Finite Group Databasebeta

← #13175881

Subgroups — #13175881

Order 1728.

1576 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×#64088C211yesmaximal, characteristic, complemented
3C32×#885C211yesmaximal, characteristic
4#11723642C211yesmaximal, characteristic, complemented
5#67⋊(C3⋊Dic3)C211yesmaximal, characteristic
6#11723600C211yesmaximal, characteristic
7#11723596C211yesmaximal, characteristic, complemented
8C4×#6293C211yesmaximal, characteristic, complemented
9#11723500C211yesmaximal, characteristic, complemented
10#11723459C211yesmaximal, characteristic, complemented
11#11723432C211yesmaximal, characteristic, complemented
12#11723432C211yesmaximal, characteristic, complemented
13#11723397C211yesmaximal, characteristic, complemented
14#11723393C211yesmaximal, characteristic, complemented
15#11723389C211yesmaximal, characteristic, complemented
16#10596142—31nomaximal
17#10594145—34nomaximal
18C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C32×#67C2211yescharacteristic
20C2×C32×Dic6C2211yescharacteristic
21C2×#6293C2211yescharacteristic
22C2×#6293C2211yescharacteristic
23C2×D3×C3⋊Dic3C2211yescharacteristic
24C33×C22⋊C4C2211yescharacteristic
25#88864C2211yescharacteristic
26#88857C2211yescharacteristic
27#88857C2211yescharacteristic
28C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
29C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
30C3×#3757C2211yescharacteristic
31C3×#3757C2211yescharacteristic
32C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
33C3×#3750C2211yescharacteristic
34C3×#3750C2211yescharacteristic
35C3×C4×C3⋊Dic3C2211yescharacteristic
36C32×#217C2211yescharacteristic
37C32×(C2×D3)⋊C4C2211yescharacteristic
38C32×(C2×D3)⋊C4C2211yescharacteristic
39C32×C4×Dic3C2211yescharacteristic
40#88807C2211yescharacteristic
41#88807C2211yescharacteristic
42#88806C2211yescharacteristic
43#88806C2211yescharacteristic
44Dic3⋊(C3⋊Dic3)C2211yescharacteristic
45Dic3⋊(C3⋊Dic3)C2211yescharacteristic
46Dic3⋊(C3⋊Dic3)C2211yescharacteristic
47(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
48(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
49(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
50Dic3×C3⋊Dic3C2211yescharacteristic
51Dic3×C3⋊Dic3C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×#124—31no—
54Q8×C3⋊Dic3—31no—
55#64153—31no—
56#64145—31no—
57D4×C3⋊Dic3—31no—
58#64088—31no—
59#64088—31no—
60#64088—31no—
61#64088D311yescharacteristic, complemented
62#64087—31no—
63#64087—31no—
64#64086—31no—
65#64086—31no—
66#64086—34no—
67#64086—64no—
68C4×#634—31no—
69C4×#634—31no—
70C3×#885D314yes—
71C3×#879—34no—
72#63982—34no—
73#67⋊Dic3—34no—
74#63959—34no—
75#63956—34no—
76C4×#627—34no—
77#63899—34no—
78#63877—34no—
79#63862—34no—
80#63862—34no—
81#63843—34no—
82#63839—34no—
83#63837—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⋊Dic3C2311yescharacteristic
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C32×#67—22no—
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101C32×Dic6—22no—
102#6293—22no—
103#6293—22no—
104D3×C3⋊Dic3—22no—
105#5702—91no—
106#5671—94no—
107C2×C32×Q8—31no—
108C2×C32×D4—31no—
109C22×C3⋊Dic3—31no—
110C22×C3⋊Dic3—34no—
111C22×C3⋊Dic3—31no—
112C22×C3⋊Dic3—64no—
113C2×#634—31no—
114C2×#634—31no—
115C2×C3×#67C2×D314yes—
116C2×C3×Dic6C2×D314yes—
117C2×#627—34no—
118C2×#627—34no—
119C2×D3×Dic3—34no—
120C32×C22⋊C4—24no—
121C32×C22⋊C4—31no—
122C32×C22⋊C4—31no—
123C32×C22⋊C4—31no—
124C32×C22⋊C4C2×D314yes—
125C32×C22⋊C4C2×D311yescharacteristic
126C32×C42—31no—
127#3757—31no—
128#3757—34no—
129#3757—31no—
130#3757—31no—
131#3757—31no—
132#3757—64no—
133#3757C2×D311yescharacteristic
134#3757C2×D311yescharacteristic
135C4⋊(C3⋊Dic3)—31no—
136C4⋊(C3⋊Dic3)—31no—
137C4⋊(C3⋊Dic3)—31no—
138C4⋊(C3⋊Dic3)C2×D311yescharacteristic
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—64no—
148#3750—64no—
149#3750C2×D311yescharacteristic
150#3750C2×D311yescharacteristic
151C4×C3⋊Dic3—31no—
152C4×C3⋊Dic3—31no—
153C4×C3⋊Dic3—34no—
154C4×C3⋊Dic3—34no—
155C4×C3⋊Dic3—31no—
156C4×C3⋊Dic3—31no—
157C4×C3⋊Dic3—31no—
158C4×C3⋊Dic3—64no—
159C4×C3⋊Dic3—64no—
160C4×C3⋊Dic3C2×D311yescharacteristic
161C3×#217—34no—
162C3×#217—34no—
163C3×#217C2×D314yes—
164C3×(C2×D3)⋊C4C2×D314yes—
165C3×(C2×D3)⋊C4C2×D314yes—
166C3×C4⋊Dic3—34no—
167C3×Dic3⋊C4—34no—
168C3×Dic3⋊C4—34no—
169C3×C4×Dic3—34no—
170C3×C4×Dic3C2×D314yes—
171#3724—34no—
172#3724—34no—
173Dic3⋊Dic3—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176Dic3⋊Dic3—34no—
177Dic3⋊Dic3—34no—
178(C2×D3)⋊Dic3—34no—
179(C2×D3)⋊Dic3—34no—
180(C2×D3)⋊Dic3—34no—
181Dic32—34no—
182Dic32—34no—
183Dic32—34no—
184C22×C33—21no—
185C22×C33—21no—
186C22×C33—21no—
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)#4112yes—
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×#126—91no—
205C3×#124—34no—
206Q8×Dic3—94no—
207#944—94no—
208#936—94no—
209D4×Dic3—94no—
210#898—91no—
211#898—91no—
212#896—91no—
213#888—91no—
214#887—91no—
215#887—91no—
216#885—91no—
217#885C3⋊D311yescharacteristic
218#67⋊C4—91no—
219#879—34no—
220#879—94no—
221#879—94no—
222#879—94no—
223#879—91no—
224#878—94no—
225#878—94no—
226#878—91no—
227#877—94no—
228#877—94no—
229#877—91no—
230#877—184no—
231#872—91no—
232C4×Dic6—94no—
233C4×Dic6—94no—
234C4×Dic6—91no—
235C23×C32—24no—
236C23×C32—31no—
237C23×C32C22×D314yes—
238C23×C32C22×D311yescharacteristic
239C22×C3×D3C22×D314yes—
240C32×Q8—62no—
241C32×D4—62no—
242C2×C32×C4—24no—
243C2×C32×C4—24no—
244C2×C32×C4—31no—
245C2×C32×C4—31no—
246C2×C32×C4—31no—
247C2×C32×C4C22×D314yes—
248C2×C32×C4C22×D314yes—
249C2×C32×C4C22×D311yescharacteristic
250C2×C32×C4C22×D311yescharacteristic
251C2×C3⋊Dic3—31no—
252C2×C3⋊Dic3—31no—
253C2×C3⋊Dic3—31no—
254C2×C3⋊Dic3—31no—
255C2×C3⋊Dic3—34no—
256C2×C3⋊Dic3—34no—
257C2×C3⋊Dic3—34no—
258C2×C3⋊Dic3—34no—
259C2×C3⋊Dic3—64no—
260C2×C3⋊Dic3—64no—
261C2×C3⋊Dic3—64no—
262C2×C3⋊Dic3—64no—
263C2×C3⋊Dic3—62no—
264C2×C3⋊Dic3—62no—
265C2×C3⋊Dic3—68no—
266C2×C3⋊Dic3—616no—
267C2×C3⋊Dic3C22×D311yescharacteristic
268C2×C3⋊Dic3C22×D311yescharacteristic
269C2×C3⋊Dic3C22×D311yescharacteristic
270C2×C3⋊Dic3C22×D311yescharacteristic
271#634—62no—
272#634—62no—
273C3×#67—28no—
274C2×C3×Dic3—34no—
275C2×C3×Dic3—34no—
276C2×C3×Dic3—34no—
277C2×C3×Dic3—34no—
278C2×C3×Dic3C22×D314yes—
279C2×C3×Dic3C22×D314yes—
280C2×C3×Dic3C22×D314yes—
281C3×Dic6—28no—
282#627—68no—
283#627—68no—
284D3×Dic3—68no—
285#556—271noSylow
286C2×C33—41no—
287C2×C33C2×#4111yescharacteristic
288C2×C33C2×#4111yescharacteristic
289C2×C332−1+411yescharacteristic
290C32×D3—41no—
291C2×C3×Q8—34no—
292C2×C3×D4—34no—
293C2×#67C2×C3⋊D311yescharacteristic
294C22×Dic3—94no—
295C22×Dic3—94no—
296C22×Dic3—91no—
297C22×Dic3—184no—
298C2×C4×D3—91no—
299C2×Dic6—94no—
300C2×Dic6—94no—
301C2×Dic6—91no—
302C2×Dic6—91no—
303C2×Dic6C2×C3⋊D311yescharacteristic
304C3×C4⋊C4—91no—
305C3×C4⋊C4—91no—
306C3×C4⋊C4—91no—
307C3×C22⋊C4—24no—
308C3×C22⋊C4—34no—
309C3×C22⋊C4—34no—
310C3×C22⋊C4—34no—
311C3×C22⋊C4—91no—
312C3×C22⋊C4—91no—
313C3×C22⋊C4D3214yes—
314C3×C22⋊C4C2×C3⋊D311yescharacteristic
315C3×C42—34no—
316C3×C42—91no—
317#217—34no—
318#217—34no—
319#217—94no—
320#217—94no—
321#217—94no—
322#217—94no—
323#217—91no—
324#217—184no—
325#217C2×C3⋊D311yescharacteristic
326(C2×D3)⋊C4—91no—
327(C2×D3)⋊C4—91no—
328(C2×D3)⋊C4—91no—
329(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
330(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
331C4⋊Dic3—34no—
332C4⋊Dic3—94no—
333C4⋊Dic3—94no—
334C4⋊Dic3—94no—
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337Dic3⋊C4—34no—
338Dic3⋊C4—34no—
339Dic3⋊C4—94no—
340Dic3⋊C4—94no—
341Dic3⋊C4—94no—
342Dic3⋊C4—94no—
343Dic3⋊C4—94no—
344Dic3⋊C4—94no—
345Dic3⋊C4—91no—
346Dic3⋊C4—91no—
347Dic3⋊C4—91no—
348Dic3⋊C4—91no—
349Dic3⋊C4—91no—
350Dic3⋊C4—91no—
351Dic3⋊C4—91no—
352Dic3⋊C4—184no—
353Dic3⋊C4—184no—
354C4×Dic3—34no—
355C4×Dic3—94no—
356C4×Dic3—94no—
357C4×Dic3—94no—
358C4×Dic3—94no—
359C4×Dic3—94no—
360C4×Dic3—91no—
361C4×Dic3—91no—
362C4×Dic3—91no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—184no—
366C4×Dic3—184no—
367C4×Dic3C2×C3⋊D311yescharacteristic
368C22×C32—28no—
369C22×C32—24no—
370C22×C32—21no—
371C22×C32—21no—
372C22×C32—21no—
373C22×C32—24no—
374C22×C32—24no—
375C22×C32—24no—
376C22×C32—44no—
377C22×C32—44no—
378C22×C32—61no—
379C22×C32—61no—
380C22×C32—61no—
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—32no—
400C3⋊Dic3—38no—
401C3⋊Dic3—68no—
402C3⋊Dic3—68no—
403C3⋊Dic3—61no—
404C3⋊Dic3—64no—
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—
418C4⋊Q8—271no—
419#126—91no—
420#126—271no—
421#126—271no—
422#126—271no—
423#124—31no—
424#124—271no—
425C22⋊Q8—271no—
426C22⋊Q8—271no—
427C4×Q8—271no—
428C4×Q8—271no—
429C4×Q8—271no—
430C4×D4—271no—
431#117—271no—
432#117—271no—
433C33#55611yescharacteristic, complemented, Sylow
434C23×C3—24no—
435C23×C3—34no—
436C23×C3C2×D3214yes—
437C23×C3C22×C3⋊D311yescharacteristic
438C22×D3C22×C3⋊D311yescharacteristic
439C3×Q8—68no—
440C3×D4—68no—
441C2×C3×C4—24no—
442C2×C3×C4—24no—
443C2×C3×C4—34no—
444C2×C3×C4—34no—
445C2×C3×C4—34no—
446C2×C3×C4—91no—
447C2×C3×C4—91no—
448C2×C3×C4—91no—
449C2×C3×C4—91no—
450C2×C3×C4C2×D3214yes—
451C2×C3×C4C2×D3214yes—
452C2×C3×C4C22×C3⋊D311yescharacteristic
453C2×C3×C4C22×C3⋊D311yescharacteristic
454#67—22no—
455C2×Dic3—34no—
456C2×Dic3—34no—
457C2×Dic3—34no—
458C2×Dic3—34no—
459C2×Dic3—94no—
460C2×Dic3—91no—
461C2×Dic3—94no—
462C2×Dic3—94no—
463C2×Dic3—94no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—184no—
468C2×Dic3—184no—
469C2×Dic3—184no—
470C2×Dic3—184no—
471C2×Dic3—188no—
472C2×Dic3—188no—
473C2×Dic3—182no—
474C2×Dic3—1816no—
475C2×Dic3C22×C3⋊D311yescharacteristic
476C2×Dic3C22×C3⋊D311yescharacteristic
477C2×Dic3C22×C3⋊D311yescharacteristic
478C4×D3—182no—
479Dic6—22no—
480Dic6—182no—
481Dic6—188no—
482Dic6—188no—
483Dic6—182no—
484C2×C32—24no—
485C2×C32—24no—
486C2×C32—24no—
487C2×C32—48no—
488C2×C32—44no—
489C2×C32—41no—
490C2×C32—121no—
491C2×C32C2×#23714yes—
492C2×C32D3×#4114yes—
493C2×C32D3×#4111yescharacteristic
494C2×C32D3×#4111yescharacteristic
495C2×C32#101014yes—
496C2×C32#101011yescharacteristic
497C3×D3—44no—
498C2×Q8—31no—
499C2×Q8—271no—
500C2×Q8—271no—
501C2×D4—31no—
502C22×C4—271no—
503C22×C4—271no—
504C4⋊C4—91no—
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—
516C22⋊C4—31no—
517C22⋊C4—31no—
518C22⋊C4—31no—
519C22⋊C4—91no—
520C22⋊C4—91no—
521C22⋊C4—271no—
522C22⋊C4—271no—
523C22⋊C4—271no—
524C22⋊C4—271no—
525C22⋊C4D3×C3⋊D311yescharacteristic
526C42—31no—
527C42—91no—
528C42—271no—
529C42—271no—
530C42—271no—
531C42—271no—
532C42—271no—
533C22×C3—28no—
534C22×C3—24no—
535C22×C3—24no—
536C22×C3—24no—
537C22×C3—21no—
538C22×C3—21no—
539C22×C3—21no—
540C22×C3—24no—
541C22×C3—44no—
542C22×C3—44no—
543C22×C3—64no—
544C22×C3—64no—
545C22×C3—64no—
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—98no—
571Dic3—92no—
572Dic3—188no—
573Dic3—188no—
574Dic3—184no—
575Dic3—184no—
576Dic3—184no—
577Dic3—181no—
578Dic3—181no—
579Dic3—181no—
580Dic3—364no—
581Dic3—364no—
582Dic3#383012yes—
583C32—24no—
584C32#567114yescomplemented
585C32#570211yescharacteristic, complemented
586C23—31no—
587C23C2×D3×C3⋊D311yescharacteristic
588Q8—62no—
589Q8—542no—
590Q8—542no—
591D4—62no—
592C2×C4—31no—
593C2×C4—31no—
594C2×C4—31no—
595C2×C4—91no—
596C2×C4—91no—
597C2×C4—91no—
598C2×C4—91no—
599C2×C4—271no—
600C2×C4—271no—
601C2×C4—271no—
602C2×C4—271no—
603C2×C4—542no—
604C2×C4—542no—
605C2×C4C2×D3×C3⋊D311yescharacteristic
606C2×C4C2×D3×C3⋊D311yescharacteristic
607C2×C3—24no—
608C2×C3—24no—
609C2×C3—24no—
610C2×C3—48no—
611C2×C3—41no—
612C2×C3—44no—
613C2×C3—124no—
614C2×C3#6430614yes—
615C2×C3#6430914yes—
616C2×C3D3×#23714yes—
617C2×C3C2×#383011yescharacteristic
618C2×C3#41×C3⋊D311yescharacteristic
619C2×C3#6436711yescharacteristic
620D3—41no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—61no—
625C22—61no—
626C22—61no—
627C22C22×D3×C3⋊D311yescharacteristic
628C4—21no—
629C4—21no—
630C4—32no—
631C4—61no—
632C4—61no—
633C4—181no—
634C4—181no—
635C4—181no—
636C4—181no—
637C4—272no—
638C4—541no—
639C4—541no—
640C4—541no—
641C3—24no—
642C3#1059414514yescomplemented
643C3#1059614211yescharacteristic, complemented
644C2—41no—
645C2—121no—
646C2#1172457811yescharacteristic
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#556
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×#64088C2semidirect
4. #11723642C2semidirect
7. #11723596C2semidirect
8. C4×#6293C2semidirect
9. #11723500C2semidirect
10. #11723459C2semidirect
11. #11723432C2semidirect
12. #11723432C2semidirect
13. #11723397C2semidirect
14. #11723393C2semidirect
15. #11723389C2semidirect
61. #64088D3semidirect
433. C33#556semidirect
584. C32#5671semidirect
585. C32#5702semidirect
642. C3#10594145semidirect
643. C3#10596142semidirect

Maximal subgroups

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

subgroupindex
#117239182
C3×#640882
C32×#8852
#117236422
#67⋊(C3⋊Dic3)2
#117236002
#117235962
C4×#62932
#117235002
#117234592
#117234322
#117233972
#117233932
#117233892
#105961423
#105941453