Finite Group Databasebeta

← #13175342

Subgroups — #13175342

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#11723913C211yesmaximal, characteristic, complemented
2C3×#64085C211yesmaximal, characteristic, complemented
3C32×C4×Dic6C211yesmaximal, characteristic, complemented
4#11723502C211yesmaximal, characteristic, complemented
5#11723472C211yesmaximal, characteristic
6#11723464C211yesmaximal, characteristic, complemented
7#11723453C211yesmaximal, characteristic, complemented
8#11723442C211yesmaximal, characteristic, complemented
9#11723433C211yesmaximal, characteristic, complemented
10#11723416C211yesmaximal, characteristic
11#11723411C211yesmaximal, characteristic, complemented
12#11723393C211yesmaximal, characteristic, complemented
13#11723392C211yesmaximal, characteristic
14#11723391C211yesmaximal, characteristic, complemented
15#11723389C211yesmaximal, characteristic, complemented
16#10596039—31nomaximal
17#10593940—34nomaximal
18C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
19C2×C32×Dic6C2211yescharacteristic
20C2×#6293C2211yescharacteristic
21C2×#6293C2211yescharacteristic
22C2×#6292C2211yescharacteristic
23C2×Dic3×C3⋊D3C2211yescharacteristic
24C33×C42C2211yescharacteristic
25#88859C2211yescharacteristic
26#88859C2211yescharacteristic
27#88857C2211yescharacteristic
28#88857C2211yescharacteristic
29C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
30C3×#3752C2211yescharacteristic
31C3×#3752C2211yescharacteristic
32C3×#3752C2211yescharacteristic
33C3×#3750C2211yescharacteristic
34C3×#3750C2211yescharacteristic
35C3×#3750C2211yescharacteristic
36C32×C4⋊Dic3C2211yescharacteristic
37C32×Dic3⋊C4C2211yescharacteristic
38C32×Dic3⋊C4C2211yescharacteristic
39C32×C4×Dic3C2211yescharacteristic
40C32×C4×Dic3C2211yescharacteristic
41#88807C2211yescharacteristic
42#88806C2211yescharacteristic
43#88806C2211yescharacteristic
44Dic3⋊(C3⋊Dic3)C2211yescharacteristic
45#88804C2211yescharacteristic
46#88804C2211yescharacteristic
47#88804C2211yescharacteristic
48#88803C2211yescharacteristic
49#88803C2211yescharacteristic
50Dic3×C3⋊Dic3C2211yescharacteristic
51Dic3×C3⋊Dic3C2211yescharacteristic
52Dic3×C3⋊Dic3C2211yescharacteristic
53C32×C4×Q8—31no—
54#64156—31no—
55#64153—31no—
56#64107—31no—
57#64107—31no—
58#64105—31no—
59#64105—31no—
60#64102—31no—
61#64096—31no—
62#64085—31no—
63#64085D311yescharacteristic, complemented
64#64084—31no—
65#64081—31no—
66#64081—31no—
67#64081—34no—
68#64081—64no—
69C4×#634—31no—
70C3×#876—34no—
71C3×C4×Dic6D314yescomplemented
72#63907—34no—
73#63899—34no—
74#63881—34no—
75#63877—34no—
76#63868—34no—
77#63865—34no—
78#63862—34no—
79#63852—34no—
80#63848—34no—
81#63839—34no—
82#63838—34no—
83#63837—34no—
84C22×C3⋊(C3⋊D3)C2311yescharacteristic
85C22×C3×C3⋊D3C2311yescharacteristic
86C2×C33×C4C2311yescharacteristic
87C2×C33×C4C2311yescharacteristic
88C2×C33×C4C2311yescharacteristic
89C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
90C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
92C4×C3⋊(C3⋊D3)—22no—
93C2×C3×C3⋊Dic3C2311yescharacteristic
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C32×Dic3C2311yescharacteristic
97C2×C32×Dic3C2311yescharacteristic
98C2×C32×Dic3C2311yescharacteristic
99C2×C32×Dic3C2311yescharacteristic
100C32×Dic6—22no—
101#6293—22no—
102#6293—22no—
103#6292—22no—
104Dic3×C3⋊D3—22no—
105#5702—91no—
106#5568—94no—
107C2×C32×Q8—31no—
108C2×C4⋊(C3⋊D3)—31no—
109C2×C4×C3⋊D3—31no—
110C2×C4×C3⋊D3—31no—
111C2×C4×C3⋊D3—34no—
112C2×C4×C3⋊D3—64no—
113C2×#634—31no—
114C2×#634—31no—
115C2×C3×Dic6C2×D314yes—
116C2×#627—34no—
117C2×#627—34no—
118C2×#626—34no—
119C2×D3×Dic3—34no—
120C32×C4⋊C4—31no—
121C32×C4⋊C4—31no—
122C32×C4⋊C4—31no—
123C32×C42—24no—
124C32×C42—31no—
125C32×C42—31no—
126C32×C42C2×D314yes—
127C32×C42C2×D311yescharacteristic
128#3752—31no—
129#3752—31no—
130#3752—31no—
131#3752—31no—
132#3752—34no—
133#3752—34no—
134#3752—31no—
135#3752—31no—
136#3752—31no—
137#3752—64no—
138#3752—64no—
139#3752C2×D311yescharacteristic
140#3752C2×D311yescharacteristic
141#3752C2×D311yescharacteristic
142C4⋊(C3⋊Dic3)—31no—
143#3750—31no—
144#3750—31no—
145#3750—34no—
146#3750—34no—
147#3750—31no—
148#3750—31no—
149#3750—31no—
150#3750—64no—
151#3750—64no—
152#3750C2×D311yescharacteristic
153#3750C2×D311yescharacteristic
154#3750C2×D311yescharacteristic
155C4×C3⋊Dic3—31no—
156C4×C3⋊Dic3—34no—
157C4×C3⋊Dic3—31no—
158C4×C3⋊Dic3—31no—
159C4×C3⋊Dic3—31no—
160C4×C3⋊Dic3—64no—
161C3×(C2×D3)⋊C4—34no—
162C3×(C2×D3)⋊C4—34no—
163C3×(C2×D3)⋊C4—34no—
164C3×C4⋊Dic3C2×D314yes—
165C3×Dic3⋊C4—34no—
166C3×Dic3⋊C4—34no—
167C3×Dic3⋊C4—34no—
168C3×Dic3⋊C4C2×D314yes—
169C3×Dic3⋊C4C2×D314yes—
170C3×C4×Dic3C2×D314yes—
171C3×C4×Dic3C2×D314yes—
172#3724—34no—
173#3724—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176#3722—34no—
177#3722—34no—
178#3722—34no—
179(C2×D3)⋊Dic3—34no—
180(C2×D3)⋊Dic3—34no—
181Dic32—34no—
182Dic32—34no—
183Dic32—34no—
184C22×C33C2411yescharacteristic
185C2×C3⋊(C3⋊D3)—21no—
186C2×C3⋊(C3⋊D3)—21no—
187C2×C3⋊(C3⋊D3)—21no—
188C2×C3×C3⋊D3—21no—
189C2×C3×C3⋊D3—21no—
190C2×C3×C3⋊D3—21no—
191C33×C4—21no—
192C33×C4—21no—
193C33×C4—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—
200C32×Dic3—21no—
201C32×Dic3—21no—
202C32×Dic3—21no—
203C32×Dic3#4112yes—
204C3×#126—91no—
205C3×C4×Q8—34no—
206#947—94no—
207#944—94no—
208#898—94no—
209#898—94no—
210#898—91no—
211#898—91no—
212#896—94no—
213#896—94no—
214#896—91no—
215#893—94no—
216#888—91no—
217#887—94no—
218#887—91no—
219#887—91no—
220#885—91no—
221#885—91no—
222#67⋊C4—91no—
223#879—91no—
224#878—91no—
225#877—91no—
226#876—34no—
227#876—94no—
228#875—94no—
229#872—94no—
230#872—94no—
231#872—91no—
232#872—184no—
233C4×Dic6—94no—
234C4×Dic6C3⋊D311yescharacteristic, complemented
235C22×C3⋊D3—31no—
236C22×C3⋊D3—34no—
237C22×C3⋊D3—64no—
238C22×C3⋊D3C22×D311yescharacteristic
239C22×C3×D3—34no—
240C32×Q8—62no—
241C2×C32×C4—24no—
242C2×C32×C4—24no—
243C2×C32×C4—24no—
244C2×C32×C4—31no—
245C2×C32×C4—31no—
246C2×C32×C4—31no—
247C2×C32×C4—31no—
248C2×C32×C4C22×D314yes—
249C2×C32×C4C22×D314yes—
250C2×C32×C4C22×D314yes—
251C2×C32×C4C22×D311yescharacteristic
252C2×C32×C4C22×D311yescharacteristic
253C2×C32×C4C22×D311yescharacteristic
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—64no—
261C2×C3⋊Dic3—64no—
262C2×C3⋊Dic3—64no—
263C2×C3⋊Dic3C22×D311yescharacteristic
264C2×C3⋊Dic3C22×D311yescharacteristic
265C2×C3⋊Dic3C22×D311yescharacteristic
266C4⋊(C3⋊D3)—62no—
267C4×C3⋊D3—62no—
268C4×C3⋊D3—68no—
269C4×C3⋊D3—62no—
270C4×C3⋊D3—616no—
271#634—62no—
272#634—62no—
273C2×C3×Dic3—34no—
274C2×C3×Dic3—34no—
275C2×C3×Dic3—34no—
276C2×C3×Dic3C22×D314yes—
277C2×C3×Dic3C22×D314yes—
278C2×C3×Dic3C22×D314yes—
279C2×C3×Dic3C22×D314yes—
280C3×Dic6—28no—
281#627—68no—
282#627—68no—
283#626—68no—
284D3×Dic3—68no—
285#556—271noSylow
286C2×C33C2×#4111yescharacteristic
287C2×C33C2×#4111yescharacteristic
288C2×C332−1+411yescharacteristic
289C3⋊(C3⋊D3)—41no—
290C3×C3⋊D3—41no—
291C2×C3×Q8—34no—
292C2×#67—91no—
293C22×Dic3—91no—
294C2×D12—94no—
295C2×C4×D3—94no—
296C2×C4×D3—94no—
297C2×C4×D3—91no—
298C2×C4×D3—184no—
299C2×Dic6—94no—
300C2×Dic6—94no—
301C2×Dic6—91no—
302C2×Dic6—91no—
303C2×Dic6C2×C3⋊D311yescharacteristic
304C3×C4⋊C4—34no—
305C3×C4⋊C4—34no—
306C3×C4⋊C4—34no—
307C3×C4⋊C4—91no—
308C3×C4⋊C4—91no—
309C3×C4⋊C4—91no—
310C3×C22⋊C4—91no—
311C3×C22⋊C4—91no—
312C3×C22⋊C4—91no—
313C3×C42—24no—
314C3×C42—34no—
315C3×C42—34no—
316C3×C42D3214yes—
317C3×C42C2×C3⋊D311yescharacteristic
318#217—91no—
319#217—91no—
320(C2×D3)⋊C4—34no—
321(C2×D3)⋊C4—34no—
322(C2×D3)⋊C4—34no—
323(C2×D3)⋊C4—94no—
324(C2×D3)⋊C4—94no—
325(C2×D3)⋊C4—94no—
326(C2×D3)⋊C4—94no—
327(C2×D3)⋊C4—94no—
328(C2×D3)⋊C4—94no—
329(C2×D3)⋊C4—94no—
330(C2×D3)⋊C4—91no—
331(C2×D3)⋊C4—91no—
332(C2×D3)⋊C4—91no—
333(C2×D3)⋊C4—91no—
334(C2×D3)⋊C4—91no—
335(C2×D3)⋊C4—184no—
336(C2×D3)⋊C4—184no—
337C4⋊Dic3—94no—
338C4⋊Dic3—91no—
339C4⋊Dic3C2×C3⋊D311yescharacteristic
340Dic3⋊C4—34no—
341Dic3⋊C4—34no—
342Dic3⋊C4—34no—
343Dic3⋊C4—94no—
344Dic3⋊C4—94no—
345Dic3⋊C4—94no—
346Dic3⋊C4—94no—
347Dic3⋊C4—94no—
348Dic3⋊C4—91no—
349Dic3⋊C4—91no—
350Dic3⋊C4—91no—
351Dic3⋊C4—91no—
352Dic3⋊C4—91no—
353Dic3⋊C4—184no—
354Dic3⋊C4—184no—
355Dic3⋊C4C2×C3⋊D311yescharacteristic
356Dic3⋊C4C2×C3⋊D311yescharacteristic
357C4×Dic3—94no—
358C4×Dic3—94no—
359C4×Dic3—94no—
360C4×Dic3—94no—
361C4×Dic3—91no—
362C4×Dic3—91no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—184no—
366C4×Dic3C2×C3⋊D311yescharacteristic
367C4×Dic3C2×C3⋊D311yescharacteristic
368C22×C32—24no—
369C22×C32C23×D314yes—
370C22×C32C23×D311yescharacteristic
371C2×C3⋊D3—21no—
372C2×C3⋊D3—21no—
373C2×C3⋊D3—21no—
374C2×C3⋊D3—68no—
375C2×C3⋊D3—64no—
376C2×C3⋊D3—61no—
377C2×C3⋊D3—61no—
378C2×C3⋊D3—61no—
379C2×C3⋊D3—64no—
380C2×C3⋊D3—64no—
381C2×C3⋊D3—124no—
382C2×C3⋊D3—124no—
383C2×C3×D3—64no—
384C2×C3×D3—64no—
385C2×C3×D3—64no—
386C32×C4—28no—
387C32×C4—24no—
388C32×C4—21no—
389C32×C4—21no—
390C32×C4—21no—
391C32×C4—24no—
392C32×C4—24no—
393C32×C4—32no—
394C32×C4—44no—
395C32×C4—44no—
396C32×C4—61no—
397C32×C4—61no—
398C32×C4—61no—
399C3⋊Dic3—21no—
400C3⋊Dic3—21no—
401C3⋊Dic3—21no—
402C3⋊Dic3—32no—
403C3⋊Dic3—38no—
404C3⋊Dic3—68no—
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—24no—
414C3×Dic3—64no—
415C3×Dic3—64no—
416C3×Dic3—64no—
417C3×Dic3#23518yes—
418C4⋊Q8—271no—
419#126—91no—
420#126—271no—
421#126—271no—
422#126—271no—
423#124—271no—
424#124—271no—
425C22⋊Q8—271no—
426C22⋊Q8—271no—
427C4×Q8—31no—
428C4×Q8—271no—
429C4×Q8—271no—
430C4×D4—271no—
431#117—271no—
432#117—271no—
433C33#55611yescharacteristic, complemented, Sylow
434C23×C3—91no—
435C22×D3—34no—
436C22×D3—94no—
437C22×D3—91no—
438C22×D3—184no—
439C3×Q8—68no—
440C2×C3×C4—24no—
441C2×C3×C4—24no—
442C2×C3×C4—24no—
443C2×C3×C4—34no—
444C2×C3×C4—34no—
445C2×C3×C4—34no—
446C2×C3×C4—34no—
447C2×C3×C4—91no—
448C2×C3×C4—91no—
449C2×C3×C4—91no—
450C2×C3×C4C2×D3214yes—
451C2×C3×C4C2×D3214yes—
452C2×C3×C4C2×D3214yes—
453C2×C3×C4C22×C3⋊D311yescharacteristic
454C2×C3×C4C22×C3⋊D311yescharacteristic
455C2×C3×C4C22×C3⋊D311yescharacteristic
456#67—182no—
457C2×Dic3—34no—
458C2×Dic3—34no—
459C2×Dic3—34no—
460C2×Dic3—94no—
461C2×Dic3—91no—
462C2×Dic3—94no—
463C2×Dic3—94no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—184no—
467C2×Dic3—184no—
468C2×Dic3—184no—
469C2×Dic3—182no—
470C2×Dic3C22×C3⋊D311yescharacteristic
471C2×Dic3C22×C3⋊D311yescharacteristic
472C2×Dic3C22×C3⋊D311yescharacteristic
473C2×Dic3C22×C3⋊D311yescharacteristic
474D12—188no—
475C4×D3—188no—
476C4×D3—182no—
477C4×D3—188no—
478C4×D3—1816no—
479Dic6—22no—
480Dic6—182no—
481Dic6—188no—
482Dic6—188no—
483Dic6—182no—
484C2×C32—24no—
485C2×C32—24no—
486C2×C32—24no—
487C2×C32C2×#23514yes—
488C2×C32#100714yes—
489C2×C32D3×#4114yes—
490C2×C32D3×#4111yescharacteristic
491C2×C32D3×#4111yescharacteristic
492C2×C32#101011yescharacteristic
493C3⋊D3—41no—
494C3⋊D3—128no—
495C3⋊D3—121no—
496C3⋊D3—124no—
497C3×D3—124no—
498C2×Q8—31no—
499C2×Q8—271no—
500C2×Q8—271no—
501C2×D4—271no—
502C22×C4—271no—
503C22×C4—271no—
504C4⋊C4—31no—
505C4⋊C4—31no—
506C4⋊C4—31no—
507C4⋊C4—91no—
508C4⋊C4—91no—
509C4⋊C4—91no—
510C4⋊C4—271no—
511C4⋊C4—271no—
512C4⋊C4—271no—
513C4⋊C4—271no—
514C4⋊C4—271no—
515C4⋊C4—271no—
516C22⋊C4—91no—
517C22⋊C4—91no—
518C22⋊C4—91no—
519C22⋊C4—271no—
520C22⋊C4—271no—
521C22⋊C4—271no—
522C22⋊C4—271no—
523C22⋊C4—271no—
524C22⋊C4—271no—
525C22⋊C4—271no—
526C42—31no—
527C42—31no—
528C42—271no—
529C42—271no—
530C42—271no—
531C42—271no—
532C42D3×C3⋊D311yescharacteristic
533C22×C3—24no—
534C22×C3—181no—
535C22×C3—181no—
536C22×C3—181no—
537C22×C3C22×D3214yes—
538C22×C3C23×C3⋊D311yescharacteristic
539C2×D3—64no—
540C2×D3—64no—
541C2×D3—64no—
542C2×D3—188no—
543C2×D3—184no—
544C2×D3—184no—
545C2×D3—184no—
546C2×D3—181no—
547C2×D3—181no—
548C2×D3—181no—
549C2×D3—364no—
550C2×D3—364no—
551C3×C4—28no—
552C3×C4—24no—
553C3×C4—24no—
554C3×C4—24no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—38no—
559C3×C4—44no—
560C3×C4—44no—
561C3×C4—64no—
562C3×C4—64no—
563C3×C4—64no—
564C3×C4—181no—
565C3×C4—181no—
566C3×C4—181no—
567Dic3—21no—
568Dic3—21no—
569Dic3—21no—
570Dic3—64no—
571Dic3—64no—
572Dic3—64no—
573Dic3—98no—
574Dic3—92no—
575Dic3—188no—
576Dic3—184no—
577Dic3—184no—
578Dic3—181no—
579Dic3—181no—
580Dic3—364no—
581Dic3—364no—
582Dic3#382812yes—
583C32—24no—
584C32#556814yescomplemented
585C32#570211yescharacteristic, complemented
586C23—91no—
587C23—271no—
588Q8—62no—
589Q8—542no—
590Q8—542no—
591D4—542no—
592C2×C4—31no—
593C2×C4—31no—
594C2×C4—31no—
595C2×C4—31no—
596C2×C4—91no—
597C2×C4—91no—
598C2×C4—91no—
599C2×C4—271no—
600C2×C4—271no—
601C2×C4—271no—
602C2×C4—542no—
603C2×C4—542no—
604C2×C4C2×D3×C3⋊D311yescharacteristic
605C2×C4C2×D3×C3⋊D311yescharacteristic
606C2×C4C2×D3×C3⋊D311yescharacteristic
607C2×C3—24no—
608C2×C3—24no—
609C2×C3—24no—
610C2×C3—361no—
611C2×C3#6429714yes—
612C2×C3D3×#23514yes—
613C2×C3#6430614yes—
614C2×C3C2×#382811yescharacteristic
615C2×C3#6436411yescharacteristic
616C2×C3#41×C3⋊D311yescharacteristic
617D3—124no—
618D3—368no—
619D3—364no—
620D3—361no—
621C22—181no—
622C22—181no—
623C22—181no—
624C22—541no—
625C22—541no—
626C22—541no—
627C22C22×D3×C3⋊D311yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—32no—
632C4—61no—
633C4—61no—
634C4—61no—
635C4—181no—
636C4—181no—
637C4—181no—
638C4—272no—
639C4—541no—
640C4—541no—
641C3—24no—
642C3#1059394014yescomplemented
643C3#1059603911yescharacteristic, complemented
644C2—361no—
645C2—1081no—
646C2#1172456611yescharacteristic
647C2D3×#382811yescharacteristic
648C2#1172457811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radicalC2×#6292C22
hypercenterC42D3×C3⋊D3
hypocenterC33#556
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723913C2semidirect
2. C3×#64085C2semidirect
3. C32×C4×Dic6C2semidirect
4. #11723502C2semidirect
6. #11723464C2semidirect
7. #11723453C2semidirect
8. #11723442C2semidirect
9. #11723433C2semidirect
11. #11723411C2semidirect
12. #11723393C2semidirect
14. #11723391C2semidirect
15. #11723389C2semidirect
63. #64085D3semidirect
71. C3×C4×Dic6D3semidirect
234. C4×Dic6C3⋊D3semidirect
433. C33#556semidirect
584. C32#5568semidirect
585. C32#5702semidirect
642. C3#10593940semidirect
643. C3#10596039semidirect

Maximal subgroups

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

subgroupindex
#117239132
C3×#640852
C32×C4×Dic62
#117235022
#117234722
#117234642
#117234532
#117234422
#117234332
#117234162
#117234112
#117233932
#117233922
#117233912
#117233892
#105960393
#105939403