Finite Group Databasebeta

← #13144026

Subgroups — #13144026

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
2#11721867C211yesmaximal, characteristic, complemented
3#11721783C211yesmaximal, characteristic, complemented
4#11721711C211yesmaximal, characteristic, complemented
5C3×C9×#104C211yesmaximal, characteristic, complemented
6#11720557C211yesmaximal, characteristic, complemented
7#11720518C211yesmaximal, characteristic, complemented
8#10592521—31nomaximal
9#10588502—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)—22no—
18C2×C4⋊(C3⋊D9)C2211yescharacteristic
19C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
20C3×C9×M16C2211yescharacteristic, complemented
21C3×C42×C9C2211yescharacteristic, complemented
22#88548—21no—
23#88544—21no—
24#88532C2211yescharacteristic, complemented
25#88525—21no—
26#88524—21no—
27#64366—31no—
28#64158—31no—
29#64117—31no—
30#64083—31no—
31#63715—33no—
32C32×#104D311yescharacteristic
33#63650—31no—
34#63632—31no—
35#63512—33no—
36#63470—33no—
37#63436—33no—
38C9×#104D313yescomplemented
39#63383—33no—
40#63364—33no—
41C22×C3⋊D9—22no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9D412yes—
44C3×Q8×C9D411yescharacteristic, complemented
45C3×D4×C9—21no—
46C3×D4×C9D411yescharacteristic, complemented
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9C2311yescharacteristic
50#6233—22no—
51#6233—21no—
52C4⋊(C3⋊D9)—22no—
53C4⋊(C3⋊D9)—21no—
54C4⋊(C3⋊D9)—21no—
55C4⋊(C3⋊D9)—21no—
56C4⋊(C3⋊D9)—21no—
57C4⋊(C3⋊D9)—44no—
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#4822—93no—
65#4822—91no—
66C32×#41C2×D311yescharacteristic
67#3832—61no—
68D4×C3⋊D3—62no—
69D4×C3⋊D3—61no—
70#3828—31no—
71#3828—61no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—31no—
74C2×C4⋊(C3⋊D3)—61no—
75C2×C4⋊(C3⋊D3)—62no—
76C32×M16C2×D311yescharacteristic
77C32×C42C2×D311yescharacteristic
78#3755—61no—
79#3753—61no—
80#3748—31no—
81#3745—61no—
82#3744—61no—
83C9×#41C2×D313yes—
84#3701—63no—
85D4×D9—66no—
86D4×D9—63no—
87#3697—33no—
88#3697—63no—
89C2×D36—33no—
90C2×D36—33no—
91C2×D36—63no—
92C2×D36—66no—
93C9×M16C2×D313yescomplemented
94C42×C9C2×D313yescomplemented
95#3675—63no—
96#3673—63no—
97#3667—33no—
98D72—63no—
99#3664—63no—
100C22×C3×C9—21no—
101C22×C3×C9C2×D411yescharacteristic
102C2×C3⋊D9—22no—
103C2×C3⋊D9—21no—
104C2×C3⋊D9—42no—
105C2×C3⋊D9—42no—
106C2×C3⋊D9—41no—
107C2×C3⋊D9—82no—
108C3×C4×C9—22no—
109C3×C4×C9—21no—
110C3×C4×C9C2×D411yescharacteristic
111C3×C4×C9C2×D411yescharacteristic
112C3⋊Dic9—21no—
113#1009—93no—
114#1009—91no—
115#949—93no—
116#949—91no—
117#908—93no—
118#908—91no—
119#874—93no—
120#874—91no—
121C3×#104D913yescomplemented
122C3×#104C3⋊D311yescharacteristic
123#823—93no—
124#823—91no—
125#805—93no—
126#805—91no—
127C22×C3⋊D3—32no—
128C22×C3⋊D3—62no—
129C22×C3⋊D3—61no—
130C32×Q8D1211yescharacteristic
131C32×D4—21no—
132C32×D4D1211yescharacteristic
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4C22×D311yescharacteristic
136#638—62no—
137#638—61no—
138C4⋊(C3⋊D3)—31no—
139C4⋊(C3⋊D3)—62no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—61no—
142C4⋊(C3⋊D3)—61no—
143C4⋊(C3⋊D3)—61no—
144C4⋊(C3⋊D3)—124no—
145C4×C3⋊D3—62no—
146C4×C3⋊D3—61no—
147#634—31no—
148C22×D9—36no—
149C22×D9—66no—
150C22×D9—63no—
151C32×C8—21no—
152C3⋊(C3⋊C8)—61no—
153Q8×C9D1213yescomplemented
154D4×C9—23no—
155D4×C9D1213yescomplemented
156C2×C4×C9—23no—
157C2×C4×C9—23no—
158C2×C4×C9C22×D313yes—
159#611—66no—
160#611—63no—
161D36—33no—
162D36—66no—
163D36—63no—
164D36—63no—
165D36—63no—
166D36—63no—
167D36—1212no—
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#12011yescharacteristic
177C3⋊D9—42no—
178C3⋊D9—41no—
179C3⋊D9—81no—
180C3×#41C2×D913yes—
181C3×#41C2×C3⋊D311yescharacteristic
182#239—183no—
183#239—181no—
184D3×D4—182no—
185D3×D4—186no—
186D3×D4—183no—
187D3×D4—181no—
188#235—93no—
189#235—91no—
190#235—181no—
191#235—183no—
192C2×D12—93no—
193C2×D12—93no—
194C2×D12—91no—
195C2×D12—91no—
196C2×D12—181no—
197C2×D12—183no—
198C2×D12—182no—
199C2×D12—186no—
200C3×M16C2×D913yescomplemented
201C3×M16C2×C3⋊D311yescharacteristic
202C3×C42C2×D913yescomplemented
203C3×C42C2×C3⋊D311yescharacteristic
204#215—183no—
205#215—181no—
206#213—183no—
207#213—181no—
208#208—93no—
209#208—91no—
210D24—181no—
211D24—183no—
212#204—181no—
213#204—183no—
214C22×C32—21no—
215C22×C32D3×D411yescharacteristic
216C2×C3⋊D3—61no—
217C2×C3⋊D3—62no—
218C2×C3⋊D3—122no—
219C2×C3⋊D3—122no—
220C2×C3⋊D3—121no—
221C2×C3⋊D3—242no—
222C32×C4—22no—
223C32×C4—21no—
224C32×C4C2×D1211yescharacteristic
225C32×C4D3×D411yescharacteristic
226C3⋊Dic3—61no—
227C22×C9—23no—
228C22×C9D3×D413yes—
229C2×D9—63no—
230C2×D9—66no—
231C2×D9—126no—
232C2×D9—126no—
233C2×D9—123no—
234C2×D9—246no—
235C4×C9—26no—
236C4×C9—23no—
237C4×C9C2×D1213yes—
238C4×C9D3×D413yes—
239Dic9—63no—
240H(5,𝔽2)—271no—
241AGL(1,ℤ/8)—271no—
242AGL(1,ℤ/8)—271no—
243#127—271no—
244#104—271no—
245#104C3⋊D911yescharacteristic, complemented
246#100—271no—
247C3×C9#45311yescharacteristic, complemented, Sylow
248C22×D3—96no—
249C22×D3—92no—
250C22×D3—182no—
251C22×D3—186no—
252C22×D3—183no—
253C22×D3—181no—
254C3×Q8D3613yescomplemented
255C3×Q8C4⋊(C3⋊D3)11yescharacteristic
256C3×D4—23no—
257C3×D4—21no—
258C3×D4D3613yescomplemented
259C3×D4C4⋊(C3⋊D3)11yescharacteristic
260C2×C3×C4—23no—
261C2×C3×C4—21no—
262C2×C3×C4—23no—
263C2×C3×C4—21no—
264C2×C3×C4C22×D913yes—
265C2×C3×C4C22×C3⋊D311yescharacteristic
266#67—182no—
267#67—186no—
268#67—183no—
269#67—181no—
270D12—93no—
271D12—91no—
272D12—182no—
273D12—186no—
274D12—183no—
275D12—183no—
276D12—183no—
277D12—183no—
278D12—181no—
279D12—181no—
280D12—181no—
281D12—181no—
282D12—364no—
283D12—3612no—
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#88211yescharacteristic
297C3⋊D3—122no—
298C3⋊D3—121no—
299C3⋊D3—241no—
300C2×C9—23no—
301C2×C9—43no—
302C2×C9#88213yes—
303D9—126no—
304D9—123no—
305D9—243no—
306#41—271no—
307#41—541no—
308#41—541no—
309#41C2×C3⋊D911yescharacteristic
310C2×D4—271no—
311C2×D4—271no—
312C2×D4—541no—
313C2×D4—542no—
314C2×D4—542no—
315C2×D4—541no—
316SD16—541no—
317SD16—541no—
318D8—541no—
319D8—541no—
320M16—271no—
321M16C2×C3⋊D911yescharacteristic, complemented
322C42C2×C3⋊D911yescharacteristic, complemented
323C22×C3—21no—
324C22×C3—23no—
325C22×C3D4×D913yes—
326C22×C3D4×C3⋊D311yescharacteristic
327C2×D3—181no—
328C2×D3—182no—
329C2×D3—183no—
330C2×D3—186no—
331C2×D3—362no—
332C2×D3—366no—
333C2×D3—362no—
334C2×D3—366no—
335C2×D3—361no—
336C2×D3—363no—
337C2×D3—722no—
338C2×D3—726no—
339C3×C4—22no—
340C3×C4—26no—
341C3×C4—21no—
342C3×C4—23no—
343C3×C4C2×D3613yes—
344C3×C4D4×D913yes—
345C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
346C3×C4D4×C3⋊D311yescharacteristic
347Dic3—181no—
348Dic3—183no—
349C32#482211yescharacteristic
350C9#482213yescomplemented
351C23—272no—
352C23—542no—
353C23—541no—
354Q8—271no—
355Q8C4⋊(C3⋊D9)11yescharacteristic, complemented
356D4—21no—
357D4—271no—
358D4—542no—
359D4—542no—
360D4—541no—
361D4—541no—
362D4—541no—
363D4—541no—
364D4—541no—
365D4—1084no—
366D4C4⋊(C3⋊D9)11yescharacteristic, complemented
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—542no—
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#6344413yes—
379C2×C3#6409111yescharacteristic
380D3—362no—
381D3—361no—
382D3—366no—
383D3—363no—
384D3—721no—
385D3—723no—
386C22—21no—
387C22—541no—
388C22—542no—
389C22—1082no—
390C22—1082no—
391C22—1081no—
392C22—2162no—
393C22D4×C3⋊D911yescharacteristic
394C4—22no—
395C4—21no—
396C4—541no—
397C4C2×C4⋊(C3⋊D9)11yescharacteristic
398C4D4×C3⋊D911yescharacteristic
399C3#1058850213yescomplemented
400C3#1059252111yescharacteristic
401C2—21no—
402C2—41no—
403C2—1082no—
404C2—1081no—
405C2—2161no—
406C2#1172172811yescharacteristic
subgroupG/H
centerC2#11721728
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#104C2
socleC2×C32#882
hypercenter#104C3⋊D9
hypocenterC3×C9#453
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
1. #11722672C2semidirect
2. #11721867C2semidirect
3. #11721783C2semidirect
4. #11721711C2semidirect
5. C3×C9×#104C2semidirect
6. #11720557C2semidirect
7. #11720518C2semidirect
19. C2×C4⋊(C3⋊D9)C22semidirect
20. C3×C9×M16C22semidirect
21. C3×C42×C9C22semidirect
24. #88532C22semidirect
38. C9×#104D3semidirect
44. C3×Q8×C9D4semidirect
46. C3×D4×C9D4semidirect
58. C4⋊(C3⋊D9)D4semidirect
61. #6226D4semidirect
93. C9×M16C2×D3semidirect
94. C42×C9C2×D3semidirect
121. C3×#104D9semidirect
153. Q8×C9D12semidirect
155. D4×C9D12semidirect
200. C3×M16C2×D9semidirect
202. C3×C42C2×D9semidirect
245. #104C3⋊D9semidirect
247. C3×C9#453semidirect
254. C3×Q8D36semidirect
258. C3×D4D36semidirect
321. M16C2×C3⋊D9semidirect
322. C42C2×C3⋊D9semidirect
350. C9#4822semidirect
355. Q8C4⋊(C3⋊D9)semidirect
366. D4C4⋊(C3⋊D9)semidirect
399. C3#10588502semidirect

Maximal subgroups

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

subgroupindex
#117226722
#117218672
#117217832
#117217112
C3×C9×#1042
#117205572
#117205182
#105925213
#105885023