Finite Group Databasebeta

← #13148121

Subgroups — #13148121

Order 1728.

666 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721732C211yesmaximal, characteristic
2C9×#879C211yesmaximal, characteristic, complemented
3C3×#63441C211yesmaximal, characteristic, complemented
4#11721275C211yesmaximal, characteristic
5#11721267C211yesmaximal, characteristic, complemented
6#11721267C211yesmaximal, characteristic, complemented
7#11721242C211yesmaximal, characteristic
8#11720995C211yesmaximal, characteristic, complemented
9#11720994C211yesmaximal, characteristic, complemented
10#11720970C211yesmaximal, characteristic, complemented
11#11720969C211yesmaximal, characteristic, complemented
12#11720934C211yesmaximal, characteristic, complemented
13#11720926C211yesmaximal, characteristic, complemented
14#11720915C211yesmaximal, characteristic, complemented
15#11720914C211yesmaximal, characteristic, complemented
16#10594036—31nomaximal
17#10589814—31nomaximal
18C2×#6233C2211yescharacteristic
19C22×C3⋊Dic9C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2211yescharacteristic
21C2×#6189C2211yescharacteristic
22C2×#6189C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C3×C9×C22⋊C4C2211yescharacteristic
25#88536C2211yescharacteristic
26#88536C2211yescharacteristic
27C4⋊(C3⋊Dic9)C2211yescharacteristic
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29C9×#217C2211yescharacteristic
30C9×#217C2211yescharacteristic
31C3×#3676C2211yescharacteristic
32C3×#3676C2211yescharacteristic
33C9×C4⋊Dic3C2211yescharacteristic
34C9×Dic3⋊C4C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C4×C9×Dic3C2211yescharacteristic
37C3×C4⋊Dic9C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39C3×Dic9⋊C4C2211yescharacteristic
40C3×C4×Dic9C2211yescharacteristic
41#88445C2211yescharacteristic
42#88445C2211yescharacteristic
43#88445C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45Dic3⋊Dic9C2211yescharacteristic
46Dic3⋊Dic9C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic9⋊Dic3C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64095—31no—
54C3×#879—31no—
55C3×#879D311yescharacteristic
56#63982—31no—
57#63982—31no—
58#63975—31no—
59#63953—31no—
60#63918—31no—
61#63918—31no—
62#63895—31no—
63#63895—31no—
64#63868—31no—
65#63868—31no—
66#63845—31no—
67#63845—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#63448—61no—
77#63447—31no—
78#63445—31no—
79#63441D311yescharacteristic, complemented
80#63440—31no—
81#63440—31no—
82#63438—31no—
83#63431—31no—
84C23×C3×C9C2311yescharacteristic
85C22×C3⋊D9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88#6233—41no—
89C2×C3⋊Dic9—41no—
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101(C3⋊D9)⋊C4—41no—
102#6189—41no—
103#6189—41no—
104#6189—41no—
105#5712—91no—
106#5712—91no—
107C2×#638—31no—
108C22×C3⋊Dic3—31no—
109C2×#627—31no—
110C2×#627—31no—
111C2×#627—31no—
112C2×#624—31no—
113C32×C22⋊C4C2×D311yescharacteristic
114#3752—31no—
115#3752—31no—
116C4⋊(C3⋊Dic3)—31no—
117C4⋊(C3⋊Dic3)—31no—
118C3×#217—31no—
119C3×#217—31no—
120C3×#217C2×D311yescharacteristic
121C3×#217C2×D311yescharacteristic
122C3×C4⋊Dic3—31no—
123C3×C4⋊Dic3C2×D311yescharacteristic
124C3×Dic3⋊C4—31no—
125C3×Dic3⋊C4—31no—
126C3×Dic3⋊C4C2×D311yescharacteristic
127C3×Dic3⋊C4C2×D311yescharacteristic
128C3×C4×Dic3—31no—
129C3×C4×Dic3C2×D311yescharacteristic
130#3724—31no—
131#3724—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134Dic3⋊Dic3—31no—
135Dic3⋊Dic3—31no—
136Dic3⋊Dic3—31no—
137Dic3⋊Dic3—31no—
138#3722—31no—
139#3722—31no—
140#3722—31no—
141Dic32—31no—
142C2×#611—31no—
143C2×#611—61no—
144C22×Dic9—31no—
145C22×Dic9—61no—
146C2×C4×D9—31no—
147C2×Dic18—31no—
148C2×Dic18—31no—
149C2×Dic18—31no—
150C9×C4⋊C4—31no—
151C9×C4⋊C4—31no—
152C9×C4⋊C4—31no—
153C9×C22⋊C4—21no—
154C9×C22⋊C4—31no—
155C9×C22⋊C4—31no—
156C9×C22⋊C4C2×D311yescharacteristic
157C42×C9—31no—
158#3676C2×D311yescharacteristic
159#3676C2×D311yescharacteristic
160(C2×D9)⋊C4—31no—
161(C2×D9)⋊C4—31no—
162(C2×D9)⋊C4—31no—
163(C2×D9)⋊C4—31no—
164(C2×D9)⋊C4—31no—
165(C2×D9)⋊C4—61no—
166(C2×D9)⋊C4—61no—
167C4⋊Dic9—31no—
168C4⋊Dic9—31no—
169C4⋊Dic9—31no—
170C4⋊Dic9—31no—
171C4⋊Dic9—31no—
172C4⋊Dic9—61no—
173C4⋊Dic9—61no—
174C4⋊Dic9C2×D311yescharacteristic
175Dic9⋊C4—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—31no—
178Dic9⋊C4—31no—
179Dic9⋊C4—31no—
180Dic9⋊C4C2×D311yescharacteristic
181Dic9⋊C4C2×D311yescharacteristic
182C4×Dic9—31no—
183C4×Dic9C2×D311yescharacteristic
184C22×C3×C9—21no—
185C22×C3×C9—21no—
186C22×C3×C9—21no—
187C22×C3×C9C2411yescharacteristic
188C2×C3⋊D9—21no—
189C2×C3⋊D9—21no—
190C2×C3⋊D9—21no—
191C3×C4×C9—21no—
192C3×C4×C9—21no—
193C3⋊Dic9—21no—
194C3⋊Dic9—21no—
195C3⋊Dic9—21no—
196C9×Dic3—21no—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3—21no—
200C3×Dic9—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#886—181no—
221#885—91no—
222#885—91no—
223#883—91no—
224#883—91no—
225#879—31no—
226#879D911yescharacteristic, complemented
227#878—91no—
228#878—91no—
229#878—91no—
230#878—91no—
231#876—91no—
232#876—91no—
233#869—91no—
234#869—91no—
235C23×C32C22×D311yescharacteristic
236C22×C3⋊D3—31no—
237C2×C32×C4C22×D311yescharacteristic
238C2×C32×C4C22×D311yescharacteristic
239#638—121no—
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242C2×C3⋊Dic3—31no—
243C2×C3⋊Dic3—121no—
244C2×C3×Dic3—31no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3—31no—
247C2×C3×Dic3—31no—
248C2×C3×Dic3C22×D311yescharacteristic
249C2×C3×Dic3C22×D311yescharacteristic
250C2×C3×Dic3C22×D311yescharacteristic
251C2×C3×Dic3C22×D311yescharacteristic
252#627—121no—
253#627—121no—
254#627—121no—
255#624—121no—
256C23×C9—21no—
257C23×C9C22×D311yescharacteristic
258C22×D9—31no—
259C22×D9—61no—
260C2×C4×C9—21no—
261C2×C4×C9—21no—
262C2×C4×C9—31no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9—31no—
266C2×C4×C9C22×D311yescharacteristic
267C2×C4×C9C22×D311yescharacteristic
268#611—121no—
269#611—122no—
270C2×Dic9—31no—
271C2×Dic9—31no—
272C2×Dic9—31no—
273C2×Dic9—61no—
274C2×Dic9—61no—
275C2×Dic9—61no—
276C2×Dic9—121no—
277C2×Dic9—122no—
278C2×Dic9C22×D311yescharacteristic
279C2×Dic9C22×D311yescharacteristic
280C2×Dic9C22×D311yescharacteristic
281C2×Dic9C22×D311yescharacteristic
282C4×D9—121no—
283Dic18—121no—
284Dic18—121no—
285Dic18—121no—
286#563—271noSylow
287C2×C3×C9—41no—
288C2×C3×C9H(5,𝔽2)11yescharacteristic
289C2×C3×C92−1+411yescharacteristic
290C2×C3×C92−1+411yescharacteristic
291C3⋊D9—41no—
292C2×#67—91no—
293C2×#67—91no—
294C2×#67—181no—
295C22×Dic3—91no—
296C22×Dic3—91no—
297C22×Dic3—181no—
298C2×C4×D3—91no—
299C2×C4×D3—91no—
300C2×Dic6—91no—
301C2×Dic6—91no—
302C2×Dic6—91no—
303C2×Dic6—91no—
304C2×Dic6—91no—
305C2×Dic6—91no—
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—21no—
313C3×C22⋊C4—31no—
314C3×C22⋊C4—31no—
315C3×C22⋊C4—91no—
316C3×C22⋊C4—91no—
317C3×C22⋊C4C2×D911yescharacteristic
318C3×C22⋊C4D3211yescharacteristic
319C3×C42—31no—
320C3×C42—91no—
321#217—31no—
322#217—31no—
323#217C2×D911yescharacteristic
324#217C2×D911yescharacteristic
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—31no—
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—91no—
348C4⋊Dic3—181no—
349C4⋊Dic3—181no—
350C4⋊Dic3C2×D911yescharacteristic
351Dic3⋊C4—31no—
352Dic3⋊C4—31no—
353Dic3⋊C4—91no—
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⋊C4—91no—
363Dic3⋊C4C2×D911yescharacteristic
364Dic3⋊C4C2×D911yescharacteristic
365C4×Dic3—31no—
366C4×Dic3—91no—
367C4×Dic3—91no—
368C4×Dic3C2×D911yescharacteristic
369C22×C32—21no—
370C22×C32—21no—
371C22×C32—21no—
372C22×C32C23×D311yescharacteristic
373C2×C3⋊D3—61no—
374C2×C3⋊D3—61no—
375C2×C3⋊D3—61no—
376C32×C4—21no—
377C32×C4—21no—
378C3⋊Dic3—61no—
379C3⋊Dic3—61no—
380C3⋊Dic3—61no—
381C3×Dic3—21no—
382C3×Dic3—21no—
383C3×Dic3—21no—
384C3×Dic3—21no—
385C3×Dic3—61no—
386C3×Dic3—61no—
387C3×Dic3—61no—
388C3×Dic3—61no—
389C22×C9—22no—
390C22×C9—21no—
391C22×C9—21no—
392C22×C9—21no—
393C22×C9—21no—
394C22×C9—41no—
395C22×C9—41no—
396C22×C9C23×D311yescharacteristic
397C2×D9—61no—
398C2×D9—61no—
399C2×D9—61no—
400C2×D9—121no—
401C2×D9—121no—
402C2×D9—121no—
403C4×C9—21no—
404C4×C9—21no—
405C4×C9—41no—
406C4×C9—41no—
407C4×C9—61no—
408C4×C9—61no—
409C4×C9—61no—
410C4×C9—61no—
411Dic9—21no—
412Dic9—21no—
413Dic9—21no—
414Dic9—21no—
415Dic9—61no—
416Dic9—61no—
417Dic9—61no—
418Dic9—121no—
419Dic9—121no—
420Dic9—121no—
421C4⋊Q8—271no—
422C4⋊Q8—271no—
423#126—31no—
424#126—91no—
425#126—271no—
426#126—271no—
427#125—271no—
428#125—271no—
429#124—271no—
430#123—271no—
431#123—271no—
432C22⋊Q8—271no—
433C22⋊Q8—271no—
434C22⋊Q8—271no—
435C22⋊Q8—271no—
436C3×C9#56311yescharacteristic, complemented, Sylow
437C23×C3—21no—
438C23×C3C22×D911yescharacteristic
439C23×C3C2×D3211yescharacteristic
440C22×D3—91no—
441C22×D3—91no—
442C22×D3—181no—
443C2×C3×C4—21no—
444C2×C3×C4—21no—
445C2×C3×C4—31no—
446C2×C3×C4—31no—
447C2×C3×C4—31no—
448C2×C3×C4—31no—
449C2×C3×C4—91no—
450C2×C3×C4—91no—
451C2×C3×C4—91no—
452C2×C3×C4—91no—
453C2×C3×C4C22×D911yescharacteristic
454C2×C3×C4C22×D911yescharacteristic
455C2×C3×C4C2×D3211yescharacteristic
456C2×C3×C4C2×D3211yescharacteristic
457#67—361no—
458#67—361no—
459#67—362no—
460C2×Dic3—31no—
461C2×Dic3—31no—
462C2×Dic3—31no—
463C2×Dic3—31no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—91no—
468C2×Dic3—91no—
469C2×Dic3—91no—
470C2×Dic3—181no—
471C2×Dic3—181no—
472C2×Dic3—181no—
473C2×Dic3—361no—
474C2×Dic3—361no—
475C2×Dic3—362no—
476C2×Dic3C22×D911yescharacteristic
477C2×Dic3C22×D911yescharacteristic
478C2×Dic3C22×D911yescharacteristic
479C2×Dic3C22×D911yescharacteristic
480C4×D3—361no—
481C4×D3—361no—
482Dic6—361no—
483Dic6—361no—
484Dic6—361no—
485Dic6—361no—
486Dic6—361no—
487Dic6—361no—
488C2×C32—41no—
489C2×C32#100911yescharacteristic
490C2×C32#101011yescharacteristic
491C2×C32#101011yescharacteristic
492C3⋊D3—121no—
493C2×C9—21no—
494C2×C9—21no—
495C2×C9—21no—
496C2×C9—42no—
497C2×C9—41no—
498C2×C9#100911yescharacteristic
499C2×C9#101011yescharacteristic
500C2×C9#101011yescharacteristic
501D9—121no—
502D9—241no—
503C2×Q8—271no—
504C2×Q8—271no—
505C2×Q8—271no—
506C2×D4—271no—
507C22×C4—271no—
508C22×C4—271no—
509C4⋊C4—31no—
510C4⋊C4—31no—
511C4⋊C4—31no—
512C4⋊C4—91no—
513C4⋊C4—91no—
514C4⋊C4—91no—
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—
524C4⋊C4—271no—
525C22⋊C4—31no—
526C22⋊C4—31no—
527C22⋊C4—91no—
528C22⋊C4—91no—
529C22⋊C4—271no—
530C22⋊C4—271no—
531C22⋊C4—271no—
532C22⋊C4—271no—
533C22⋊C4—271no—
534C22⋊C4D3×D911yescharacteristic
535C42—31no—
536C42—91no—
537C42—271no—
538C22×C3—22no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—21no—
544C22×C3—21no—
545C22×C3—21no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3C23×D911yescharacteristic
549C22×C3C22×D3211yescharacteristic
550C2×D3—181no—
551C2×D3—181no—
552C2×D3—181no—
553C2×D3—181no—
554C2×D3—181no—
555C2×D3—181no—
556C2×D3—361no—
557C2×D3—361no—
558C2×D3—361no—
559C3×C4—21no—
560C3×C4—21no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—41no—
564C3×C4—41no—
565C3×C4—61no—
566C3×C4—61no—
567C3×C4—61no—
568C3×C4—61no—
569C3×C4—181no—
570C3×C4—181no—
571C3×C4—181no—
572C3×C4—181no—
573Dic3—21no—
574Dic3—21no—
575Dic3—21no—
576Dic3—21no—
577Dic3—61no—
578Dic3—61no—
579Dic3—61no—
580Dic3—61no—
581Dic3—181no—
582Dic3—181no—
583Dic3—181no—
584Dic3—181no—
585Dic3—181no—
586Dic3—181no—
587Dic3—361no—
588Dic3—361no—
589Dic3—361no—
590C32#571211yescharacteristic
591C9—21no—
592C9#571211yescharacteristic, complemented
593C23—271no—
594C23C2×D3×D911yescharacteristic
595Q8—1081no—
596Q8—1081no—
597Q8—1081no—
598D4—1081no—
599C2×C4—31no—
600C2×C4—31no—
601C2×C4—31no—
602C2×C4—31no—
603C2×C4—91no—
604C2×C4—91no—
605C2×C4—91no—
606C2×C4—91no—
607C2×C4—271no—
608C2×C4—271no—
609C2×C4—271no—
610C2×C4—1081no—
611C2×C4—1081no—
612C2×C4C2×D3×D911yescharacteristic
613C2×C4C2×D3×D911yescharacteristic
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—42no—
618C2×C3—41no—
619C2×C3—41no—
620C2×C3#6371511yescharacteristic
621C2×C3#6371611yescharacteristic
622C2×C3#6371611yescharacteristic
623C2×C3#6430811yescharacteristic
624C2×C3#6430911yescharacteristic
625C2×C3#6430911yescharacteristic
626D3—361no—
627D3—361no—
628D3—721no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—541no—
633C22—541no—
634C22—541no—
635C22C22×D3×D911yescharacteristic
636C4—21no—
637C4—21no—
638C4—61no—
639C4—61no—
640C4—61no—
641C4—61no—
642C4—181no—
643C4—181no—
644C4—181no—
645C4—181no—
646C4—541no—
647C4—541no—
648C4—541no—
649C3—21no—
650C3#1058981411yescharacteristic, complemented
651C3#1059403611yescharacteristic
652C2—41no—
653C2—1081no—
654C2#1172247811yescharacteristic
655C2#1172249411yescharacteristic
656C2#1172249411yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#6233C22
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#563
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C9×#879C2semidirect
3. C3×#63441C2semidirect
5. #11721267C2semidirect
6. #11721267C2semidirect
8. #11720995C2semidirect
9. #11720994C2semidirect
10. #11720970C2semidirect
11. #11720969C2semidirect
12. #11720934C2semidirect
13. #11720926C2semidirect
14. #11720915C2semidirect
15. #11720914C2semidirect
79. #63441D3semidirect
226. #879D9semidirect
436. C3×C9#563semidirect
592. C9#5712semidirect
650. C3#10589814semidirect

Maximal subgroups

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

subgroupindex
#117217322
C9×#8792
C3×#634412
#117212752
#117212672
#117212422
#117209952
#117209942
#117209702
#117209692
#117209342
#117209262
#117209152
#117209142
#105940363
#105898143