Finite Group Databasebeta

← #13161357

Subgroups — #13161357

Order 1728.

1672 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1672 conjugacy classes of subgroups — too many to draw. Only its 144 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#11724684C211yesmaximal, characteristic, complemented
2C3×#64158C211yesmaximal, characteristic, complemented
3#11723517C211yesmaximal, characteristic, complemented
4#11723370C211yesmaximal, characteristic, complemented
5C32×#837C211yesmaximal, characteristic, complemented
6#11722048C211yesmaximal, characteristic, complemented
7#11722027C211yesmaximal, characteristic, complemented
8#10592521—31nomaximal
9#10591667—34nomaximal
10C33×#41C2211yescharacteristic
11#89080—21no—
12D4×C3⋊(C3⋊D3)—22no—
13D4×C3⋊(C3⋊D3)—21no—
14#89076—21no—
15#89076C2211yescharacteristic
16C2×C4⋊(C3⋊(C3⋊D3))—21no—
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
19C2×#6292—22no—
20C3×#3755—21no—
21C3×#3753—21no—
22C3×#3748C2211yescharacteristic, complemented
23C32×C4×Dic3C2211yescharacteristic, complemented
24C32×#208C2211yescharacteristic, complemented
25#88796—21no—
26#88791—21no—
27#64366—31no—
28#64366—34no—
29#64366—64no—
30#64158D311yescharacteristic, complemented
31#64117—31no—
32#64083—31no—
33C3×#949—34no—
34#63911—34no—
35#63830—34no—
36C32×#104—31no—
37#63650—31no—
38#63632—31no—
39C3×#837D314yescomplemented
40#63570—34no—
41#63559—34no—
42C22×C3⋊(C3⋊D3)—22no—
43C22×C3⋊(C3⋊D3)D412yes—
44C22×C3×C3⋊D3—21no—
45C33×Q8D411yescharacteristic, complemented
46C33×D4—21no—
47C33×D4D411yescharacteristic, complemented
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50#6312—22no—
51#6312—21no—
52C4⋊(C3⋊(C3⋊D3))—22no—
53C4⋊(C3⋊(C3⋊D3))—21no—
54C4⋊(C3⋊(C3⋊D3))—21no—
55C4⋊(C3⋊(C3⋊D3))D411yescharacteristic, complemented
56C4×C3⋊(C3⋊D3)—22no—
57C4×C3⋊(C3⋊D3)—21no—
58#6308D411yescharacteristic, complemented
59C3×C4⋊(C3⋊D3)—21no—
60C3×C4⋊(C3⋊D3)—21no—
61C2×C32×Dic3—21no—
62#6292—44no—
63C3×C3⋊(C3⋊C8)—21no—
64C32×C3⋊C8—21no—
65#5198—91no—
66#4822—94no—
67C32×#41—24no—
68C32×#41C2×D314yes—
69C32×#41C2×D311yescharacteristic
70#3832—68no—
71#3832—64no—
72#3832—61no—
73D4×C3⋊D3—62no—
74D4×C3⋊D3—68no—
75D4×C3⋊D3—616no—
76D4×C3⋊D3—68no—
77D4×C3⋊D3—64no—
78D4×C3⋊D3—61no—
79#3828—31no—
80#3828—34no—
81#3828—61no—
82#3828—64no—
83#3828—64no—
84#3828—68no—
85C2×C4⋊(C3⋊D3)—31no—
86C2×C4⋊(C3⋊D3)—34no—
87C2×C4⋊(C3⋊D3)—61no—
88C2×C4⋊(C3⋊D3)—64no—
89C2×C4⋊(C3⋊D3)—64no—
90C2×C4⋊(C3⋊D3)—68no—
91C2×C4⋊(C3⋊D3)—62no—
92C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
93C2×C3×D12—34no—
94C2×#626—68no—
95C32×M16—31no—
96C32×C42—31no—
97#3755—21no—
98#3753—21no—
99#3748C2×D311yescharacteristic, complemented
100#3745—61no—
101#3744—61no—
102C3×#215—64no—
103C3×#213—64no—
104C3×C4×Dic3C2×D314yescomplemented
105C3×#208—34no—
106C3×#208C2×D314yescomplemented
107#3716—64no—
108#3714—64no—
109C22×C33—21no—
110C22×C33C2×D411yescharacteristic
111C2×C3⋊(C3⋊D3)—22no—
112C2×C3⋊(C3⋊D3)—21no—
113C2×C3⋊(C3⋊D3)—42no—
114C2×C3⋊(C3⋊D3)—82no—
115C2×C3×C3⋊D3—42no—
116C2×C3×C3⋊D3—41no—
117C33×C4—21no—
118C33×C4C2×D411yescharacteristic
119C33×C4C2×D411yescharacteristic
120C3⋊(C3⋊Dic3)—21no—
121C32×Dic3—22no—
122#1009—94no—
123#1009—91no—
124#1009—184no—
125C3×AGL(1,ℤ/8)—91no—
126#949—34no—
127#940—91no—
128#932—91no—
129#908—94no—
130#874—94no—
131C3×#104—34no—
132#837C3⋊D311yescharacteristic, complemented
133#825—91no—
134#823—94no—
135#823—91no—
136#805—94no—
137C22×C3⋊D3—21no—
138C22×C3⋊D3—32no—
139C22×C3⋊D3—38no—
140C22×C3⋊D3—68no—
141C22×C3⋊D3—616no—
142C22×C3⋊D3—68no—
143C22×C3⋊D3—62no—
144C22×C3×D3—64no—
145C32×Q8—24no—
146C32×Q8D1214yescomplemented
147C32×Q8#6711yescharacteristic, complemented
148C32×D4—28no—
149C32×D4—24no—
150C32×D4—21no—
151C32×D4—24no—
152C32×D4D1214yescomplemented
153C32×D4#6711yescharacteristic, complemented
154C2×C32×C4—28no—
155C2×C32×C4—24no—
156C2×C32×C4—21no—
157C2×C32×C4—24no—
158C2×C32×C4—61no—
159C2×C32×C4C22×D314yes—
160C2×C32×C4C22×D311yescharacteristic
161#638—616no—
162#638—68no—
163#638—62no—
164#638—68no—
165#638—64no—
166#638—61no—
167C4⋊(C3⋊D3)—21no—
168C4⋊(C3⋊D3)—21no—
169C4⋊(C3⋊D3)—31no—
170C4⋊(C3⋊D3)—34no—
171C4⋊(C3⋊D3)—64no—
172C4⋊(C3⋊D3)—68no—
173C4⋊(C3⋊D3)—616no—
174C4⋊(C3⋊D3)—68no—
175C4⋊(C3⋊D3)—62no—
176C4⋊(C3⋊D3)—68no—
177C4⋊(C3⋊D3)—64no—
178C4⋊(C3⋊D3)—64no—
179C4⋊(C3⋊D3)—61no—
180C4⋊(C3⋊D3)—61no—
181C4⋊(C3⋊D3)—124no—
182C4×C3⋊D3—616no—
183C4×C3⋊D3—68no—
184C4×C3⋊D3—62no—
185C4×C3⋊D3—68no—
186C4×C3⋊D3—64no—
187C4×C3⋊D3—61no—
188#634—31no—
189#634—34no—
190#634—64no—
191C2×C3×Dic3—24no—
192C3×D12—64no—
193C3×D12—64no—
194#626—1216no—
195C32×C8—61no—
196C3⋊(C3⋊C8)—21no—
197C3×C3⋊C8—24no—
198C3×C3⋊C8—64no—
199#453—271noSylow
200C2×C33—21no—
201C2×C33—41no—
202C2×C33#12011yescharacteristic
203C3⋊(C3⋊D3)—42no—
204C3⋊(C3⋊D3)—41no—
205C3×C3⋊D3—81no—
206C3×#41—24no—
207C3×#41D3214yes—
208C3×#41C2×C3⋊D311yescharacteristic
209C2×C3×D4—91no—
210C2×#67—182no—
211#239—188no—
212#239—184no—
213#239—181no—
214D3×D4—188no—
215D3×D4—182no—
216D3×D4—1816no—
217D3×D4—188no—
218D3×D4—184no—
219D3×D4—181no—
220#235—94no—
221#235—91no—
222#235—184no—
223#235—184no—
224#235—181no—
225#235—188no—
226C2×D12—34no—
227C2×D12—94no—
228C2×D12—91no—
229C2×D12—184no—
230C2×D12—184no—
231C2×D12—181no—
232C2×D12—188no—
233C2×D12—188no—
234C3×SD16—181no—
235C3×D8—181no—
236C3×M16—34no—
237C3×M16—91no—
238C3×C42—34no—
239#215—64no—
240#214—181no—
241#213—64no—
242#213—181no—
243C4×Dic3C2×C3⋊D311yescharacteristic, complemented
244#208—34no—
245#208C2×C3⋊D311yescharacteristic, complemented
246D24—184no—
247#204—184no—
248C22×C32—28no—
249C22×C32—24no—
250C22×C32—21no—
251C22×C32—24no—
252C22×C32D3×D414yes—
253C22×C32D3×D411yescharacteristic
254C2×C3⋊D3—42no—
255C2×C3⋊D3—41no—
256C2×C3⋊D3—68no—
257C2×C3⋊D3—616no—
258C2×C3⋊D3—64no—
259C2×C3⋊D3—68no—
260C2×C3⋊D3—61no—
261C2×C3⋊D3—62no—
262C2×C3⋊D3—122no—
263C2×C3⋊D3—128no—
264C2×C3⋊D3—248no—
265C2×C3⋊D3—2416no—
266C2×C3⋊D3—242no—
267C2×C3⋊D3—248no—
268C2×C3×D3—128no—
269C2×C3×D3—124no—
270C32×C4—28no—
271C32×C4—24no—
272C32×C4—21no—
273C32×C4—24no—
274C32×C4—24no—
275C32×C4—62no—
276C32×C4C2×D1214yes—
277C32×C4D3×D414yes—
278C32×C4D3×D411yescharacteristic
279C32×C4C2×#6711yescharacteristic
280C3⋊Dic3—68no—
281C3⋊Dic3—64no—
282C3⋊Dic3—61no—
283C3×Dic3—28no—
284H(5,𝔽2)—271no—
285AGL(1,ℤ/8)—91no—
286AGL(1,ℤ/8)—271no—
287#127—271no—
288#104—31no—
289#104—271no—
290#100—271no—
291C33#45311yescharacteristic, complemented, Sylow
292C23×C3—181no—
293C22×D3—64no—
294C22×D3—98no—
295C22×D3—92no—
296C22×D3—188no—
297C22×D3—1816no—
298C22×D3—182no—
299C22×D3—188no—
300C3×Q8—24no—
301C3×Q8#62614yescomplemented
302C3×Q8C4⋊(C3⋊D3)11yescharacteristic, complemented
303C3×D4—28no—
304C3×D4—24no—
305C3×D4—21no—
306C3×D4—24no—
307C3×D4—181no—
308C3×D4—181no—
309C3×D4#62614yescomplemented
310C3×D4C4⋊(C3⋊D3)11yescharacteristic, complemented
311C2×C3×C4—28no—
312C2×C3×C4—24no—
313C2×C3×C4—21no—
314C2×C3×C4—24no—
315C2×C3×C4—64no—
316C2×C3×C4C2×D3214yes—
317C2×C3×C4C22×C3⋊D311yescharacteristic
318#67—1816no—
319#67—182no—
320#67—188no—
321#67—188no—
322#67—184no—
323#67—181no—
324#67—364no—
325C2×Dic3—21no—
326D12—64no—
327D12—64no—
328D12—94no—
329D12—91no—
330D12—184no—
331D12—188no—
332D12—1816no—
333D12—182no—
334D12—188no—
335D12—188no—
336D12—184no—
337D12—184no—
338D12—181no—
339D12—181no—
340D12—3616no—
341C4×D3—1816no—
342C4×D3—182no—
343C4×D3—188no—
344C4×D3—188no—
345C4×D3—184no—
346C4×D3—181no—
347Dic6—94no—
348Dic6—91no—
349Dic6—184no—
350C3×C8—64no—
351C3×C8—181no—
352C3⋊C8—21no—
353C3⋊C8—64no—
354C2×C32—21no—
355C2×C32—24no—
356C2×C32—24no—
357C2×C32—44no—
358C2×C32—48no—
359C2×C32—41no—
360C2×C32—44no—
361C2×C32#88214yes—
362C2×C32#93711yescharacteristic
363C3⋊D3—81no—
364C3⋊D3—1216no—
365C3⋊D3—128no—
366C3⋊D3—122no—
367C3⋊D3—121no—
368C3⋊D3—128no—
369C3⋊D3—124no—
370C3×D3—244no—
371#41—271no—
372#41—541no—
373#41—541no—
374#41D3×C3⋊D311yescharacteristic
375C2×D4—91no—
376C2×D4—271no—
377C2×D4—541no—
378C2×D4—542no—
379C2×D4—542no—
380C2×D4—541no—
381SD16—181no—
382SD16—541no—
383D8—181no—
384D8—541no—
385M16—31no—
386M16—91no—
387C42—31no—
388C22×C3—28no—
389C22×C3—21no—
390C22×C3—24no—
391C22×C3—24no—
392C22×C3—362no—
393C22×C3—361no—
394C22×C3#381114yes—
395C22×C3D4×C3⋊D311yescharacteristic
396C2×D3—128no—
397C2×D3—124no—
398C2×D3—188no—
399C2×D3—1816no—
400C2×D3—181no—
401C2×D3—182no—
402C2×D3—184no—
403C2×D3—188no—
404C2×D3—368no—
405C2×D3—362no—
406C2×D3—728no—
407C2×D3—7216no—
408C2×D3—722no—
409C2×D3—728no—
410C3×C4—28no—
411C3×C4—21no—
412C3×C4—24no—
413C3×C4—24no—
414C3×C4—24no—
415C3×C4—68no—
416C3×C4C2×#62614yes—
417C3×C4#381114yes—
418C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
419C3×C4D4×C3⋊D311yescharacteristic
420Dic3—22no—
421Dic3—188no—
422Dic3—181no—
423Dic3—184no—
424C32—24no—
425C32#482214yescomplemented
426C32#519811yescharacteristic, complemented
427C23—181no—
428C23—272no—
429C23—542no—
430Q8—271no—
431Q8#629211yescharacteristic, complemented
432D4—21no—
433D4—181no—
434D4—181no—
435D4—271no—
436D4—542no—
437D4—542no—
438D4—541no—
439D4—541no—
440D4—541no—
441D4—1084no—
442D4#629211yescharacteristic, complemented
443C2×C4—21no—
444C2×C4—61no—
445C2×C4—542no—
446C2×C4—541no—
447C2×C4C2×D3×C3⋊D311yescharacteristic
448C8—61no—
449C8—181no—
450C2×C3—21no—
451C2×C3—24no—
452C2×C3—24no—
453C2×C3—44no—
454C2×C3—48no—
455C2×C3—41no—
456C2×C3—44no—
457C2×C3—721no—
458C2×C3#6398114yes—
459C2×C3#6409111yescharacteristic
460D3—244no—
461D3—3616no—
462D3—368no—
463D3—362no—
464D3—361no—
465D3—368no—
466D3—364no—
467C22—21no—
468C22—362no—
469C22—361no—
470C22—541no—
471C22—542no—
472C22—1082no—
473C22—2162no—
474C22#8903911yescharacteristic
475C4—21no—
476C4—62no—
477C4—541no—
478C4C2×#629211yescharacteristic
479C4#8903911yescharacteristic
480C3—24no—
481C3#1059166714yescomplemented
482C3#1059252111yescharacteristic, complemented
483C2—21no—
484C2—41no—
485C2—721no—
486C2—1082no—
487C2—1081no—
488C2#1172363911yescharacteristic
subgroupG/H
centerC2#11723639
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33#120
hypercenter#41D3×C3⋊D3
hypocenterC33#453
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11724684C2semidirect
2. C3×#64158C2semidirect
3. #11723517C2semidirect
4. #11723370C2semidirect
5. C32×#837C2semidirect
6. #11722048C2semidirect
7. #11722027C2semidirect
18. C2×C3×C4⋊(C3⋊D3)C22semidirect
22. C3×#3748C22semidirect
23. C32×C4×Dic3C22semidirect
24. C32×#208C22semidirect
30. #64158D3semidirect
39. C3×#837D3semidirect
45. C33×Q8D4semidirect
47. C33×D4D4semidirect
55. C4⋊(C3⋊(C3⋊D3))D4semidirect
58. #6308D4semidirect
92. C2×C4⋊(C3⋊D3)C2×D3semidirect
99. #3748C2×D3semidirect
104. C3×C4×Dic3C2×D3semidirect
106. C3×#208C2×D3semidirect
132. #837C3⋊D3semidirect
146. C32×Q8D12semidirect
147. C32×Q8#67semidirect
152. C32×D4D12semidirect
153. C32×D4#67semidirect
243. C4×Dic3C2×C3⋊D3semidirect
245. #208C2×C3⋊D3semidirect
291. C33#453semidirect
301. C3×Q8#626semidirect
302. C3×Q8C4⋊(C3⋊D3)semidirect
309. C3×D4#626semidirect
310. C3×D4C4⋊(C3⋊D3)semidirect
425. C32#4822semidirect
426. C32#5198semidirect
431. Q8#6292semidirect
442. D4#6292semidirect
481. C3#10591667semidirect
482. C3#10592521semidirect

Maximal subgroups

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

subgroupindex
#117246842
C3×#641582
#117235172
#117233702
C32×#8372
#117220482
#117220272
#105925213
#105916673