Finite Group Databasebeta

← #13161118

Subgroups — #13161118

Order 1728.

1440 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1440 conjugacy classes of subgroups — too many to draw. Only its 134 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#11724581C211yesmaximal, characteristic, complemented
2#11723981C211yesmaximal, characteristic, complemented
3C3×#64141C211yesmaximal, characteristic, complemented
4C32×#932C211yesmaximal, characteristic, complemented
5#11722045C211yesmaximal, characteristic, complemented
6#11722043C211yesmaximal, characteristic, complemented
7#11722027C211yesmaximal, characteristic, complemented
8#10592897—31nomaximal
9#10591522—34nomaximal
10C2×C33×D4C2211yescharacteristic, complemented
11C2×#6312—22no—
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C3×#3828C2211yescharacteristic
14C32×#235C2211yescharacteristic
15#89039—22no—
16C3⋊D3×D12—21no—
17D3×C4⋊(C3⋊D3)—21no—
18#89022—21no—
19#89018—21no—
20C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
21C3×#3754—21no—
22C3×#3753—21no—
23C3×#3748C2211yescharacteristic, complemented
24C32×#214—21no—
25C32×#213—21no—
26C32×#208C2211yescharacteristic, complemented
27#64366—31no—
28#64308—34no—
29C32×AGL(1,ℤ/8)—31no—
30#64149—31no—
31#64149—34no—
32#64149—64no—
33#64141D311yescharacteristic, complemented
34C3×#932—34no—
35C3×#932D314yescomplemented
36#63664—31no—
37#63652—31no—
38#63650—31no—
39#63568—34no—
40#63568—34no—
41#63559—34no—
42C23×C33—21no—
43C22×C3⋊(C3⋊D3)D412yes—
44C2×D3×C3⋊D3—22no—
45C33×D4—21no—
46C33×D4—21no—
47C2×C33×C4C2311yescharacteristic
48#6312—44no—
49C2×C3⋊(C3⋊Dic3)—21no—
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))—21no—
52C3×#638—21no—
53C3×C4⋊(C3⋊D3)D411yescharacteristic, complemented
54C3×C4×C3⋊D3—21no—
55C3×#634D411yescharacteristic, complemented
56C32×#67—21no—
57C32×D12D411yescharacteristic, complemented
58C32×C4×D3—21no—
59C32×Dic6D411yescharacteristic, complemented
60#6292—22no—
61#6291—22no—
62#6289—22no—
63C3×C3⋊(C3⋊C8)—21no—
64C32×C3⋊C8—21no—
65#5198—94no—
66#5198—91no—
67C32×#41—31no—
68C2×C32×D4—24no—
69C2×C32×D4C2×D314yescomplemented
70C2×C32×D4C2×D311yescharacteristic, complemented
71C2×#638—62no—
72C2×#638—68no—
73C2×#638—616no—
74#3832—61no—
75D4×C3⋊D3—61no—
76D4×C3⋊D3—62no—
77#3828—61no—
78#3828C2×D311yescharacteristic
79C2×C4⋊(C3⋊D3)—31no—
80C2×C4⋊(C3⋊D3)—34no—
81C2×C4⋊(C3⋊D3)—61no—
82C2×C4⋊(C3⋊D3)—64no—
83C3×#235—34no—
84C3×#235C2×D314yes—
85#3811—68no—
86D3×D12—64no—
87D3×D12—64no—
88#3799—64no—
89#3799—64no—
90C32×SD16—61no—
91C32×D8—61no—
92C32×M16—31no—
93#3754—21no—
94#3753—21no—
95C4×C3⋊Dic3—31no—
96C4×C3⋊Dic3—34no—
97C4×C3⋊Dic3—64no—
98#3748C2×D311yescharacteristic, complemented
99C3×#214—24no—
100C3×#214—64no—
101C3×#213—24no—
102C3×#213—64no—
103C3×#208—34no—
104C3×#208C2×D314yescomplemented
105C22×C33—42no—
106C22×C33—41no—
107C22×C33C2×D411yescharacteristic
108C2×C3⋊(C3⋊D3)—22no—
109C2×C3⋊(C3⋊D3)—42no—
110C2×C3×C3⋊D3—21no—
111C2×C32×D3—21no—
112D3×C3⋊D3—82no—
113C33×C4C2×D411yescharacteristic
114C33×C4C2×D411yescharacteristic
115C3⋊(C3⋊Dic3)—22no—
116C3×C3⋊Dic3—21no—
117C32×Dic3—21no—
118#1009—94no—
119#1009—91no—
120C3×AGL(1,ℤ/8)—34no—
121C3×AGL(1,ℤ/8)—91no—
122#940—94no—
123#940—91no—
124#940—184no—
125#932—34no—
126#932C3⋊D311yescharacteristic, complemented
127#837—94no—
128#837—91no—
129#825—94no—
130#825—91no—
131#823—94no—
132#823—91no—
133C23×C32—28no—
134C23×C32—24no—
135C23×C32—21no—
136C22×C3⋊D3—32no—
137C22×C3⋊D3—38no—
138C22×C3⋊D3—68no—
139C22×C3⋊D3—62no—
140C2×D32—68no—
141C32×Q8—31no—
142C32×D4—21no—
143C32×D4—21no—
144C32×D4—24no—
145C32×D4—24no—
146C32×D4—31no—
147C32×D4—44no—
148C32×D4—44no—
149C32×D4—61no—
150C2×C32×C4—24no—
151C2×C32×C4—61no—
152C2×C32×C4C22×D314yes—
153C2×C32×C4C22×D311yescharacteristic
154#638—21no—
155#638—62no—
156#638—124no—
157#638—1216no—
158#638—1232no—
159C2×C3⋊Dic3—68no—
160C2×C3⋊Dic3—64no—
161C2×C3⋊Dic3—61no—
162C4⋊(C3⋊D3)—62no—
163C4⋊(C3⋊D3)—61no—
164C4⋊(C3⋊D3)—61no—
165C4⋊(C3⋊D3)—64no—
166C4⋊(C3⋊D3)—64no—
167C4⋊(C3⋊D3)—124no—
168C4⋊(C3⋊D3)—124no—
169C4⋊(C3⋊D3)#6711yescharacteristic, complemented
170C4×C3⋊D3—21no—
171C4×C3⋊D3—62no—
172#634#6711yescharacteristic, complemented
173C3×#67—24no—
174C3×#67—64no—
175C3×D12—34no—
176C3×D12#6714yescomplemented
177C3×C4×D3—24no—
178C3×C4×D3—64no—
179C3×Dic6—34no—
180C3×Dic6#6714yescomplemented
181#626—68no—
182#626—68no—
183#624—68no—
184C32×C8—61no—
185C3⋊(C3⋊C8)—21no—
186C3×C3⋊C8—24no—
187C3×C3⋊C8—64no—
188#453—271noSylow
189C2×C33—21no—
190C2×C33—81no—
191C2×C33#12011yescharacteristic
192C3⋊(C3⋊D3)—42no—
193C3×C3⋊D3—41no—
194C32×D3—41no—
195C3×#41—34no—
196C3×#41—91no—
197C2×C3×D4—24no—
198C2×C3×D4D3214yescomplemented
199C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
200C2×#67—188no—
201C2×#67—182no—
202C2×#67—1816no—
203#239—181no—
204#239—184no—
205D3×D4—181no—
206D3×D4—184no—
207D3×D4—182no—
208D3×D4—188no—
209#235—34no—
210#235—184no—
211#235—181no—
212#235C2×C3⋊D311yescharacteristic
213C2×D12—94no—
214C2×D12—91no—
215C2×D12—184no—
216C2×D12—184no—
217C2×D12—181no—
218C3×SD16—64no—
219C3×SD16—181no—
220C3×D8—64no—
221C3×D8—181no—
222C3×M16—34no—
223C3×M16—91no—
224#214—21no—
225#214—64no—
226#213—21no—
227#213—64no—
228C4×Dic3—94no—
229C4×Dic3—91no—
230C4×Dic3—184no—
231#208—34no—
232#208C2×C3⋊D311yescharacteristic, complemented
233C22×C32—24no—
234C22×C32—416no—
235C22×C32—48no—
236C22×C32—42no—
237C22×C32—48no—
238C22×C32—44no—
239C22×C32—41no—
240C22×C32—61no—
241C22×C32D3×D414yes—
242C22×C32D3×D411yescharacteristic
243C2×C3⋊D3—21no—
244C2×C3⋊D3—62no—
245C2×C3⋊D3—68no—
246C2×C3⋊D3—128no—
247C2×C3⋊D3—122no—
248C2×C3⋊D3—128no—
249C2×C3⋊D3—1216no—
250C2×C3⋊D3—242no—
251C2×C3×D3—24no—
252C2×C3×D3—64no—
253D32—248no—
254C32×C4—24no—
255C32×C4—24no—
256C32×C4—61no—
257C32×C4D3×D414yes—
258C32×C4D3×D411yescharacteristic
259C32×C4C2×#6714yes—
260C32×C4C2×#6711yescharacteristic
261C3⋊Dic3—21no—
262C3⋊Dic3—616no—
263C3⋊Dic3—68no—
264C3⋊Dic3—62no—
265C3×Dic3—24no—
266C3×Dic3—64no—
267H(5,𝔽2)—271no—
268AGL(1,ℤ/8)—31no—
269AGL(1,ℤ/8)—91no—
270#127—271no—
271#104—271no—
272#104—271no—
273#100—271no—
274C33#45311yescharacteristic, complemented, Sylow
275C23×C3—28no—
276C23×C3—24no—
277C23×C3—21no—
278C22×D3—98no—
279C22×D3—92no—
280C22×D3—188no—
281C22×D3—182no—
282C22×D3—188no—
283C3×Q8—34no—
284C3×Q8—91no—
285C3×D4—24no—
286C3×D4—24no—
287C3×D4—21no—
288C3×D4—21no—
289C3×D4—34no—
290C3×D4—44no—
291C3×D4—44no—
292C3×D4—64no—
293C3×D4—91no—
294C3×D4—181no—
295C2×C3×C4—24no—
296C2×C3×C4—64no—
297C2×C3×C4—181no—
298C2×C3×C4C2×D3214yes—
299C2×C3×C4C22×C3⋊D311yescharacteristic
300#67—21no—
301#67—64no—
302#67—182no—
303#67—188no—
304#67—3616no—
305#67—364no—
306#67—3632no—
307C2×Dic3—188no—
308C2×Dic3—184no—
309C2×Dic3—181no—
310D12—34no—
311D12—188no—
312D12—182no—
313D12—184no—
314D12—184no—
315D12—181no—
316D12—181no—
317D12—364no—
318D12—364no—
319D12#63811yescharacteristic, complemented
320C4×D3—21no—
321C4×D3—64no—
322C4×D3—182no—
323C4×D3—188no—
324Dic6—34no—
325Dic6#63811yescharacteristic, complemented
326C3×C8—64no—
327C3×C8—181no—
328C3⋊C8—21no—
329C3⋊C8—64no—
330C2×C32—28no—
331C2×C32—24no—
332C2×C32—21no—
333C2×C32—24no—
334C2×C32—88no—
335C2×C32—84no—
336C2×C32—81no—
337C2×C32—121no—
338C2×C32#93714yes—
339C2×C32#93711yescharacteristic
340C3⋊D3—41no—
341C3⋊D3—122no—
342C3⋊D3—128no—
343C3⋊D3—248no—
344C3×D3—44no—
345C3×D3—124no—
346#41—31no—
347#41—91no—
348#41—541no—
349#41—541no—
350C2×D4—271no—
351C2×D4—541no—
352C2×D4—541no—
353C2×D4—542no—
354C2×D4—542no—
355C2×D4D3×C3⋊D311yescharacteristic, complemented
356SD16—61no—
357SD16—181no—
358D8—61no—
359D8—181no—
360M16—31no—
361M16—91no—
362C42—271no—
363C22×C3—24no—
364C22×C3—416no—
365C22×C3—48no—
366C22×C3—42no—
367C22×C3—48no—
368C22×C3—41no—
369C22×C3—44no—
370C22×C3—64no—
371C22×C3—181no—
372C22×C3#381114yes—
373C22×C3D4×C3⋊D311yescharacteristic
374C2×D3—21no—
375C2×D3—64no—
376C2×D3—182no—
377C2×D3—188no—
378C2×D3—368no—
379C2×D3—368no—
380C2×D3—362no—
381C2×D3—3616no—
382C2×D3—722no—
383C2×D3—728no—
384C3×C4—24no—
385C3×C4—24no—
386C3×C4—64no—
387C3×C4—181no—
388C3×C4D3×#6714yes—
389C3×C4D3×#6714yes—
390C3×C4D4×C3⋊D311yescharacteristic
391C3×C4C2×#63811yescharacteristic
392Dic3—21no—
393Dic3—64no—
394Dic3—1816no—
395Dic3—188no—
396Dic3—182no—
397C32—24no—
398C32#519814yescomplemented
399C32#519811yescharacteristic, complemented
400C23—21no—
401C23—272no—
402C23—542no—
403Q8—31no—
404Q8—91no—
405D4—21no—
406D4—21no—
407D4—31no—
408D4—61no—
409D4—91no—
410D4—181no—
411D4—542no—
412D4—542no—
413D4—541no—
414D4—541no—
415D4—1084no—
416C2×C4—61no—
417C2×C4—181no—
418C2×C4—542no—
419C2×C4—541no—
420C2×C4C2×D3×C3⋊D311yescharacteristic
421C8—61no—
422C8—181no—
423C2×C3—28no—
424C2×C3—24no—
425C2×C3—21no—
426C2×C3—24no—
427C2×C3—88no—
428C2×C3—84no—
429C2×C3—81no—
430C2×C3—124no—
431C2×C3—361no—
432C2×C3#6398314yes—
433C2×C3#6414611yescharacteristic
434D3—41no—
435D3—124no—
436D3—368no—
437D3—362no—
438D3—728no—
439C22—42no—
440C22—41no—
441C22—61no—
442C22—181no—
443C22—542no—
444C22—1082no—
445C22—2162no—
446C22#8903911yescharacteristic
447C4—61no—
448C4—181no—
449C4—542no—
450C4D3×#63811yescharacteristic
451C4C3⋊D3×#6711yescharacteristic
452C3—24no—
453C3#1059152214yescomplemented
454C3#1059289711yescharacteristic, complemented
455C2—21no—
456C2—81no—
457C2—121no—
458C2—361no—
459C2—1082no—
460C2#1172364311yescharacteristic
subgroupG/H
centerC2#11723643
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33#120
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#453
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11724581C2semidirect
2. #11723981C2semidirect
3. C3×#64141C2semidirect
4. C32×#932C2semidirect
5. #11722045C2semidirect
6. #11722043C2semidirect
7. #11722027C2semidirect
10. C2×C33×D4C22semidirect
20. C4×C3⋊(C3⋊Dic3)C22semidirect
23. C3×#3748C22semidirect
26. C32×#208C22semidirect
33. #64141D3semidirect
35. C3×#932D3semidirect
53. C3×C4⋊(C3⋊D3)D4semidirect
55. C3×#634D4semidirect
57. C32×D12D4semidirect
59. C32×Dic6D4semidirect
69. C2×C32×D4C2×D3semidirect
70. C2×C32×D4C2×D3semidirect
98. #3748C2×D3semidirect
104. C3×#208C2×D3semidirect
126. #932C3⋊D3semidirect
169. C4⋊(C3⋊D3)#67semidirect
172. #634#67semidirect
176. C3×D12#67semidirect
180. C3×Dic6#67semidirect
198. C2×C3×D4D32semidirect
199. C2×C3×D4C2×C3⋊D3semidirect
232. #208C2×C3⋊D3semidirect
274. C33#453semidirect
319. D12#638semidirect
325. Dic6#638semidirect
355. C2×D4D3×C3⋊D3semidirect
398. C32#5198semidirect
399. C32#5198semidirect
453. C3#10591522semidirect
454. C3#10592897semidirect

Maximal subgroups

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

subgroupindex
#117245812
#117239812
C3×#641412
C32×#9322
#117220452
#117220432
#117220272
#105928973
#105915223