Finite Group Databasebeta

← #13144782

Subgroups — #13144782

Order 1728.

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

normal subgroups only

This group has 840 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, complemented
2C3×C9×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#11721840C211yesmaximal, characteristic, complemented
4#11721833C211yesmaximal, characteristic, complemented
5#11720585C211yesmaximal, characteristic, complemented
6#11720558C211yesmaximal, characteristic, complemented
7#11720557C211yesmaximal, characteristic, complemented
8#10592897—31nomaximal
9#10588879—33nomaximal
10C3×C9×#41C2211yescharacteristic
11C2×C3×D4×C9C2211yescharacteristic, complemented
12C2×#6233—22no—
13#88746—21no—
14D4×C3⋊D9—22no—
15D4×C3⋊D9—21no—
16#88736—21no—
17#88736C2211yescharacteristic
18C2×C4⋊(C3⋊D9)—21no—
19C2×C4⋊(C3⋊D9)C2211yescharacteristic
20C3×C9×SD16—21no—
21C3×C9×D8—21no—
22C3×C9×M16C2211yescharacteristic, complemented
23#88544—21no—
24#88543—21no—
25C4×C3⋊Dic9C2211yescharacteristic, complemented
26#88532C2211yescharacteristic, complemented
27#64366—31no—
28C32×AGL(1,ℤ/8)D311yescharacteristic
29#64149—31no—
30#64141—31no—
31#63715—33no—
32#63664—31no—
33#63652—31no—
34#63650—31no—
35C9×AGL(1,ℤ/8)D313yescomplemented
36#63502—33no—
37#63495—33no—
38#63396—33no—
39#63384—33no—
40#63383—33no—
41C23×C3×C9—21no—
42C22×C3⋊D9—22no—
43C22×C3⋊D9D412yes—
44C3×Q8×C9D411yescharacteristic, complemented
45C3×D4×C9—21no—
46C3×D4×C9—21no—
47C3×D4×C9—21no—
48C3×D4×C9D411yescharacteristic, complemented
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9C2311yescharacteristic
51#6233—22no—
52#6233—21no—
53#6233—44no—
54C2×C3⋊Dic9—21no—
55C4⋊(C3⋊D9)—22no—
56C4⋊(C3⋊D9)—21no—
57C4⋊(C3⋊D9)—21no—
58C4⋊(C3⋊D9)D411yescharacteristic, complemented
59C4×C3⋊D9—22no—
60C4×C3⋊D9—21no—
61#6226D411yescharacteristic, complemented
62C3×C8×C9—21no—
63C3⋊(C9⋊C8)—21no—
64#5198—93no—
65#5198—91no—
66C32×#41C2×D311yescharacteristic
67C2×C32×D4C2×D311yescharacteristic
68C2×#638—62no—
69#3832—61no—
70D4×C3⋊D3—62no—
71D4×C3⋊D3—61no—
72#3828—31no—
73#3828—61no—
74C2×C4⋊(C3⋊D3)—31no—
75C2×C4⋊(C3⋊D3)—61no—
76C32×SD16—21no—
77C32×D8—21no—
78C32×M16C2×D311yescharacteristic
79#3754—61no—
80#3753—61no—
81C4×C3⋊Dic3—31no—
82#3748—31no—
83C9×#41C2×D313yes—
84C2×D4×C9C2×D313yescomplemented
85C2×#611—66no—
86#3701—63no—
87D4×D9—66no—
88D4×D9—63no—
89#3697—33no—
90#3697—63no—
91C2×D36—33no—
92C2×D36—63no—
93C9×SD16—23no—
94C9×D8—23no—
95C9×M16C2×D313yescomplemented
96#3673—63no—
97#3672—63no—
98C4×Dic9—33no—
99#3667—33no—
100C22×C3×C9—21no—
101C22×C3×C9—42no—
102C22×C3×C9—41no—
103C22×C3×C9C2×D411yescharacteristic
104C2×C3⋊D9—22no—
105C2×C3⋊D9—21no—
106C2×C3⋊D9—42no—
107C2×C3⋊D9—82no—
108C3×C4×C9—21no—
109C3×C4×C9C2×D411yescharacteristic
110C3×C4×C9C2×D411yescharacteristic
111C3⋊Dic9—22no—
112C3⋊Dic9—21no—
113#1009—93no—
114#1009—91no—
115C3×AGL(1,ℤ/8)D913yescomplemented
116C3×AGL(1,ℤ/8)C3⋊D311yescharacteristic
117#940—93no—
118#940—91no—
119#932—93no—
120#932—91no—
121#837—93no—
122#837—91no—
123#825—93no—
124#825—91no—
125#823—93no—
126#823—91no—
127C23×C32—21no—
128C22×C3⋊D3—32no—
129C22×C3⋊D3—62no—
130C32×Q8#6711yescharacteristic
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—21no—
134C32×D4#6711yescharacteristic
135C2×C32×C4—21no—
136C2×C32×C4C22×D311yescharacteristic
137#638—62no—
138#638—61no—
139#638—124no—
140C2×C3⋊Dic3—61no—
141C4⋊(C3⋊D3)—31no—
142C4⋊(C3⋊D3)—62no—
143C4⋊(C3⋊D3)—61no—
144C4⋊(C3⋊D3)—61no—
145C4×C3⋊D3—62no—
146C4×C3⋊D3—61no—
147#634—31no—
148C23×C9—23no—
149C22×D9—36no—
150C22×D9—66no—
151C32×C8—21no—
152C3⋊(C3⋊C8)—61no—
153Q8×C9#6713yescomplemented
154D4×C9—23no—
155D4×C9—23no—
156D4×C9—23no—
157D4×C9#6713yescomplemented
158C2×C4×C9—23no—
159C2×C4×C9C22×D313yes—
160#611—66no—
161#611—63no—
162#611—1212no—
163C2×Dic9—63no—
164D36—33no—
165D36—66no—
166D36—63no—
167D36—63no—
168C4×D9—66no—
169C4×D9—63no—
170Dic18—33no—
171C8×C9—23no—
172C9⋊C8—63no—
173#453—271noSylow
174C2×C3×C9—21no—
175C2×C3×C9—41no—
176C2×C3×C9—81no—
177C2×C3×C9#12011yescharacteristic
178C3⋊D9—42no—
179C3⋊D9—41no—
180C3×#41C2×D913yes—
181C3×#41C2×C3⋊D311yescharacteristic
182C2×C3×D4C2×D913yescomplemented
183C2×C3×D4C2×C3⋊D311yescharacteristic
184C2×#67—182no—
185C2×#67—186no—
186#239—183no—
187#239—181no—
188D3×D4—182no—
189D3×D4—186no—
190D3×D4—183no—
191D3×D4—181no—
192#235—93no—
193#235—91no—
194#235—181no—
195#235—183no—
196C2×D12—93no—
197C2×D12—91no—
198C2×D12—181no—
199C2×D12—183no—
200C3×SD16—23no—
201C3×SD16—21no—
202C3×D8—23no—
203C3×D8—21no—
204C3×M16C2×D913yescomplemented
205C3×M16C2×C3⋊D311yescharacteristic
206#214—181no—
207#214—183no—
208#213—181no—
209#213—183no—
210C4×Dic3—93no—
211C4×Dic3—91no—
212#208—93no—
213#208—91no—
214C22×C32—21no—
215C22×C32—42no—
216C22×C32—41no—
217C22×C32D3×D411yescharacteristic
218C2×C3⋊D3—61no—
219C2×C3⋊D3—62no—
220C2×C3⋊D3—122no—
221C2×C3⋊D3—242no—
222C32×C4—21no—
223C32×C4D3×D411yescharacteristic
224C32×C4C2×#6711yescharacteristic
225C3⋊Dic3—62no—
226C3⋊Dic3—61no—
227C22×C9—23no—
228C22×C9—46no—
229C22×C9—43no—
230C22×C9D3×D413yes—
231C2×D9—63no—
232C2×D9—66no—
233C2×D9—126no—
234C2×D9—246no—
235C4×C9—23no—
236C4×C9D3×D413yes—
237C4×C9C2×#6713yes—
238Dic9—66no—
239Dic9—63no—
240H(5,𝔽2)—271no—
241AGL(1,ℤ/8)—271no—
242AGL(1,ℤ/8)C3⋊D911yescharacteristic, complemented
243#127—271no—
244#104—271no—
245#104—271no—
246#100—271no—
247C3×C9#45311yescharacteristic, complemented, Sylow
248C23×C3—23no—
249C23×C3—21no—
250C22×D3—96no—
251C22×D3—92no—
252C22×D3—182no—
253C22×D3—186no—
254C3×Q8#61113yescomplemented
255C3×Q8#63811yescharacteristic
256C3×D4—23no—
257C3×D4—21no—
258C3×D4—23no—
259C3×D4—23no—
260C3×D4—21no—
261C3×D4—21no—
262C3×D4#61113yescomplemented
263C3×D4#63811yescharacteristic
264C2×C3×C4—23no—
265C2×C3×C4—21no—
266C2×C3×C4C22×D913yes—
267C2×C3×C4C22×C3⋊D311yescharacteristic
268#67—182no—
269#67—186no—
270#67—183no—
271#67—181no—
272#67—364no—
273#67—3612no—
274C2×Dic3—183no—
275C2×Dic3—181no—
276D12—93no—
277D12—91no—
278D12—182no—
279D12—186no—
280D12—183no—
281D12—183no—
282D12—181no—
283D12—181no—
284C4×D3—182no—
285C4×D3—186no—
286C4×D3—183no—
287C4×D3—181no—
288Dic6—93no—
289Dic6—91no—
290C3×C8—23no—
291C3×C8—21no—
292C3⋊C8—183no—
293C3⋊C8—181no—
294C2×C32—21no—
295C2×C32—41no—
296C2×C32—81no—
297C2×C32#93711yescharacteristic
298C3⋊D3—122no—
299C3⋊D3—121no—
300C2×C9—23no—
301C2×C9—43no—
302C2×C9—83no—
303C2×C9#93713yes—
304D9—126no—
305D9—123no—
306#41—271no—
307#41—541no—
308#41—541no—
309#41C2×C3⋊D911yescharacteristic
310C2×D4—271no—
311C2×D4—541no—
312C2×D4—542no—
313C2×D4—541no—
314C2×D4—542no—
315C2×D4C2×C3⋊D911yescharacteristic, complemented
316SD16—21no—
317SD16—541no—
318D8—21no—
319D8—541no—
320M16—271no—
321M16C2×C3⋊D911yescharacteristic, complemented
322C42—271no—
323C22×C3—23no—
324C22×C3—21no—
325C22×C3—42no—
326C22×C3—46no—
327C22×C3—43no—
328C22×C3—41no—
329C22×C3D4×D913yes—
330C22×C3D4×C3⋊D311yescharacteristic
331C2×D3—183no—
332C2×D3—186no—
333C2×D3—181no—
334C2×D3—182no—
335C2×D3—362no—
336C2×D3—366no—
337C2×D3—722no—
338C2×D3—726no—
339C3×C4—23no—
340C3×C4—21no—
341C3×C4D4×D913yes—
342C3×C4C2×#61113yes—
343C3×C4D4×C3⋊D311yescharacteristic
344C3×C4C2×#63811yescharacteristic
345Dic3—182no—
346Dic3—186no—
347Dic3—183no—
348Dic3—181no—
349C32#519811yescharacteristic
350C9#519813yescomplemented
351C23—21no—
352C23—272no—
353C23—542no—
354Q8—271no—
355Q8#623311yescharacteristic, complemented
356D4—21no—
357D4—21no—
358D4—21no—
359D4—271no—
360D4—542no—
361D4—542no—
362D4—541no—
363D4—541no—
364D4—541no—
365D4—1084no—
366D4#623311yescharacteristic, complemented
367C2×C4—21no—
368C2×C4—542no—
369C2×C4—541no—
370C2×C4—541no—
371C2×C4C22×C3⋊D911yescharacteristic
372C8—21no—
373C8—541no—
374C2×C3—21no—
375C2×C3—23no—
376C2×C3—41no—
377C2×C3—43no—
378C2×C3—81no—
379C2×C3—83no—
380C2×C3#6349913yes—
381C2×C3#6414611yescharacteristic
382D3—362no—
383D3—361no—
384D3—366no—
385D3—363no—
386C22—21no—
387C22—42no—
388C22—41no—
389C22—541no—
390C22—542no—
391C22—1082no—
392C22—2162no—
393C22D4×C3⋊D911yescharacteristic
394C4—21no—
395C4—542no—
396C4—541no—
397C4D4×C3⋊D911yescharacteristic
398C4C2×#623311yescharacteristic
399C3#1058887913yescomplemented
400C3#1059289711yescharacteristic
401C2—21no—
402C2—41no—
403C2—81no—
404C2—1082no—
405C2—1081no—
406C2#1172183711yescharacteristic
subgroupG/H
centerC2#11721837
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×AGL(1,ℤ/8)C2
socleC2×C32#937
hypercenterAGL(1,ℤ/8)C3⋊D9
hypocenterC3×C9#453
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #11722672C2semidirect
2. C3×C9×AGL(1,ℤ/8)C2semidirect
3. #11721840C2semidirect
4. #11721833C2semidirect
5. #11720585C2semidirect
6. #11720558C2semidirect
7. #11720557C2semidirect
11. C2×C3×D4×C9C22semidirect
22. C3×C9×M16C22semidirect
25. C4×C3⋊Dic9C22semidirect
26. #88532C22semidirect
35. C9×AGL(1,ℤ/8)D3semidirect
44. C3×Q8×C9D4semidirect
48. C3×D4×C9D4semidirect
58. C4⋊(C3⋊D9)D4semidirect
61. #6226D4semidirect
84. C2×D4×C9C2×D3semidirect
95. C9×M16C2×D3semidirect
115. C3×AGL(1,ℤ/8)D9semidirect
153. Q8×C9#67semidirect
157. D4×C9#67semidirect
182. C2×C3×D4C2×D9semidirect
204. C3×M16C2×D9semidirect
242. AGL(1,ℤ/8)C3⋊D9semidirect
247. C3×C9#453semidirect
254. C3×Q8#611semidirect
262. C3×D4#611semidirect
315. C2×D4C2×C3⋊D9semidirect
321. M16C2×C3⋊D9semidirect
350. C9#5198semidirect
355. Q8#6233semidirect
366. D4#6233semidirect
399. C3#10588879semidirect

Maximal subgroups

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

subgroupindex
#117226722
C3×C9×AGL(1,ℤ/8)2
#117218402
#117218332
#117205852
#117205582
#117205572
#105928973
#105888793