Finite Group Databasebeta

← #13147577

Subgroups — #13147577

Order 1728.

712 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#11721709C211yesmaximal, characteristic, complemented
2C3×#63438C211yesmaximal, characteristic, complemented
3C4×C9×Dic6C211yesmaximal, characteristic, complemented
4#11721003C211yesmaximal, characteristic
5#11720999C211yesmaximal, characteristic, complemented
6#11720978C211yesmaximal, characteristic
7#11720974C211yesmaximal, characteristic, complemented
8#11720950C211yesmaximal, characteristic, complemented
9#11720940C211yesmaximal, characteristic
10#11720926C211yesmaximal, characteristic, complemented
11#11720923C211yesmaximal, characteristic, complemented
12#11720920C211yesmaximal, characteristic, complemented
13#11720903C211yesmaximal, characteristic, complemented
14#11720902C211yesmaximal, characteristic, complemented
15#11720900C211yesmaximal, characteristic, complemented
16#10593940—31nomaximal
17#10589670—31nomaximal
18C2×C4×C3⋊D9C2211yescharacteristic
19C2×C9×Dic6C2211yescharacteristic
20C2×#6192C2211yescharacteristic
21C2×Dic3×D9C2211yescharacteristic
22C2×#6189C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C3×C42×C9C2211yescharacteristic
25#88536C2211yescharacteristic
26#88536C2211yescharacteristic
27#88534C2211yescharacteristic
28#88534C2211yescharacteristic
29C4×C3⋊Dic9C2211yescharacteristic
30C3×(C2×D9)⋊C4C2211yescharacteristic
31C3×(C2×D9)⋊C4C2211yescharacteristic
32C3×(C2×D9)⋊C4C2211yescharacteristic
33C9×C4⋊Dic3C2211yescharacteristic
34C9×Dic3⋊C4C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C4×C9×Dic3C2211yescharacteristic
37C4×C9×Dic3C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39C3×Dic9⋊C4C2211yescharacteristic
40C3×Dic9⋊C4C2211yescharacteristic
41#88445C2211yescharacteristic
42#88445C2211yescharacteristic
43#88445C2211yescharacteristic
44(C2×D9)⋊Dic3C2211yescharacteristic
45(C2×D9)⋊Dic3C2211yescharacteristic
46Dic3⋊Dic9C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic3×Dic9C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64081—31no—
54C3×#876—31no—
55C3×C4×Dic6D311yescharacteristic
56#63907—31no—
57#63899—31no—
58#63881—31no—
59#63877—31no—
60#63868—31no—
61#63865—31no—
62#63862—31no—
63#63852—31no—
64#63848—31no—
65#63839—31no—
66#63838—31no—
67#63837—31no—
68C4×Q8×C9—31no—
69#63509—31no—
70#63506—31no—
71#63460—31no—
72#63460—31no—
73#63458—31no—
74#63458—31no—
75#63455—31no—
76#63449—31no—
77#63438—31no—
78#63438D311yescharacteristic, complemented
79#63437—31no—
80#63434—31no—
81#63434—31no—
82#63434—61no—
83C4×Dic18—31no—
84C22×C3⋊D9C2311yescharacteristic
85C22×C3×D9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3×C4×C9C2311yescharacteristic
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C4×C3⋊D9—22no—
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C9×Dic6—22no—
101#6192—22no—
102Dic3×D9—22no—
103#6189—22no—
104#6189—22no—
105#5702—91no—
106#5568—91no—
107C2×C4×C3⋊D3—31no—
108C2×C3×Dic6C2×D311yescharacteristic
109C2×#627—31no—
110C2×#627—31no—
111C2×#626—31no—
112C2×D3×Dic3—31no—
113C32×C42C2×D311yescharacteristic
114#3752—31no—
115#3752—31no—
116#3750—31no—
117#3750—31no—
118C4×C3⋊Dic3—31no—
119C3×(C2×D3)⋊C4—31no—
120C3×(C2×D3)⋊C4—31no—
121C3×(C2×D3)⋊C4—31no—
122C3×C4⋊Dic3C2×D311yescharacteristic
123C3×Dic3⋊C4—31no—
124C3×Dic3⋊C4—31no—
125C3×Dic3⋊C4—31no—
126C3×Dic3⋊C4C2×D311yescharacteristic
127C3×Dic3⋊C4C2×D311yescharacteristic
128C3×C4×Dic3C2×D311yescharacteristic
129C3×C4×Dic3C2×D311yescharacteristic
130#3724—31no—
131#3724—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134#3722—31no—
135#3722—31no—
136#3722—31no—
137(C2×D3)⋊Dic3—31no—
138(C2×D3)⋊Dic3—31no—
139Dic32—31no—
140Dic32—31no—
141Dic32—31no—
142C2×Q8×C9—31no—
143C2×D36—31no—
144C2×C4×D9—31no—
145C2×C4×D9—31no—
146C2×C4×D9—61no—
147C2×Dic18—31no—
148C2×Dic18—31no—
149C9×C4⋊C4—31no—
150C9×C4⋊C4—31no—
151C9×C4⋊C4—31no—
152C42×C9—21no—
153C42×C9—31no—
154C42×C9—31no—
155C42×C9C2×D311yescharacteristic
156(C2×D9)⋊C4—31no—
157(C2×D9)⋊C4—31no—
158(C2×D9)⋊C4—31no—
159(C2×D9)⋊C4—31no—
160(C2×D9)⋊C4—31no—
161(C2×D9)⋊C4—31no—
162(C2×D9)⋊C4—31no—
163(C2×D9)⋊C4—61no—
164(C2×D9)⋊C4—61no—
165(C2×D9)⋊C4C2×D311yescharacteristic
166(C2×D9)⋊C4C2×D311yescharacteristic
167(C2×D9)⋊C4C2×D311yescharacteristic
168C4⋊Dic9—31no—
169Dic9⋊C4—31no—
170Dic9⋊C4—31no—
171Dic9⋊C4—31no—
172Dic9⋊C4—31no—
173Dic9⋊C4—31no—
174Dic9⋊C4—61no—
175Dic9⋊C4—61no—
176Dic9⋊C4C2×D311yescharacteristic
177Dic9⋊C4C2×D311yescharacteristic
178Dic9⋊C4C2×D311yescharacteristic
179C4×Dic9—31no—
180C4×Dic9—31no—
181C4×Dic9—31no—
182C4×Dic9—31no—
183C4×Dic9—61no—
184C22×C3×C9C2411yescharacteristic
185C2×C3⋊D9—21no—
186C2×C3⋊D9—21no—
187C2×C3⋊D9—21no—
188C2×C3×D9—21no—
189C2×C3×D9—21no—
190C2×C3×D9—21no—
191C3×C4×C9—21no—
192C3×C4×C9—21no—
193C3×C4×C9—21no—
194C3⋊Dic9—21no—
195C3⋊Dic9—21no—
196C3⋊Dic9#4112yes—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3—21no—
200C9×Dic3#4112yes—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×Dic9—21no—
204C3×#126—91no—
205C3×C4×Q8—31no—
206#947—91no—
207#944—91no—
208#898—91no—
209#898—91no—
210#898—91no—
211#898—91no—
212#896—91no—
213#896—91no—
214#896—91no—
215#893—91no—
216#888—91no—
217#887—91no—
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—31no—
227#876—91no—
228#875—91no—
229#872—91no—
230#872—91no—
231#872—91no—
232#872—181no—
233C4×Dic6—91no—
234C4×Dic6D911yescharacteristic, complemented
235C22×C3⋊D3—31no—
236C22×C3×D3—31no—
237C2×C32×C4C22×D311yescharacteristic
238C2×C32×C4C22×D311yescharacteristic
239C2×C32×C4C22×D311yescharacteristic
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242C2×C3⋊Dic3—31no—
243C4×C3⋊D3—62no—
244C2×C3×Dic3—31no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3—31no—
247C2×C3×Dic3C22×D311yescharacteristic
248C2×C3×Dic3C22×D311yescharacteristic
249C2×C3×Dic3C22×D311yescharacteristic
250C2×C3×Dic3C22×D311yescharacteristic
251C3×Dic6—22no—
252#627—62no—
253#627—62no—
254#626—62no—
255D3×Dic3—62no—
256C22×D9—31no—
257C22×D9—61no—
258C22×D9C22×D311yescharacteristic
259Q8×C9—62no—
260C2×C4×C9—21no—
261C2×C4×C9—21no—
262C2×C4×C9—21no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9—31no—
266C2×C4×C9—31no—
267C2×C4×C9C22×D311yescharacteristic
268C2×C4×C9C22×D311yescharacteristic
269C2×C4×C9C22×D311yescharacteristic
270C2×Dic9—31no—
271C2×Dic9—31no—
272C2×Dic9—31no—
273C2×Dic9—61no—
274C2×Dic9—61no—
275C2×Dic9—61no—
276C2×Dic9C22×D311yescharacteristic
277C2×Dic9C22×D311yescharacteristic
278C2×Dic9C22×D311yescharacteristic
279D36—62no—
280C4×D9—62no—
281C4×D9—62no—
282C4×D9—64no—
283Dic18—62no—
284Dic18—62no—
285#556—271noSylow
286C2×C3×C9C2×#4111yescharacteristic
287C2×C3×C9C2×#4111yescharacteristic
288C2×C3×C92−1+411yescharacteristic
289C3⋊D9—41no—
290C3×D9—41no—
291C2×C3×Q8—31no—
292C2×#67—91no—
293C22×Dic3—91no—
294C2×D12—91no—
295C2×C4×D3—91no—
296C2×C4×D3—91no—
297C2×C4×D3—91no—
298C2×C4×D3—181no—
299C2×Dic6—91no—
300C2×Dic6—91no—
301C2×Dic6—91no—
302C2×Dic6—91no—
303C2×Dic6C2×D911yescharacteristic
304C3×C4⋊C4—31no—
305C3×C4⋊C4—31no—
306C3×C4⋊C4—31no—
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—21no—
314C3×C42—31no—
315C3×C42—31no—
316C3×C42C2×D911yescharacteristic
317C3×C42D3211yescharacteristic
318#217—91no—
319#217—91no—
320(C2×D3)⋊C4—31no—
321(C2×D3)⋊C4—31no—
322(C2×D3)⋊C4—31no—
323(C2×D3)⋊C4—91no—
324(C2×D3)⋊C4—91no—
325(C2×D3)⋊C4—91no—
326(C2×D3)⋊C4—91no—
327(C2×D3)⋊C4—91no—
328(C2×D3)⋊C4—91no—
329(C2×D3)⋊C4—91no—
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—181no—
336(C2×D3)⋊C4—181no—
337C4⋊Dic3—91no—
338C4⋊Dic3—91no—
339C4⋊Dic3C2×D911yescharacteristic
340Dic3⋊C4—31no—
341Dic3⋊C4—31no—
342Dic3⋊C4—31no—
343Dic3⋊C4—91no—
344Dic3⋊C4—91no—
345Dic3⋊C4—91no—
346Dic3⋊C4—91no—
347Dic3⋊C4—91no—
348Dic3⋊C4—91no—
349Dic3⋊C4—91no—
350Dic3⋊C4—91no—
351Dic3⋊C4—91no—
352Dic3⋊C4—91no—
353Dic3⋊C4—181no—
354Dic3⋊C4—181no—
355Dic3⋊C4C2×D911yescharacteristic
356Dic3⋊C4C2×D911yescharacteristic
357C4×Dic3—91no—
358C4×Dic3—91no—
359C4×Dic3—91no—
360C4×Dic3—91no—
361C4×Dic3—91no—
362C4×Dic3—91no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—181no—
366C4×Dic3C2×D911yescharacteristic
367C4×Dic3C2×D911yescharacteristic
368C22×C32C23×D311yescharacteristic
369C2×C3⋊D3—61no—
370C2×C3⋊D3—61no—
371C2×C3⋊D3—61no—
372C2×C3×D3—61no—
373C2×C3×D3—61no—
374C2×C3×D3—61no—
375C32×C4—21no—
376C32×C4—21no—
377C32×C4—21no—
378C3⋊Dic3—32no—
379C3⋊Dic3—61no—
380C3⋊Dic3—61no—
381C3×Dic3—21no—
382C3×Dic3—21no—
383C3×Dic3—21no—
384C3×Dic3—61no—
385C3×Dic3—61no—
386C3×Dic3—61no—
387C3×Dic3#23512yes—
388C22×C9—21no—
389C22×C9C23×D311yescharacteristic
390C2×D9—21no—
391C2×D9—21no—
392C2×D9—21no—
393C2×D9—62no—
394C2×D9—61no—
395C2×D9—61no—
396C2×D9—61no—
397C2×D9—121no—
398C2×D9—121no—
399C4×C9—22no—
400C4×C9—21no—
401C4×C9—21no—
402C4×C9—21no—
403C4×C9—32no—
404C4×C9—41no—
405C4×C9—41no—
406C4×C9—61no—
407C4×C9—61no—
408C4×C9—61no—
409Dic9—21no—
410Dic9—21no—
411Dic9—21no—
412Dic9—32no—
413Dic9—62no—
414Dic9—61no—
415Dic9—61no—
416Dic9—121no—
417Dic9—121no—
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—
433C3×C9#55611yescharacteristic, complemented, Sylow
434C23×C3—91no—
435C22×D3—31no—
436C22×D3—91no—
437C22×D3—91no—
438C22×D3—181no—
439C3×Q8—62no—
440C2×C3×C4—21no—
441C2×C3×C4—21no—
442C2×C3×C4—21no—
443C2×C3×C4—31no—
444C2×C3×C4—31no—
445C2×C3×C4—31no—
446C2×C3×C4—31no—
447C2×C3×C4—91no—
448C2×C3×C4—91no—
449C2×C3×C4—91no—
450C2×C3×C4C22×D911yescharacteristic
451C2×C3×C4C22×D911yescharacteristic
452C2×C3×C4C22×D911yescharacteristic
453C2×C3×C4C2×D3211yescharacteristic
454C2×C3×C4C2×D3211yescharacteristic
455C2×C3×C4C2×D3211yescharacteristic
456#67—182no—
457C2×Dic3—31no—
458C2×Dic3—31no—
459C2×Dic3—31no—
460C2×Dic3—91no—
461C2×Dic3—91no—
462C2×Dic3—91no—
463C2×Dic3—91no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—181no—
467C2×Dic3—181no—
468C2×Dic3—181no—
469C2×Dic3—182no—
470C2×Dic3C22×D911yescharacteristic
471C2×Dic3C22×D911yescharacteristic
472C2×Dic3C22×D911yescharacteristic
473C2×Dic3C22×D911yescharacteristic
474D12—182no—
475C4×D3—182no—
476C4×D3—182no—
477C4×D3—182no—
478C4×D3—184no—
479Dic6—22no—
480Dic6—182no—
481Dic6—182no—
482Dic6—182no—
483Dic6—182no—
484C2×C32C2×#23511yescharacteristic
485C2×C32#100711yescharacteristic
486C2×C32D3×#4111yescharacteristic
487C3⋊D3—121no—
488C3×D3—121no—
489C2×C9—21no—
490C2×C9—21no—
491C2×C9—21no—
492C2×C9D3×#4111yescharacteristic
493C2×C9D3×#4111yescharacteristic
494C2×C9#101011yescharacteristic
495D9—41no—
496D9—122no—
497D9—121no—
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×D911yescharacteristic
533C22×C3—21no—
534C22×C3—181no—
535C22×C3—181no—
536C22×C3—181no—
537C22×C3C23×D911yescharacteristic
538C22×C3C22×D3211yescharacteristic
539C2×D3—61no—
540C2×D3—61no—
541C2×D3—61no—
542C2×D3—182no—
543C2×D3—181no—
544C2×D3—181no—
545C2×D3—181no—
546C2×D3—181no—
547C2×D3—181no—
548C2×D3—181no—
549C2×D3—361no—
550C2×D3—361no—
551C3×C4—22no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—32no—
559C3×C4—41no—
560C3×C4—41no—
561C3×C4—61no—
562C3×C4—61no—
563C3×C4—61no—
564C3×C4—181no—
565C3×C4—181no—
566C3×C4—181no—
567Dic3—21no—
568Dic3—21no—
569Dic3—21no—
570Dic3—61no—
571Dic3—61no—
572Dic3—61no—
573Dic3—92no—
574Dic3—92no—
575Dic3—182no—
576Dic3—181no—
577Dic3—181no—
578Dic3—181no—
579Dic3—181no—
580Dic3—361no—
581Dic3—361no—
582Dic3#369712yes—
583C32#556811yescharacteristic
584C9—21no—
585C9#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×D911yescharacteristic
605C2×C4C2×D3×D911yescharacteristic
606C2×C4C2×D3×D911yescharacteristic
607C2×C3—21no—
608C2×C3—21no—
609C2×C3—21no—
610C2×C3—361no—
611C2×C3C2×#369711yescharacteristic
612C2×C3#6371311yescharacteristic
613C2×C3#41×D911yescharacteristic
614C2×C3#6429711yescharacteristic
615C2×C3D3×#23511yescharacteristic
616C2×C3#6430611yescharacteristic
617D3—121no—
618D3—362no—
619D3—361no—
620D3—361no—
621C22—181no—
622C22—181no—
623C22—181no—
624C22—541no—
625C22—541no—
626C22—541no—
627C22C22×D3×D911yescharacteristic
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—21no—
642C3#1058967011yescharacteristic, complemented
643C3#1059394011yescharacteristic
644C2—361no—
645C2—1081no—
646C2#1172245811yescharacteristic
647C2D3×#369711yescharacteristic
648C2#1172247511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#6192C22
hypercenterC42D3×D9
hypocenterC3×C9#556
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721709C2semidirect
2. C3×#63438C2semidirect
3. C4×C9×Dic6C2semidirect
5. #11720999C2semidirect
7. #11720974C2semidirect
8. #11720950C2semidirect
10. #11720926C2semidirect
11. #11720923C2semidirect
12. #11720920C2semidirect
13. #11720903C2semidirect
14. #11720902C2semidirect
15. #11720900C2semidirect
78. #63438D3semidirect
234. C4×Dic6D9semidirect
433. C3×C9#556semidirect
585. C9#5702semidirect
642. C3#10589670semidirect

Maximal subgroups

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

subgroupindex
#117217092
C3×#634382
C4×C9×Dic62
#117210032
#117209992
#117209782
#117209742
#117209502
#117209402
#117209262
#117209232
#117209202
#117209032
#117209022
#117209002
#105939403
#105896703