Finite Group Databasebeta

← #13160038

Subgroups — #13160038

Order 1728.

1388 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1388 conjugacy classes of subgroups — too many to draw. Only its 170 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×D3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×C3×#3745C211yesmaximal, characteristic, complemented
3#11723515C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5#11722191C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592431—31nomaximal
9#10590944—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
14C2×C32×C4×D3C2211yescharacteristic
15C2×#6292—21no—
16C2×#6290—21no—
17D3×C4⋊(C3⋊D3)—22no—
18D3×C4⋊(C3⋊D3)—22no—
19C2×C33×C8C2211yescharacteristic, complemented
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
21C3×#3745—22no—
22C2×C32×C3⋊C8C2211yescharacteristic, complemented
23#88803—21no—
24#88791—22no—
25C22×C4⋊(C3⋊D3)—31no—
26C2×D3×D12—34no—
27#64139—31no—
28C2×#3745—31no—
29C2×#3745D311yescharacteristic, complemented
30C2×C3×D24—34no—
31#63908—34no—
32C2×#3714—34no—
33C32×C22⋊C8—31no—
34#63648—31no—
35#63648—31no—
36#63648—34no—
37#63648—64no—
38C3×(C2×D3)⋊C8D314yescomplemented
39#63563—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3—21no—
43C22×C32×D3D411yescharacteristic
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
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)D412yescomplemented
55C2×C32×Dic3D411yescharacteristic
56C32×C4×D3—21no—
57C32×C4×D3—21no—
58#6292—24no—
59#6290—42no—
60C33×C8—21no—
61C32×C3⋊C8—21no—
62#5156—91no—
63#4732—94no—
64C23×C3⋊D3—61no—
65C22×D32—64no—
66C22×C32×C4—31no—
67C2×#638—61no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—34no—
70C2×C4⋊(C3⋊D3)—62no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)—62no—
73C2×C4⋊(C3⋊D3)—64no—
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
76C2×C3×D12—34no—
77C2×C3×D12—34no—
78C2×C3×C4×D3C2×D314yes—
79C2×#626—64no—
80C2×#625—64no—
81D3×D12—68no—
82D3×D12—68no—
83C2×C32×C8—24no—
84C2×C32×C8—31no—
85C2×C32×C8C2×D314yescomplemented
86C2×C32×C8C2×D311yescharacteristic, complemented
87#3752—61no—
88C4⋊(C3⋊Dic3)—31no—
89C4⋊(C3⋊Dic3)—34no—
90C4⋊(C3⋊Dic3)—64no—
91#3745—22no—
92#3745—62no—
93C2×C3×C3⋊C8C2×D314yescomplemented
94C3×D24—68no—
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—22no—
102C2×C3×C3⋊D3—41no—
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3—41no—
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3—41no—
107C2×C32×D3—21no—
108C2×C32×D3—21no—
109C2×C32×D3D812yescomplemented
110D3×C3⋊D3—48no—
111C33×C4C2×D411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3⋊(C3⋊Dic3)—41no—
114C32×Dic3—21no—
115C2×D3×D4—91no—
116C22×D12—94no—
117C2×C3×D8—91no—
118#938—91no—
119C2×#213—91no—
120#930—94no—
121C2×D24—34no—
122C2×D24—94no—
123C3×C22⋊C8—34no—
124#832—91no—
125#821—94no—
126#821—94no—
127#821—91no—
128#821—184no—
129(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
130C23×C32—31no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3—21no—
133C22×C3⋊D3—64no—
134C22×C3⋊D3—61no—
135C22×C3⋊D3—64no—
136C22×C3⋊D3—124no—
137C22×C3⋊D3—122no—
138C22×C3⋊D3—122no—
139C22×C3×D3—64no—
140C22×C3×D3—64no—
141C22×C3×D3D1214yes—
142C2×D32—616no—
143C2×D32—128no—
144C2×D32—128no—
145C2×C32×C4—24no—
146C2×C32×C4—31no—
147C2×C32×C4—61no—
148C2×C32×C4—61no—
149C2×C32×C4C22×D314yes—
150C2×C32×C4C22×D311yescharacteristic
151#638—122no—
152C2×C3⋊Dic3—61no—
153C2×C3⋊Dic3—64no—
154C2×C3⋊Dic3—124no—
155C4⋊(C3⋊D3)—21no—
156C4⋊(C3⋊D3)—21no—
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)—32no—
159C4⋊(C3⋊D3)—38no—
160C4⋊(C3⋊D3)—68no—
161C4⋊(C3⋊D3)—64no—
162C4⋊(C3⋊D3)—61no—
163C4⋊(C3⋊D3)—64no—
164C4⋊(C3⋊D3)—124no—
165C4⋊(C3⋊D3)#6712yescomplemented
166C2×C3×Dic3D1214yes—
167C3×D12—38no—
168C3×D12—64no—
169C3×D12—64no—
170C3×D12—64no—
171C3×C4×D3—24no—
172C3×C4×D3—24no—
173#626—616no—
174#625—128no—
175C32×C8—21no—
176C32×C8—24no—
177C32×C8—44no—
178C32×C8—61no—
179C3×C3⋊C8—24no—
180#447—271noSylow
181C2×C33#12011yescharacteristic
182C2×C33C2×D811yescharacteristic
183C2×C33AGL(1,ℤ/8)11yescharacteristic
184C3⋊(C3⋊D3)—42no—
185C3×C3⋊D3—42no—
186C3×C3⋊D3—81no—
187C32×D3—22no—
188C23×D3—184no—
189C23×D3—181no—
190C2×C3×D4—91no—
191C2×C3×D4—91no—
192C22×C3×C4—34no—
193C2×#67—184no—
194C2×#67—181no—
195C2×#67—181no—
196D3×D4—182no—
197D3×D4—182no—
198C2×D12—34no—
199C2×D12—34no—
200C2×D12—94no—
201C2×D12—91no—
202C2×D12—184no—
203C2×D12—188no—
204C2×D12—188no—
205C2×D12—184no—
206C2×C4×D3C2×C3⋊D311yescharacteristic
207C3×D8—182no—
208C2×C3×C8—24no—
209C2×C3×C8—34no—
210C2×C3×C8D3214yescomplemented
211C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
212#217—181no—
213#213—182no—
214(C2×D3)⋊C4—184no—
215C4⋊Dic3—94no—
216C4⋊Dic3—91no—
217C4⋊Dic3—184no—
218C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
219D24—68no—
220D24—188no—
221C22×C32—24no—
222C22×C32—32no—
223C22×C32—61no—
224C22×C32—61no—
225C22×C32C2×D1214yes—
226C22×C32D3×D411yescharacteristic
227C2×C3⋊D3—22no—
228C2×C3⋊D3—41no—
229C2×C3⋊D3—41no—
230C2×C3⋊D3—41no—
231C2×C3⋊D3—41no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3—62no—
234C2×C3⋊D3—68no—
235C2×C3⋊D3—128no—
236C2×C3⋊D3—128no—
237C2×C3⋊D3—121no—
238C2×C3⋊D3—121no—
239C2×C3⋊D3—124no—
240C2×C3⋊D3—124no—
241C2×C3⋊D3—244no—
242C2×C3⋊D3—244no—
243C2×C3×D3—24no—
244C2×C3×D3—24no—
245C2×C3×D3—68no—
246C2×C3×D3—124no—
247C2×C3×D3—124no—
248C2×C3×D3—124no—
249C2×C3×D3—124no—
250C2×C3×D3—124no—
251C2×C3×D3D2418yescomplemented
252D32—1232no—
253C32×C4—24no—
254C32×C4—24no—
255C32×C4—61no—
256C32×C4D3×D414yes—
257C32×C4D3×D414yes—
258C32×C4D3×D411yescharacteristic
259C32×C4C2×#6711yescharacteristic
260C3⋊Dic3—121no—
261C3⋊Dic3—124no—
262C3⋊Dic3—244no—
263C3×Dic3—24no—
264C22×D4—271no—
265C2×D8—91no—
266C2×D8—271no—
267#121—271no—
268#102—271no—
269#102—271no—
270C22⋊C8—31no—
271C33#44711yescharacteristic, complemented, Sylow
272C23×C3—34no—
273C23×C3—181no—
274C23×C3—181no—
275C22×D3—64no—
276C22×D3—64no—
277C22×D3—184no—
278C22×D3—1816no—
279C22×D3—184no—
280C22×D3—181no—
281C22×D3—364no—
282C22×D3—362no—
283C22×D3—368no—
284C22×D3—368no—
285C22×D3—362no—
286C22×D3C4⋊(C3⋊D3)11yescharacteristic
287C3×D4—92no—
288C3×D4—181no—
289C3×D4—181no—
290C3×D4—181no—
291C2×C3×C4—24no—
292C2×C3×C4—34no—
293C2×C3×C4—64no—
294C2×C3×C4—64no—
295C2×C3×C4C2×D3214yes—
296C2×C3×C4C22×C3⋊D311yescharacteristic
297#67—184no—
298#67—368no—
299#67—362no—
300C2×Dic3—184no—
301C2×Dic3—181no—
302C2×Dic3—364no—
303C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
304D12—38no—
305D12—64no—
306D12—64no—
307D12—64no—
308D12—98no—
309D12—92no—
310D12—188no—
311D12—1816no—
312D12—184no—
313D12—181no—
314D12—364no—
315C4×D3—21no—
316C4×D3—21no—
317C3×C8—24no—
318C3×C8—21no—
319C3×C8—44no—
320C3×C8—64no—
321C3⋊C8—21no—
322C2×C32—24no—
323C2×C32—24no—
324C2×C32—24no—
325C2×C32—62no—
326C2×C32#88214yes—
327C2×C32C2×D2414yes—
328C2×C32#90814yes—
329C2×C32D3×D811yescharacteristic
330C2×C32#91111yescharacteristic
331C2×C32#93711yescharacteristic
332C3⋊D3—42no—
333C3⋊D3—81no—
334C3⋊D3—122no—
335C3⋊D3—128no—
336C3⋊D3—248no—
337C3×D3—28no—
338C3×D3—128no—
339C3×D3—244no—
340C24—541no—
341C2×D4—91no—
342C2×D4—91no—
343C2×D4—271no—
344C2×D4—541no—
345C2×D4—542no—
346C2×D4—542no—
347C2×D4—541no—
348C22×C4—31no—
349D8—182no—
350D8—542no—
351C2×C8—31no—
352C2×C8D3×C3⋊D311yescharacteristic, complemented
353C4⋊C4—271no—
354C22⋊C4—541no—
355C22×C3—24no—
356C22×C3—38no—
357C22×C3—64no—
358C22×C3—64no—
359C22×C3—182no—
360C22×C3—361no—
361C22×C3—361no—
362C22×C3—361no—
363C22×C3—361no—
364C22×C3—361no—
365C22×C3D3×D1214yes—
366C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
367C2×D3—21no—
368C2×D3—21no—
369C2×D3—68no—
370C2×D3—124no—
371C2×D3—124no—
372C2×D3—124no—
373C2×D3—124no—
374C2×D3—124no—
375C2×D3—182no—
376C2×D3—188no—
377C2×D3—368no—
378C2×D3—368no—
379C2×D3—3632no—
380C2×D3—364no—
381C2×D3—364no—
382C2×D3—361no—
383C2×D3—361no—
384C2×D3—724no—
385C2×D3—724no—
386C2×D3#374512yescomplemented
387C3×C4—24no—
388C3×C4—24no—
389C3×C4—64no—
390C3×C4#380214yes—
391C3×C4D3×#6714yes—
392C3×C4D4×C3⋊D311yescharacteristic
393C3×C4D4×C3⋊D311yescharacteristic
394Dic3—21no—
395Dic3—364no—
396Dic3—361no—
397Dic3—724no—
398C32—24no—
399C32#473214yescomplemented
400C32#515611yescharacteristic, complemented
401C23—31no—
402C23—181no—
403C23—181no—
404C23—544no—
405C23—541no—
406C23—1082no—
407C23—1082no—
408D4—92no—
409D4—181no—
410D4—181no—
411D4—181no—
412D4—272no—
413D4—544no—
414D4—541no—
415D4—1082no—
416C2×C4—31no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—541no—
420C2×C4C2×D3×C3⋊D311yescharacteristic
421C8—21no—
422C8—61no—
423C2×C3—24no—
424C2×C3—24no—
425C2×C3—24no—
426C2×C3—68no—
427C2×C3—362no—
428C2×C3—721no—
429C2×C3D3×D2414yes—
430C2×C3#6379514yes—
431C2×C3#6392214yes—
432C2×C3#6409111yescharacteristic
433C2×C3C2×#374511yescharacteristic
434C2×C3#6411711yescharacteristic
435D3—22no—
436D3—128no—
437D3—244no—
438D3—362no—
439D3—368no—
440D3—728no—
441C22—32no—
442C22—61no—
443C22—61no—
444C22—182no—
445C22—361no—
446C22—361no—
447C22—361no—
448C22—361no—
449C22—361no—
450C22—542no—
451C22—1088no—
452C22—1081no—
453C22—1081no—
454C22D3×C4⋊(C3⋊D3)11yescharacteristic
455C4—61no—
456C4—1081no—
457C4#8902611yescharacteristic
458C4C3⋊D3×#6711yescharacteristic
459C3—24no—
460C3#1059094414yescomplemented
461C3#1059243111yescharacteristic, complemented
462C2—62no—
463C2—362no—
464C2—721no—
465C2—1082no—
466C2D3×#374511yescharacteristic
467C2#1172331611yescharacteristic
468C2#1172354111yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
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×D3×C4⋊(C3⋊D3)C2semidirect
2. C2×C3×#3745C2semidirect
3. #11723515C2semidirect
4. C2×#88791C2semidirect
5. #11722191C2semidirect
6. C32×(C2×D3)⋊C8C2semidirect
7. #11722034C2semidirect
13. C2×C3×C4⋊(C3⋊D3)C22semidirect
19. C2×C33×C8C22semidirect
20. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
22. C2×C32×C3⋊C8C22semidirect
29. C2×#3745D3semidirect
38. C3×(C2×D3)⋊C8D3semidirect
50. C4⋊(C3⋊(C3⋊D3))D4semidirect
54. C3×C4⋊(C3⋊D3)D4semidirect
75. C2×C4⋊(C3⋊D3)C2×D3semidirect
85. C2×C32×C8C2×D3semidirect
86. C2×C32×C8C2×D3semidirect
93. C2×C3×C3⋊C8C2×D3semidirect
109. C2×C32×D3D8semidirect
129. (C2×D3)⋊C8C3⋊D3semidirect
165. C4⋊(C3⋊D3)#67semidirect
210. C2×C3×C8D32semidirect
211. C2×C3×C8C2×C3⋊D3semidirect
218. C2×C3⋊C8C2×C3⋊D3semidirect
251. C2×C3×D3D24semidirect
271. C33#447semidirect
352. C2×C8D3×C3⋊D3semidirect
386. C2×D3#3745semidirect
399. C32#4732semidirect
400. C32#5156semidirect
460. C3#10590944semidirect
461. C3#10592431semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊(C3⋊D3)2
C2×C3×#37452
#117235152
C2×#887912
#117221912
C32×(C2×D3)⋊C82
#117220342
#105924313
#105909443