Finite Group Databasebeta

← #13150569

Subgroups — #13150569

Order 1728.

1166 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic, complemented
2C2×#88646C211yesmaximal, characteristic, complemented
3C9×#930C211yesmaximal, characteristic, complemented
4C3×#63492C211yesmaximal, characteristic, complemented
5#11721292C212yesmaximal, complemented
6#11721289C212yesmaximal, complemented
7#11721051C212yesmaximal, complemented
8#11721045C211yesmaximal, characteristic, complemented
9#11721042C212yesmaximal, complemented
10#11721007C211yesmaximal, characteristic, complemented
11#11720922C211yesmaximal, characteristic, complemented
12#10595082—31nomaximal
13#10589709—31nomaximal
14C22×D3×D9C2212yes—
15C22×C3×C4×C9C2211yescharacteristic, complemented
16C2×#6233C2211yescharacteristic
17C2×#6233C2211yescharacteristic, complemented
18#88736—24no—
19C2×C4⋊(C3⋊D9)C2211yescharacteristic
20C2×C4×C3⋊D9C2211yescharacteristic
21C2×C4×C3⋊D9C2211yescharacteristic, complemented
22C2×#6226C2211yescharacteristic, complemented
23C2×C9×#67C2212yescomplemented
24C2×C3×#611C2212yescomplemented
25C2×C9×D12C2211yescharacteristic
26C2×C3×D36C2211yescharacteristic
27C2×#6195C2212yescomplemented
28C2×#6194C2212yes—
29C2×#6192C2212yescomplemented
30#88646—24no—
31C9×(C2×D3)⋊C4C2212yescomplemented
32C3×(C2×D9)⋊C4C2212yescomplemented
33C9×C4⋊Dic3C2211yescharacteristic, complemented
34C3×C4⋊Dic9C2211yescharacteristic, complemented
35(C2×D3)⋊Dic9C2212yescomplemented
36#88445C2212yes—
37(C2×D9)⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C2×#3828—31no—
40C2×#3802—31no—
41C3×#930—31no—
42C3×#930D311yescharacteristic
43#63983—32no—
44#63979—32no—
45#63923—32no—
46#63916—31no—
47#63916—31no—
48#63913—32no—
49#63849—31no—
50C2×D4×D9—31no—
51C2×#3697—31no—
52C2×#3697—61no—
53C9×#121—31no—
54#63502—32no—
55#63500—31no—
56#63499—32no—
57#63492D311yescharacteristic, complemented
58#63457—31no—
59#63454—31no—
60#63447—32no—
61#63444—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×C3⋊D9C2311yescharacteristic
65C22×D3×C9C2312yes—
66C22×C3×D9C2312yes—
67C2×D3×D9—24no—
68C2×D3×D9—42no—
69C2×D3×D9—42no—
70C2×C3×C4×C9—22no—
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3×C4×C9C2311yescharacteristic
73#6233—22no—
74#6233—22no—
75C2×C3⋊Dic9C2311yescharacteristic
76C2×C3⋊Dic9C2311yescharacteristic
77C4⋊(C3⋊D9)—22no—
78C4×C3⋊D9—22no—
79C4×C3⋊D9D414yescomplemented
80#6226—22no—
81C9×#67—42no—
82C3×#611—42no—
83C2×C9×Dic3C2312yes—
84C2×C3×Dic9C2312yes—
85C9×D12—22no—
86C3×D36—22no—
87#6195—42no—
88#6194—24no—
89#6192—42no—
90#5607—91no—
91#5607—91no—
92C22×D32—32no—
93C22×C32×C4C2×D311yescharacteristic
94C2×#638—31no—
95C2×#638—31no—
96#3828—64no—
97C2×C4⋊(C3⋊D3)—31no—
98C2×C4×C3⋊D3—31no—
99C2×C4×C3⋊D3—31no—
100C2×#634—31no—
101C2×C3×#67—32no—
102C2×C3×#67C2×D312yes—
103C2×C3×D12—31no—
104C2×C3×D12C2×D311yescharacteristic
105C2×#626—32no—
106C2×#626—32no—
107C2×#625—32no—
108#3802—64no—
109C23×D9—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4C2×D312yes—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3C2×D311yescharacteristic
114#3722—32no—
115(C2×D3)⋊Dic3—32no—
116(C2×D3)⋊Dic3—32no—
117Dic32—32no—
118C2×D4×C9—32no—
119C2×D4×C9—31no—
120C22×C4×C9—21no—
121C22×C4×C9C2×D311yescharacteristic, complemented
122C2×#611—31no—
123C2×#611—31no—
124C2×#611—32no—
125C2×#611—32no—
126C2×#611—61no—
127C2×#611—61no—
128C2×#611C2×D312yescomplemented
129D4×D9—64no—
130#3697—64no—
131#3697—68no—
132C2×D36—31no—
133C2×D36—32no—
134C2×D36—61no—
135C2×D36C2×D311yescharacteristic
136C2×C4×D9—31no—
137C2×C4×D9—31no—
138C2×C4×D9—61no—
139C2×C4×D9—61no—
140C2×Dic18—31no—
141C2×Dic18—61no—
142C9×C4⋊C4—31no—
143C9×C22⋊C4—32no—
144#3676—32no—
145(C2×D9)⋊C4—32no—
146(C2×D9)⋊C4—32no—
147(C2×D9)⋊C4C2×D312yescomplemented
148C4⋊Dic9C2×D311yescharacteristic, complemented
149C4×Dic9—32no—
150C22×C3×C9—21no—
151C22×C3×C9—21no—
152C22×C3×C9—21no—
153C22×C3×C9C2411yescharacteristic
154C2×C3⋊D9—21no—
155C2×C3⋊D9—21no—
156C2×C3⋊D9—21no—
157C2×C3⋊D9—21no—
158C2×C3⋊D9—21no—
159C2×C3⋊D9C2×D412yes—
160C2×D3×C9—22no—
161C2×D3×C9—22no—
162C2×D3×C9—22no—
163C2×C3×D9—22no—
164C2×C3×D9—22no—
165C2×C3×D9—22no—
166D3×D9—48no—
167C3×C4×C9—21no—
168C3×C4×C9C2×D412yes—
169C3⋊Dic9—21no—
170C3⋊Dic9C2×D412yes—
171C9×Dic3—22no—
172C3×Dic9—22no—
173C2×D3×D4—91no—
174C2×D3×D4—91no—
175C2×#235—91no—
176C2×#235—91no—
177C2×#235—181no—
178C3×#121—31no—
179C3×#121—91no—
180#940—92no—
181#940—92no—
182#938—91no—
183#938—91no—
184#937—92no—
185#937—92no—
186#930—31no—
187#930D911yescharacteristic, complemented
188#895—91no—
189#895—91no—
190#892—91no—
191#892—91no—
192#885—92no—
193#885—92no—
194#882—92no—
195#882—92no—
196C23×C32C22×D311yescharacteristic
197C22×C3⋊D3—31no—
198C22×C3⋊D3—31no—
199C22×C3×D3—32no—
200C22×C3×D3C22×D312yes—
201C2×D32—64no—
202C2×D32—122no—
203C2×D32—122no—
204C2×C32×C4—22no—
205C2×C32×C4C22×D311yescharacteristic
206C2×C32×C4C22×D311yescharacteristic
207#638—62no—
208#638—62no—
209C2×C3⋊Dic3—31no—
210C2×C3⋊Dic3—31no—
211C4⋊(C3⋊D3)—62no—
212C4×C3⋊D3—34no—
213C4×C3⋊D3—62no—
214#634—62no—
215C3×#67—42no—
216C3×#67—122no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3C22×D312yes—
219C3×D12—22no—
220C3×D12—62no—
221#626—122no—
222#626—122no—
223#625—64no—
224C23×C9—21no—
225C23×C9—32no—
226C23×C9C22×D311yescharacteristic
227C22×D9—31no—
228C22×D9—31no—
229C22×D9—61no—
230C22×D9—61no—
231C22×D9—64no—
232C22×D9—122no—
233C22×D9—122no—
234C22×D9C22×D312yes—
235D4×C9—62no—
236D4×C9—122no—
237C2×C4×C9—24no—
238C2×C4×C9—22no—
239C2×C4×C9—21no—
240C2×C4×C9—21no—
241C2×C4×C9—32no—
242C2×C4×C9C22×D311yescharacteristic
243C2×C4×C9C22×D311yescharacteristic
244#611—42no—
245#611—64no—
246#611—62no—
247#611—62no—
248#611—64no—
249#611—64no—
250#611—122no—
251C2×Dic9—31no—
252C2×Dic9—31no—
253C2×Dic9—61no—
254C2×Dic9—61no—
255C2×Dic9C22×D312yes—
256D36—22no—
257D36—62no—
258D36—64no—
259D36—122no—
260C4×D9—34no—
261C4×D9—64no—
262C4×D9—64no—
263C4×D9—62no—
264Dic18—62no—
265Dic18—64no—
266#535—271noSylow
267C2×C3×C9—41no—
268C2×C3×C9C22×D411yescharacteristic
269C2×C3×C9H(5,𝔽2)11yescharacteristic
270C2×C3×C9H(5,𝔽2)11yescharacteristic
271C3⋊D9—22no—
272C3⋊D9—41no—
273D3×C9—42no—
274C3×D9—42no—
275C23×D3—92no—
276C23×D3—92no—
277C2×C3×D4—32no—
278C2×C3×D4—31no—
279C2×C3×D4—92no—
280C2×C3×D4—91no—
281C22×C3×C4—21no—
282C22×C3×C4C2×D911yescharacteristic, complemented
283C22×C3×C4D3211yescharacteristic
284C2×#67—32no—
285C2×#67—92no—
286C2×#67—92no—
287C2×#67—91no—
288C2×#67—91no—
289C2×#67—91no—
290C2×#67—91no—
291C2×#67—92no—
292C2×#67—92no—
293C2×#67—181no—
294C2×#67—181no—
295C2×#67C2×D912yescomplemented
296D3×D4—184no—
297D3×D4—184no—
298#235—184no—
299#235—184no—
300#235—188no—
301C2×D12—31no—
302C2×D12—91no—
303C2×D12—91no—
304C2×D12—92no—
305C2×D12—92no—
306C2×D12—181no—
307C2×D12C2×D911yescharacteristic
308C2×C4×D3—91no—
309C2×C4×D3—91no—
310C2×C4×D3—91no—
311C2×C4×D3—91no—
312C2×C4×D3—181no—
313C2×C4×D3—181no—
314C2×Dic6—91no—
315C2×Dic6—91no—
316C2×Dic6—181no—
317C3×C4⋊C4—31no—
318C3×C4⋊C4—91no—
319C3×C22⋊C4—32no—
320C3×C22⋊C4—92no—
321#217—92no—
322#217—92no—
323(C2×D3)⋊C4—32no—
324(C2×D3)⋊C4—92no—
325(C2×D3)⋊C4—92no—
326(C2×D3)⋊C4—92no—
327(C2×D3)⋊C4—92no—
328(C2×D3)⋊C4C2×D912yescomplemented
329C4⋊Dic3—31no—
330C4⋊Dic3C2×D911yescharacteristic, complemented
331C4×Dic3—92no—
332C4×Dic3—92no—
333C22×C32—21no—
334C22×C32—21no—
335C22×C32—21no—
336C22×C32C23×D311yescharacteristic
337C2×C3⋊D3—32no—
338C2×C3⋊D3—61no—
339C2×C3⋊D3—61no—
340C2×C3⋊D3—61no—
341C2×C3⋊D3—61no—
342C2×C3⋊D3—61no—
343C2×C3×D3—22no—
344C2×C3×D3—22no—
345C2×C3×D3—22no—
346C2×C3×D3—62no—
347C2×C3×D3—62no—
348C2×C3×D3—62no—
349D32—128no—
350C32×C4—21no—
351C32×C4D3×D412yes—
352C3⋊Dic3—32no—
353C3⋊Dic3—61no—
354C3×Dic3—22no—
355C3×Dic3—62no—
356C22×C9—22no—
357C22×C9—21no—
358C22×C9—21no—
359C22×C9—21no—
360C22×C9—21no—
361C22×C9—41no—
362C22×C9—41no—
363C22×C9—62no—
364C22×C9—62no—
365C22×C9—62no—
366C22×C9C23×D311yescharacteristic
367C2×D9—22no—
368C2×D9—22no—
369C2×D9—22no—
370C2×D9—32no—
371C2×D9—62no—
372C2×D9—61no—
373C2×D9—61no—
374C2×D9—62no—
375C2×D9—61no—
376C2×D9—61no—
377C2×D9—61no—
378C2×D9—121no—
379C2×D9—121no—
380C2×D9—121no—
381C2×D9—121no—
382C2×D9—128no—
383C4×C9—22no—
384C4×C9—21no—
385C4×C9—22no—
386C4×C9—62no—
387C4×C9D3×D412yes—
388Dic9—22no—
389Dic9—32no—
390Dic9—62no—
391Dic9—62no—
392Dic9—61no—
393C2×#41—271no—
394C22×D4—271no—
395#127—272no—
396#124—272no—
397#121—31no—
398#121—91no—
399#121—271no—
400#121—271no—
401#120—272no—
402#120—272no—
403#117—271no—
404C3×C9#53511yescharacteristic, complemented, Sylow
405C23×C3—21no—
406C23×C3—32no—
407C23×C3—92no—
408C23×C3C22×D911yescharacteristic
409C23×C3C2×D3211yescharacteristic
410C22×D3—32no—
411C22×D3—91no—
412C22×D3—91no—
413C22×D3—91no—
414C22×D3—91no—
415C22×D3—181no—
416C22×D3—181no—
417C22×D3—184no—
418C22×D3—184no—
419C22×D3—362no—
420C22×D3—362no—
421C22×D3—362no—
422C22×D3—362no—
423C22×D3C22×D912yes—
424C3×D4—62no—
425C3×D4—122no—
426C3×D4—182no—
427C3×D4—362no—
428C2×C3×C4—24no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—21no—
432C2×C3×C4—21no—
433C2×C3×C4—32no—
434C2×C3×C4—92no—
435C2×C3×C4C22×D911yescharacteristic
436C2×C3×C4C22×D911yescharacteristic
437C2×C3×C4C2×D3211yescharacteristic
438C2×C3×C4C2×D3211yescharacteristic
439#67—42no—
440#67—122no—
441#67—184no—
442#67—184no—
443#67—182no—
444#67—182no—
445#67—182no—
446#67—182no—
447#67—184no—
448#67—184no—
449#67—362no—
450#67—362no—
451C2×Dic3—32no—
452C2×Dic3—91no—
453C2×Dic3—91no—
454C2×Dic3—91no—
455C2×Dic3—91no—
456C2×Dic3—181no—
457C2×Dic3—181no—
458C2×Dic3C22×D912yes—
459D12—22no—
460D12—62no—
461D12—182no—
462D12—182no—
463D12—184no—
464D12—362no—
465D12—362no—
466C4×D3—94no—
467C4×D3—94no—
468C4×D3—184no—
469C4×D3—184no—
470C4×D3—182no—
471C4×D3—182no—
472Dic6—182no—
473Dic6—182no—
474Dic6—184no—
475C2×C32—41no—
476C2×C32C2×D3×D411yescharacteristic
477C2×C32#100411yescharacteristic
478C2×C32#100911yescharacteristic
479C3⋊D3—62no—
480C3⋊D3—121no—
481C3×D3—42no—
482C3×D3—122no—
483C2×C9—21no—
484C2×C9—21no—
485C2×C9—21no—
486C2×C9—42no—
487C2×C9—41no—
488C2×C9—122no—
489C2×C9C2×D3×D411yescharacteristic
490C2×C9#100411yescharacteristic
491C2×C9#100911yescharacteristic
492D9—42no—
493D9—62no—
494D9—122no—
495D9—122no—
496D9—121no—
497C24—272no—
498#41—544no—
499C2×Q8—271no—
500C2×D4—32no—
501C2×D4—31no—
502C2×D4—92no—
503C2×D4—91no—
504C2×D4—272no—
505C2×D4—272no—
506C2×D4—272no—
507C2×D4—271no—
508C2×D4—271no—
509C2×D4—271no—
510C2×D4—544no—
511C22×C4—271no—
512C22×C4—271no—
513C22×C4D3×D911yescharacteristic, complemented
514C4⋊C4—31no—
515C4⋊C4—91no—
516C22⋊C4—32no—
517C22⋊C4—92no—
518C22⋊C4—272no—
519C22⋊C4—272no—
520C22⋊C4—272no—
521C42—272no—
522C22×C3—22no—
523C22×C3—21no—
524C22×C3—21no—
525C22×C3—21no—
526C22×C3—21no—
527C22×C3—21no—
528C22×C3—21no—
529C22×C3—21no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—62no—
533C22×C3—62no—
534C22×C3—62no—
535C22×C3—182no—
536C22×C3—182no—
537C22×C3—182no—
538C22×C3C23×D911yescharacteristic
539C22×C3C22×D3211yescharacteristic
540C2×D3—22no—
541C2×D3—22no—
542C2×D3—22no—
543C2×D3—62no—
544C2×D3—62no—
545C2×D3—62no—
546C2×D3—92no—
547C2×D3—92no—
548C2×D3—182no—
549C2×D3—181no—
550C2×D3—181no—
551C2×D3—181no—
552C2×D3—181no—
553C2×D3—182no—
554C2×D3—181no—
555C2×D3—181no—
556C2×D3—181no—
557C2×D3—181no—
558C2×D3—181no—
559C2×D3—181no—
560C2×D3—361no—
561C2×D3—361no—
562C2×D3—361no—
563C2×D3—361no—
564C2×D3—368no—
565C2×D3—368no—
566C3×C4—22no—
567C3×C4—21no—
568C3×C4—21no—
569C3×C4—22no—
570C3×C4—62no—
571C3×C4—182no—
572C3×C4D4×D912yes—
573C3×C4#381112yes—
574Dic3—22no—
575Dic3—62no—
576Dic3—92no—
577Dic3—92no—
578Dic3—182no—
579Dic3—182no—
580Dic3—181no—
581Dic3—181no—
582C32#560711yescharacteristic
583C9—21no—
584C9#560711yescharacteristic, complemented
585C23—32no—
586C23—92no—
587C23—271no—
588C23—271no—
589C23—544no—
590C23—1082no—
591C23—1082no—
592C23C2×D3×D911yescharacteristic
593Q8—542no—
594D4—62no—
595D4—122no—
596D4—182no—
597D4—362no—
598D4—542no—
599D4—542no—
600D4—544no—
601D4—542no—
602D4—1082no—
603D4—1082no—
604C2×C4—22no—
605C2×C4—32no—
606C2×C4—92no—
607C2×C4—271no—
608C2×C4—274no—
609C2×C4—271no—
610C2×C4—542no—
611C2×C4C2×D3×D911yescharacteristic
612C2×C4C2×D3×D911yescharacteristic
613C2×C3—21no—
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—42no—
617C2×C3—41no—
618C2×C3—41no—
619C2×C3—122no—
620C2×C3—362no—
621C2×C3C2×D4×D911yescharacteristic
622C2×C3#6371011yescharacteristic
623C2×C3#6371511yescharacteristic
624C2×C3#6430711yescharacteristic
625C2×C3#6430711yescharacteristic
626C2×C3C2×#381111yescharacteristic
627D3—42no—
628D3—122no—
629D3—182no—
630D3—182no—
631D3—362no—
632D3—362no—
633D3—361no—
634D3—361no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—62no—
639C22—62no—
640C22—62no—
641C22—182no—
642C22—182no—
643C22—182no—
644C22—272no—
645C22—541no—
646C22—541no—
647C22—541no—
648C22—541no—
649C22—541no—
650C22—1088no—
651C22C22×D3×D911yescharacteristic
652C4—21no—
653C4—62no—
654C4—182no—
655C4—272no—
656C4—541no—
657C4#8866812yes—
658C3—21no—
659C3#1058970911yescharacteristic, complemented
660C3#1059508211yescharacteristic
661C2—41no—
662C2—122no—
663C2—362no—
664C2—542no—
665C2—1081no—
666C2#1172247611yescharacteristic
667C2#1172247711yescharacteristic
668C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#535
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×#88736C2semidirect
2. C2×#88646C2semidirect
3. C9×#930C2semidirect
4. C3×#63492C2semidirect
5. #11721292C2semidirect
6. #11721289C2semidirect
7. #11721051C2semidirect
8. #11721045C2semidirect
9. #11721042C2semidirect
10. #11721007C2semidirect
11. #11720922C2semidirect
15. C22×C3×C4×C9C22semidirect
17. C2×#6233C22semidirect
21. C2×C4×C3⋊D9C22semidirect
22. C2×#6226C22semidirect
23. C2×C9×#67C22semidirect
24. C2×C3×#611C22semidirect
27. C2×#6195C22semidirect
29. C2×#6192C22semidirect
31. C9×(C2×D3)⋊C4C22semidirect
32. C3×(C2×D9)⋊C4C22semidirect
33. C9×C4⋊Dic3C22semidirect
34. C3×C4⋊Dic9C22semidirect
35. (C2×D3)⋊Dic9C22semidirect
37. (C2×D9)⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
57. #63492D3semidirect
79. C4×C3⋊D9D4semidirect
121. C22×C4×C9C2×D3semidirect
128. C2×#611C2×D3semidirect
147. (C2×D9)⋊C4C2×D3semidirect
148. C4⋊Dic9C2×D3semidirect
187. #930D9semidirect
282. C22×C3×C4C2×D9semidirect
295. C2×#67C2×D9semidirect
328. (C2×D3)⋊C4C2×D9semidirect
330. C4⋊Dic3C2×D9semidirect
404. C3×C9#535semidirect
513. C22×C4D3×D9semidirect
584. C9#5607semidirect
659. C3#10589709semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C2×#886462
C9×#9302
C3×#634922
#117212922
#117212892
#117210512
#117210452
#117210422
#117210072
#117209222
#105950823
#105897093