Finite Group Databasebeta

← #13150417

Subgroups — #13150417

Order 1728.

734 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88534C211yesmaximal, characteristic, complemented
2C9×#929C211yesmaximal, characteristic
3C3×#63488C211yesmaximal, characteristic, complemented
4#67⋊Dic9C211yesmaximal, characteristic
5#11721267C211yesmaximal, characteristic, complemented
6#11721266C211yesmaximal, characteristic
7#11721248C211yesmaximal, characteristic, complemented
8#11721026C211yesmaximal, characteristic, complemented
9#11721010C211yesmaximal, characteristic, complemented
10#11720963C211yesmaximal, characteristic, complemented
11#11720962C211yesmaximal, characteristic, complemented
12#11720933C211yesmaximal, characteristic, complemented
13#11720919C211yesmaximal, characteristic, complemented
14#11720916C211yesmaximal, characteristic, complemented
15#11720901C211yesmaximal, characteristic, complemented
16#10594960—31nomaximal
17#10589752—31nomaximal
18C22×C3×C4×C9C2211yescharacteristic
19C22×C3⋊Dic9C2211yescharacteristic
20C22×C3⋊Dic9C2211yescharacteristic
21C2×C9×#67C2211yescharacteristic
22C2×D3×Dic9C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24#88534C2211yescharacteristic
25#88534C2211yescharacteristic
26#88534C2211yescharacteristic
27#88534C2211yescharacteristic
28C9×#217C2211yescharacteristic
29C3×#3676C2211yescharacteristic
30C3×#3676C2211yescharacteristic
31C9×(C2×D3)⋊C4C2211yescharacteristic
32C9×(C2×D3)⋊C4C2211yescharacteristic
33C9×Dic3⋊C4C2211yescharacteristic
34C9×Dic3⋊C4C2211yescharacteristic
35C3×C4⋊Dic9C2211yescharacteristic
36C3×C4⋊Dic9C2211yescharacteristic
37C3×C4×Dic9C2211yescharacteristic
38C3×C4×Dic9C2211yescharacteristic
39(C2×D3)⋊Dic9C2211yescharacteristic
40(C2×D3)⋊Dic9C2211yescharacteristic
41(C2×D3)⋊Dic9C2211yescharacteristic
42Dic3⋊Dic9C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44#88442C2211yescharacteristic
45#88442C2211yescharacteristic
46#88442C2211yescharacteristic
47Dic9⋊Dic3C2211yescharacteristic
48Dic9⋊Dic3C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic3×Dic9C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53C2×#3750—31no—
54C3×#929D311yescharacteristic
55C3×#926—31no—
56#63982—31no—
57#67⋊Dic3—31no—
58#63962—31no—
59#63957—31no—
60#63899—31no—
61#63877—31no—
62#63867—31no—
63#63859—31no—
64#63858—31no—
65#63847—31no—
66#63846—31no—
67#63837—31no—
68C9×#123—31no—
69#63497—31no—
70D4×Dic9—31no—
71#63488D311yescharacteristic, complemented
72C2×Dic9⋊C4—31no—
73C2×Dic9⋊C4—61no—
74#63452—31no—
75#63451—31no—
76#63451—31no—
77#63449—31no—
78#63441—31no—
79#63441—31no—
80#63440—31no—
81#63440—31no—
82#63439—31no—
83#63439—31no—
84C23×C3×C9C2311yescharacteristic
85C22×D3×C9C2311yescharacteristic
86C2×C3×C4×C9—22no—
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3×C4×C9C2311yescharacteristic
89C2×C3⋊Dic9—22no—
90C2×C3⋊Dic9—22no—
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93C2×C3⋊Dic9C2311yescharacteristic
94C2×C3⋊Dic9C2311yescharacteristic
95C9×#67—22no—
96C2×C9×Dic3C2311yescharacteristic
97C2×C9×Dic3C2311yescharacteristic
98C2×C9×Dic3C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C2×C3×Dic9C2311yescharacteristic
102C2×C3×Dic9C2311yescharacteristic
103D3×Dic9—22no—
104#6189—22no—
105#5650—91no—
106#5531—91no—
107C22×C32×C4C2×D311yescharacteristic
108C22×C3⋊Dic3—31no—
109C22×C3⋊Dic3—31no—
110C2×C3×#67C2×D311yescharacteristic
111C2×#627—31no—
112C2×D3×Dic3—31no—
113#3750—31no—
114#3750—31no—
115#3750—31no—
116#3750—31no—
117C3×#217—31no—
118C3×#217—31no—
119C3×#217C2×D311yescharacteristic
120C3×(C2×D3)⋊C4C2×D311yescharacteristic
121C3×(C2×D3)⋊C4C2×D311yescharacteristic
122C3×C4⋊Dic3—31no—
123C3×C4⋊Dic3—31no—
124C3×Dic3⋊C4C2×D311yescharacteristic
125C3×Dic3⋊C4C2×D311yescharacteristic
126C3×C4×Dic3—31no—
127C3×C4×Dic3—31no—
128#3724—31no—
129#3724—31no—
130#3724—31no—
131Dic3⋊Dic3—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134Dic3⋊Dic3—31no—
135Dic3⋊Dic3—31no—
136(C2×D3)⋊Dic3—31no—
137(C2×D3)⋊Dic3—31no—
138(C2×D3)⋊Dic3—31no—
139Dic32—31no—
140Dic32—31no—
141Dic32—31no—
142C2×D4×C9—31no—
143C22×C4×C9—21no—
144C22×C4×C9C2×D311yescharacteristic
145C22×Dic9—31no—
146C22×Dic9—31no—
147C22×Dic9—31no—
148C22×Dic9—61no—
149C22×Dic9—61no—
150C2×Dic18—31no—
151C9×C4⋊C4—31no—
152C9×C4⋊C4—31no—
153C9×C22⋊C4—31no—
154C9×C22⋊C4—31no—
155C9×C22⋊C4—31no—
156#3676—31no—
157#3676—31no—
158#3676—31no—
159#3676C2×D311yescharacteristic
160#3676C2×D311yescharacteristic
161C4⋊Dic9—31no—
162C4⋊Dic9—31no—
163C4⋊Dic9C2×D311yescharacteristic
164C4⋊Dic9C2×D311yescharacteristic
165Dic9⋊C4—31no—
166Dic9⋊C4—31no—
167Dic9⋊C4—31no—
168Dic9⋊C4—31no—
169Dic9⋊C4—31no—
170Dic9⋊C4—31no—
171Dic9⋊C4—31no—
172Dic9⋊C4—31no—
173Dic9⋊C4—31no—
174Dic9⋊C4—31no—
175Dic9⋊C4—61no—
176Dic9⋊C4—61no—
177Dic9⋊C4—61no—
178Dic9⋊C4—61no—
179C4×Dic9—31no—
180C4×Dic9—31no—
181C4×Dic9—31no—
182C4×Dic9C2×D311yescharacteristic
183C4×Dic9C2×D311yescharacteristic
184C22×C3×C9—21no—
185C22×C3×C9—21no—
186C22×C3×C9#4112yes—
187C22×C3×C9C2411yescharacteristic
188C2×D3×C9—21no—
189C2×D3×C9—21no—
190C2×D3×C9—21no—
191C3×C4×C9—21no—
192C3×C4×C9—21no—
193C3⋊Dic9—21no—
194C3⋊Dic9—21no—
195C3⋊Dic9—21no—
196C3⋊Dic9#4112yes—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3—21no—
200C3×Dic9—21no—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×Dic9—21no—
204C3×#123—31no—
205C3×#117—91no—
206#935—91no—
207D4×Dic3—91no—
208#929D911yescharacteristic
209#926—31no—
210C2×Dic3⋊C4—91no—
211C2×Dic3⋊C4—91no—
212C2×Dic3⋊C4—181no—
213#898—91no—
214#896—91no—
215#890—91no—
216#889—91no—
217#889—91no—
218#889—91no—
219#889—91no—
220#887—91no—
221#886—91no—
222#67⋊C4—91no—
223#879—91no—
224#879—91no—
225#878—91no—
226#878—91no—
227#878—91no—
228#877—91no—
229#877—91no—
230#877—91no—
231#876—91no—
232#872—91no—
233#870—91no—
234C4×Dic6—91no—
235C23×C32C22×D311yescharacteristic
236C22×C3×D3C22×D311yescharacteristic
237C2×C32×C4—22no—
238C2×C32×C4C22×D311yescharacteristic
239C2×C32×C4C22×D311yescharacteristic
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242C2×C3⋊Dic3—31no—
243C2×C3⋊Dic3—31no—
244C2×C3⋊Dic3—62no—
245C2×C3⋊Dic3—62no—
246C3×#67—22no—
247C2×C3×Dic3—31no—
248C2×C3×Dic3—31no—
249C2×C3×Dic3—31no—
250C2×C3×Dic3—31no—
251C2×C3×Dic3C22×D311yescharacteristic
252C2×C3×Dic3C22×D311yescharacteristic
253C2×C3×Dic3C22×D311yescharacteristic
254#627—62no—
255D3×Dic3—62no—
256C23×C9—21no—
257C23×C9—31no—
258C23×C9C22×D311yescharacteristic
259D4×C9—62no—
260C2×C4×C9—22no—
261C2×C4×C9—21no—
262C2×C4×C9—21no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9—31no—
266C2×C4×C9—42no—
267C2×C4×C9C22×D311yescharacteristic
268C2×C4×C9C22×D311yescharacteristic
269C2×Dic9—31no—
270C2×Dic9—31no—
271C2×Dic9—31no—
272C2×Dic9—31no—
273C2×Dic9—61no—
274C2×Dic9—61no—
275C2×Dic9—61no—
276C2×Dic9—61no—
277C2×Dic9—62no—
278C2×Dic9—62no—
279C2×Dic9—64no—
280C2×Dic9—62no—
281C2×Dic9—122no—
282C2×Dic9C22×D311yescharacteristic
283C2×Dic9C22×D311yescharacteristic
284C2×Dic9C22×D311yescharacteristic
285C2×Dic9C22×D311yescharacteristic
286Dic18—62no—
287#552—271noSylow
288C2×C3×C9—22no—
289C2×C3×C9C2×#4111yescharacteristic
290C2×C3×C9C2×#4111yescharacteristic
291C2×C3×C92−1+411yescharacteristic
292D3×C9—41no—
293C2×C3×D4—31no—
294C22×C3×C4—21no—
295C22×C3×C4C2×D911yescharacteristic
296C22×C3×C4D3211yescharacteristic
297C2×#67C2×D911yescharacteristic
298C22×Dic3—91no—
299C22×Dic3—91no—
300C22×Dic3—91no—
301C22×Dic3—91no—
302C22×Dic3—91no—
303C22×Dic3—181no—
304C22×Dic3—181no—
305C2×C4×D3—91no—
306C2×Dic6—91no—
307C2×Dic6—91no—
308C3×C4⋊C4—31no—
309C3×C4⋊C4—31no—
310C3×C4⋊C4—91no—
311C3×C4⋊C4—91no—
312C3×C22⋊C4—31no—
313C3×C22⋊C4—31no—
314C3×C22⋊C4—31no—
315C3×C22⋊C4—91no—
316C3×C22⋊C4—91no—
317C3×C42—91no—
318C3×C42—91no—
319#217—31no—
320#217—31no—
321#217—91no—
322#217—91no—
323#217—91no—
324#217C2×D911yescharacteristic
325(C2×D3)⋊C4—91no—
326(C2×D3)⋊C4—91no—
327(C2×D3)⋊C4—91no—
328(C2×D3)⋊C4C2×D911yescharacteristic
329(C2×D3)⋊C4C2×D911yescharacteristic
330C4⋊Dic3—31no—
331C4⋊Dic3—31no—
332C4⋊Dic3—91no—
333C4⋊Dic3—91no—
334C4⋊Dic3—91no—
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337Dic3⋊C4—91no—
338Dic3⋊C4—91no—
339Dic3⋊C4—91no—
340Dic3⋊C4—91no—
341Dic3⋊C4—91no—
342Dic3⋊C4—91no—
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—91no—
354Dic3⋊C4—91no—
355Dic3⋊C4—91no—
356Dic3⋊C4—181no—
357Dic3⋊C4—181no—
358Dic3⋊C4—181no—
359Dic3⋊C4—181no—
360Dic3⋊C4C2×D911yescharacteristic
361Dic3⋊C4C2×D911yescharacteristic
362C4×Dic3—31no—
363C4×Dic3—31no—
364C4×Dic3—91no—
365C4×Dic3—91no—
366C4×Dic3—91no—
367C4×Dic3—91no—
368C4×Dic3—91no—
369C4×Dic3—91no—
370C22×C32—21no—
371C22×C32—21no—
372C22×C32#23712yes—
373C22×C32C23×D311yescharacteristic
374C2×C3×D3—21no—
375C2×C3×D3—21no—
376C2×C3×D3—21no—
377C32×C4—21no—
378C32×C4—21no—
379C3⋊Dic3—32no—
380C3⋊Dic3—61no—
381C3⋊Dic3—61no—
382C3⋊Dic3—61no—
383C3×Dic3—21no—
384C3×Dic3—21no—
385C3×Dic3—21no—
386C3×Dic3—61no—
387C3×Dic3—61no—
388C3×Dic3—61no—
389C3×Dic3—61no—
390C22×C9—22no—
391C22×C9—22no—
392C22×C9—21no—
393C22×C9—21no—
394C22×C9—21no—
395C22×C9—22no—
396C22×C9—61no—
397C22×C9—61no—
398C22×C9—61no—
399C22×C9#23512yes—
400C22×C9C23×D311yescharacteristic
401C4×C9—21no—
402C4×C9—21no—
403C4×C9—41no—
404C4×C9—41no—
405C4×C9—61no—
406C4×C9—61no—
407C4×C9—61no—
408Dic9—21no—
409Dic9—21no—
410Dic9—21no—
411Dic9—21no—
412Dic9—32no—
413Dic9—62no—
414Dic9—62no—
415Dic9—61no—
416Dic9—61no—
417Dic9—61no—
418Dic9—121no—
419Dic9—121no—
420#126—271no—
421#126—271no—
422#125—271no—
423#125—271no—
424#125—271no—
425#123—31no—
426#123—271no—
427C22⋊Q8—271no—
428C22⋊Q8—271no—
429C4×Q8—271no—
430C4×D4—271no—
431#117—91no—
432#117—271no—
433#117—271no—
434C2×C4⋊C4—271no—
435C3×C9#55211yescharacteristic, complemented, Sylow
436C23×C3—21no—
437C23×C3—31no—
438C23×C3C22×D911yescharacteristic
439C23×C3C2×D3211yescharacteristic
440C22×D3C22×D911yescharacteristic
441C3×D4—62no—
442C2×C3×C4—22no—
443C2×C3×C4—22no—
444C2×C3×C4—21no—
445C2×C3×C4—21no—
446C2×C3×C4—31no—
447C2×C3×C4—31no—
448C2×C3×C4—31no—
449C2×C3×C4—42no—
450C2×C3×C4—91no—
451C2×C3×C4—91no—
452C2×C3×C4—91no—
453C2×C3×C4—91no—
454C2×C3×C4C22×D911yescharacteristic
455C2×C3×C4C22×D911yescharacteristic
456C2×C3×C4C2×D3211yescharacteristic
457C2×C3×C4C2×D3211yescharacteristic
458#67—22no—
459C2×Dic3—31no—
460C2×Dic3—31no—
461C2×Dic3—31no—
462C2×Dic3—31no—
463C2×Dic3—91no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—91no—
468C2×Dic3—91no—
469C2×Dic3—91no—
470C2×Dic3—91no—
471C2×Dic3—181no—
472C2×Dic3—181no—
473C2×Dic3—181no—
474C2×Dic3—181no—
475C2×Dic3—182no—
476C2×Dic3—182no—
477C2×Dic3—182no—
478C2×Dic3—184no—
479C2×Dic3—182no—
480C2×Dic3—182no—
481C2×Dic3—362no—
482C2×Dic3C22×D911yescharacteristic
483C2×Dic3C22×D911yescharacteristic
484C2×Dic3C22×D911yescharacteristic
485C4×D3—182no—
486Dic6—182no—
487Dic6—182no—
488C2×C32—22no—
489C2×C32C2×#23711yescharacteristic
490C2×C32D3×#4111yescharacteristic
491C2×C32#101011yescharacteristic
492C3×D3—41no—
493C2×C9—24no—
494C2×C9—22no—
495C2×C9—21no—
496C2×C9—21no—
497C2×C9—21no—
498C2×C9—121no—
499C2×C9C2×#23511yescharacteristic
500C2×C9D3×#4111yescharacteristic
501C2×C9#101011yescharacteristic
502C2×Q8—271no—
503C2×D4—31no—
504C22×C4—271no—
505C22×C4—271no—
506C22×C4—271no—
507C22×C4D3×D911yescharacteristic
508C4⋊C4—31no—
509C4⋊C4—31no—
510C4⋊C4—91no—
511C4⋊C4—91no—
512C4⋊C4—271no—
513C4⋊C4—271no—
514C4⋊C4—271no—
515C4⋊C4—271no—
516C4⋊C4—271no—
517C4⋊C4—271no—
518C4⋊C4—271no—
519C4⋊C4—271no—
520C4⋊C4—271no—
521C4⋊C4—271no—
522C4⋊C4—271no—
523C4⋊C4—271no—
524C22⋊C4—31no—
525C22⋊C4—31no—
526C22⋊C4—31no—
527C22⋊C4—91no—
528C22⋊C4—91no—
529C22⋊C4—271no—
530C22⋊C4—271no—
531C22⋊C4—271no—
532C42—91no—
533C42—91no—
534C42—271no—
535C42—271no—
536C42—271no—
537C22×C3—22no—
538C22×C3—22no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—21no—
544C22×C3—22no—
545C22×C3—61no—
546C22×C3—61no—
547C22×C3—61no—
548C22×C3#369912yes—
549C22×C3C23×D911yescharacteristic
550C22×C3#380412yes—
551C22×C3C22×D3211yescharacteristic
552C2×D3—21no—
553C2×D3—21no—
554C2×D3—21no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—21no—
559C3×C4—41no—
560C3×C4—41no—
561C3×C4—61no—
562C3×C4—61no—
563C3×C4—61no—
564C3×C4—181no—
565C3×C4—181no—
566C3×C4—181no—
567C3×C4—181no—
568Dic3—21no—
569Dic3—21no—
570Dic3—21no—
571Dic3—61no—
572Dic3—61no—
573Dic3—61no—
574Dic3—61no—
575Dic3—92no—
576Dic3—92no—
577Dic3—182no—
578Dic3—182no—
579Dic3—181no—
580Dic3—181no—
581Dic3—181no—
582Dic3—181no—
583Dic3—181no—
584Dic3—181no—
585Dic3—361no—
586Dic3—361no—
587C32#565011yescharacteristic
588C9—21no—
589C9#553111yescharacteristic, complemented
590C23—31no—
591C23C2×D3×D911yescharacteristic
592Q8—542no—
593D4—62no—
594C2×C4—22no—
595C2×C4—31no—
596C2×C4—31no—
597C2×C4—31no—
598C2×C4—91no—
599C2×C4—91no—
600C2×C4—91no—
601C2×C4—91no—
602C2×C4—271no—
603C2×C4—271no—
604C2×C4—271no—
605C2×C4—271no—
606C2×C4—542no—
607C2×C4—542no—
608C2×C4—542no—
609C2×C4C2×D3×D911yescharacteristic
610C2×C4C2×D3×D911yescharacteristic
611C2×C3—24no—
612C2×C3—22no—
613C2×C3—22no—
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—121no—
618C2×C3C2×#369911yescharacteristic
619C2×C3#41×D911yescharacteristic
620C2×C3#6371611yescharacteristic
621C2×C3#6429811yescharacteristic
622C2×C3#6430611yescharacteristic
623C2×C3C2×#380411yescharacteristic
624D3—41no—
625C22—21no—
626C22—21no—
627C22—61no—
628C22—61no—
629C22—61no—
630C22#8866212yes—
631C22C22×D3×D911yescharacteristic
632C4—21no—
633C4—21no—
634C4—61no—
635C4—61no—
636C4—61no—
637C4—181no—
638C4—181no—
639C4—181no—
640C4—181no—
641C4—272no—
642C4—541no—
643C4—541no—
644C4—541no—
645C3—21no—
646C3#1058975211yescharacteristic, complemented
647C3#1059496011yescharacteristic
648C2—22no—
649C2—121no—
650C2#1172246511yescharacteristic
651C2#1172247511yescharacteristic
652C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×#67C2×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#552
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88534C2semidirect
3. C3×#63488C2semidirect
5. #11721267C2semidirect
7. #11721248C2semidirect
8. #11721026C2semidirect
9. #11721010C2semidirect
10. #11720963C2semidirect
11. #11720962C2semidirect
12. #11720933C2semidirect
13. #11720919C2semidirect
14. #11720916C2semidirect
15. #11720901C2semidirect
71. #63488D3semidirect
435. C3×C9#552semidirect
589. C9#5531semidirect
646. C3#10589752semidirect

Maximal subgroups

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

subgroupindex
C2×#885342
C9×#9292
C3×#634882
#67⋊Dic92
#117212672
#117212662
#117212482
#117210262
#117210102
#117209632
#117209622
#117209332
#117209192
#117209162
#117209012
#105949603
#105897523