Finite Group Databasebeta

← #13160042

Subgroups — #13160042

Order 1728.

1388 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1388 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C2×C3⋊D3×D12C211yesmaximal, characteristic, complemented
2C2×C32×D24C211yesmaximal, characteristic, complemented
3#11723522C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5#11722191C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592855—31nomaximal
9#10590944—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic, complemented
15C2×#6291—21no—
16C2×#6290—21no—
17C3⋊D3×D12—22no—
18C3⋊D3×D12—22no—
19C2×C33×C8C2211yescharacteristic, complemented
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
22C32×D24—22no—
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88790—22no—
25C2×D4×C3⋊D3—31no—
26C2×D3×D12—34no—
27C2×C32×D8—31no—
28#64147—31no—
29C2×#3753—31no—
30C2×C3×D24D314yescomplemented
31#63908—34no—
32C2×#3714—34no—
33#63659—31no—
34#63648—31no—
35#63648—34no—
36#63648—64no—
37#63647D311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—34no—
39#63563—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—21no—
43C22×C32×D3—21no—
44C2×D3×C3⋊D3—24no—
45C2×D3×C3⋊D3—42no—
46C2×D3×C3⋊D3—42no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊(C3⋊D3))—21no—
50C4⋊(C3⋊(C3⋊D3))D412yescomplemented
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C32×D12—21no—
55C32×D12—21no—
56C32×D12—21no—
57C32×D12D412yescomplemented
58#6291—24no—
59#6290—42no—
60C33×C8—21no—
61C3×C3⋊(C3⋊C8)—21no—
62#5156—94no—
63#4732—91no—
64C23×C3⋊D3—61no—
65C22×D32—64no—
66C2×C32×D4—31no—
67C2×C32×D4—31no—
68C2×#638—61no—
69C2×#638—61no—
70D4×C3⋊D3—62no—
71D4×C3⋊D3—62no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—34no—
74C2×C4⋊(C3⋊D3)—64no—
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×C3×D12C2×D314yes—
77C2×C3×D12C2×D314yescomplemented
78C2×C3×C4×D3—34no—
79C2×#626—64no—
80C2×#625—64no—
81D3×D12—68no—
82D3×D12—68no—
83C32×D8—62no—
84C2×C32×C8—24no—
85C2×C32×C8C2×D314yescomplemented
86C2×C32×C8C2×D311yescharacteristic, complemented
87#3757—61no—
88#3753—62no—
89C4⋊(C3⋊Dic3)—31no—
90C4⋊(C3⋊Dic3)—34no—
91C4⋊(C3⋊Dic3)—64no—
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
93C2×C3×C3⋊C8—34no—
94C3×D24—28no—
95(C2×D3)⋊Dic3—64no—
96#3714—68no—
97C22×C33C2×D411yescharacteristic
98C2×C3⋊(C3⋊D3)—22no—
99C2×C3⋊(C3⋊D3)—41no—
100C2×C3⋊(C3⋊D3)—41no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—21no—
103C2×C3×C3⋊D3D812yescomplemented
104C2×C32×D3—22no—
105C2×C32×D3—41no—
106C2×C32×D3—41no—
107C2×C32×D3—41no—
108C2×C32×D3—41no—
109C2×C32×D3—41no—
110D3×C3⋊D3—48no—
111C33×C4C2×D411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3⋊(C3⋊Dic3)—41no—
114C3×C3⋊Dic3—21no—
115C2×D3×D4—94no—
116C22×D12—91no—
117C2×C3×D8—34no—
118#938—94no—
119C2×#213—94no—
120#930—91no—
121C2×D24—91no—
122C2×D24C3⋊D311yescharacteristic, complemented
123C3×C22⋊C8—91no—
124#832—94no—
125#821—91no—
126#821—94no—
127#821—91no—
128#821—184no—
129(C2×D3)⋊C8—34no—
130C23×C32—61no—
131C23×C32—61no—
132C22×C3⋊D3—64no—
133C22×C3⋊D3—61no—
134C22×C3⋊D3—64no—
135C22×C3⋊D3—124no—
136C22×C3⋊D3—122no—
137C22×C3⋊D3—122no—
138C22×C3⋊D3D1211yescharacteristic
139C22×C3×D3—24no—
140C22×C3×D3—24no—
141C22×C3×D3—34no—
142C2×D32—616no—
143C2×D32—128no—
144C2×D32—128no—
145C32×D4—32no—
146C32×D4—61no—
147C32×D4—61no—
148C32×D4—61no—
149C2×C32×C4—24no—
150C2×C32×C4C22×D314yes—
151C2×C32×C4C22×D311yescharacteristic
152#638—64no—
153#638—122no—
154C2×C3⋊Dic3—61no—
155C2×C3⋊Dic3—64no—
156C2×C3⋊Dic3—124no—
157C2×C3⋊Dic3D1211yescharacteristic
158C4⋊(C3⋊D3)—32no—
159C4⋊(C3⋊D3)—38no—
160C4⋊(C3⋊D3)—68no—
161C4⋊(C3⋊D3)—61no—
162C4⋊(C3⋊D3)—64no—
163C4⋊(C3⋊D3)—124no—
164C4×C3⋊D3—21no—
165C4×C3⋊D3—21no—
166C2×C3×Dic3—34no—
167C3×D12—24no—
168C3×D12—24no—
169C3×D12—24no—
170C3×D12#6718yescomplemented
171C3×C4×D3—64no—
172C3×C4×D3—64no—
173#626—616no—
174#625—128no—
175C32×C8—21no—
176C32×C8—24no—
177C32×C8—44no—
178C3⋊(C3⋊C8)—21no—
179C3×C3⋊C8—64no—
180#447—271noSylow
181C2×C33#12011yescharacteristic
182C2×C33C2×D811yescharacteristic
183C2×C33AGL(1,ℤ/8)11yescharacteristic
184C3⋊(C3⋊D3)—42no—
185C3×C3⋊D3—22no—
186C32×D3—42no—
187C32×D3—81no—
188C23×D3—181no—
189C23×D3—184no—
190C2×C3×D4—34no—
191C2×C3×D4—34no—
192C22×C3×C4—91no—
193C2×#67—184no—
194C2×#67—181no—
195C2×#67—184no—
196D3×D4—188no—
197D3×D4—188no—
198C2×D12—94no—
199C2×D12—91no—
200C2×D12—184no—
201C2×D12—182no—
202C2×D12—181no—
203C2×D12—182no—
204C2×D12C2×C3⋊D311yescharacteristic
205C2×D12C2×C3⋊D311yescharacteristic, complemented
206C2×C4×D3—34no—
207C3×D8—68no—
208C2×C3×C8—24no—
209C2×C3×C8—91no—
210C2×C3×C8D3214yescomplemented
211C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
212#217—184no—
213#213—188no—
214(C2×D3)⋊C4—181no—
215C4⋊Dic3—94no—
216C4⋊Dic3—91no—
217C4⋊Dic3—184no—
218C2×C3⋊C8—34no—
219D24—22no—
220D24—182no—
221C22×C32—24no—
222C22×C32—62no—
223C22×C32—121no—
224C22×C32—121no—
225C22×C32—121no—
226C22×C32—121no—
227C22×C32—121no—
228C22×C32C2×D1211yescharacteristic
229C22×C32D3×D414yes—
230C2×C3⋊D3—21no—
231C2×C3⋊D3—21no—
232C2×C3⋊D3—62no—
233C2×C3⋊D3—68no—
234C2×C3⋊D3—128no—
235C2×C3⋊D3—128no—
236C2×C3⋊D3—121no—
237C2×C3⋊D3—121no—
238C2×C3⋊D3—124no—
239C2×C3⋊D3—124no—
240C2×C3⋊D3—244no—
241C2×C3⋊D3—244no—
242C2×C3⋊D3D2412yescomplemented
243C2×C3×D3—28no—
244C2×C3×D3—38no—
245C2×C3×D3—44no—
246C2×C3×D3—44no—
247C2×C3×D3—44no—
248C2×C3×D3—44no—
249C2×C3×D3—44no—
250C2×C3×D3—64no—
251C2×C3×D3—64no—
252D32—1232no—
253C32×C4—24no—
254C32×C4—24no—
255C32×C4D3×D414yes—
256C32×C4D3×D411yescharacteristic
257C32×C4D3×D411yescharacteristic
258C32×C4C2×#6714yes—
259C3⋊Dic3—21no—
260C3⋊Dic3—121no—
261C3⋊Dic3—124no—
262C3⋊Dic3—244no—
263C3×Dic3—64no—
264C22×D4—271no—
265C2×D8—31no—
266C2×D8—271no—
267#121—271no—
268#102—271no—
269#102—271no—
270C22⋊C8—91no—
271C33#44711yescharacteristic, complemented, Sylow
272C23×C3—64no—
273C23×C3—64no—
274C23×C3—91no—
275C22×D3—21no—
276C22×D3—21no—
277C22×D3—34no—
278C22×D3—184no—
279C22×D3—1816no—
280C22×D3—184no—
281C22×D3—181no—
282C22×D3—364no—
283C22×D3—368no—
284C22×D3—362no—
285C22×D3—362no—
286C22×D3—368no—
287C3×D4—38no—
288C3×D4—64no—
289C3×D4—64no—
290C3×D4—64no—
291C2×C3×C4—24no—
292C2×C3×C4—91no—
293C2×C3×C4—181no—
294C2×C3×C4—181no—
295C2×C3×C4C2×D3214yes—
296C2×C3×C4C22×C3⋊D311yescharacteristic
297#67—1816no—
298#67—368no—
299#67—362no—
300C2×Dic3—34no—
301C2×Dic3—184no—
302C2×Dic3—181no—
303C2×Dic3—364no—
304D12—21no—
305D12—21no—
306D12—21no—
307D12—98no—
308D12—92no—
309D12—188no—
310D12—184no—
311D12—184no—
312D12—181no—
313D12—364no—
314D12#63812yescomplemented
315C4×D3—64no—
316C4×D3—64no—
317C3×C8—24no—
318C3×C8—21no—
319C3×C8—44no—
320C3×C8—181no—
321C3⋊C8—64no—
322C2×C32—24no—
323C2×C32—24no—
324C2×C32—24no—
325C2×C32—122no—
326C2×C32—241no—
327C2×C32#88211yescharacteristic
328C2×C32C2×D2411yescharacteristic
329C2×C32#90811yescharacteristic
330C2×C32D3×D814yes—
331C2×C32#91114yes—
332C2×C32#93714yes—
333C3⋊D3—22no—
334C3⋊D3—122no—
335C3⋊D3—128no—
336C3⋊D3—248no—
337C3×D3—48no—
338C3×D3—68no—
339C3×D3—84no—
340C24—541no—
341C2×D4—31no—
342C2×D4—31no—
343C2×D4—271no—
344C2×D4—541no—
345C2×D4—542no—
346C2×D4—542no—
347C2×D4—541no—
348C22×C4—91no—
349D8—62no—
350D8—542no—
351C2×C8—91no—
352C2×C8D3×C3⋊D311yescharacteristic, complemented
353C4⋊C4—271no—
354C22⋊C4—541no—
355C22×C3—24no—
356C22×C3—68no—
357C22×C3—92no—
358C22×C3—124no—
359C22×C3—124no—
360C22×C3—124no—
361C22×C3—124no—
362C22×C3—124no—
363C22×C3—181no—
364C22×C3—181no—
365C22×C3D3×D1214yes—
366C22×C3D4×C3⋊D311yescharacteristic
367C2×D3—22no—
368C2×D3—38no—
369C2×D3—41no—
370C2×D3—41no—
371C2×D3—41no—
372C2×D3—41no—
373C2×D3—41no—
374C2×D3—64no—
375C2×D3—64no—
376C2×D3—182no—
377C2×D3—188no—
378C2×D3—368no—
379C2×D3—368no—
380C2×D3—3632no—
381C2×D3—364no—
382C2×D3—364no—
383C2×D3—361no—
384C2×D3—361no—
385C2×D3—724no—
386C2×D3—724no—
387C3×C4—24no—
388C3×C4—24no—
389C3×C4—181no—
390C3×C4#380214yes—
391C3×C4D3×#6714yes—
392C3×C4D4×C3⋊D311yescharacteristic
393C3×C4C2×#63811yescharacteristic
394Dic3—64no—
395Dic3—364no—
396Dic3—361no—
397Dic3—724no—
398C32—24no—
399C32#473211yescharacteristic, complemented
400C32#515614yescomplemented
401C23—61no—
402C23—61no—
403C23—91no—
404C23—544no—
405C23—541no—
406C23—1082no—
407C23—1082no—
408D4—32no—
409D4—61no—
410D4—61no—
411D4—61no—
412D4—272no—
413D4—544no—
414D4—541no—
415D4—1082no—
416C2×C4—91no—
417C2×C4—181no—
418C2×C4—181no—
419C2×C4—541no—
420C2×C4C2×D3×C3⋊D311yescharacteristic
421C8—21no—
422C8—181no—
423C2×C3—24no—
424C2×C3—24no—
425C2×C3—24no—
426C2×C3—128no—
427C2×C3—182no—
428C2×C3—244no—
429C2×C3D3×D2414yes—
430C2×C3#6379514yes—
431C2×C3#6392214yes—
432C2×C3D8×C3⋊D311yescharacteristic
433C2×C3#6412011yescharacteristic
434C2×C3#6414611yescharacteristic
435D3—42no—
436D3—68no—
437D3—81no—
438D3—362no—
439D3—368no—
440D3—728no—
441C22—62no—
442C22—92no—
443C22—121no—
444C22—121no—
445C22—121no—
446C22—121no—
447C22—121no—
448C22—181no—
449C22—181no—
450C22—542no—
451C22—1088no—
452C22—1081no—
453C22—1081no—
454C22C3⋊D3×D1211yescharacteristic
455C4—181no—
456C4—1081no—
457C4#8902611yescharacteristic
458C4D3×#63811yescharacteristic
459C3—24no—
460C3#1059094414yescomplemented
461C3#1059285511yescharacteristic, complemented
462C2—122no—
463C2—182no—
464C2—241no—
465C2—1082no—
466C2C3⋊D3×D2411yescharacteristic
467C2#1172332011yescharacteristic
468C2#1172354011yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#447
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×D12C2semidirect
2. C2×C32×D24C2semidirect
3. #11723522C2semidirect
4. C2×#88790C2semidirect
5. #11722191C2semidirect
6. C3×#63647C2semidirect
7. #11722032C2semidirect
14. C2×C32×D12C22semidirect
19. C2×C33×C8C22semidirect
20. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
30. C2×C3×D24D3semidirect
37. #63647D3semidirect
50. C4⋊(C3⋊(C3⋊D3))D4semidirect
57. C32×D12D4semidirect
77. C2×C3×D12C2×D3semidirect
85. C2×C32×C8C2×D3semidirect
86. C2×C32×C8C2×D3semidirect
92. C2×C3⋊(C3⋊C8)C2×D3semidirect
103. C2×C3×C3⋊D3D8semidirect
122. C2×D24C3⋊D3semidirect
170. C3×D12#67semidirect
205. C2×D12C2×C3⋊D3semidirect
210. C2×C3×C8D32semidirect
211. C2×C3×C8C2×C3⋊D3semidirect
242. C2×C3⋊D3D24semidirect
271. C33#447semidirect
314. D12#638semidirect
352. C2×C8D3×C3⋊D3semidirect
399. C32#4732semidirect
400. C32#5156semidirect
460. C3#10590944semidirect
461. C3#10592855semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
C2×C32×D242
#117235222
C2×#887902
#117221912
C3×#636472
#117220322
#105928553
#105909443