Finite Group Databasebeta

← #13155242

Subgroups — #13155242

Order 1728.

450 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 70 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#11723654C211yesmaximal, characteristic, complemented
2C3×#63797C211yesmaximal, characteristic, complemented
3#11722007C211yesmaximal, characteristic, complemented
4#11722004C211yesmaximal, characteristic, complemented
5#11722002C211yesmaximal, characteristic, complemented
6C3×#63548C211yesmaximal, characteristic, complemented
7C3×#63546C211yesmaximal, characteristic, complemented
8#10591033—31nomaximal
9#10591031—31nomaximal
10#10590236—31nomaximal
11#89041—21no—
12C3×#3802—21no—
13C3×#3746C2211yescharacteristic, complemented
14C3×#3745C2211yescharacteristic, complemented
15C3×C8×C3⋊D3C2211yescharacteristic
16C32×D24C2211yescharacteristic
17C32×D24C2211yescharacteristic
18#88811—21no—
19C3×#3713—21no—
20C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic, complemented
21C32×C3⋊C16C2211yescharacteristic, complemented
22C3×D3×D8—31no—
23C3×D3×D8—31no—
24#63810—31no—
25#63808—31no—
26#63807—31no—
27#63797D311yescharacteristic, complemented
28#63654—31no—
29#63653—31no—
30#63625—31no—
31C3×#827—31no—
32C3×#826—31no—
33C3×#800—31no—
34C3×D48—31no—
35#63548D311yescharacteristic, complemented
36#63547—31no—
37#63547—31no—
38#63547—61no—
39#63546D311yescharacteristic, complemented
40#63545—31no—
41#63545—31no—
42#63545—61no—
43#63543—31no—
44#63543—31no—
45#63543—61no—
46C2×C3×D32—41no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4×C3⋊D3D411yescharacteristic
49C3×#634—21no—
50C32×D12—21no—
51C32×D12—21no—
52#6295—41no—
53#6294—41no—
54C3×#625—41no—
55C33×C8C2311yescharacteristic
56C3×C3⋊(C3⋊C8)D411yescharacteristic
57#5147—91no—
58#4914—91no—
59#4911—91no—
60C3×D3×D4—61no—
61C3×D3×D4—61no—
62#3802—21no—
63#3799—61no—
64#3796—61no—
65#3795—61no—
66C32×D8—31no—
67C32×D8—31no—
68#3746C2×D311yescharacteristic, complemented
69#3745C2×D311yescharacteristic, complemented
70C8×C3⋊D3C2×D311yescharacteristic
71C3×#213—61no—
72C3×#213—61no—
73C3×Dic12—31no—
74C3×Dic12—31no—
75C3×Dic12—61no—
76C3×D24—21no—
77C3×D24—21no—
78C3×D24—31no—
79C3×D24—31no—
80C3×D24—61no—
81C3×D24C2×D311yescharacteristic
82C3×D24C2×D311yescharacteristic
83C3×D24C2×D311yescharacteristic
84C3×D24C2×D311yescharacteristic
85C3×D3×C8—31no—
86C3×D3×C8—31no—
87C3×D3×C8—61no—
88#3717—61no—
89#3716—61no—
90#3715—61no—
91#3713—21no—
92C32×C16—31no—
93C3⋊(C3⋊C16)—31no—
94C3⋊(C3⋊C16)—31no—
95C3⋊(C3⋊C16)—31no—
96C3⋊(C3⋊C16)—61no—
97C3⋊(C3⋊C16)—61no—
98C3⋊(C3⋊C16)—61no—
99C3⋊(C3⋊C16)—121no—
100C3×C3⋊C16—21no—
101C3×C3⋊C16C2×D311yescharacteristic, complemented
102C3×C3⋊C16C2×D311yescharacteristic, complemented
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3D811yescharacteristic
105C2×C32×D3—41no—
106C2×C32×D3—41no—
107C3×D32—82no—
108C33×C4C2×D411yescharacteristic
109C3×C3⋊Dic3—41no—
110C3×C3⋊Dic3D811yescharacteristic
111C2×C3×D8—91no—
112#919—91no—
113#912—91no—
114D3×D8—31no—
115D3×D8—31no—
116#904—91no—
117C3×SD32—91no—
118C3×D16—91no—
119#828—91no—
120#827—31no—
121#827—91no—
122#827—91no—
123#827—181no—
124#826—31no—
125#826—91no—
126#826—91no—
127#826—91no—
128#826—181no—
129#812—91no—
130#800—31no—
131D48—31no—
132#798—91no—
133#798—91no—
134#798—181no—
135C22×C3×D3—121no—
136C22×C3×D3—121no—
137C2×D32—41no—
138C32×D4—61no—
139C32×D4—61no—
140C4⋊(C3⋊D3)—21no—
141C4×C3⋊D3#6711yescharacteristic
142#634—21no—
143C3×#67—121no—
144C3×#67—121no—
145C3×D12—21no—
146C3×D12—21no—
147C3×D12—21no—
148C3×D12—21no—
149C3×D12—41no—
150C3×D12—41no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—121no—
154C3×C4×D3—31no—
155C3×C4×D3—31no—
156C3×C4×D3—61no—
157C3×Dic6—61no—
158C3×Dic6—61no—
159C3×Dic6—121no—
160#626—121no—
161#626—121no—
162#625—41no—
163#625—121no—
164#624—121no—
165D3×Dic3—121no—
166D3×Dic3—121no—
167C32×C8—21no—
168C32×C8—21no—
169C32×C8—21no—
170C32×C8—41no—
171C32×C8C22×D311yescharacteristic
172C32×C8C22×D311yescharacteristic
173C32×C8C22×D311yescharacteristic
174C3⋊(C3⋊C8)#6711yescharacteristic
175C3×C3⋊C8—31no—
176C3×C3⋊C8—31no—
177C3×C3⋊C8—61no—
178#509—271noSylow
179C2×C33C2×D811yescharacteristic
180C3×C3⋊D3—21no—
181C3×C3⋊D3—81no—
182C32×D3—81no—
183C32×D3—81no—
184C2×C3×D4—181no—
185#239—181no—
186#237—181no—
187D3×D4—61no—
188D3×D4—61no—
189#235—181no—
190C3×Q16—91no—
191C3×D8—31no—
192C3×D8—31no—
193C3×D8—31no—
194C3×D8—31no—
195C3×D8—61no—
196C3×D8—61no—
197C3×D8—91no—
198C3×D8—181no—
199C2×C3×C8—91no—
200#215—181no—
201#214—181no—
202#213—61no—
203#213—61no—
204Dic12—31no—
205Dic12—31no—
206Dic12—61no—
207D24—31no—
208D24—31no—
209D24—61no—
210D24D3211yescharacteristic
211D24D3211yescharacteristic
212#204—181no—
213D3×C8—31no—
214D3×C8—31no—
215D3×C8—61no—
216C3×C16—31no—
217C3×C16—31no—
218C3×C16—61no—
219C3⋊C16—91no—
220C3⋊C16—91no—
221C3⋊C16—91no—
222C3⋊C16—181no—
223C3⋊C16—181no—
224C3⋊C16—181no—
225C3⋊C16—361no—
226C3⋊C16D3211yescharacteristic, complemented
227C22×C32—121no—
228C22×C32—121no—
229C2×C3⋊D3—41no—
230C2×C3⋊D3#21311yescharacteristic
231C2×C3×D3—31no—
232C2×C3×D3—31no—
233C2×C3×D3—41no—
234C2×C3×D3—41no—
235C2×C3×D3—41no—
236C2×C3×D3—41no—
237C2×C3×D3—61no—
238C2×C3×D3—81no—
239C2×C3×D3—81no—
240C2×C3×D3—121no—
241C2×C3×D3—121no—
242C2×C3×D3—241no—
243C2×C3×D3—242no—
244C2×C3×D3—242no—
245D32—82no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—41no—
250C32×C4D3×D411yescharacteristic
251C32×C4D3×D411yescharacteristic
252C32×C4C2×#6711yescharacteristic
253C3⋊Dic3—41no—
254C3⋊Dic3#21311yescharacteristic
255C3×Dic3—31no—
256C3×Dic3—31no—
257C3×Dic3—61no—
258C3×Dic3—121no—
259C3×Dic3—121no—
260C3×Dic3—241no—
261#135—271no—
262C2×D8—91no—
263SD32—91no—
264SD32—271no—
265D16—91no—
266D16—271no—
267M32—271no—
268C33#50911yescharacteristic, complemented, Sylow
269C23×C3—361no—
270C22×D3—121no—
271C22×D3—121no—
272C3×Q8—181no—
273C3×D4—61no—
274C3×D4—61no—
275C3×D4—61no—
276C3×D4—61no—
277C3×D4—121no—
278C3×D4—121no—
279C3×D4—181no—
280C3×D4—361no—
281C2×C3×C4—91no—
282#67—121no—
283#67—121no—
284#67—361no—
285#67—361no—
286C2×Dic3—361no—
287D12—21no—
288D12—21no—
289D12—61no—
290D12—61no—
291D12—121no—
292D12—361no—
293C4×D3—31no—
294C4×D3—31no—
295C4×D3—61no—
296C4×D3—361no—
297C4×D3—361no—
298Dic6—61no—
299Dic6—61no—
300Dic6—121no—
301C3×C8—21no—
302C3×C8—21no—
303C3×C8—21no—
304C3×C8—41no—
305C3×C8—91no—
306C3×C8C2×D3211yescharacteristic
307C3×C8C2×D3211yescharacteristic
308C3×C8C2×D3211yescharacteristic
309C3⋊C8—31no—
310C3⋊C8—31no—
311C3⋊C8—61no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—41no—
316C2×C32—241no—
317C2×C32—241no—
318C2×C32D3×D811yescharacteristic
319C2×C32D3×D811yescharacteristic
320C2×C32C2×#21311yescharacteristic
321C3⋊D3—21no—
322C3⋊D3—81no—
323C3×D3—61no—
324C3×D3—61no—
325C3×D3—81no—
326C3×D3—81no—
327C3×D3—81no—
328C3×D3—81no—
329C3×D3—121no—
330C3×D3—161no—
331C3×D3—161no—
332C3×D3—241no—
333C3×D3—241no—
334C3×D3—481no—
335#41—541no—
336C2×D4—181no—
337Q16—91no—
338SD16—541no—
339D8—31no—
340D8—31no—
341D8—91no—
342D8—181no—
343C2×C8—91no—
344C16—31no—
345C16—271no—
346C22×C3—91no—
347C22×C3—121no—
348C22×C3—121no—
349C22×C3—121no—
350C22×C3—121no—
351C22×C3—241no—
352C22×C3—241no—
353C22×C3—361no—
354C22×C3—722no—
355C2×D3—31no—
356C2×D3—31no—
357C2×D3—41no—
358C2×D3—41no—
359C2×D3—61no—
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—241no—
363C2×D3—242no—
364C2×D3—242no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—41no—
369C3×C4—91no—
370C3×C4—361no—
371C3×C4#380211yescharacteristic
372C3×C4D3×#6711yescharacteristic
373C3×C4D3×#6711yescharacteristic
374Dic3—31no—
375Dic3—31no—
376Dic3—61no—
377Dic3—121no—
378Dic3—121no—
379Dic3—241no—
380C32—21no—
381C32—21no—
382C32—21no—
383C32—41no—
384C32#491111yescharacteristic, complemented
385C32#491411yescharacteristic, complemented
386C32#514711yescharacteristic, complemented
387C23—361no—
388Q8—181no—
389D4—61no—
390D4—61no—
391D4—181no—
392D4—361no—
393D4—1081no—
394C2×C4—91no—
395C2×C4—1081no—
396C8—91no—
397C8D3311yescharacteristic
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—41no—
402C2×C3—181no—
403C2×C3—241no—
404C2×C3—241no—
405C2×C3—241no—
406C2×C3—241no—
407C2×C3—481no—
408C2×C3—481no—
409C2×C3—721no—
410C2×C3#6379711yescharacteristic
411C2×C3D3×#21311yescharacteristic
412C2×C3D3×#21311yescharacteristic
413D3—61no—
414D3—61no—
415D3—81no—
416D3—81no—
417D3—121no—
418D3—241no—
419D3—241no—
420D3—481no—
421C22—91no—
422C22—121no—
423C22—121no—
424C22—361no—
425C22—722no—
426C4—91no—
427C4—361no—
428C4#8895311yescharacteristic
429C3—21no—
430C3—21no—
431C3—21no—
432C3—41no—
433C3#1059023611yescharacteristic, complemented
434C3#1059103111yescharacteristic, complemented
435C3#1059103311yescharacteristic, complemented
436C2—181no—
437C2—241no—
438C2—241no—
439C2—721no—
440C2#1172283111yescharacteristic
subgroupG/H
centerC2#11722831
commutator, Fitting, cosocleC33×C8C23
Frattini, hypercenterC8D33
socleC2×C33C2×D8
hypocenterC33#509
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11723654C2semidirect
2. C3×#63797C2semidirect
3. #11722007C2semidirect
4. #11722004C2semidirect
5. #11722002C2semidirect
6. C3×#63548C2semidirect
7. C3×#63546C2semidirect
13. C3×#3746C22semidirect
14. C3×#3745C22semidirect
20. C3⋊(C3⋊(C3⋊C16))C22semidirect
21. C32×C3⋊C16C22semidirect
27. #63797D3semidirect
35. #63548D3semidirect
39. #63546D3semidirect
68. #3746C2×D3semidirect
69. #3745C2×D3semidirect
101. C3×C3⋊C16C2×D3semidirect
102. C3×C3⋊C16C2×D3semidirect
226. C3⋊C16D32semidirect
268. C33#509semidirect
384. C32#4911semidirect
385. C32#4914semidirect
386. C32#5147semidirect
433. C3#10590236semidirect
434. C3#10591031semidirect
435. C3#10591033semidirect

Maximal subgroups

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

subgroupindex
#117236542
C3×#637972
#117220072
#117220042
#117220022
C3×#635482
C3×#635462
#105910333
#105910313
#105902363