Finite Group Databasebeta

← #13175886

Subgroups — #13175886

Order 1728.

1568 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723919C211yesmaximal, characteristic, complemented
2C3×#64088C211yesmaximal, characteristic, complemented
3C32×#885C211yesmaximal, characteristic
4#67⋊(C3⋊Dic3)C211yesmaximal, characteristic
5#11723600C211yesmaximal, characteristic
6#11723598C211yesmaximal, characteristic, complemented
7#11723597C211yesmaximal, characteristic, complemented
8#11723496C211yesmaximal, characteristic, complemented
9#11723459C211yesmaximal, characteristic, complemented
10#11723430C211yesmaximal, characteristic, complemented
11#11723427C211yesmaximal, characteristic, complemented
12#11723399C211yesmaximal, characteristic, complemented
13Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
14#11723393C211yesmaximal, characteristic, complemented
15#11723389C211yesmaximal, characteristic, complemented
16#10596142—31nomaximal
17#10594147—34nomaximal
18C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×#6308C2211yescharacteristic
20C2×C32×#67C2211yescharacteristic
21C2×C32×Dic6C2211yescharacteristic
22C2×#6293C2211yescharacteristic
23C2×D3×C3⋊Dic3C2211yescharacteristic
24C33×C22⋊C4C2211yescharacteristic
25#88864C2211yescharacteristic
26C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
27#88857C2211yescharacteristic
28#88857C2211yescharacteristic
29C3×#3757C2211yescharacteristic
30C3×#3757C2211yescharacteristic
31C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
32C3×#3750C2211yescharacteristic
33C3×#3750C2211yescharacteristic
34C3×C4×C3⋊Dic3C2211yescharacteristic
35C32×#217C2211yescharacteristic
36C32×(C2×D3)⋊C4C2211yescharacteristic
37C32×(C2×D3)⋊C4C2211yescharacteristic
38C32×C4×Dic3C2211yescharacteristic
39#88807C2211yescharacteristic
40#88806C2211yescharacteristic
41#88806C2211yescharacteristic
42#88806C2211yescharacteristic
43Dic3⋊(C3⋊Dic3)C2211yescharacteristic
44Dic3⋊(C3⋊Dic3)C2211yescharacteristic
45(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
46(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
47(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
48Dic3×C3⋊Dic3C2211yescharacteristic
49Dic3×C3⋊Dic3C2211yescharacteristic
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#64087—34no—
65#64087—64no—
66#64086—31no—
67#64086—31no—
68C4×#634—31no—
69C4×#634—31no—
70C3×#885D314yes—
71C3×#879—34no—
72#67⋊Dic3—34no—
73#63959—34no—
74#63957—34no—
75#63956—34no—
76#63896—34no—
77#63877—34no—
78#63861—34no—
79#63858—34no—
80#63844—34no—
81Dic3×Dic6—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
93#6308—22no—
94C2×C3×C3⋊Dic3C2311yescharacteristic
95C2×C3×C3⋊Dic3C2311yescharacteristic
96C2×C3×C3⋊Dic3C2311yescharacteristic
97C2×C3×C3⋊Dic3C2311yescharacteristic
98C32×#67—22no—
99C2×C32×Dic3C2311yescharacteristic
100C2×C32×Dic3C2311yescharacteristic
101C2×C32×Dic3C2311yescharacteristic
102C32×Dic6—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—34no—
115C2×#634—31no—
116C2×#634—64no—
117C2×C3×#67C2×D314yes—
118C2×C3×Dic6C2×D314yes—
119C2×#627—34no—
120C2×D3×Dic3—34no—
121C32×C22⋊C4—24no—
122C32×C22⋊C4—31no—
123C32×C22⋊C4—31no—
124C32×C22⋊C4—31no—
125C32×C22⋊C4C2×D314yes—
126C32×C22⋊C4C2×D311yescharacteristic
127C32×C42—31no—
128#3757—31no—
129#3757—34no—
130#3757—31no—
131#3757—31no—
132#3757—31no—
133#3757—64no—
134#3757C2×D311yescharacteristic
135#3757C2×D311yescharacteristic
136C4⋊(C3⋊Dic3)—31no—
137C4⋊(C3⋊Dic3)—34no—
138C4⋊(C3⋊Dic3)—31no—
139C4⋊(C3⋊Dic3)—31no—
140C4⋊(C3⋊Dic3)—64no—
141C4⋊(C3⋊Dic3)C2×D311yescharacteristic
142#3750—31no—
143#3750—31no—
144#3750—34no—
145#3750—34no—
146#3750—31no—
147#3750—31no—
148#3750—31no—
149#3750—31no—
150#3750—64no—
151#3750—64no—
152#3750C2×D311yescharacteristic
153#3750C2×D311yescharacteristic
154C4×C3⋊Dic3—31no—
155C4×C3⋊Dic3—31no—
156C4×C3⋊Dic3—31no—
157C4×C3⋊Dic3—31no—
158C4×C3⋊Dic3—31no—
159C4×C3⋊Dic3C2×D311yescharacteristic
160C3×#217—34no—
161C3×#217—34no—
162C3×#217C2×D314yes—
163C3×(C2×D3)⋊C4C2×D314yes—
164C3×(C2×D3)⋊C4C2×D314yes—
165C3×C4⋊Dic3—34no—
166C3×Dic3⋊C4—34no—
167C3×Dic3⋊C4—34no—
168C3×C4×Dic3—34no—
169C3×C4×Dic3C2×D314yes—
170#3724—34no—
171#3724—34no—
172#3724—34no—
173Dic3⋊Dic3—34no—
174Dic3⋊Dic3—34no—
175Dic3⋊Dic3—34no—
176(C2×D3)⋊Dic3—34no—
177(C2×D3)⋊Dic3—34no—
178(C2×D3)⋊Dic3—34no—
179Dic32—34no—
180Dic32—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#878—184no—
228#877—94no—
229#877—94no—
230#877—91no—
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—128no—
267C2×C3⋊Dic3C22×D311yescharacteristic
268C2×C3⋊Dic3C22×D311yescharacteristic
269C2×C3⋊Dic3C22×D311yescharacteristic
270C2×C3⋊Dic3C22×D311yescharacteristic
271#634—62no—
272#634—62no—
273#634—68no—
274#634—128no—
275C3×#67—28no—
276C2×C3×Dic3—34no—
277C2×C3×Dic3—34no—
278C2×C3×Dic3—34no—
279C2×C3×Dic3—34no—
280C2×C3×Dic3C22×D314yes—
281C2×C3×Dic3C22×D314yes—
282C2×C3×Dic3C22×D314yes—
283C3×Dic6—28no—
284#627—68no—
285D3×Dic3—68no—
286#556—271noSylow
287C2×C33—41no—
288C2×C33C2×#4111yescharacteristic
289C2×C33C2×#4111yescharacteristic
290C2×C332−1+411yescharacteristic
291C32×D3—41no—
292C2×C3×Q8—34no—
293C2×C3×D4—34no—
294C2×#67C2×C3⋊D311yescharacteristic
295C22×Dic3—94no—
296C22×Dic3—94no—
297C22×Dic3—91no—
298C22×Dic3—184no—
299C2×C4×D3—91no—
300C2×Dic6—94no—
301C2×Dic6—94no—
302C2×Dic6—91no—
303C2×Dic6—91no—
304C2×Dic6—184no—
305C2×Dic6C2×C3⋊D311yescharacteristic
306C3×C4⋊C4—91no—
307C3×C4⋊C4—91no—
308C3×C4⋊C4—91no—
309C3×C22⋊C4—24no—
310C3×C22⋊C4—34no—
311C3×C22⋊C4—34no—
312C3×C22⋊C4—34no—
313C3×C22⋊C4—91no—
314C3×C22⋊C4—91no—
315C3×C22⋊C4D3214yes—
316C3×C22⋊C4C2×C3⋊D311yescharacteristic
317C3×C42—34no—
318C3×C42—91no—
319#217—34no—
320#217—34no—
321#217—94no—
322#217—94no—
323#217—94no—
324#217—94no—
325#217—91no—
326#217—184no—
327#217C2×C3⋊D311yescharacteristic
328(C2×D3)⋊C4—91no—
329(C2×D3)⋊C4—91no—
330(C2×D3)⋊C4—91no—
331(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
332(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
333C4⋊Dic3—34no—
334C4⋊Dic3—94no—
335C4⋊Dic3—94no—
336C4⋊Dic3—94no—
337C4⋊Dic3—91no—
338C4⋊Dic3—91no—
339C4⋊Dic3—184no—
340Dic3⋊C4—34no—
341Dic3⋊C4—34no—
342Dic3⋊C4—94no—
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—91no—
354Dic3⋊C4—91no—
355Dic3⋊C4—184no—
356Dic3⋊C4—184no—
357C4×Dic3—34no—
358C4×Dic3—94no—
359C4×Dic3—94no—
360C4×Dic3—94no—
361C4×Dic3—94no—
362C4×Dic3—94no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—91no—
366C4×Dic3—91no—
367C4×Dic3—91no—
368C4×Dic3C2×C3⋊D311yescharacteristic
369C22×C32—28no—
370C22×C32—21no—
371C22×C32—21no—
372C22×C32—24no—
373C22×C32—21no—
374C22×C32—24no—
375C22×C32—24no—
376C22×C32—24no—
377C22×C32—44no—
378C22×C32—44no—
379C22×C32—61no—
380C22×C32—61no—
381C22×C32—61no—
382C22×C32C23×D314yes—
383C22×C32C23×D311yescharacteristic
384C2×C3×D3—24no—
385C2×C3×D3—24no—
386C2×C3×D3—24no—
387C32×C4—21no—
388C32×C4—21no—
389C32×C4—24no—
390C32×C4—24no—
391C32×C4—32no—
392C32×C4—44no—
393C32×C4—44no—
394C32×C4—61no—
395C32×C4—61no—
396C3⋊Dic3—21no—
397C3⋊Dic3—21no—
398C3⋊Dic3—21no—
399C3⋊Dic3—21no—
400C3⋊Dic3—32no—
401C3⋊Dic3—38no—
402C3⋊Dic3—68no—
403C3⋊Dic3—61no—
404C3⋊Dic3—61no—
405C3⋊Dic3—61no—
406C3⋊Dic3—64no—
407C3⋊Dic3—64no—
408C3⋊Dic3—64no—
409C3⋊Dic3—124no—
410C3⋊Dic3—124no—
411C3⋊Dic3—124no—
412C3×Dic3—24no—
413C3×Dic3—24no—
414C3×Dic3—64no—
415C3×Dic3—64no—
416C3×Dic3—64no—
417C3×Dic3—64no—
418C3×Dic3#23718yes—
419C4⋊Q8—271no—
420#126—91no—
421#126—271no—
422#126—271no—
423#126—271no—
424#124—31no—
425#124—271no—
426C22⋊Q8—271no—
427C22⋊Q8—271no—
428C4×Q8—271no—
429C4×Q8—271no—
430C4×Q8—271no—
431C4×D4—271no—
432#117—271no—
433#117—271no—
434C33#55611yescharacteristic, complemented, Sylow
435C23×C3—24no—
436C23×C3—34no—
437C23×C3C2×D3214yes—
438C23×C3C22×C3⋊D311yescharacteristic
439C22×D3C22×C3⋊D311yescharacteristic
440C3×Q8—68no—
441C3×D4—68no—
442C2×C3×C4—24no—
443C2×C3×C4—24no—
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×C4—91no—
451C2×C3×C4C2×D3214yes—
452C2×C3×C4C2×D3214yes—
453C2×C3×C4C22×C3⋊D311yescharacteristic
454C2×C3×C4C22×C3⋊D311yescharacteristic
455#67—22no—
456C2×Dic3—34no—
457C2×Dic3—34no—
458C2×Dic3—34no—
459C2×Dic3—34no—
460C2×Dic3—94no—
461C2×Dic3—91no—
462C2×Dic3—94no—
463C2×Dic3—94no—
464C2×Dic3—94no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—91no—
468C2×Dic3—184no—
469C2×Dic3—184no—
470C2×Dic3—184no—
471C2×Dic3—184no—
472C2×Dic3—188no—
473C2×Dic3—182no—
474C2×Dic3—188no—
475C2×Dic3—368no—
476C2×Dic3C22×C3⋊D311yescharacteristic
477C2×Dic3C22×C3⋊D311yescharacteristic
478C2×Dic3C22×C3⋊D311yescharacteristic
479C4×D3—182no—
480Dic6—22no—
481Dic6—188no—
482Dic6—182no—
483Dic6—188no—
484Dic6—182no—
485Dic6—368no—
486C2×C32—24no—
487C2×C32—24no—
488C2×C32—24no—
489C2×C32—48no—
490C2×C32—44no—
491C2×C32—41no—
492C2×C32—121no—
493C2×C32C2×#23714yes—
494C2×C32D3×#4114yes—
495C2×C32D3×#4111yescharacteristic
496C2×C32D3×#4111yescharacteristic
497C2×C32#101014yes—
498C2×C32#101011yescharacteristic
499C3×D3—44no—
500C2×Q8—31no—
501C2×Q8—271no—
502C2×Q8—271no—
503C2×D4—31no—
504C22×C4—271no—
505C22×C4—271no—
506C4⋊C4—91no—
507C4⋊C4—91no—
508C4⋊C4—91no—
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—
518C22⋊C4—31no—
519C22⋊C4—31no—
520C22⋊C4—31no—
521C22⋊C4—91no—
522C22⋊C4—91no—
523C22⋊C4—271no—
524C22⋊C4—271no—
525C22⋊C4—271no—
526C22⋊C4—271no—
527C22⋊C4D3×C3⋊D311yescharacteristic
528C42—31no—
529C42—91no—
530C42—271no—
531C42—271no—
532C42—271no—
533C42—271no—
534C42—271no—
535C22×C3—28no—
536C22×C3—24no—
537C22×C3—24no—
538C22×C3—24no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—24no—
543C22×C3—44no—
544C22×C3—44no—
545C22×C3—64no—
546C22×C3—64no—
547C22×C3—64no—
548C22×C3C22×D3214yes—
549C22×C3C23×C3⋊D311yescharacteristic
550C2×D3—21no—
551C2×D3—21no—
552C2×D3—21no—
553C3×C4—24no—
554C3×C4—24no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—38no—
558C3×C4—44no—
559C3×C4—44no—
560C3×C4—64no—
561C3×C4—64no—
562C3×C4—181no—
563C3×C4—181no—
564C3×C4—181no—
565C3×C4—181no—
566Dic3—21no—
567Dic3—21no—
568Dic3—64no—
569Dic3—64no—
570Dic3—64no—
571Dic3—64no—
572Dic3—98no—
573Dic3—92no—
574Dic3—188no—
575Dic3—184no—
576Dic3—184no—
577Dic3—184no—
578Dic3—181no—
579Dic3—181no—
580Dic3—181no—
581Dic3—364no—
582Dic3—364no—
583Dic3—364no—
584Dic3#383012yes—
585C32—24no—
586C32#567114yescomplemented
587C32#570211yescharacteristic, complemented
588C23—31no—
589C23C2×D3×C3⋊D311yescharacteristic
590Q8—62no—
591Q8—542no—
592Q8—542no—
593D4—62no—
594C2×C4—31no—
595C2×C4—31no—
596C2×C4—31no—
597C2×C4—91no—
598C2×C4—91no—
599C2×C4—91no—
600C2×C4—91no—
601C2×C4—271no—
602C2×C4—271no—
603C2×C4—271no—
604C2×C4—271no—
605C2×C4—542no—
606C2×C4—542no—
607C2×C4C2×D3×C3⋊D311yescharacteristic
608C2×C4C2×D3×C3⋊D311yescharacteristic
609C2×C3—24no—
610C2×C3—24no—
611C2×C3—24no—
612C2×C3—48no—
613C2×C3—41no—
614C2×C3—44no—
615C2×C3—124no—
616C2×C3#6429814yes—
617C2×C3D3×#23714yes—
618C2×C3#6431214yes—
619C2×C3C2×#383011yescharacteristic
620C2×C3#41×C3⋊D311yescharacteristic
621C2×C3#6436711yescharacteristic
622D3—41no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—61no—
627C22—61no—
628C22—61no—
629C22C22×D3×C3⋊D311yescharacteristic
630C4—21no—
631C4—21no—
632C4—32no—
633C4—61no—
634C4—61no—
635C4—181no—
636C4—181no—
637C4—181no—
638C4—181no—
639C4—272no—
640C4—541no—
641C4—541no—
642C4—541no—
643C3—24no—
644C3#1059414714yescomplemented
645C3#1059614211yescharacteristic, complemented
646C2—41no—
647C2—121no—
648C2#1172456311yescharacteristic
649C2D3×#383011yescharacteristic
650C2#1172458611yescharacteristic
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. #11723919C2semidirect
2. C3×#64088C2semidirect
6. #11723598C2semidirect
7. #11723597C2semidirect
8. #11723496C2semidirect
9. #11723459C2semidirect
10. #11723430C2semidirect
11. #11723427C2semidirect
12. #11723399C2semidirect
13. Dic6×C3⋊Dic3C2semidirect
14. #11723393C2semidirect
15. #11723389C2semidirect
61. #64088D3semidirect
434. C33#556semidirect
586. C32#5671semidirect
587. C32#5702semidirect
644. C3#10594147semidirect
645. C3#10596142semidirect

Maximal subgroups

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

subgroupindex
#117239192
C3×#640882
C32×#8852
#67⋊(C3⋊Dic3)2
#117236002
#117235982
#117235972
#117234962
#117234592
#117234302
#117234272
#117233992
Dic6×C3⋊Dic32
#117233932
#117233892
#105961423
#105941473