Finite Group Databasebeta

← #13161469

Subgroups — #13161469

Order 1728.

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

normal subgroups only

This group has 450 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
1#11723654C211yesmaximal, characteristic, complemented
2#11723647C211yesmaximal, characteristic, complemented
3C3×#63627C211yesmaximal, characteristic, complemented
4C3×#63626C211yesmaximal, characteristic, complemented
5C3×#63625C211yesmaximal, characteristic, complemented
6#11722068C211yesmaximal, characteristic, complemented
7#11722067C211yesmaximal, characteristic, complemented
8#10590236—31nomaximal
9#10590233—31nomaximal
10#10590232—31nomaximal
11C4⋊#1113—21no—
12#89041—21no—
13C3×#3746C2211yescharacteristic, complemented
14C3×#3745C2211yescharacteristic
15C3×#3745C2211yescharacteristic
16C3×#3745C2211yescharacteristic, complemented
17C3×C8×C3⋊D3C2211yescharacteristic
18#88811—21no—
19#88810—21no—
20C33×C16C2211yescharacteristic, complemented
21C3×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
22#63810—31no—
23#63808—31no—
24#63807—31no—
25#63797—31no—
26D3×D24—31no—
27D3×D24—31no—
28#63627D311yescharacteristic, complemented
29#63626D311yescharacteristic, complemented
30#63625D311yescharacteristic, complemented
31C3×#800—31no—
32C3×#800—31no—
33C3×#800—61no—
34C3×D48—31no—
35C3×D48—31no—
36C3×D48—61no—
37C3×#798—31no—
38C3×#798—31no—
39C3×#798—61no—
40#63549—31no—
41#63548—31no—
42#63547—31no—
43#63546—31no—
44#63546—31no—
45#63545—31no—
46C2×#1113—41no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4×C3⋊D3D411yescharacteristic
51C3×#634—21no—
52#6295—41no—
53#6295—41no—
54#6294—41no—
55C33×C8C2311yescharacteristic
56C3×C3⋊(C3⋊C8)D411yescharacteristic
57#4914—91no—
58#4911—91no—
59#4908—91no—
60#3802—61no—
61D3×D12—61no—
62D3×D12—61no—
63#3799—61no—
64#3796—61no—
65#3795—61no—
66#3746C2×D311yescharacteristic, complemented
67#3745C2×D311yescharacteristic
68#3745C2×D311yescharacteristic
69#3745C2×D311yescharacteristic, complemented
70C8×C3⋊D3C2×D311yescharacteristic
71C3×Dic12—31no—
72C3×Dic12—31no—
73C3×Dic12—61no—
74C3×D24—31no—
75C3×D24—31no—
76C3×D24—31no—
77C3×D24—31no—
78C3×D24—31no—
79C3×D24—31no—
80C3×D24—61no—
81C3×D24—61no—
82C3×D24—61no—
83C3×D3×C8—31no—
84C3×D3×C8—31no—
85C3×D3×C8—61no—
86#3717—61no—
87#3716—61no—
88#3715—61no—
89#3714—61no—
90#3714—61no—
91#3713—61no—
92C32×C16—21no—
93C32×C16—21no—
94C32×C16—21no—
95C32×C16—41no—
96C32×C16C2×D311yescharacteristic, complemented
97C32×C16C2×D311yescharacteristic, complemented
98C32×C16C2×D311yescharacteristic, complemented
99C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
100C3×C3⋊C16—31no—
101C3×C3⋊C16—31no—
102C3×C3⋊C16—61no—
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3—41no—
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3D811yescharacteristic
107#1113—82no—
108C33×C4C2×D411yescharacteristic
109C3×C3⋊Dic3—41no—
110C3×C3⋊Dic3D811yescharacteristic
111#919—91no—
112#912—91no—
113D3×D8—91no—
114D3×D8—91no—
115#904—91no—
116C2×D24—91no—
117C3×SD32—91no—
118C3×D16—91no—
119C3×M32—91no—
120#828—91no—
121#827—91no—
122#826—91no—
123#826—91no—
124#800—31no—
125#800—31no—
126#800—61no—
127#800—91no—
128D48—31no—
129D48—31no—
130D48—61no—
131D48—91no—
132#798—31no—
133#798—31no—
134#798—61no—
135C2×D32—121no—
136C2×D32—121no—
137C2×D32—121no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)—21no—
141C4×C3⋊D3D1211yescharacteristic
142#634—21no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—121no—
150C3×D12—121no—
151C3×D12—121no—
152C3×C4×D3—31no—
153C3×C4×D3—31no—
154C3×C4×D3—61no—
155C3×Dic6—61no—
156C3×Dic6—61no—
157C3×Dic6—121no—
158#626—121no—
159#626—121no—
160#626—121no—
161#626—121no—
162#625—121no—
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)D1211yescharacteristic
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—
182C3×C3⋊D3—81no—
183C3×C3⋊D3—81no—
184#239—181no—
185#237—181no—
186D3×D4—181no—
187D3×D4—181no—
188#235—181no—
189C2×D12—181no—
190C3×Q16—91no—
191C3×D8—91no—
192C3×D8—91no—
193C3×D8—91no—
194C2×C3×C8—91no—
195#215—181no—
196#214—181no—
197#213—181no—
198#213—181no—
199Dic12—31no—
200Dic12—31no—
201Dic12—61no—
202D24—31no—
203D24—31no—
204D24—31no—
205D24—31no—
206D24—31no—
207D24—31no—
208D24—61no—
209D24—61no—
210D24—61no—
211D24—181no—
212#204—181no—
213D3×C8—31no—
214D3×C8—31no—
215D3×C8—61no—
216C3×C16—21no—
217C3×C16—21no—
218C3×C16—21no—
219C3×C16—41no—
220C3×C16—91no—
221C3×C16D3211yescharacteristic, complemented
222C3×C16D3211yescharacteristic, complemented
223C3×C16D3211yescharacteristic, complemented
224C3⋊C16—31no—
225C3⋊C16—31no—
226C3⋊C16—61no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—41no—
229C2×C3⋊D3—41no—
230C2×C3⋊D3D2411yescharacteristic
231C2×C3×D3—31no—
232C2×C3×D3—31no—
233C2×C3×D3—61no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C2×C3×D3—241no—
241C2×C3×D3—241no—
242C2×C3×D3—241no—
243D32—242no—
244D32—242no—
245D32—242no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—41no—
250C32×C4C2×D1211yescharacteristic
251C32×C4D3×D411yescharacteristic
252C32×C4D3×D411yescharacteristic
253C3⋊Dic3—41no—
254C3⋊Dic3D2411yescharacteristic
255C3×Dic3—31no—
256C3×Dic3—31no—
257C3×Dic3—61no—
258C3×Dic3—121no—
259C3×Dic3—121no—
260C3×Dic3—241no—
261#135—271no—
262C2×D8—271no—
263SD32—91no—
264SD32—271no—
265D16—91no—
266D16—271no—
267M32—91no—
268C33#50911yescharacteristic, complemented, Sylow
269C22×D3—361no—
270C22×D3—361no—
271C22×D3—361no—
272C3×Q8—181no—
273C3×D4—181no—
274C3×D4—181no—
275C3×D4—181no—
276C2×C3×C4—91no—
277#67—361no—
278#67—361no—
279#67—361no—
280#67—361no—
281C2×Dic3—361no—
282D12—61no—
283D12—61no—
284D12—61no—
285D12—61no—
286D12—61no—
287D12—61no—
288D12—121no—
289D12—121no—
290D12—121no—
291D12—361no—
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×C32C2×D2411yescharacteristic
317C2×C32D3×D811yescharacteristic
318C2×C32D3×D811yescharacteristic
319C3⋊D3—21no—
320C3⋊D3—81no—
321C3⋊D3—81no—
322C3⋊D3—81no—
323C3×D3—61no—
324C3×D3—61no—
325C3×D3—121no—
326C3×D3—241no—
327C3×D3—241no—
328C3×D3—241no—
329C3×D3—241no—
330C3×D3—241no—
331C3×D3—241no—
332C3×D3—481no—
333C3×D3—481no—
334C3×D3—481no—
335#41—541no—
336C2×D4—541no—
337Q16—91no—
338SD16—541no—
339D8—91no—
340D8—91no—
341D8—91no—
342D8—541no—
343C2×C8—91no—
344C16—91no—
345C16#111311yescharacteristic, complemented
346C22×C3—91no—
347C22×C3—361no—
348C22×C3—361no—
349C22×C3—361no—
350C2×D3—31no—
351C2×D3—31no—
352C2×D3—61no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—121no—
358C2×D3—121no—
359C2×D3—241no—
360C2×D3—241no—
361C2×D3—241no—
362C2×D3—722no—
363C2×D3—722no—
364C2×D3—722no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—41no—
369C3×C4—91no—
370C3×C4—361no—
371C3×C4D3×D1211yescharacteristic
372C3×C4D3×D1211yescharacteristic
373C3×C4#380211yescharacteristic
374Dic3—31no—
375Dic3—31no—
376Dic3—61no—
377Dic3—121no—
378Dic3—121no—
379Dic3—241no—
380C32—21no—
381C32—21no—
382C32—21no—
383C32—41no—
384C32#490811yescharacteristic, complemented
385C32#491111yescharacteristic, complemented
386C32#491411yescharacteristic, complemented
387C23—1081no—
388Q8—181no—
389D4—181no—
390D4—181no—
391D4—181no—
392D4—1081no—
393D4—1081no—
394C2×C4—91no—
395C2×C4—1081no—
396C8—91no—
397C8C2×#111311yescharacteristic
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—41no—
402C2×C3—181no—
403C2×C3—721no—
404C2×C3—721no—
405C2×C3—721no—
406C2×C3D3×D2411yescharacteristic
407C2×C3D3×D2411yescharacteristic
408C2×C3#6379711yescharacteristic
409D3—61no—
410D3—61no—
411D3—121no—
412D3—241no—
413D3—241no—
414D3—241no—
415D3—241no—
416D3—241no—
417D3—241no—
418D3—481no—
419D3—481no—
420D3—481no—
421C22—91no—
422C22—361no—
423C22—361no—
424C22—361no—
425C22—2162no—
426C4—91no—
427C4—361no—
428C4C4⋊#111311yescharacteristic
429C3—21no—
430C3—21no—
431C3—21no—
432C3—41no—
433C3#1059023211yescharacteristic, complemented
434C3#1059023311yescharacteristic, complemented
435C3#1059023611yescharacteristic, complemented
436C2—181no—
437C2—721no—
438C2—721no—
439C2—721no—
440C2#1172364711yescharacteristic
subgroupG/H
centerC2#11723647
commutator, cosocleC33×C8C23
FrattiniC8C2×#1113
FittingC33×C16C22
socleC2×C33C2×D8
hypercenterC16#1113
hypocenterC33#509
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

Complements

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

NH ≅ G/Ntype
1. #11723654C2semidirect
2. #11723647C2semidirect
3. C3×#63627C2semidirect
4. C3×#63626C2semidirect
5. C3×#63625C2semidirect
6. #11722068C2semidirect
7. #11722067C2semidirect
13. C3×#3746C22semidirect
16. C3×#3745C22semidirect
20. C33×C16C22semidirect
21. C3×C3⋊(C3⋊C16)C22semidirect
28. #63627D3semidirect
29. #63626D3semidirect
30. #63625D3semidirect
66. #3746C2×D3semidirect
69. #3745C2×D3semidirect
96. C32×C16C2×D3semidirect
97. C32×C16C2×D3semidirect
98. C32×C16C2×D3semidirect
99. C3⋊(C3⋊C16)C2×D3semidirect
221. C3×C16D32semidirect
222. C3×C16D32semidirect
223. C3×C16D32semidirect
268. C33#509semidirect
345. C16#1113semidirect
384. C32#4908semidirect
385. C32#4911semidirect
386. C32#4914semidirect
433. C3#10590232semidirect
434. C3#10590233semidirect
435. C3#10590236semidirect

Maximal subgroups

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

subgroupindex
#117236542
#117236472
C3×#636272
C3×#636262
C3×#636252
#117220682
#117220672
#105902363
#105902333
#105902323