Finite Group Databasebeta

← #13160458

Subgroups — #13160458

Order 1728.

1508 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(C3⋊(C3⋊Dic3))C211yesmaximal, characteristic, complemented
2C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic
4C2×(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic
5C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
6C3×#63658C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592650—31nomaximal
9#10591316—34nomaximal
10C23×C32×D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C4⋊(C3⋊(C3⋊Dic3))—22no—
18C32×(C2×D3)⋊C4C414yescomplemented
19(C2×D3)⋊(C3⋊Dic3)—22no—
20(C2×D3)⋊(C3⋊Dic3)—22no—
21Dic3×C3⋊Dic3—22no—
22C2×C32×C22⋊C4—31no—
23C2×#3757—31no—
24C2×#3757—31no—
25C2×C4⋊(C3⋊Dic3)—31no—
26C2×C4⋊(C3⋊Dic3)—34no—
27C2×C4⋊(C3⋊Dic3)—64no—
28C2×C4×C3⋊Dic3—31no—
29C2×C3×(C2×D3)⋊C4D314yes—
30C2×(C2×D3)⋊Dic3—34no—
31C2×(C2×D3)⋊Dic3—34no—
32C2×Dic32—34no—
33#63658—31no—
34#63658D311yescharacteristic, complemented
35C3×C3⋊#95—34no—
36#63579—34no—
37C23×C33C2311yescharacteristic
38C22×C32×D3—21no—
39C22×C32×D3—21no—
40C22×C32×D3—21no—
41C22×C32×D3—21no—
42C22×C32×D3—21no—
43C22×C32×D3—21no—
44C22×C32×D3C2×C412yes—
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C33×C4C2×C412yes—
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3⋊(C3⋊Dic3)—21no—
51C2×C3⋊(C3⋊Dic3)—21no—
52C2×C3⋊(C3⋊Dic3)—21no—
53C2×C3⋊(C3⋊Dic3)D412yes—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C2×C3×C3⋊Dic3—21no—
58C2×C3×C3⋊Dic3—21no—
59C2×C3×C3⋊Dic3D412yes—
60C2×C32×Dic3—21no—
61C2×C32×Dic3—21no—
62C2×C32×Dic3C2×C412yes—
63#4951—94no—
64#4670—91no—
65C24×C32—31no—
66C23×C3×D3C2×D314yes—
67C22×C32×C4—24no—
68C22×C32×C4—31no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D314yes—
71C22×C3⋊Dic3—31no—
72C22×C3⋊Dic3—34no—
73C22×C3⋊Dic3—31no—
74C22×C3⋊Dic3—34no—
75C22×C3⋊Dic3—64no—
76C22×C3⋊Dic3—64no—
77C22×C3⋊Dic3C2×D311yescharacteristic
78C22×C3⋊Dic3C2×D311yescharacteristic
79C22×C3×Dic3—34no—
80C22×C3×Dic3—34no—
81C22×C3×Dic3C2×D314yes—
82C32×C22⋊C4—34no—
83#3757—62no—
84#3757—62no—
85C4⋊(C3⋊Dic3)—62no—
86C4⋊(C3⋊Dic3)—68no—
87C4⋊(C3⋊Dic3)—616no—
88C4×C3⋊Dic3—62no—
89C3×(C2×D3)⋊C4Dic3116yescomplemented
90(C2×D3)⋊Dic3—68no—
91(C2×D3)⋊Dic3—68no—
92Dic32—68no—
93C22×C33C22×C411yescharacteristic
94C22×C33C2×D411yescharacteristic
95C22×C33C2×D411yescharacteristic
96C22×C33#4111yescharacteristic
97C22×C33#4111yescharacteristic
98C22×C33#4111yescharacteristic
99C22×C33#4111yescharacteristic
100C2×C32×D3—22no—
101C2×C32×D3—22no—
102C2×C32×D3—22no—
103C2×C32×D3—41no—
104C2×C32×D3—41no—
105C2×C32×D3—41no—
106C2×C32×D3—41no—
107C33×C4—22no—
108C3⋊(C3⋊Dic3)—22no—
109C3⋊(C3⋊Dic3)—41no—
110C3×C3⋊Dic3—22no—
111C3×C3⋊Dic3—41no—
112C32×Dic3—22no—
113C2×C3×C22⋊C4—34no—
114C2×#217—94no—
115C2×#217—94no—
116C2×(C2×D3)⋊C4—91no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4C3⋊D311yescharacteristic
119C2×C4⋊Dic3—94no—
120C2×C4⋊Dic3—91no—
121C2×C4⋊Dic3—184no—
122C2×C4×Dic3—94no—
123C2×C4×Dic3—91no—
124C3×#95—91no—
125C3⋊#95—34no—
126C3⋊#95—94no—
127C3⋊#95—91no—
128C23×C32—24no—
129C23×C32—32no—
130C23×C32—61no—
131C23×C32—61no—
132C23×C32—61no—
133C23×C32—61no—
134C23×C32—61no—
135C23×C32—61no—
136C23×C32C22×D314yes—
137C23×C32C22×D311yescharacteristic
138C22×C3×D3—24no—
139C22×C3×D3—24no—
140C22×C3×D3—24no—
141C22×C3×D3—24no—
142C22×C3×D3—24no—
143C22×C3×D3—24no—
144C22×C3×D3C2×Dic318yes—
145C2×C32×C4—28no—
146C2×C32×C4—28no—
147C2×C32×C4—24no—
148C2×C32×C4—21no—
149C2×C32×C4—24no—
150C2×C32×C4—21no—
151C2×C32×C4—28no—
152C2×C32×C4—32no—
153C2×C32×C4—61no—
154C2×C32×C4—61no—
155C2×C32×C4C4×D312yes—
156C2×C32×C4C2×Dic318yes—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—32no—
163C2×C3⋊Dic3—38no—
164C2×C3⋊Dic3—68no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—68no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—64no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—64no—
173C2×C3⋊Dic3—64no—
174C2×C3⋊Dic3—64no—
175C2×C3⋊Dic3—64no—
176C2×C3⋊Dic3—124no—
177C2×C3⋊Dic3—124no—
178C2×C3⋊Dic3—124no—
179C2×C3⋊Dic3—124no—
180C2×C3⋊Dic3D1212yes—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—38no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3—64no—
186C2×C3×Dic3—64no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3C2×Dic318yes—
190#388—271noSylow
191C2×C33#11711yescharacteristic
192C2×C33C4×D411yescharacteristic
193C2×C33C4×D411yescharacteristic
194C2×C33#12111yescharacteristic
195C2×C33#12311yescharacteristic
196C2×C33#12411yescharacteristic
197C2×C33#12611yescharacteristic
198C32×D3—42no—
199C24×C3—34no—
200C23×D3C2×C3⋊D311yescharacteristic
201C22×C3×C4—24no—
202C22×C3×C4—34no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C22×C3×C4D3214yes—
206C22×C3×C4C2×C3⋊D311yescharacteristic
207C22×Dic3—34no—
208C22×Dic3—34no—
209C22×Dic3—94no—
210C22×Dic3—91no—
211C22×Dic3—94no—
212C22×Dic3—91no—
213C22×Dic3—184no—
214C22×Dic3—184no—
215C22×Dic3C2×C3⋊D311yescharacteristic
216C3×C22⋊C4—316no—
217#217—188no—
218#217—188no—
219(C2×D3)⋊C4—182no—
220(C2×D3)⋊C4—182no—
221(C2×D3)⋊C4C3⋊Dic314yescomplemented
222C4⋊Dic3—188no—
223C4⋊Dic3—182no—
224C4⋊Dic3—1816no—
225C4×Dic3—182no—
226C4×Dic3—188no—
227C22×C32—24no—
228C22×C32—24no—
229C22×C32—24no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—24no—
233C22×C32—24no—
234C22×C32—62no—
235C22×C32—62no—
236C22×C32—62no—
237C22×C32—121no—
238C22×C32—121no—
239C22×C32—121no—
240C22×C32—121no—
241C22×C32C2×C4×D311yescharacteristic
242C22×C32C2×D1211yescharacteristic
243C22×C32#23514yes—
244C22×C32#23514yes—
245C22×C32#23511yescharacteristic
246C22×C32D3×D414yes—
247C22×C32D3×D414yes—
248C22×C32D3×D411yescharacteristic
249C22×C32#23714yes—
250C22×C32#23714yes—
251C22×C32#23711yescharacteristic
252C22×C32#23711yescharacteristic
253C22×C32#23911yescharacteristic
254C22×C32C22×Dic314yes—
255C2×C3×D3—28no—
256C2×C3×D3—28no—
257C2×C3×D3—28no—
258C2×C3×D3—44no—
259C2×C3×D3—44no—
260C2×C3×D3—44no—
261C2×C3×D3—44no—
262C32×C4—216no—
263C32×C4—22no—
264C32×C4—28no—
265C32×C4—62no—
266C3⋊Dic3—22no—
267C3⋊Dic3—41no—
268C3⋊Dic3—62no—
269C3⋊Dic3—68no—
270C3⋊Dic3—128no—
271C3⋊Dic3—128no—
272C3⋊Dic3—124no—
273C3⋊Dic3—121no—
274C3×Dic3—28no—
275C3×Dic3—68no—
276C3×Dic3—124no—
277C2×C4⋊C4—271no—
278C2×C22⋊C4—31no—
279C2×C22⋊C4—271no—
280C2×C22⋊C4—271no—
281C2×C42—271no—
282#95—91no—
283#95—271no—
284C33#38811yescharacteristic, complemented, Sylow
285C23×C3—24no—
286C23×C3—38no—
287C23×C3—64no—
288C23×C3—64no—
289C23×C3—64no—
290C23×C3—64no—
291C23×C3—64no—
292C23×C3—64no—
293C23×C3C2×D3214yes—
294C23×C3C22×C3⋊D311yescharacteristic
295C22×D3—21no—
296C22×D3—21no—
297C22×D3—21no—
298C22×D3—21no—
299C22×D3—21no—
300C22×D3—21no—
301C22×D3C2×C3⋊Dic312yes—
302C2×C3×C4—28no—
303C2×C3×C4—28no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—28no—
309C2×C3×C4—38no—
310C2×C3×C4—64no—
311C2×C3×C4—64no—
312C2×C3×C4—92no—
313C2×C3×C4—181no—
314C2×C3×C4—181no—
315C2×C3×C4—181no—
316C2×C3×C4—181no—
317C2×C3×C4—181no—
318C2×C3×C4D3×Dic318yes—
319C2×C3×C4C2×C3⋊Dic312yes—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—38no—
323C2×Dic3—64no—
324C2×Dic3—64no—
325C2×Dic3—64no—
326C2×Dic3—64no—
327C2×Dic3—64no—
328C2×Dic3—98no—
329C2×Dic3—92no—
330C2×Dic3—188no—
331C2×Dic3—184no—
332C2×Dic3—184no—
333C2×Dic3—181no—
334C2×Dic3—181no—
335C2×Dic3—188no—
336C2×Dic3—184no—
337C2×Dic3—184no—
338C2×Dic3—184no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—181no—
342C2×Dic3—364no—
343C2×Dic3—364no—
344C2×Dic3—364no—
345C2×Dic3—364no—
346C2×Dic3C2×C3⋊Dic312yes—
347C2×C32—24no—
348C2×C32—24no—
349C2×C32—24no—
350C2×C32—24no—
351C2×C32—24no—
352C2×C32—24no—
353C2×C32—24no—
354C2×C32—122no—
355C2×C32C4×D1211yescharacteristic
356C2×C32#87914yes—
357C2×C32#67⋊C411yescharacteristic
358C2×C32#88314yes—
359C2×C32#88414yes—
360C2×C32#88514yes—
361C2×C32#88511yescharacteristic
362C2×C32#88611yescharacteristic
363C2×C32#89211yescharacteristic
364C2×C32#89511yescharacteristic
365C2×C32#89811yescharacteristic
366C2×C32#92614yes—
367C2×C32D4×Dic314yes—
368C2×C32D4×Dic314yes—
369C3×D3—48no—
370C24—31no—
371C22×C4—31no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—271no—
375C22×C4—271no—
376C22×C4D3×C3⋊D311yescharacteristic
377C4⋊C4—542no—
378C22⋊C4—34no—
379C22⋊C4—542no—
380C22⋊C4—542no—
381C42—542no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—24no—
385C22×C3—24no—
386C22×C3—24no—
387C22×C3—24no—
388C22×C3—24no—
389C22×C3—68no—
390C22×C3—68no—
391C22×C3—68no—
392C22×C3—124no—
393C22×C3—124no—
394C22×C3—124no—
395C22×C3—124no—
396C22×C3#379514yes—
397C22×C3#379514yes—
398C22×C3#379914yes—
399C22×C3D3×D1214yes—
400C22×C3C2×D3×Dic314yes—
401C22×C3#380514yes—
402C22×C3#381114yes—
403C22×C3#382811yescharacteristic
404C22×C3#382811yescharacteristic
405C22×C3D4×C3⋊D311yescharacteristic
406C22×C3D4×C3⋊D311yescharacteristic
407C22×C3#383011yescharacteristic
408C22×C3#383011yescharacteristic
409C22×C3C22×C3⋊Dic311yescharacteristic
410C2×D3—22no—
411C2×D3—22no—
412C2×D3—22no—
413C2×D3—41no—
414C2×D3—41no—
415C2×D3—41no—
416C2×D3—41no—
417C3×C4—216no—
418C3×C4—22no—
419C3×C4—28no—
420C3×C4—68no—
421C3×C4—182no—
422C3×C4—361no—
423Dic3—22no—
424Dic3—68no—
425Dic3—124no—
426Dic3—182no—
427Dic3—188no—
428Dic3—368no—
429Dic3—368no—
430Dic3—361no—
431Dic3—364no—
432C32—24no—
433C32#467011yescharacteristic, complemented
434C32#495114yescomplemented
435C23—32no—
436C23—61no—
437C23—61no—
438C23—61no—
439C23—61no—
440C23—61no—
441C23—61no—
442C23C2×D3×C3⋊D311yescharacteristic
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—32no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—92no—
449C2×C4—181no—
450C2×C4—181no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—272no—
455C2×C4—541no—
456C2×C4—541no—
457C2×C4—541no—
458C2×C4—541no—
459C2×C4—541no—
460C2×C4D3×C3⋊Dic312yes—
461C2×C3—24no—
462C2×C3—24no—
463C2×C3—24no—
464C2×C3—24no—
465C2×C3—24no—
466C2×C3—24no—
467C2×C3—24no—
468C2×C3—128no—
469C2×C3#6384114yes—
470C2×C3#6386114yes—
471C2×C3#6389114yes—
472C2×C3Dic3×D1214yes—
473C2×C3#6391014yes—
474C2×C3#6391314yes—
475C2×C3#67⋊Dic314yes—
476C2×C3#6408811yescharacteristic
477C2×C3#6409211yescharacteristic
478C2×C3#6409311yescharacteristic
479C2×C3#6409411yescharacteristic
480C2×C3#6413511yescharacteristic
481C2×C3D4×C3⋊Dic311yescharacteristic
482C2×C3D4×C3⋊Dic311yescharacteristic
483D3—42no—
484C22—62no—
485C22—62no—
486C22—62no—
487C22—121no—
488C22—121no—
489C22—121no—
490C22—121no—
491C22#8901411yescharacteristic
492C22#8902111yescharacteristic
493C22#8902211yescharacteristic
494C22C3⋊D3×D1211yescharacteristic
495C22C2×D3×C3⋊Dic311yescharacteristic
496C22#8902911yescharacteristic
497C22#8903911yescharacteristic
498C4—22no—
499C4—62no—
500C4—182no—
501C4—361no—
502C4—542no—
503C4—1081no—
504C3—24no—
505C3#1059131614yescomplemented
506C3#1059265011yescharacteristic, complemented
507C2—122no—
508C2#1172342411yescharacteristic
509C2#1172343011yescharacteristic
510C2D12×C3⋊Dic311yescharacteristic
511C2#1172352311yescharacteristic
512C2#1172352511yescharacteristic
513C2#1172352811yescharacteristic
514C2#67⋊(C3⋊Dic3)11yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
FittingC22×C33×C4C22
socle, cosocleC23×C33C23
prime radicalC23×C32×D3C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#388
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(C3⋊(C3⋊Dic3))C2semidirect
5. C2×Dic3×C3⋊Dic3C2semidirect
6. C3×#63658C2semidirect
7. #11722061C2semidirect
18. C32×(C2×D3)⋊C4C4semidirect
34. #63658D3semidirect
89. C3×(C2×D3)⋊C4Dic3semidirect
221. (C2×D3)⋊C4C3⋊Dic3semidirect
284. C33#388semidirect
433. C32#4670semidirect
434. C32#4951semidirect
505. C3#10591316semidirect
506. C3#10592650semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊(C3⋊Dic3))2
C2×C32×(C2×D3)⋊C42
C2×(C2×D3)⋊(C3⋊Dic3)2
C2×Dic3×C3⋊Dic32
C3×#636582
#117220612
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