Finite Group Databasebeta

← #13161392

Subgroups — #13161392

Order 1728.

1184 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1184 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
2C3×#64158C211yesmaximal, characteristic, complemented
3#11723529C211yesmaximal, characteristic, complemented
4#11723365C211yesmaximal, characteristic, complemented
5#11722207C211yesmaximal, characteristic, complemented
6#11722043C211yesmaximal, characteristic, complemented
7#11722025C211yesmaximal, characteristic, complemented
8#10592940—31nomaximal
9#10591665—34nomaximal
10C33×#41C2211yescharacteristic
11C2×C3×C4⋊(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×#6290—22no—
20C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
21#88855C2211yescharacteristic, complemented
22C3×#3755—21no—
23C3×#3753—21no—
24C3×#3748C2211yescharacteristic, complemented
25#88793—21no—
26#88789—21no—
27C32×H(5,𝔽2)—31no—
28C3×#1009D314yescomplemented
29#64158—31no—
30#64158D311yescharacteristic, complemented
31#64149—31no—
32C3×#949—34no—
33#63914—34no—
34#63823—34no—
35#63664—31no—
36#63664—31no—
37#63664—34no—
38#63664—64no—
39#63660—31no—
40#63568—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
50C2×C3⋊(C3⋊Dic3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—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#6290—44no—
63C3⋊(C3⋊(C3⋊C8))—21no—
64C3×C3⋊(C3⋊C8)—21no—
65#5241—94no—
66#5198—91no—
67C32×#41—24no—
68C32×#41—31no—
69C32×#41—61no—
70C32×#41—61no—
71C32×#41C2×D314yes—
72C32×#41C2×D311yescharacteristic
73C2×C32×D4—31no—
74C2×C32×D4—61no—
75C2×C32×D4—62no—
76C2×C32×D4—61no—
77C2×#638—62no—
78C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
79C3×#239—24no—
80C3×D3×D4—28no—
81C3×D3×D4—24no—
82C3×#235—24no—
83C3×#235C2×D314yes—
84C2×C3×D12—24no—
85C2×C3×D12—34no—
86C2×C3×D12C2×D314yes—
87C2×#625—68no—
88#3755—21no—
89#3755—61no—
90#3753—21no—
91#3753—61no—
92C4×C3⋊Dic3—31no—
93C4×C3⋊Dic3—34no—
94C4×C3⋊Dic3—64no—
95#3748—31no—
96#3748—34no—
97#3748—64no—
98#3748C2×D311yescharacteristic, complemented
99C3×#215—64no—
100C3×#213—64no—
101C3×#208—34no—
102#3715—64no—
103#3713—64no—
104C22×C33—21no—
105C22×C33C2×D411yescharacteristic
106C2×C3×C3⋊D3—42no—
107C2×C3×C3⋊D3—41no—
108C2×C32×D3—22no—
109C2×C32×D3—21no—
110C2×C32×D3—42no—
111C2×C32×D3—82no—
112C33×C4—21no—
113C33×C4C2×D411yescharacteristic
114C33×C4C2×D411yescharacteristic
115C3⋊(C3⋊Dic3)—22no—
116C32×Dic3—21no—
117C3×H(5,𝔽2)—34no—
118#1009C3⋊D311yescharacteristic, complemented
119C3×AGL(1,ℤ/8)—91no—
120#949—34no—
121#949—94no—
122#940—94no—
123#940—91no—
124#932—91no—
125#837—94no—
126#837—94no—
127#837—91no—
128#837—184no—
129#833—94no—
130#825—91no—
131#823—91no—
132C23×C32—32no—
133C23×C32—62no—
134C22×C3⋊D3—21no—
135C22×C3×D3—28no—
136C22×C3×D3—64no—
137C22×C3×D3#6718yes—
138C32×Q8—24no—
139C32×Q8—31no—
140C32×Q8#6714yescomplemented
141C32×Q8#6711yescharacteristic, complemented
142C32×D4—21no—
143C32×D4—24no—
144C32×D4—24no—
145C32×D4—31no—
146C32×D4—44no—
147C32×D4—62no—
148C32×D4—62no—
149C32×D4—61no—
150C32×D4—61no—
151C32×D4—61no—
152C32×D4#6714yescomplemented
153C32×D4#6711yescharacteristic, complemented
154C2×C32×C4—21no—
155C2×C32×C4—24no—
156C2×C32×C4—24no—
157C2×C32×C4—44no—
158C2×C32×C4—62no—
159C2×C32×C4—61no—
160C2×C32×C4C22×D314yes—
161C2×C32×C4C22×D311yescharacteristic
162#638—124no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—64no—
165C2×C3⋊Dic3—124no—
166C4⋊(C3⋊D3)—21no—
167C4⋊(C3⋊D3)—21no—
168C3×#67—28no—
169C3×#67—24no—
170C3×D12—28no—
171C3×D12—24no—
172C3×D12—24no—
173C3×D12—64no—
174C3×D12—64no—
175C3×D12#6714yescomplemented
176C3×C4×D3—28no—
177C3×C4×D3—24no—
178C3×Dic6#6714yescomplemented
179#625—1216no—
180C3⋊(C3⋊C8)—21no—
181C3⋊(C3⋊C8)—61no—
182C3⋊(C3⋊C8)—64no—
183C3⋊(C3⋊C8)—124no—
184C3×C3⋊C8—64no—
185#453—271noSylow
186C2×C33—21no—
187C2×C33—41no—
188C2×C33#12011yescharacteristic
189C3×C3⋊D3—81no—
190C32×D3—42no—
191C32×D3—41no—
192C3×#41—24no—
193C3×#41—34no—
194C3×#41—64no—
195C3×#41—64no—
196C3×#41D3214yes—
197C3×#41C2×C3⋊D311yescharacteristic
198C2×C3×D4—34no—
199C2×C3×D4—64no—
200C2×C3×D4—68no—
201C2×C3×D4—64no—
202C2×C3×D4—91no—
203C2×#67—188no—
204C2×#67—182no—
205#239—21no—
206D3×D4—22no—
207D3×D4—21no—
208#235—21no—
209#235C2×C3⋊D311yescharacteristic
210C2×D12—21no—
211C2×D12—34no—
212C2×D12C2×C3⋊D311yescharacteristic
213C3×SD16—181no—
214C3×D8—181no—
215C3×M16—91no—
216#215—64no—
217#215—184no—
218#214—181no—
219#213—64no—
220#213—184no—
221#213—181no—
222C4×Dic3—94no—
223C4×Dic3—91no—
224C4×Dic3—184no—
225#208—34no—
226#208—94no—
227#208—91no—
228#208—184no—
229C22×C32—21no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—44no—
233C22×C32—61no—
234C22×C32—62no—
235C22×C32—122no—
236C22×C32—242no—
237C22×C32D3×D411yescharacteristic
238C22×C32C2×#6714yes—
239C2×C3⋊D3—42no—
240C2×C3⋊D3—41no—
241C2×C3×D3—28no—
242C2×C3×D3—24no—
243C2×C3×D3—48no—
244C2×C3×D3—88no—
245C2×C3×D3—128no—
246C2×C3×D3—124no—
247C32×C4—21no—
248C32×C4—24no—
249C32×C4—24no—
250C32×C4—24no—
251C32×C4—44no—
252C32×C4—61no—
253C32×C4D3×D411yescharacteristic
254C32×C4C2×#6714yes—
255C32×C4C2×#6714yes—
256C32×C4C2×#6711yescharacteristic
257C3⋊Dic3—62no—
258C3⋊Dic3—68no—
259C3⋊Dic3—128no—
260C3×Dic3—24no—
261H(5,𝔽2)—31no—
262AGL(1,ℤ/8)—91no—
263AGL(1,ℤ/8)—271no—
264#127—271no—
265#104—271no—
266#104—271no—
267#100—271no—
268C33#45311yescharacteristic, complemented, Sylow
269C23×C3—38no—
270C23×C3—68no—
271C23×C3—181no—
272C22×D3—22no—
273C22×D3—64no—
274C22×D3#63812yes—
275C3×Q8—24no—
276C3×Q8—34no—
277C3×Q8#62514yescomplemented
278C3×Q8#63811yescharacteristic, complemented
279C3×D4—24no—
280C3×D4—21no—
281C3×D4—24no—
282C3×D4—34no—
283C3×D4—44no—
284C3×D4—68no—
285C3×D4—68no—
286C3×D4—64no—
287C3×D4—64no—
288C3×D4—64no—
289C3×D4—181no—
290C3×D4—181no—
291C3×D4#62514yescomplemented
292C3×D4#63811yescharacteristic, complemented
293C2×C3×C4—24no—
294C2×C3×C4—21no—
295C2×C3×C4—24no—
296C2×C3×C4—44no—
297C2×C3×C4—68no—
298C2×C3×C4—64no—
299C2×C3×C4C2×D3214yes—
300C2×C3×C4C22×C3⋊D311yescharacteristic
301#67—22no—
302#67—21no—
303#67—364no—
304#67—3616no—
305C2×Dic3—184no—
306C2×Dic3—181no—
307C2×Dic3—364no—
308D12—22no—
309D12—21no—
310D12—21no—
311D12—64no—
312D12—64no—
313D12#63811yescharacteristic, complemented
314C4×D3—22no—
315C4×D3—21no—
316Dic6#63811yescharacteristic, complemented
317C3×C8—181no—
318C3⋊C8—64no—
319C3⋊C8—184no—
320C3⋊C8—181no—
321C3⋊C8—364no—
322C2×C32—21no—
323C2×C32—24no—
324C2×C32—24no—
325C2×C32—44no—
326C2×C32—41no—
327C2×C32—44no—
328C2×C32—84no—
329C2×C32—122no—
330C2×C32—121no—
331C2×C32#93711yescharacteristic
332C2×C32#95314yes—
333C3⋊D3—81no—
334C3×D3—48no—
335C3×D3—44no—
336C3×D3—244no—
337#41—31no—
338#41—61no—
339#41—61no—
340#41D3×C3⋊D311yescharacteristic
341C2×D4—31no—
342C2×D4—61no—
343C2×D4—62no—
344C2×D4—61no—
345C2×D4—91no—
346C2×D4—542no—
347SD16—181no—
348SD16—541no—
349D8—181no—
350D8—541no—
351M16—91no—
352M16—271no—
353C42—271no—
354C22×C3—21no—
355C22×C3—24no—
356C22×C3—24no—
357C22×C3—44no—
358C22×C3—64no—
359C22×C3—68no—
360C22×C3—128no—
361C22×C3—248no—
362C22×C3—362no—
363C22×C3—361no—
364C22×C3D3×#6714yes—
365C22×C3C2×#63811yescharacteristic
366C2×D3—22no—
367C2×D3—21no—
368C2×D3—42no—
369C2×D3—82no—
370C2×D3—128no—
371C2×D3—124no—
372C3×C4—21no—
373C3×C4—24no—
374C3×C4—24no—
375C3×C4—24no—
376C3×C4—44no—
377C3×C4—64no—
378C3×C4C2×#62514yes—
379C3×C4D3×#6714yes—
380C3×C4C2×#63811yescharacteristic
381C3×C4C2×#63811yescharacteristic
382Dic3—21no—
383Dic3—182no—
384Dic3—188no—
385Dic3—368no—
386C32—24no—
387C32#519811yescharacteristic, complemented
388C32#524114yescomplemented
389C23—32no—
390C23—62no—
391C23—181no—
392Q8—31no—
393Q8#629011yescharacteristic, complemented
394D4—21no—
395D4—31no—
396D4—62no—
397D4—62no—
398D4—61no—
399D4—61no—
400D4—61no—
401D4—181no—
402D4—181no—
403D4—1084no—
404D4#629011yescharacteristic, complemented
405C2×C4—21no—
406C2×C4—62no—
407C2×C4—61no—
408C2×C4—541no—
409C2×C4C2×D3×C3⋊D311yescharacteristic
410C8—181no—
411C8—541no—
412C2×C3—21no—
413C2×C3—24no—
414C2×C3—24no—
415C2×C3—44no—
416C2×C3—41no—
417C2×C3—44no—
418C2×C3—84no—
419C2×C3—128no—
420C2×C3—124no—
421C2×C3—721no—
422C2×C3#6397614yes—
423C2×C3#6416211yescharacteristic
424D3—42no—
425D3—41no—
426D3—244no—
427C22—21no—
428C22—61no—
429C22—62no—
430C22—122no—
431C22—242no—
432C22—362no—
433C22—361no—
434C22D3×#63811yescharacteristic
435C4—21no—
436C4—61no—
437C4—542no—
438C4C2×#629011yescharacteristic
439C4D3×#63811yescharacteristic
440C3—24no—
441C3#1059166514yescomplemented
442C3#1059294011yescharacteristic, complemented
443C2—21no—
444C2—41no—
445C2—122no—
446C2—121no—
447C2—721no—
448C2#1172362911yescharacteristic
subgroupG/H
centerC2#11723629
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. C3×#64158C2semidirect
3. #11723529C2semidirect
4. #11723365C2semidirect
5. #11722207C2semidirect
6. #11722043C2semidirect
7. #11722025C2semidirect
11. C2×C3×C4⋊(C3⋊D3)C22semidirect
20. C4×C3⋊(C3⋊Dic3)C22semidirect
21. #88855C22semidirect
24. C3×#3748C22semidirect
28. C3×#1009D3semidirect
30. #64158D3semidirect
45. C33×Q8D4semidirect
47. C33×D4D4semidirect
58. C32×D12D4semidirect
61. C32×Dic6D4semidirect
78. C2×C4⋊(C3⋊D3)C2×D3semidirect
98. #3748C2×D3semidirect
118. #1009C3⋊D3semidirect
140. C32×Q8#67semidirect
141. C32×Q8#67semidirect
152. C32×D4#67semidirect
153. C32×D4#67semidirect
175. C3×D12#67semidirect
178. C3×Dic6#67semidirect
268. C33#453semidirect
277. C3×Q8#625semidirect
278. C3×Q8#638semidirect
291. C3×D4#625semidirect
292. C3×D4#638semidirect
313. D12#638semidirect
316. Dic6#638semidirect
387. C32#5198semidirect
388. C32#5241semidirect
393. Q8#6290semidirect
404. D4#6290semidirect
441. C3#10591665semidirect
442. C3#10592940semidirect

Maximal subgroups

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

subgroupindex
C32×#10092
C3×#641582
#117235292
#117233652
#117222072
#117220432
#117220252
#105929403
#105916653