Finite Group Databasebeta

← #13147565

Subgroups — #13147565

Order 1728.

656 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721706C211yesmaximal, characteristic, complemented
2C9×#876C211yesmaximal, characteristic, complemented
3C3×#63438C211yesmaximal, characteristic, complemented
4#11721042C211yesmaximal, characteristic
5#11721024C211yesmaximal, characteristic, complemented
6#11721023C211yesmaximal, characteristic, complemented
7#11721021C211yesmaximal, characteristic
8#11720996C211yesmaximal, characteristic, complemented
9#11720991C211yesmaximal, characteristic, complemented
10#11720989C211yesmaximal, characteristic
11#11720965C211yesmaximal, characteristic, complemented
12#11720949C211yesmaximal, characteristic, complemented
13#11720915C211yesmaximal, characteristic, complemented
14#11720914C211yesmaximal, characteristic, complemented
15#11720905C211yesmaximal, characteristic, complemented
16#10593962—31nomaximal
17#10589814—31nomaximal
18C2×#6226C2211yescharacteristic
19C2×#6226C2211yescharacteristic
20C2×#6194C2211yescharacteristic
21C2×D3×Dic9C2211yescharacteristic
22C2×Dic3×D9C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C3×C42×C9C2211yescharacteristic
25C4⋊(C3⋊Dic9)C2211yescharacteristic
26C4⋊(C3⋊Dic9)C2211yescharacteristic
27C4⋊(C3⋊Dic9)C2211yescharacteristic
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29C9×(C2×D3)⋊C4C2211yescharacteristic
30C9×(C2×D3)⋊C4C2211yescharacteristic
31C9×(C2×D3)⋊C4C2211yescharacteristic
32C3×(C2×D9)⋊C4C2211yescharacteristic
33C3×(C2×D9)⋊C4C2211yescharacteristic
34C3×(C2×D9)⋊C4C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C9×Dic3⋊C4C2211yescharacteristic
37C9×Dic3⋊C4C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39C3×Dic9⋊C4C2211yescharacteristic
40C3×Dic9⋊C4C2211yescharacteristic
41(C2×D3)⋊Dic9C2211yescharacteristic
42(C2×D3)⋊Dic9C2211yescharacteristic
43(C2×D9)⋊Dic3C2211yescharacteristic
44(C2×D9)⋊Dic3C2211yescharacteristic
45Dic3⋊Dic9C2211yescharacteristic
46Dic3⋊Dic9C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64078—31no—
54C3×#876—31no—
55C3×#876D311yescharacteristic
56#63913—31no—
57#63896—31no—
58#63896—31no—
59#63893—31no—
60#63893—31no—
61#63891—31no—
62#63891—31no—
63#63861—31no—
64#63861—31no—
65#63845—31no—
66#63845—31no—
67#63840—31no—
68C9×#126—31no—
69#63460—31no—
70#63459—31no—
71#63458—31no—
72#63452—31no—
73#63451—31no—
74#63450—31no—
75#63448—31no—
76#63447—31no—
77#63445—31no—
78#63441—31no—
79#63440—31no—
80#63440—31no—
81#63438D311yescharacteristic, complemented
82#63431—31no—
83#63431—61no—
84C22×D3×C9C2311yescharacteristic
85C22×C3×D9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3×C4×C9C2311yescharacteristic
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93#6226—41no—
94#6226—41no—
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C9×Dic3C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101#6194—41no—
102D3×Dic9—41no—
103Dic3×D9—41no—
104#6189—41no—
105#5712—91no—
106#5712—91no—
107C2×#634—31no—
108C2×#634—31no—
109C2×#627—31no—
110C2×#625—31no—
111C2×D3×Dic3—31no—
112C2×D3×Dic3—31no—
113C32×C42C2×D311yescharacteristic
114C4⋊(C3⋊Dic3)—31no—
115C4⋊(C3⋊Dic3)—31no—
116C4⋊(C3⋊Dic3)—31no—
117C4⋊(C3⋊Dic3)—31no—
118C3×(C2×D3)⋊C4—31no—
119C3×(C2×D3)⋊C4—31no—
120C3×(C2×D3)⋊C4—31no—
121C3×(C2×D3)⋊C4C2×D311yescharacteristic
122C3×(C2×D3)⋊C4C2×D311yescharacteristic
123C3×(C2×D3)⋊C4C2×D311yescharacteristic
124C3×Dic3⋊C4—31no—
125C3×Dic3⋊C4—31no—
126C3×Dic3⋊C4—31no—
127C3×Dic3⋊C4C2×D311yescharacteristic
128C3×Dic3⋊C4C2×D311yescharacteristic
129C3×Dic3⋊C4C2×D311yescharacteristic
130#3724—31no—
131#3724—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—
139(C2×D3)⋊Dic3—31no—
140Dic32—31no—
141Dic32—31no—
142C2×#611—31no—
143C22×Dic9—31no—
144C2×C4×D9—31no—
145C2×Dic18—31no—
146C2×Dic18—31no—
147C2×Dic18—31no—
148C2×Dic18—61no—
149C2×Dic18—61no—
150C9×C4⋊C4—31no—
151C9×C4⋊C4—31no—
152C9×C4⋊C4—31no—
153C9×C22⋊C4—31no—
154C9×C22⋊C4—31no—
155C9×C22⋊C4—31no—
156C42×C9—21no—
157C42×C9C2×D311yescharacteristic
158#3676—31no—
159#3676—31no—
160(C2×D9)⋊C4—31no—
161(C2×D9)⋊C4—31no—
162(C2×D9)⋊C4C2×D311yescharacteristic
163(C2×D9)⋊C4C2×D311yescharacteristic
164(C2×D9)⋊C4C2×D311yescharacteristic
165C4⋊Dic9—31no—
166C4⋊Dic9—31no—
167C4⋊Dic9—31no—
168C4⋊Dic9—31no—
169C4⋊Dic9—31no—
170C4⋊Dic9—31no—
171C4⋊Dic9—61no—
172C4⋊Dic9—61no—
173C4⋊Dic9—61no—
174C4⋊Dic9—61no—
175Dic9⋊C4—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—31no—
178Dic9⋊C4—31no—
179Dic9⋊C4C2×D311yescharacteristic
180Dic9⋊C4C2×D311yescharacteristic
181Dic9⋊C4C2×D311yescharacteristic
182C4×Dic9—31no—
183C4×Dic9—31no—
184C22×C3×C9C2411yescharacteristic
185C2×D3×C9—21no—
186C2×D3×C9—21no—
187C2×D3×C9—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—21no—
197C3⋊Dic9—21no—
198C9×Dic3—21no—
199C9×Dic3—21no—
200C9×Dic3—21no—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×Dic9—21no—
204C3×#126—31no—
205C3×#126—91no—
206#898—91no—
207#898—91no—
208#897—91no—
209#897—91no—
210#896—91no—
211#896—91no—
212#890—91no—
213#890—91no—
214#889—91no—
215#889—91no—
216#888—91no—
217#888—91no—
218#886—91no—
219#886—91no—
220#885—91no—
221#885—91no—
222#883—91no—
223#883—91no—
224#879—91no—
225#879—91no—
226#878—91no—
227#878—91no—
228#878—91no—
229#878—91no—
230#876—31no—
231#876D911yescharacteristic, complemented
232#869—91no—
233#869—91no—
234#869—181no—
235C22×C3×D3—31no—
236C22×C3×D3C22×D311yescharacteristic
237C2×C32×C4C22×D311yescharacteristic
238C2×C32×C4C22×D311yescharacteristic
239C2×C32×C4C22×D311yescharacteristic
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242C2×C3⋊Dic3—31no—
243C2×C3⋊Dic3—31no—
244#634—121no—
245#634—121no—
246C2×C3×Dic3—31no—
247C2×C3×Dic3—31no—
248C2×C3×Dic3—31no—
249C2×C3×Dic3C22×D311yescharacteristic
250C2×C3×Dic3C22×D311yescharacteristic
251C2×C3×Dic3C22×D311yescharacteristic
252#627—121no—
253#625—121no—
254D3×Dic3—121no—
255D3×Dic3—121no—
256C23×C9—31no—
257C22×D9C22×D311yescharacteristic
258C2×C4×C9—21no—
259C2×C4×C9—21no—
260C2×C4×C9—21no—
261C2×C4×C9—31no—
262C2×C4×C9—31no—
263C2×C4×C9—31no—
264C2×C4×C9C22×D311yescharacteristic
265C2×C4×C9C22×D311yescharacteristic
266C2×C4×C9C22×D311yescharacteristic
267#611—121no—
268C2×Dic9—31no—
269C2×Dic9—31no—
270C2×Dic9—31no—
271C2×Dic9—31no—
272C2×Dic9—61no—
273C2×Dic9—61no—
274C2×Dic9—61no—
275C2×Dic9—61no—
276C2×Dic9—121no—
277C2×Dic9C22×D311yescharacteristic
278C2×Dic9C22×D311yescharacteristic
279C2×Dic9C22×D311yescharacteristic
280C4×D9—121no—
281Dic18—121no—
282Dic18—121no—
283Dic18—121no—
284Dic18—122no—
285Dic18—122no—
286#563—271noSylow
287C2×C3×C9H(5,𝔽2)11yescharacteristic
288C2×C3×C92−1+411yescharacteristic
289C2×C3×C92−1+411yescharacteristic
290D3×C9—41no—
291C3×D9—41no—
292C2×#67—91no—
293C2×#67—91no—
294C22×Dic3—91no—
295C22×Dic3—91no—
296C2×C4×D3—91no—
297C2×C4×D3—91no—
298C2×Dic6—91no—
299C2×Dic6—91no—
300C2×Dic6—91no—
301C2×Dic6—91no—
302C2×Dic6—91no—
303C2×Dic6—91no—
304C2×Dic6—181no—
305C2×Dic6—181no—
306C3×C4⋊C4—31no—
307C3×C4⋊C4—31no—
308C3×C4⋊C4—31no—
309C3×C4⋊C4—91no—
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×C22⋊C4—91no—
318C3×C42—21no—
319C3×C42C2×D911yescharacteristic
320C3×C42D3211yescharacteristic
321#217—91no—
322#217—91no—
323#217—91no—
324#217—91no—
325(C2×D3)⋊C4—31no—
326(C2×D3)⋊C4—31no—
327(C2×D3)⋊C4—31no—
328(C2×D3)⋊C4—91no—
329(C2×D3)⋊C4—91no—
330(C2×D3)⋊C4—91no—
331(C2×D3)⋊C4—91no—
332(C2×D3)⋊C4C2×D911yescharacteristic
333(C2×D3)⋊C4C2×D911yescharacteristic
334(C2×D3)⋊C4C2×D911yescharacteristic
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337C4⋊Dic3—91no—
338C4⋊Dic3—91no—
339C4⋊Dic3—91no—
340C4⋊Dic3—91no—
341C4⋊Dic3—91no—
342C4⋊Dic3—91no—
343C4⋊Dic3—91no—
344C4⋊Dic3—91no—
345C4⋊Dic3—91no—
346C4⋊Dic3—91no—
347C4⋊Dic3—181no—
348C4⋊Dic3—181no—
349C4⋊Dic3—181no—
350C4⋊Dic3—181no—
351Dic3⋊C4—31no—
352Dic3⋊C4—31no—
353Dic3⋊C4—31no—
354Dic3⋊C4—91no—
355Dic3⋊C4—91no—
356Dic3⋊C4—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—91no—
360Dic3⋊C4—91no—
361Dic3⋊C4—91no—
362Dic3⋊C4C2×D911yescharacteristic
363Dic3⋊C4C2×D911yescharacteristic
364Dic3⋊C4C2×D911yescharacteristic
365C4×Dic3—91no—
366C4×Dic3—91no—
367C4×Dic3—91no—
368C4×Dic3—91no—
369C22×C32C23×D311yescharacteristic
370C2×C3×D3—21no—
371C2×C3×D3—21no—
372C2×C3×D3—21no—
373C2×C3×D3—61no—
374C2×C3×D3—61no—
375C2×C3×D3—61no—
376C32×C4—21no—
377C32×C4—21no—
378C32×C4—21no—
379C3⋊Dic3—61no—
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—
389C22×C9—21no—
390C22×C9—61no—
391C22×C9—61no—
392C22×C9—61no—
393C22×C9C23×D311yescharacteristic
394C2×D9—21no—
395C2×D9—21no—
396C2×D9—21no—
397C4×C9—22no—
398C4×C9—22no—
399C4×C9—22no—
400C4×C9—21no—
401C4×C9—21no—
402C4×C9—21no—
403C4×C9—61no—
404C4×C9—61no—
405C4×C9—61no—
406Dic9—21no—
407Dic9—21no—
408Dic9—21no—
409Dic9—61no—
410Dic9—61no—
411Dic9—61no—
412Dic9—61no—
413Dic9—121no—
414Dic9—121no—
415Dic9—121no—
416Dic9—121no—
417C4⋊Q8—271no—
418C4⋊Q8—271no—
419#126—31no—
420#126—91no—
421#126—271no—
422#126—271no—
423#125—271no—
424#125—271no—
425#124—271no—
426#123—271no—
427#123—271no—
428C22⋊Q8—271no—
429C22⋊Q8—271no—
430C22⋊Q8—271no—
431C22⋊Q8—271no—
432C3×C9#56311yescharacteristic, complemented, Sylow
433C23×C3—31no—
434C23×C3—91no—
435C22×D3—31no—
436C22×D3C22×D911yescharacteristic
437C2×C3×C4—21no—
438C2×C3×C4—21no—
439C2×C3×C4—21no—
440C2×C3×C4—31no—
441C2×C3×C4—31no—
442C2×C3×C4—31no—
443C2×C3×C4—91no—
444C2×C3×C4—91no—
445C2×C3×C4—91no—
446C2×C3×C4C22×D911yescharacteristic
447C2×C3×C4C22×D911yescharacteristic
448C2×C3×C4C22×D911yescharacteristic
449C2×C3×C4C2×D3211yescharacteristic
450C2×C3×C4C2×D3211yescharacteristic
451C2×C3×C4C2×D3211yescharacteristic
452#67—361no—
453#67—361no—
454C2×Dic3—31no—
455C2×Dic3—31no—
456C2×Dic3—31no—
457C2×Dic3—91no—
458C2×Dic3—91no—
459C2×Dic3—91no—
460C2×Dic3—91no—
461C2×Dic3—91no—
462C2×Dic3—91no—
463C2×Dic3—91no—
464C2×Dic3—91no—
465C2×Dic3—181no—
466C2×Dic3—181no—
467C2×Dic3—181no—
468C2×Dic3—181no—
469C2×Dic3—361no—
470C2×Dic3—361no—
471C2×Dic3C22×D911yescharacteristic
472C2×Dic3C22×D911yescharacteristic
473C2×Dic3C22×D911yescharacteristic
474C4×D3—361no—
475C4×D3—361no—
476Dic6—361no—
477Dic6—361no—
478Dic6—361no—
479Dic6—361no—
480Dic6—361no—
481Dic6—361no—
482Dic6—362no—
483Dic6—362no—
484C2×C32#100911yescharacteristic
485C2×C32#101011yescharacteristic
486C2×C32#101011yescharacteristic
487C3×D3—41no—
488C3×D3—121no—
489C2×C9—21no—
490C2×C9—21no—
491C2×C9—21no—
492C2×C9—121no—
493C2×C9#100911yescharacteristic
494C2×C9#101011yescharacteristic
495C2×C9#101011yescharacteristic
496D9—41no—
497C2×Q8—271no—
498C2×Q8—271no—
499C2×Q8—271no—
500C2×D4—271no—
501C22×C4—271no—
502C22×C4—271no—
503C4⋊C4—31no—
504C4⋊C4—31no—
505C4⋊C4—31no—
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—
518C4⋊C4—271no—
519C22⋊C4—31no—
520C22⋊C4—31no—
521C22⋊C4—31no—
522C22⋊C4—91no—
523C22⋊C4—91no—
524C22⋊C4—91no—
525C22⋊C4—271no—
526C22⋊C4—271no—
527C22⋊C4—271no—
528C22⋊C4—271no—
529C42—271no—
530C42—271no—
531C42D3×D911yescharacteristic
532C22×C3—21no—
533C22×C3—61no—
534C22×C3—61no—
535C22×C3—61no—
536C22×C3—181no—
537C22×C3—181no—
538C22×C3—181no—
539C22×C3C23×D911yescharacteristic
540C22×C3C22×D3211yescharacteristic
541C2×D3—21no—
542C2×D3—21no—
543C2×D3—21no—
544C2×D3—61no—
545C2×D3—61no—
546C2×D3—61no—
547C3×C4—22no—
548C3×C4—22no—
549C3×C4—22no—
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—61no—
557C3×C4—61no—
558C3×C4—61no—
559C3×C4—181no—
560C3×C4—181no—
561C3×C4—181no—
562Dic3—21no—
563Dic3—21no—
564Dic3—21no—
565Dic3—61no—
566Dic3—61no—
567Dic3—61no—
568Dic3—181no—
569Dic3—181no—
570Dic3—181no—
571Dic3—181no—
572Dic3—181no—
573Dic3—181no—
574Dic3—181no—
575Dic3—181no—
576Dic3—361no—
577Dic3—361no—
578Dic3—361no—
579Dic3—361no—
580C32#571211yescharacteristic
581C9—21no—
582C9#571211yescharacteristic, complemented
583C23—31no—
584C23—91no—
585Q8—1081no—
586Q8—1081no—
587Q8—1081no—
588D4—1081no—
589C2×C4—31no—
590C2×C4—31no—
591C2×C4—31no—
592C2×C4—91no—
593C2×C4—91no—
594C2×C4—91no—
595C2×C4—271no—
596C2×C4—271no—
597C2×C4—271no—
598C2×C4—271no—
599C2×C4—1081no—
600C2×C4—1081no—
601C2×C4C2×D3×D911yescharacteristic
602C2×C4C2×D3×D911yescharacteristic
603C2×C4C2×D3×D911yescharacteristic
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—121no—
608C2×C3—361no—
609C2×C3#6371511yescharacteristic
610C2×C3#6371611yescharacteristic
611C2×C3#6371611yescharacteristic
612C2×C3#6429811yescharacteristic
613C2×C3#6429811yescharacteristic
614C2×C3#6430811yescharacteristic
615D3—41no—
616D3—121no—
617C22—61no—
618C22—61no—
619C22—61no—
620C22—181no—
621C22—181no—
622C22—181no—
623C22C22×D3×D911yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—61no—
628C4—61no—
629C4—61no—
630C4—181no—
631C4—181no—
632C4—181no—
633C4—541no—
634C4—541no—
635C4—541no—
636C4—541no—
637C3—21no—
638C3#1058981411yescharacteristic, complemented
639C3#1059396211yescharacteristic
640C2—121no—
641C2—361no—
642C2#1172246511yescharacteristic
643C2#1172246511yescharacteristic
644C2#1172247811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#6194C22
hypercenterC42D3×D9
hypocenterC3×C9#563
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721706C2semidirect
2. C9×#876C2semidirect
3. C3×#63438C2semidirect
5. #11721024C2semidirect
6. #11721023C2semidirect
8. #11720996C2semidirect
9. #11720991C2semidirect
11. #11720965C2semidirect
12. #11720949C2semidirect
13. #11720915C2semidirect
14. #11720914C2semidirect
15. #11720905C2semidirect
81. #63438D3semidirect
231. #876D9semidirect
432. C3×C9#563semidirect
582. C9#5712semidirect
638. C3#10589814semidirect

Maximal subgroups

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

subgroupindex
#117217062
C9×#8762
C3×#634382
#117210422
#117210242
#117210232
#117210212
#117209962
#117209912
#117209892
#117209652
#117209492
#117209152
#117209142
#117209052
#105939623
#105898143