Finite Group Databasebeta

← #13175880

Subgroups — #13175880

Order 1728.

1588 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723918C211yesmaximal, characteristic, complemented
2C3×#64086C211yesmaximal, characteristic, complemented
3C32×#67⋊C4C211yesmaximal, characteristic
4#11723640C211yesmaximal, characteristic, complemented
5C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6#11723603C211yesmaximal, characteristic
7#11723602C211yesmaximal, characteristic
8C4×#6293C211yesmaximal, characteristic, complemented
9#11723448C211yesmaximal, characteristic, complemented
10#11723445C211yesmaximal, characteristic, complemented
11#11723444C211yesmaximal, characteristic, complemented
12#11723432C211yesmaximal, characteristic, complemented
13#11723428C211yesmaximal, characteristic, complemented
14#11723423C211yesmaximal, characteristic, complemented
15#11723415C211yesmaximal, characteristic, complemented
16#10596012—31nomaximal
17#10594039—34nomaximal
18C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C22×C3×C3⋊Dic3C2211yescharacteristic
20C2×C32×#67C2211yescharacteristic
21C22×C32×Dic3C2211yescharacteristic
22C2×C32×C4×D3C2211yescharacteristic
23C2×#6293C2211yescharacteristic
24C33×C22⋊C4C2211yescharacteristic
25#88864C2211yescharacteristic
26#88857C2211yescharacteristic
27#88857C2211yescharacteristic
28C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
29C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
30C3×#3757C2211yescharacteristic
31C3×#3750C2211yescharacteristic
32C3×#3750C2211yescharacteristic
33C3×C4×C3⋊Dic3C2211yescharacteristic
34C3×C4×C3⋊Dic3C2211yescharacteristic
35C32×(C2×D3)⋊C4C2211yescharacteristic
36C32×Dic3⋊C4C2211yescharacteristic
37C32×C4×Dic3C2211yescharacteristic
38#88807C2211yescharacteristic
39#88807C2211yescharacteristic
40#88807C2211yescharacteristic
41#88806C2211yescharacteristic
42#88806C2211yescharacteristic
43#88806C2211yescharacteristic
44Dic3⋊(C3⋊Dic3)C2211yescharacteristic
45Dic3⋊(C3⋊Dic3)C2211yescharacteristic
46Dic3⋊(C3⋊Dic3)C2211yescharacteristic
47Dic3⋊(C3⋊Dic3)C2211yescharacteristic
48Dic3⋊(C3⋊Dic3)C2211yescharacteristic
49(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
50(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
51(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
52(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
53C32×C4×D4—31no—
54#64144—31no—
55#64144—31no—
56#64135—31no—
57C2×C4⋊(C3⋊Dic3)—31no—
58#64133—31no—
59#64133—31no—
60#64098—31no—
61#64098—31no—
62#64088—31no—
63#64088—31no—
64#64086—31no—
65#64086—34no—
66#64086—64no—
67#64086D311yescharacteristic, complemented
68#64079—31no—
69C4×#634—31no—
70C3×#67⋊C4D314yes—
71C3×#877—34no—
72C2×Dic3⋊Dic3—34no—
73#63964—34no—
74#63962—34no—
75#63961—34no—
76C4×#627—34no—
77#63870—34no—
78#63867—34no—
79#63867—34no—
80#63862—34no—
81#63859—34no—
82#63855—34no—
83#63851—34no—
84C23×C33C2311yescharacteristic
85C22×C32×D3C2311yescharacteristic
86C2×C33×C4C2311yescharacteristic
87C2×C33×C4C2311yescharacteristic
88C2×C3⋊(C3⋊Dic3)—22no—
89C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
92C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
93C2×C3×C3⋊Dic3—22no—
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C2×C3×C3⋊Dic3C2311yescharacteristic
98C32×#67—22no—
99C2×C32×Dic3—22no—
100C2×C32×Dic3C2311yescharacteristic
101C2×C32×Dic3C2311yescharacteristic
102C2×C32×Dic3C2311yescharacteristic
103C32×C4×D3—22no—
104#6293—22no—
105#5541—94no—
106#5531—91no—
107C2×C32×D4—31no—
108C22×C32×C4—31no—
109C22×C32×C4—31no—
110C22×C3⋊Dic3—31no—
111C22×C3⋊Dic3—34no—
112C22×C3⋊Dic3—64no—
113C22×C3⋊Dic3C2×D311yescharacteristic
114C2×#634—31no—
115C2×C3×#67C2×D314yes—
116C22×C3×Dic3—34no—
117C22×C3×Dic3C2×D314yes—
118C2×C3×C4×D3C2×D314yes—
119C2×#627—34no—
120C32×C4⋊C4—31no—
121C32×C22⋊C4—24no—
122C32×C22⋊C4—31no—
123C32×C22⋊C4C2×D314yes—
124C32×C22⋊C4C2×D311yescharacteristic
125C32×C42—31no—
126#3757—31no—
127#3757—34no—
128#3757—31no—
129#3757—31no—
130#3757—31no—
131#3757—31no—
132#3757—64no—
133#3757C2×D311yescharacteristic
134C4⋊(C3⋊Dic3)—31no—
135C4⋊(C3⋊Dic3)—31no—
136C4⋊(C3⋊Dic3)—31no—
137C4⋊(C3⋊Dic3)—31no—
138C4⋊(C3⋊Dic3)—31no—
139#3750—31no—
140#3750—31no—
141#3750—34no—
142#3750—34no—
143#3750—31no—
144#3750—31no—
145#3750—31no—
146#3750—31no—
147#3750—31no—
148#3750—31no—
149#3750—64no—
150#3750—64no—
151#3750C2×D311yescharacteristic
152#3750C2×D311yescharacteristic
153C4×C3⋊Dic3—31no—
154C4×C3⋊Dic3—31no—
155C4×C3⋊Dic3—34no—
156C4×C3⋊Dic3—34no—
157C4×C3⋊Dic3—64no—
158C4×C3⋊Dic3—64no—
159C4×C3⋊Dic3C2×D311yescharacteristic
160C4×C3⋊Dic3C2×D311yescharacteristic
161C3×#217—34no—
162C3×(C2×D3)⋊C4C2×D314yes—
163C3×Dic3⋊C4—34no—
164C3×Dic3⋊C4—34no—
165C3×Dic3⋊C4C2×D314yes—
166C3×C4×Dic3—34no—
167C3×C4×Dic3—34no—
168C3×C4×Dic3C2×D314yes—
169#3724—34no—
170#3724—34no—
171#3724—34no—
172Dic3⋊Dic3—34no—
173Dic3⋊Dic3—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176Dic3⋊Dic3—34no—
177Dic3⋊Dic3—34no—
178Dic3⋊Dic3—34no—
179Dic3⋊Dic3—34no—
180(C2×D3)⋊Dic3—34no—
181(C2×D3)⋊Dic3—34no—
182(C2×D3)⋊Dic3—34no—
183(C2×D3)⋊Dic3—34no—
184C22×C33—21no—
185C22×C33—21no—
186C22×C33#4112yes—
187C22×C33C2411yescharacteristic
188C2×C32×D3—21no—
189C2×C32×D3—21no—
190C2×C32×D3—21no—
191C33×C4—21no—
192C33×C4—21no—
193C3⋊(C3⋊Dic3)—21no—
194C3⋊(C3⋊Dic3)—21no—
195C3⋊(C3⋊Dic3)—21no—
196C3⋊(C3⋊Dic3)—21no—
197C3×C3⋊Dic3—21no—
198C3×C3⋊Dic3—21no—
199C3×C3⋊Dic3—21no—
200C3×C3⋊Dic3—21no—
201C32×Dic3—21no—
202C32×Dic3—21no—
203C32×Dic3#4112yes—
204C3×C4×D4—34no—
205C3×#117—91no—
206#935—94no—
207#935—94no—
208#929—91no—
209#926—94no—
210C2×C4⋊Dic3—94no—
211#924—94no—
212#924—94no—
213C2×Dic3⋊C4—91no—
214#898—91no—
215#896—91no—
216#889—94no—
217#889—94no—
218#889—91no—
219#889—91no—
220#886—91no—
221#67⋊C4C3⋊D311yescharacteristic
222#879—94no—
223#879—94no—
224#878—91no—
225#877—34no—
226#877—94no—
227#877—91no—
228#877—184no—
229#876—91no—
230#872—91no—
231#870—94no—
232#870—91no—
233C4×Dic6—94no—
234C4×Dic6—91no—
235C23×C32—24no—
236C23×C32—31no—
237C23×C32C22×D314yes—
238C23×C32C22×D311yescharacteristic
239C22×C3×D3C22×D314yes—
240C32×D4—62no—
241C2×C32×C4—24no—
242C2×C32×C4—24no—
243C2×C32×C4—31no—
244C2×C32×C4—31no—
245C2×C32×C4—31no—
246C2×C32×C4—62no—
247C2×C32×C4—62no—
248C2×C32×C4C22×D314yes—
249C2×C32×C4C22×D314yes—
250C2×C32×C4C22×D311yescharacteristic
251C2×C32×C4C22×D311yescharacteristic
252C2×C3⋊Dic3—22no—
253C2×C3⋊Dic3—31no—
254C2×C3⋊Dic3—31no—
255C2×C3⋊Dic3—31no—
256C2×C3⋊Dic3—31no—
257C2×C3⋊Dic3—34no—
258C2×C3⋊Dic3—34no—
259C2×C3⋊Dic3—34no—
260C2×C3⋊Dic3—34no—
261C2×C3⋊Dic3—64no—
262C2×C3⋊Dic3—64no—
263C2×C3⋊Dic3—64no—
264C2×C3⋊Dic3—64no—
265C2×C3⋊Dic3—616no—
266C2×C3⋊Dic3—68no—
267C2×C3⋊Dic3—62no—
268C2×C3⋊Dic3C22×D311yescharacteristic
269C2×C3⋊Dic3C22×D311yescharacteristic
270C2×C3⋊Dic3C22×D311yescharacteristic
271C2×C3⋊Dic3C22×D311yescharacteristic
272#634—62no—
273C3×#67—28no—
274C2×C3×Dic3—28no—
275C2×C3×Dic3—34no—
276C2×C3×Dic3—34no—
277C2×C3×Dic3—34no—
278C2×C3×Dic3—34no—
279C2×C3×Dic3—68no—
280C2×C3×Dic3C22×D314yes—
281C2×C3×Dic3C22×D314yes—
282C2×C3×Dic3C22×D314yes—
283C3×C4×D3—28no—
284#627—68no—
285#552—271noSylow
286C2×C33—22no—
287C2×C33C2×#4111yescharacteristic
288C2×C33C2×#4111yescharacteristic
289C2×C332−1+411yescharacteristic
290C32×D3—41no—
291C2×C3×D4—34no—
292C22×C3×C4—34no—
293C22×C3×C4—34no—
294C22×C3×C4—91no—
295C2×#67C2×C3⋊D311yescharacteristic
296C22×Dic3—34no—
297C22×Dic3—94no—
298C22×Dic3—91no—
299C22×Dic3—184no—
300C22×Dic3C2×C3⋊D311yescharacteristic
301C2×C4×D3C2×C3⋊D311yescharacteristic
302C2×Dic6—94no—
303C2×Dic6—91no—
304C3×C4⋊C4—34no—
305C3×C4⋊C4—91no—
306C3×C4⋊C4—91no—
307C3×C22⋊C4—24no—
308C3×C22⋊C4—34no—
309C3×C22⋊C4—91no—
310C3×C22⋊C4D3214yes—
311C3×C22⋊C4C2×C3⋊D311yescharacteristic
312C3×C42—34no—
313C3×C42—91no—
314C3×C42—91no—
315#217—34no—
316#217—94no—
317#217—94no—
318#217—94no—
319#217—94no—
320#217—94no—
321#217—91no—
322#217—184no—
323(C2×D3)⋊C4—91no—
324(C2×D3)⋊C4—91no—
325(C2×D3)⋊C4—91no—
326(C2×D3)⋊C4—91no—
327(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
328C4⋊Dic3—94no—
329C4⋊Dic3—94no—
330C4⋊Dic3—94no—
331C4⋊Dic3—94no—
332C4⋊Dic3—94no—
333C4⋊Dic3—91no—
334C4⋊Dic3—91no—
335C4⋊Dic3—91no—
336Dic3⋊C4—34no—
337Dic3⋊C4—34no—
338Dic3⋊C4—94no—
339Dic3⋊C4—94no—
340Dic3⋊C4—94no—
341Dic3⋊C4—94no—
342Dic3⋊C4—94no—
343Dic3⋊C4—94no—
344Dic3⋊C4—94no—
345Dic3⋊C4—94no—
346Dic3⋊C4—91no—
347Dic3⋊C4—91no—
348Dic3⋊C4—91no—
349Dic3⋊C4—91no—
350Dic3⋊C4—91no—
351Dic3⋊C4—91no—
352Dic3⋊C4—91no—
353Dic3⋊C4—91no—
354Dic3⋊C4—91no—
355Dic3⋊C4—91no—
356Dic3⋊C4—184no—
357Dic3⋊C4—184no—
358Dic3⋊C4C2×C3⋊D311yescharacteristic
359C4×Dic3—34no—
360C4×Dic3—34no—
361C4×Dic3—94no—
362C4×Dic3—94no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—184no—
366C4×Dic3—184no—
367C4×Dic3C2×C3⋊D311yescharacteristic
368C22×C32—21no—
369C22×C32—21no—
370C22×C32—24no—
371C22×C32—28no—
372C22×C32—24no—
373C22×C32—24no—
374C22×C32—44no—
375C22×C32—44no—
376C22×C32—61no—
377C22×C32—61no—
378C22×C32—61no—
379C22×C32#23518yes—
380C22×C32#23512yes—
381C22×C32C23×D314yes—
382C22×C32C23×D311yescharacteristic
383C2×C3×D3—24no—
384C2×C3×D3—24no—
385C2×C3×D3—24no—
386C32×C4—21no—
387C32×C4—21no—
388C32×C4—24no—
389C32×C4—24no—
390C32×C4—32no—
391C32×C4—44no—
392C32×C4—44no—
393C32×C4—61no—
394C32×C4—61no—
395C3⋊Dic3—21no—
396C3⋊Dic3—21no—
397C3⋊Dic3—21no—
398C3⋊Dic3—21no—
399C3⋊Dic3—68no—
400C3⋊Dic3—68no—
401C3⋊Dic3—64no—
402C3⋊Dic3—64no—
403C3⋊Dic3—61no—
404C3⋊Dic3—61no—
405C3⋊Dic3—61no—
406C3⋊Dic3—61no—
407C3⋊Dic3—64no—
408C3⋊Dic3—64no—
409C3⋊Dic3—124no—
410C3⋊Dic3—124no—
411C3×Dic3—24no—
412C3×Dic3—24no—
413C3×Dic3—64no—
414C3×Dic3—64no—
415C3×Dic3—64no—
416C3×Dic3—64no—
417C3×Dic3#23718yes—
418#126—271no—
419#126—271no—
420#125—271no—
421#125—271no—
422#125—271no—
423#123—271no—
424#123—271no—
425C22⋊Q8—271no—
426C22⋊Q8—271no—
427C4×Q8—271no—
428C4×D4—31no—
429#117—91no—
430#117—271no—
431#117—271no—
432C2×C4⋊C4—271no—
433C33#55211yescharacteristic, complemented, Sylow
434C23×C3—24no—
435C23×C3—34no—
436C23×C3C2×D3214yes—
437C23×C3C22×C3⋊D311yescharacteristic
438C22×D3C22×C3⋊D311yescharacteristic
439C3×D4—68no—
440C2×C3×C4—24no—
441C2×C3×C4—24no—
442C2×C3×C4—34no—
443C2×C3×C4—34no—
444C2×C3×C4—34no—
445C2×C3×C4—68no—
446C2×C3×C4—68no—
447C2×C3×C4—91no—
448C2×C3×C4—91no—
449C2×C3×C4—91no—
450C2×C3×C4—91no—
451C2×C3×C4—182no—
452C2×C3×C4C2×D3214yes—
453C2×C3×C4C2×D3214yes—
454C2×C3×C4C22×C3⋊D311yescharacteristic
455C2×C3×C4C22×C3⋊D311yescharacteristic
456#67—22no—
457C2×Dic3—22no—
458C2×Dic3—34no—
459C2×Dic3—34no—
460C2×Dic3—34no—
461C2×Dic3—34no—
462C2×Dic3—68no—
463C2×Dic3—94no—
464C2×Dic3—94no—
465C2×Dic3—94no—
466C2×Dic3—94no—
467C2×Dic3—91no—
468C2×Dic3—91no—
469C2×Dic3—91no—
470C2×Dic3—91no—
471C2×Dic3—184no—
472C2×Dic3—184no—
473C2×Dic3—184no—
474C2×Dic3—184no—
475C2×Dic3—1816no—
476C2×Dic3—188no—
477C2×Dic3—182no—
478C2×Dic3C22×C3⋊D311yescharacteristic
479C2×Dic3C22×C3⋊D311yescharacteristic
480C2×Dic3C22×C3⋊D311yescharacteristic
481C4×D3—22no—
482Dic6—188no—
483Dic6—182no—
484C2×C32—22no—
485C2×C32—24no—
486C2×C32—24no—
487C2×C32—24no—
488C2×C32—28no—
489C2×C32—48no—
490C2×C32—121no—
491C2×C32C2×#23514yes—
492C2×C32C2×#23511yescharacteristic
493C2×C32C2×#23714yes—
494C2×C32D3×#4111yescharacteristic
495C2×C32#101014yes—
496C2×C32#101011yescharacteristic
497C3×D3—44no—
498C2×Q8—271no—
499C2×D4—31no—
500C22×C4—31no—
501C22×C4—31no—
502C22×C4—91no—
503C22×C4—271no—
504C4⋊C4—31no—
505C4⋊C4—91no—
506C4⋊C4—91no—
507C4⋊C4—271no—
508C4⋊C4—271no—
509C4⋊C4—271no—
510C4⋊C4—271no—
511C4⋊C4—271no—
512C4⋊C4—271no—
513C4⋊C4—271no—
514C4⋊C4—271no—
515C4⋊C4—271no—
516C4⋊C4—271no—
517C4⋊C4—271no—
518C4⋊C4—271no—
519C4⋊C4—271no—
520C22⋊C4—31no—
521C22⋊C4—91no—
522C22⋊C4—271no—
523C22⋊C4—271no—
524C22⋊C4—271no—
525C22⋊C4—271no—
526C22⋊C4—271no—
527C22⋊C4D3×C3⋊D311yescharacteristic
528C42—31no—
529C42—91no—
530C42—91no—
531C42—271no—
532C42—271no—
533C22×C3—24no—
534C22×C3—24no—
535C22×C3—21no—
536C22×C3—21no—
537C22×C3—24no—
538C22×C3—28no—
539C22×C3—44no—
540C22×C3—44no—
541C22×C3—64no—
542C22×C3—64no—
543C22×C3—64no—
544C22×C3#379818yes—
545C22×C3#382812yes—
546C22×C3C22×D3214yes—
547C22×C3C23×C3⋊D311yescharacteristic
548C2×D3—21no—
549C2×D3—21no—
550C2×D3—21no—
551C3×C4—24no—
552C3×C4—24no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—38no—
556C3×C4—44no—
557C3×C4—44no—
558C3×C4—64no—
559C3×C4—64no—
560C3×C4—181no—
561C3×C4—181no—
562C3×C4—181no—
563C3×C4—181no—
564Dic3—21no—
565Dic3—21no—
566Dic3—64no—
567Dic3—64no—
568Dic3—64no—
569Dic3—64no—
570Dic3—188no—
571Dic3—188no—
572Dic3—184no—
573Dic3—184no—
574Dic3—184no—
575Dic3—184no—
576Dic3—181no—
577Dic3—181no—
578Dic3—181no—
579Dic3—181no—
580Dic3—364no—
581Dic3—364no—
582Dic3#383012yes—
583C32—24no—
584C32#553111yescharacteristic, complemented
585C32#554114yescomplemented
586C23—31no—
587C23C2×D3×C3⋊D311yescharacteristic
588Q8—542no—
589D4—62no—
590C2×C4—31no—
591C2×C4—31no—
592C2×C4—31no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—91no—
596C2×C4—91no—
597C2×C4—91no—
598C2×C4—91no—
599C2×C4—182no—
600C2×C4—271no—
601C2×C4—271no—
602C2×C4—271no—
603C2×C4—271no—
604C2×C4—542no—
605C2×C4C2×D3×C3⋊D311yescharacteristic
606C2×C4C2×D3×C3⋊D311yescharacteristic
607C2×C3—22no—
608C2×C3—28no—
609C2×C3—24no—
610C2×C3—24no—
611C2×C3—24no—
612C2×C3—48no—
613C2×C3—124no—
614C2×C3C2×#379814yes—
615C2×C3#6430914yes—
616C2×C3D3×#23714yes—
617C2×C3C2×#382811yescharacteristic
618C2×C3C2×#383011yescharacteristic
619C2×C3#6436711yescharacteristic
620D3—41no—
621C22—21no—
622C22—21no—
623C22—61no—
624C22—61no—
625C22—61no—
626C22#8902012yes—
627C22C22×D3×C3⋊D311yescharacteristic
628C4—21no—
629C4—21no—
630C4—32no—
631C4—61no—
632C4—61no—
633C4—181no—
634C4—181no—
635C4—181no—
636C4—181no—
637C4—541no—
638C4—541no—
639C4—541no—
640C4—541no—
641C3—24no—
642C3#1059403914yescomplemented
643C3#1059601211yescharacteristic, complemented
644C2—22no—
645C2—121no—
646C2C2×#8902011yescharacteristic
647C2#1172458211yescharacteristic
648C2D3×#383011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×C32×#67C22
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#552
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723918C2semidirect
2. C3×#64086C2semidirect
4. #11723640C2semidirect
5. C2×Dic3⋊(C3⋊Dic3)C2semidirect
8. C4×#6293C2semidirect
9. #11723448C2semidirect
10. #11723445C2semidirect
11. #11723444C2semidirect
12. #11723432C2semidirect
13. #11723428C2semidirect
14. #11723423C2semidirect
15. #11723415C2semidirect
67. #64086D3semidirect
433. C33#552semidirect
584. C32#5531semidirect
585. C32#5541semidirect
642. C3#10594039semidirect
643. C3#10596012semidirect

Maximal subgroups

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

subgroupindex
#117239182
C3×#640862
C32×#67⋊C42
#117236402
C2×Dic3⋊(C3⋊Dic3)2
#117236032
#117236022
C4×#62932
#117234482
#117234452
#117234442
#117234322
#117234282
#117234232
#117234152
#105960123
#105940393