3×C9—21no—
171C2×C3×C9—41no—
172C2×C3×C9#12011yescharacteristic
173C3⋊D9—42no—
174C3⋊D9—41no—
175C3×#41C2×D913yes—
176C3×#41C2×C3⋊D311yescharacteristic
177#239—183no—
178#239—181no—
179D3×D4—182no—
180D3×D4—186no—
181D3×D4—183no—
182D3×D4—181no—
183#235—93no—
184#235—91no—
185#235—181no—
186#235—183no—
187C2×D12—93no—
188C2×D12—91no—
189C2×D12—181no—
190C2×D12—183no—
191C2×Dic6—93no—
192C2×Dic6—91no—
193C3×M16C2×D913yescomplemented
194C3×M16C2×C3⋊D311yescharacteristic
195C3×C42C2×D913yescomplemented
196C3×C42C2×C3⋊D311yescharacteristic
197#216—183no—
198#216—181no—
199#214—183no—
200#214—181no—
201(C2×D3)⋊C4—182no—
202(C2×D3)⋊C4—186no—
203#208—93no—
204#208—91no—
205Dic12—181no—
206Dic12—183no—
207#204—181no—
208#204—183no—
209C22×C32—21no—
210C22×C32D3×D411yescharacteristic
211C2×C3⋊D3—61no—
212C2×C3⋊D3—62no—
213C2×C3⋊D3—122no—
214C2×C3⋊D3—242no—
215C32×C4—21no—
216C32×C4—41no—
217C32×C4C2×D1211yescharacteristic
218C32×C4D3×D411yescharacteristic
219C3⋊Dic3—61no—
220C3⋊Dic3—121no—
221C22×C9—23no—
222C22×C9D3×D413yes—
223C2×D9—63no—
224C2×D9—66no—
225C2×D9—126no—
226C2×D9—246no—
227C4×C9—23no—
228C4×C9—43no—
229C4×C9C2×D1213yes—
230C4×C9D3×D413yes—
231Dic9—63no—
232Dic9—123no—
233H(5,𝔽2)—271no—
234#137—271no—
235#137—271no—
236#124—271no—
237#104—271no—
238#104C3⋊D911yescharacteristic, complemented
239#100—271no—
240C3×C9#45511yescharacteristic, complemented, Sylow
241C22×D3—96no—
242C22×D3—92no—
243C22×D3—182no—
244C22×D3—186no—
245C3×Q8D3613yes—
246C3×Q8C4⋊(C3⋊D3)11yescharacteristic
247C3×D4—23no—
248C3×D4—21no—
249C3×D4D3613yes—
250C3×D4C4⋊(C3⋊D3)11yescharacteristic
251C2×C3×C4—23no—
252C2×C3×C4—21no—
253C2×C3×C4—23no—
254C2×C3×C4—21no—
255C2×C3×C4C22×D913yes—
256C2×C3×C4C22×C3⋊D311yescharacteristic
257#67—182no—
258#67—186no—
259#67—183no—
260#67—181no—
261C2×Dic3—183no—
262C2×Dic3—181no—
263D12—93no—
264D12—91no—
265D12—182no—
266D12—186no—
267D12—183no—
268D12—183no—
269D12—181no—
270D12—181no—
271C4×D3—182no—
272C4×D3—186no—
273C4×D3—183no—
274C4×D3—181no—
275Dic6—93no—
276Dic6—91no—
277Dic6—183no—
278Dic6—183no—
279Dic6—181no—
280Dic6—181no—
281C3×C8—23no—
282C3×C8—21no—
283C3⋊C8—183no—
284C3⋊C8—181no—
285C2×C32—21no—
286C2×C32—41no—
287C2×C32#88211yescharacteristic
288C3⋊D3—122no—
289C3⋊D3—121no—
290C2×C9—23no—
291C2×C9—43no—
292C2×C9#88213yes—
293D9—126no—
294D9—123no—
295#41—271no—
296#41—541no—
297#41—541no—
298#41C2×C3⋊D911yescharacteristic
299C2×Q8—271no—
300C2×D4—271no—
301C2×D4—541no—
302C2×D4—542no—
303C2×D4—541no—
304Q16—541no—
305Q16—541no—
306SD16—541no—
307SD16—541no—
308M16—271no—
309M16C2×C3⋊D911yescharacteristic, complemented
310C22⋊C4—542no—
311C42C2×C3⋊D911yescharacteristic, complemented
312C22×C3—21no—
313C22×C3—23no—
314C22×C3D4×D913yes—
315C22×C3D4×C3⋊D311yescharacteristic
316C2×D3—181no—
317C2×D3—182no—
318C2×D3—183no—
319C2×D3—186no—
320C2×D3—362no—
321C2×D3—366no—
322C2×D3—722no—
323C2×D3—726no—
324C3×C4—21no—
325C3×C4—23no—
326C3×C4—41no—
327C3×C4—43no—
328C3×C4C2×D3613yes—
329C3×C4D4×D913yes—
330C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
331C3×C4D4×C3⋊D311yescharacteristic
332Dic3—181no—
333Dic3—183no—
334Dic3—361no—
335Dic3—363no—
336C32#482511yescharacteristic
337C9#482513yescomplemented
338C23—272no—
339C23—542no—
340Q8—271no—
341Q8—541no—
342Q8—541no—
343Q8C4⋊(C3⋊D9)11yescharacteristic
344D4—21no—
345D4—271no—
346D4—542no—
347D4—542no—
348D4—541no—
349D4—541no—
350D4—541no—
351D4C4⋊(C3⋊D9)11yescharacteristic
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—542no—
355C2×C4—541no—
356C2×C4—541no—
357C2×C4C22×C3⋊D911yescharacteristic
358C8—21no—
359C8—541no—
360C2×C3—21no—
361C2×C3—23no—
362C2×C3—41no—
363C2×C3—43no—
364C2×C3#6344413yes—
365C2×C3#6409111yescharacteristic
366D3—362no—
367D3—361no—
368D3—366no—
369D3—363no—
370C22—21no—
371C22—541no—
372C22—542no—
373C22—1082no—
374C22—2162no—
375C22D4×C3⋊D911yescharacteristic
376C4—21no—
377C4—41no—
378C4—541no—
379C4—1081no—
380C4C2×C4⋊(C3⋊D9)11yescharacteristic
381C4D4×C3⋊D911yescharacteristic
382C3#1058850513yescomplemented
383C3#1059252411yescharacteristic
384C2—21no—
385C2—41no—
386C2—1082no—
387C2—1081no—
388C2#1172172811yescharacteristic
subgroupG/H
centerC2#11721728
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#104C2
socleC2×C32#882
prime radical#11722672C2
hypercenter#104C3⋊D9
hypocenterC3×C9#455
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #11721866C2semidirect
3. #11721784C2semidirect
4. #11721712C2semidirect
5. C3×C9×#104C2semidirect
6. #11720557C2semidirect
7. #11720518C2semidirect
18. C2×#6226C22semidirect
19. C3×C9×M16C22semidirect
20. C3×C42×C9C22semidirect
24. #88532C22semidirect
38. C9×#104D3semidirect
91. C9×M16C2×D3semidirect
92. C42×C9C2×D3semidirect
119. C3×#104D9semidirect
193. C3×M16C2×D9semidirect
195. C3×C42C2×D9semidirect
238. #104C3⋊D9semidirect
240. C3×C9#455semidirect
309. M16C2×C3⋊D9semidirect
311. C42C2×C3⋊D9semidirect
337. C9#4825semidirect
382. C3#10588505semidirect

Maximal subgroups

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

subgroupindex
#117226722
#117218662
#117217842
#117217122
C3×C9×#1042
#117205572
#117205182
#105925243
#105885053