Finite Group Databasebeta

← #13161356

Subgroups — #13161356

Order 1728.

1280 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1280 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C32×#1009C211yesmaximal, characteristic, complemented
2#11723990C211yesmaximal, characteristic, complemented
3#11723524C211yesmaximal, characteristic, complemented
4#11723367C211yesmaximal, characteristic, complemented
5C3×#63664C211yesmaximal, characteristic, complemented
6#11722044C211yesmaximal, characteristic, complemented
7#11722025C211yesmaximal, characteristic, complemented
8#10592940—31nomaximal
9#10591666—34nomaximal
10C33×#41C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
12C32×#239—21no—
13C32×D3×D4—22no—
14C32×D3×D4—21no—
15C32×#235—21no—
16C32×#235C2211yescharacteristic
17C2×C32×D12—21no—
18C2×C32×D12C2211yescharacteristic
19C2×#6291—22no—
20#88862—21no—
21#88860—21no—
22#88855C2211yescharacteristic, complemented
23C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
24C3×#3748C2211yescharacteristic, complemented
25#88795—21no—
26#88790—21no—
27C32×H(5,𝔽2)—31no—
28C3×#1009D314yescomplemented
29#64158—31no—
30#64158—34no—
31#64158—31no—
32#64158—64no—
33#64149—31no—
34#63911—34no—
35#63825—34no—
36#63664—31no—
37#63664D311yescharacteristic, complemented
38#63660—31no—
39C3×#837—34no—
40#63569—34no—
41#63558—34no—
42C22×C3⋊(C3⋊D3)—21no—
43C22×C32×D3—22no—
44C22×C32×D3D412yes—
45C33×Q8D411yescharacteristic, complemented
46C33×D4—21no—
47C33×D4D411yescharacteristic, complemented
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))—21no—
52C2×C3×C3⋊Dic3—21no—
53C32×#67—22no—
54C32×#67—21no—
55C32×D12—22no—
56C32×D12—21no—
57C32×D12—21no—
58C32×D12D411yescharacteristic, complemented
59C32×C4×D3—22no—
60C32×C4×D3—21no—
61C32×Dic6D411yescharacteristic, complemented
62#6291—44no—
63C3⋊(C3⋊(C3⋊C8))—21no—
64C3×C3⋊(C3⋊C8)—21no—
65#5241—94no—
66#4822—91no—
67C32×#41—24no—
68C32×#41—31no—
69C32×#41—61no—
70C32×#41—61no—
71C32×#41C2×D311yescharacteristic
72C32×#41C2×D314yes—
73C2×C32×D4—31no—
74C2×C32×D4—61no—
75C2×C32×D4—62no—
76C2×C32×D4—61no—
77C2×#638—62no—
78C2×C4⋊(C3⋊D3)—31no—
79C2×C4⋊(C3⋊D3)—34no—
80C2×C4⋊(C3⋊D3)—64no—
81C3×#239—24no—
82C3×D3×D4—28no—
83C3×D3×D4—24no—
84C3×#235—24no—
85C3×#235C2×D314yes—
86C2×C3×D12—24no—
87C2×C3×D12C2×D314yes—
88C2×#626—68no—
89#3755—61no—
90#3755—68no—
91#3755—61no—
92#3755—64no—
93#3753—61no—
94#3753—68no—
95#3753—61no—
96#3753—64no—
97C4×C3⋊Dic3C2×D311yescharacteristic, complemented
98#3748—31no—
99#3748—34no—
100#3748—64no—
101#3748C2×D311yescharacteristic, complemented
102C3×C4×Dic3—34no—
103C3×#208—34no—
104#3717—64no—
105#3714—64no—
106C22×C33—21no—
107C22×C33C2×D411yescharacteristic
108C2×C3⋊(C3⋊D3)—42no—
109C2×C3⋊(C3⋊D3)—41no—
110C2×C32×D3—22no—
111C2×C32×D3—21no—
112C2×C32×D3—42no—
113C2×C32×D3—82no—
114C33×C4—21no—
115C33×C4C2×D411yescharacteristic
116C33×C4C2×D411yescharacteristic
117C3×C3⋊Dic3—22no—
118C32×Dic3—21no—
119C3×H(5,𝔽2)—34no—
120#1009C3⋊D311yescharacteristic, complemented
121#949—94no—
122#949—94no—
123#949—91no—
124#949—184no—
125#940—94no—
126#908—91no—
127#874—91no—
128C3×#104—91no—
129#837—34no—
130#837—94no—
131#833—94no—
132#823—91no—
133#805—91no—
134C23×C32—32no—
135C23×C32—62no—
136C22×C3⋊D3—61no—
137C22×C3⋊D3—64no—
138C22×C3⋊D3—124no—
139C22×C3×D3—28no—
140C22×C3×D3#6718yes—
141C32×Q8—24no—
142C32×Q8—31no—
143C32×Q8D1211yescharacteristic, complemented
144C32×Q8#6714yescomplemented
145C32×D4—21no—
146C32×D4—24no—
147C32×D4—24no—
148C32×D4—31no—
149C32×D4—44no—
150C32×D4—62no—
151C32×D4—62no—
152C32×D4—61no—
153C32×D4—61no—
154C32×D4—61no—
155C32×D4D1211yescharacteristic, complemented
156C32×D4#6714yescomplemented
157C2×C32×C4—21no—
158C2×C32×C4—24no—
159C2×C32×C4—24no—
160C2×C32×C4—44no—
161C2×C32×C4—62no—
162C2×C32×C4—61no—
163C2×C32×C4C22×D314yes—
164C2×C32×C4C22×D311yescharacteristic
165#638—124no—
166C2×C3⋊Dic3—21no—
167C4⋊(C3⋊D3)—68no—
168C4⋊(C3⋊D3)—61no—
169C4⋊(C3⋊D3)—61no—
170C4⋊(C3⋊D3)—64no—
171C4⋊(C3⋊D3)—64no—
172C4⋊(C3⋊D3)—124no—
173C3×#67—28no—
174C3×#67—24no—
175C2×C3×Dic3—64no—
176C3×D12—28no—
177C3×D12—24no—
178C3×D12—24no—
179C3×D12#6714yescomplemented
180C3×C4×D3—28no—
181C3×C4×D3—24no—
182C3×Dic6#6714yescomplemented
183#626—1216no—
184C3⋊(C3⋊C8)—21no—
185C3⋊(C3⋊C8)—68no—
186C3⋊(C3⋊C8)—61no—
187C3⋊(C3⋊C8)—64no—
188C3×C3⋊C8—64no—
189#453—271noSylow
190C2×C33—21no—
191C2×C33—41no—
192C2×C33#12011yescharacteristic
193C3⋊(C3⋊D3)—81no—
194C32×D3—42no—
195C32×D3—41no—
196C3×#41—24no—
197C3×#41—34no—
198C3×#41—64no—
199C3×#41—64no—
200C3×#41D3214yes—
201C3×#41C2×C3⋊D311yescharacteristic
202C2×C3×D4—34no—
203C2×C3×D4—64no—
204C2×C3×D4—68no—
205C2×C3×D4—64no—
206C2×#67—188no—
207#239—21no—
208D3×D4—22no—
209D3×D4—21no—
210#235—21no—
211#235C2×C3⋊D311yescharacteristic
212C2×D12—21no—
213C2×D12—94no—
214C2×D12—91no—
215C2×D12—184no—
216C2×D12—182no—
217C2×D12C2×C3⋊D311yescharacteristic
218C3×M16—91no—
219C3×C42—91no—
220#215—184no—
221#215—188no—
222#215—184no—
223#215—181no—
224#213—184no—
225#213—188no—
226#213—184no—
227#213—181no—
228C4×Dic3—34no—
229#208—34no—
230#208—94no—
231#208—91no—
232#208—184no—
233D24—181no—
234#204—181no—
235C22×C32—21no—
236C22×C32—24no—
237C22×C32—24no—
238C22×C32—44no—
239C22×C32—61no—
240C22×C32—62no—
241C22×C32—122no—
242C22×C32—242no—
243C22×C32D3×D411yescharacteristic
244C22×C32C2×#6714yes—
245C2×C3⋊D3—128no—
246C2×C3⋊D3—122no—
247C2×C3⋊D3—121no—
248C2×C3⋊D3—124no—
249C2×C3⋊D3—128no—
250C2×C3⋊D3—248no—
251C2×C3×D3—28no—
252C2×C3×D3—24no—
253C2×C3×D3—48no—
254C2×C3×D3—88no—
255C32×C4—21no—
256C32×C4—24no—
257C32×C4—24no—
258C32×C4—24no—
259C32×C4—44no—
260C32×C4—61no—
261C32×C4C2×D1211yescharacteristic
262C32×C4D3×D411yescharacteristic
263C32×C4C2×#6714yes—
264C32×C4C2×#6714yes—
265C3⋊Dic3—22no—
266C3×Dic3—24no—
267C3×Dic3—68no—
268H(5,𝔽2)—31no—
269AGL(1,ℤ/8)—271no—
270AGL(1,ℤ/8)—271no—
271#127—271no—
272#104—91no—
273#104—271no—
274#100—271no—
275C33#45311yescharacteristic, complemented, Sylow
276C23×C3—38no—
277C23×C3—68no—
278C22×D3—22no—
279C22×D3—184no—
280C22×D3—181no—
281C22×D3—364no—
282C22×D3#63812yes—
283C3×Q8—24no—
284C3×Q8—34no—
285C3×Q8#62614yescomplemented
286C3×Q8#63811yescharacteristic, complemented
287C3×D4—24no—
288C3×D4—21no—
289C3×D4—24no—
290C3×D4—34no—
291C3×D4—44no—
292C3×D4—68no—
293C3×D4—68no—
294C3×D4—64no—
295C3×D4—64no—
296C3×D4—64no—
297C3×D4#62614yescomplemented
298C3×D4#63811yescharacteristic, complemented
299C2×C3×C4—24no—
300C2×C3×C4—21no—
301C2×C3×C4—24no—
302C2×C3×C4—44no—
303C2×C3×C4—68no—
304C2×C3×C4—64no—
305C2×C3×C4—181no—
306C2×C3×C4C2×D3214yes—
307C2×C3×C4C22×C3⋊D311yescharacteristic
308#67—22no—
309#67—21no—
310#67—3616no—
311C2×Dic3—64no—
312D12—22no—
313D12—21no—
314D12—21no—
315D12—188no—
316D12—184no—
317D12—184no—
318D12—181no—
319D12—181no—
320D12—364no—
321D12—364no—
322D12#63811yescharacteristic, complemented
323C4×D3—22no—
324C4×D3—21no—
325Dic6#63811yescharacteristic, complemented
326C3×C8—181no—
327C3⋊C8—64no—
328C3⋊C8—188no—
329C3⋊C8—184no—
330C3⋊C8—181no—
331C2×C32—21no—
332C2×C32—24no—
333C2×C32—24no—
334C2×C32—44no—
335C2×C32—41no—
336C2×C32—44no—
337C2×C32—84no—
338C2×C32—122no—
339C2×C32—121no—
340C2×C32#88211yescharacteristic
341C2×C32#95314yes—
342C3⋊D3—248no—
343C3⋊D3—244no—
344C3⋊D3—241no—
345C3×D3—48no—
346C3×D3—44no—
347#41—31no—
348#41—61no—
349#41—61no—
350#41D3×C3⋊D311yescharacteristic
351C2×D4—31no—
352C2×D4—61no—
353C2×D4—62no—
354C2×D4—61no—
355C2×D4—271no—
356C2×D4—542no—
357SD16—541no—
358SD16—541no—
359D8—541no—
360D8—541no—
361M16—91no—
362M16—271no—
363C42—91no—
364C22×C3—21no—
365C22×C3—24no—
366C22×C3—24no—
367C22×C3—44no—
368C22×C3—64no—
369C22×C3—68no—
370C22×C3—128no—
371C22×C3—248no—
372C22×C3D3×#6714yes—
373C22×C3C2×#63811yescharacteristic
374C2×D3—22no—
375C2×D3—21no—
376C2×D3—42no—
377C2×D3—82no—
378C2×D3—368no—
379C2×D3—362no—
380C2×D3—368no—
381C2×D3—361no—
382C2×D3—364no—
383C2×D3—728no—
384C3×C4—21no—
385C3×C4—24no—
386C3×C4—24no—
387C3×C4—24no—
388C3×C4—44no—
389C3×C4—64no—
390C3×C4—182no—
391C3×C4C2×#62614yes—
392C3×C4D3×#6714yes—
393C3×C4C2×#63811yescharacteristic
394C3×C4C2×#63811yescharacteristic
395Dic3—21no—
396Dic3—68no—
397C32—24no—
398C32#482211yescharacteristic, complemented
399C32#524114yescomplemented
400C23—32no—
401C23—62no—
402C23—541no—
403Q8—31no—
404Q8#629111yescharacteristic, complemented
405D4—21no—
406D4—31no—
407D4—62no—
408D4—62no—
409D4—61no—
410D4—61no—
411D4—61no—
412D4—541no—
413D4—541no—
414D4—1084no—
415D4#629111yescharacteristic, complemented
416C2×C4—21no—
417C2×C4—62no—
418C2×C4—61no—
419C2×C4—181no—
420C2×C4C2×D3×C3⋊D311yescharacteristic
421C8—181no—
422C8—541no—
423C2×C3—21no—
424C2×C3—24no—
425C2×C3—24no—
426C2×C3—44no—
427C2×C3—41no—
428C2×C3—44no—
429C2×C3—84no—
430C2×C3—128no—
431C2×C3—124no—
432C2×C3#6397714yes—
433C2×C3#6416211yescharacteristic
434D3—42no—
435D3—41no—
436D3—728no—
437D3—721no—
438D3—724no—
439C22—21no—
440C22—61no—
441C22—62no—
442C22—122no—
443C22—242no—
444C22—1082no—
445C22—1081no—
446C22D3×#63811yescharacteristic
447C4—21no—
448C4—61no—
449C4—182no—
450C4C2×#629111yescharacteristic
451C4D3×#63811yescharacteristic
452C3—24no—
453C3#1059166614yescomplemented
454C3#1059294011yescharacteristic, complemented
455C2—21no—
456C2—41no—
457C2—122no—
458C2—121no—
459C2—2161no—
460C2#1172363011yescharacteristic
subgroupG/H
centerC2#11723630
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. C32×#1009C2semidirect
2. #11723990C2semidirect
3. #11723524C2semidirect
4. #11723367C2semidirect
5. C3×#63664C2semidirect
6. #11722044C2semidirect
7. #11722025C2semidirect
11. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
22. #88855C22semidirect
23. C3×C4×C3⋊Dic3C22semidirect
24. C3×#3748C22semidirect
28. C3×#1009D3semidirect
37. #63664D3semidirect
45. C33×Q8D4semidirect
47. C33×D4D4semidirect
58. C32×D12D4semidirect
61. C32×Dic6D4semidirect
97. C4×C3⋊Dic3C2×D3semidirect
101. #3748C2×D3semidirect
120. #1009C3⋊D3semidirect
143. C32×Q8D12semidirect
144. C32×Q8#67semidirect
155. C32×D4D12semidirect
156. C32×D4#67semidirect
179. C3×D12#67semidirect
182. C3×Dic6#67semidirect
275. C33#453semidirect
285. C3×Q8#626semidirect
286. C3×Q8#638semidirect
297. C3×D4#626semidirect
298. C3×D4#638semidirect
322. D12#638semidirect
325. Dic6#638semidirect
398. C32#4822semidirect
399. C32#5241semidirect
404. Q8#6291semidirect
415. D4#6291semidirect
453. C3#10591666semidirect
454. C3#10592940semidirect

Maximal subgroups

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

subgroupindex
C32×#10092
#117239902
#117235242
#117233672
C3×#636642
#117220442
#117220252
#105929403
#105916663