Finite Group Databasebeta

← #13147974

Subgroups — #13147974

Order 1728.

662 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#11721724C211yesmaximal, characteristic, complemented
2C3×#63445C211yesmaximal, characteristic
3C9×#879C211yesmaximal, characteristic, complemented
4#11721267C211yesmaximal, characteristic, complemented
5#11721260C211yesmaximal, characteristic
6#11721257C211yesmaximal, characteristic
7#11721246C211yesmaximal, characteristic, complemented
8#11720970C211yesmaximal, characteristic, complemented
9#11720949C211yesmaximal, characteristic, complemented
10#11720947C211yesmaximal, characteristic, complemented
11#11720945C211yesmaximal, characteristic, complemented
12#11720939C211yesmaximal, characteristic, complemented
13#11720918C211yesmaximal, characteristic, complemented
14#11720916C211yesmaximal, characteristic, complemented
15#11720905C211yesmaximal, characteristic, complemented
16#10594044—31nomaximal
17#10589814—31nomaximal
18C22×C3⋊Dic9C2211yescharacteristic
19C2×#6226C2211yescharacteristic
20C2×C3×#611C2211yescharacteristic
21C2×C3×C4×D9C2211yescharacteristic
22C2×#6189C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C3×C9×C22⋊C4C2211yescharacteristic
25#88551C2211yescharacteristic
26C4⋊(C3⋊Dic9)C2211yescharacteristic
27#88534C2211yescharacteristic
28#88534C2211yescharacteristic
29C9×#217C2211yescharacteristic
30C9×#217C2211yescharacteristic
31C3×#3676C2211yescharacteristic
32C3×(C2×D9)⋊C4C2211yescharacteristic
33C9×C4⋊Dic3C2211yescharacteristic
34C9×Dic3⋊C4C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C4×C9×Dic3C2211yescharacteristic
37C3×C4⋊Dic9C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39(C2×D9)⋊Dic3C2211yescharacteristic
40(C2×D9)⋊Dic3C2211yescharacteristic
41(C2×D9)⋊Dic3C2211yescharacteristic
42(C2×D9)⋊Dic3C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45Dic3⋊Dic9C2211yescharacteristic
46Dic3⋊Dic9C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64087—31no—
54C3×#883—31no—
55C3×#879D311yescharacteristic
56#63982—31no—
57#63962—31no—
58#63959—31no—
59#63956—31no—
60#63918—31no—
61#63861—31no—
62#63859—31no—
63#63857—31no—
64#63851—31no—
65#63847—31no—
66#63846—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#63445D311yescharacteristic
78#63441—31no—
79#63440—31no—
80#63440—31no—
81#63440—61no—
82#63438—31no—
83#63431—31no—
84C23×C3×C9C2311yescharacteristic
85C22×C3×D9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3⋊Dic9—41no—
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93#6226—41no—
94C3×#611—41no—
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C9×Dic3C2311yescharacteristic
98C2×C9×Dic3C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C2×C3×Dic9C2311yescharacteristic
102C3×C4×D9—41no—
103#6189—41no—
104#6189—41no—
105#5712—91no—
106#5651—91no—
107C22×C3⋊Dic3—31no—
108C2×#634—31no—
109C2×C3×#67—31no—
110C2×C3×C4×D3—31no—
111C2×#627—31no—
112C2×#627—31no—
113C32×C22⋊C4C2×D311yescharacteristic
114#3757—31no—
115C4⋊(C3⋊Dic3)—31no—
116#3750—31no—
117#3750—31no—
118C3×#217—31no—
119C3×#217C2×D311yescharacteristic
120C3×#217C2×D311yescharacteristic
121C3×(C2×D3)⋊C4—31no—
122C3×C4⋊Dic3—31no—
123C3×C4⋊Dic3C2×D311yescharacteristic
124C3×Dic3⋊C4—31no—
125C3×Dic3⋊C4C2×D311yescharacteristic
126C3×Dic3⋊C4C2×D311yescharacteristic
127C3×C4×Dic3C2×D311yescharacteristic
128#3724—31no—
129#3724—31no—
130Dic3⋊Dic3—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—
139(C2×D3)⋊Dic3—31no—
140Dic32—31no—
141Dic32—31no—
142C2×#611C2×D311yescharacteristic
143C22×Dic9—31no—
144C22×Dic9—61no—
145C2×C4×D9C2×D311yescharacteristic
146C2×Dic18—31no—
147C2×Dic18—31no—
148C2×Dic18—31no—
149C2×Dic18—61no—
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#3676—31no—
159#3676—61no—
160#3676C2×D311yescharacteristic
161(C2×D9)⋊C4—31no—
162(C2×D9)⋊C4—31no—
163(C2×D9)⋊C4—31no—
164(C2×D9)⋊C4—31no—
165(C2×D9)⋊C4C2×D311yescharacteristic
166C4⋊Dic9—31no—
167C4⋊Dic9—31no—
168C4⋊Dic9—31no—
169C4⋊Dic9—31no—
170C4⋊Dic9—31no—
171C4⋊Dic9—61no—
172C4⋊Dic9C2×D311yescharacteristic
173Dic9⋊C4—31no—
174Dic9⋊C4—31no—
175Dic9⋊C4—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—31no—
178Dic9⋊C4—31no—
179Dic9⋊C4—61no—
180Dic9⋊C4—61no—
181Dic9⋊C4C2×D311yescharacteristic
182C4×Dic9—31no—
183C4×Dic9—31no—
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—
196C3⋊Dic9—21no—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3—21no—
200C9×Dic3—21no—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×Dic9—21no—
204C3×#126—31no—
205C3×#123—91no—
206#936—91no—
207#935—91no—
208#924—91no—
209#924—91no—
210#898—91no—
211#897—91no—
212#896—91no—
213#890—91no—
214#889—91no—
215#889—91no—
216#889—91no—
217#888—91no—
218#888—91no—
219#888—91no—
220#886—91no—
221#885—91no—
222#883—31no—
223#879—91no—
224#879—91no—
225#879—91no—
226#879—91no—
227#879D911yescharacteristic, complemented
228#878—91no—
229#878—91no—
230#878—91no—
231#878—91no—
232#878—181no—
233#876—91no—
234#869—91no—
235C23×C32C22×D311yescharacteristic
236C22×C3×D3—31no—
237C2×C32×C4C22×D311yescharacteristic
238C2×C32×C4C22×D311yescharacteristic
239C2×C3⋊Dic3—31no—
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242C2×C3⋊Dic3—31no—
243C2×C3⋊Dic3—121no—
244#634—121no—
245C3×#67—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
252C2×C3×Dic3C22×D311yescharacteristic
253C3×C4×D3—121no—
254#627—121no—
255#627—121no—
256C23×C9—21no—
257C23×C9C22×D311yescharacteristic
258C22×D9C22×D311yescharacteristic
259C2×C4×C9—21no—
260C2×C4×C9—21no—
261C2×C4×C9—31no—
262C2×C4×C9—31no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9C22×D311yescharacteristic
266C2×C4×C9C22×D311yescharacteristic
267#611—41no—
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×Dic9—122no—
278C2×Dic9C22×D311yescharacteristic
279C2×Dic9C22×D311yescharacteristic
280C2×Dic9C22×D311yescharacteristic
281C4×D9—41no—
282Dic18—121no—
283Dic18—121no—
284Dic18—121no—
285Dic18—122no—
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×C3×D4—91no—
293C22×C3×C4—91no—
294C2×#67—31no—
295C22×Dic3—91no—
296C22×Dic3—91no—
297C22×Dic3—181no—
298C2×C4×D3—31no—
299C2×Dic6—91no—
300C2×Dic6—91no—
301C2×Dic6—91no—
302C2×Dic6—91no—
303C2×Dic6—91no—
304C2×Dic6—91no—
305C2×Dic6—181no—
306C3×C4⋊C4—31no—
307C3×C4⋊C4—31no—
308C3×C4⋊C4—31no—
309C3×C4⋊C4—91no—
310C3×C4⋊C4—91no—
311C3×C22⋊C4—21no—
312C3×C22⋊C4—31no—
313C3×C22⋊C4—31no—
314C3×C22⋊C4—91no—
315C3×C22⋊C4—91no—
316C3×C22⋊C4C2×D911yescharacteristic
317C3×C22⋊C4D3211yescharacteristic
318C3×C42—31no—
319#217—31no—
320#217—91no—
321#217—91no—
322#217—91no—
323#217—91no—
324#217—91no—
325#217—91no—
326#217—181no—
327#217C2×D911yescharacteristic
328#217C2×D911yescharacteristic
329(C2×D3)⋊C4—31no—
330(C2×D3)⋊C4—91no—
331(C2×D3)⋊C4—91no—
332(C2×D3)⋊C4—91no—
333(C2×D3)⋊C4—91no—
334C4⋊Dic3—31no—
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—181no—
344C4⋊Dic3C2×D911yescharacteristic
345Dic3⋊C4—31no—
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—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—91no—
360Dic3⋊C4—181no—
361Dic3⋊C4—181no—
362Dic3⋊C4C2×D911yescharacteristic
363Dic3⋊C4C2×D911yescharacteristic
364C4×Dic3—91no—
365C4×Dic3—91no—
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—61no—
382C3×Dic3—21no—
383C3×Dic3—21no—
384C3×Dic3—21no—
385C3×Dic3—21no—
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—21no—
398C2×D9—21no—
399C2×D9—21no—
400C4×C9—21no—
401C4×C9—21no—
402C4×C9—41no—
403C4×C9—41no—
404C4×C9—61no—
405C4×C9—61no—
406C4×C9—61no—
407C4×C9—61no—
408Dic9—21no—
409Dic9—21no—
410Dic9—21no—
411Dic9—62no—
412Dic9—61no—
413Dic9—61no—
414Dic9—61no—
415Dic9—61no—
416Dic9—121no—
417Dic9—121no—
418Dic9—121no—
419C4⋊Q8—271no—
420C4⋊Q8—271no—
421#126—31no—
422#126—271no—
423#126—271no—
424#126—271no—
425#125—271no—
426#125—271no—
427#124—271no—
428#123—91no—
429#123—271no—
430C22⋊Q8—271no—
431C22⋊Q8—271no—
432C22⋊Q8—271no—
433C22⋊Q8—271no—
434C3×C9#56311yescharacteristic, complemented, Sylow
435C23×C3—21no—
436C23×C3—91no—
437C23×C3C22×D911yescharacteristic
438C23×C3C2×D3211yescharacteristic
439C22×D3—31no—
440C3×D4—361no—
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×C4—361no—
451C2×C3×C4C22×D911yescharacteristic
452C2×C3×C4C22×D911yescharacteristic
453C2×C3×C4C2×D3211yescharacteristic
454C2×C3×C4C2×D3211yescharacteristic
455#67—121no—
456C2×Dic3—31no—
457C2×Dic3—31no—
458C2×Dic3—31no—
459C2×Dic3—91no—
460C2×Dic3—91no—
461C2×Dic3—91no—
462C2×Dic3—91no—
463C2×Dic3—91no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—181no—
468C2×Dic3—181no—
469C2×Dic3—181no—
470C2×Dic3—181no—
471C2×Dic3—361no—
472C2×Dic3—361no—
473C2×Dic3—362no—
474C2×Dic3C22×D911yescharacteristic
475C2×Dic3C22×D911yescharacteristic
476C2×Dic3C22×D911yescharacteristic
477C2×Dic3C22×D911yescharacteristic
478C4×D3—121no—
479Dic6—361no—
480Dic6—361no—
481Dic6—361no—
482Dic6—361no—
483Dic6—361no—
484Dic6—361no—
485Dic6—362no—
486C2×C32—41no—
487C2×C32#100911yescharacteristic
488C2×C32#101011yescharacteristic
489C2×C32#101011yescharacteristic
490C3×D3—121no—
491C2×C9—21no—
492C2×C9—21no—
493C2×C9—21no—
494C2×C9—42no—
495C2×C9—41no—
496C2×C9#100411yescharacteristic
497C2×C9#101011yescharacteristic
498C2×C9#101011yescharacteristic
499D9—41no—
500C2×Q8—271no—
501C2×Q8—271no—
502C2×Q8—271no—
503C2×D4—91no—
504C22×C4—91no—
505C22×C4—271no—
506C4⋊C4—31no—
507C4⋊C4—31no—
508C4⋊C4—31no—
509C4⋊C4—91no—
510C4⋊C4—91no—
511C4⋊C4—271no—
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—
522C22⋊C4—31no—
523C22⋊C4—31no—
524C22⋊C4—91no—
525C22⋊C4—91no—
526C22⋊C4—271no—
527C22⋊C4—271no—
528C22⋊C4—271no—
529C22⋊C4—271no—
530C22⋊C4—271no—
531C22⋊C4D3×D911yescharacteristic
532C42—31no—
533C42—271no—
534C42—271no—
535C22×C3—22no—
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—21no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3—181no—
546C22×C3—181no—
547C22×C3—181no—
548C22×C3C23×D911yescharacteristic
549C22×C3C22×D3211yescharacteristic
550C2×D3—61no—
551C2×D3—61no—
552C2×D3—61no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—41no—
558C3×C4—41no—
559C3×C4—61no—
560C3×C4—61no—
561C3×C4—61no—
562C3×C4—61no—
563C3×C4—181no—
564C3×C4—181no—
565C3×C4—181no—
566Dic3—21no—
567Dic3—21no—
568Dic3—21no—
569Dic3—21no—
570Dic3—61no—
571Dic3—61no—
572Dic3—61no—
573Dic3—182no—
574Dic3—181no—
575Dic3—181no—
576Dic3—181no—
577Dic3—181no—
578Dic3—181no—
579Dic3—181no—
580Dic3—181no—
581Dic3—181no—
582Dic3—361no—
583Dic3—361no—
584Dic3—361no—
585C32#571211yescharacteristic
586C9—21no—
587C9#565111yescharacteristic, complemented
588C23—91no—
589C23C2×D3×D911yescharacteristic
590Q8—1081no—
591Q8—1081no—
592Q8—1081no—
593D4—361no—
594C2×C4—31no—
595C2×C4—31no—
596C2×C4—31no—
597C2×C4—31no—
598C2×C4—91no—
599C2×C4—91no—
600C2×C4—91no—
601C2×C4—271no—
602C2×C4—271no—
603C2×C4—271no—
604C2×C4—271no—
605C2×C4—361no—
606C2×C4—1081no—
607C2×C4C2×D3×D911yescharacteristic
608C2×C4C2×D3×D911yescharacteristic
609C2×C3—21no—
610C2×C3—21no—
611C2×C3—21no—
612C2×C3—42no—
613C2×C3—41no—
614C2×C3—41no—
615C2×C3—361no—
616C2×C3#6371511yescharacteristic
617C2×C3#6371611yescharacteristic
618C2×C3#6371611yescharacteristic
619C2×C3#6429811yescharacteristic
620C2×C3#6430911yescharacteristic
621C2×C3#6431411yescharacteristic
622D3—121no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—181no—
627C22—181no—
628C22—181no—
629C22C22×D3×D911yescharacteristic
630C4—21no—
631C4—21no—
632C4—61no—
633C4—61no—
634C4—61no—
635C4—61no—
636C4—181no—
637C4—181no—
638C4—181no—
639C4—541no—
640C4—541no—
641C4—541no—
642C4—541no—
643C3—21no—
644C3#1058981411yescharacteristic, complemented
645C3#1059404411yescharacteristic
646C2—41no—
647C2—361no—
648C2#1172246511yescharacteristic
649C2#1172249411yescharacteristic
650C2#1172250111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC2×C3×#611C22
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
1. #11721724C2semidirect
3. C9×#879C2semidirect
4. #11721267C2semidirect
7. #11721246C2semidirect
8. #11720970C2semidirect
9. #11720949C2semidirect
10. #11720947C2semidirect
11. #11720945C2semidirect
12. #11720939C2semidirect
13. #11720918C2semidirect
14. #11720916C2semidirect
15. #11720905C2semidirect
227. #879D9semidirect
434. C3×C9#563semidirect
587. C9#5651semidirect
644. C3#10589814semidirect

Maximal subgroups

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

subgroupindex
#117217242
C3×#634452
C9×#8792
#117212672
#117212602
#117212572
#117212462
#117209702
#117209492
#117209472
#117209452
#117209392
#117209182
#117209162
#117209052
#105940443
#105898143