Finite Group Databasebeta

← #11909738

Subgroups — #11909738

Order 1152.

1656 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#559C211yesmaximal, characteristic, complemented
2#10589820C211yesmaximal, characteristic, complemented
3#10589818C211yesmaximal, characteristic, complemented
4#10589784C212yesmaximal, complemented
5#10589783C212yesmaximal, complemented
6#10589782C212yesmaximal, complemented
7#10589779C212yesmaximal, complemented
8#10589764C214yesmaximal, complemented
9#10589757C214yesmaximal, complemented
10#10589682C211yesmaximal, characteristic, complemented
11#10589681C211yesmaximal, characteristic, complemented
12C4×Q8×D9C211yesmaximal, characteristic, complemented
13Q8×Dic18C211yesmaximal, characteristic, complemented
14#10589653C212yesmaximal, complemented
15#10589650C212yesmaximal, complemented
16#10589648C212yesmaximal, complemented
17#10589641C212yesmaximal, complemented
18#87055—31nomaximal
19C2×Q8×D9C2211yescharacteristic
20C2×#3699C2212yes—
21C2×#3699C2212yes—
22C2×#3697C2214yes—
23C9×#127C2211yescharacteristic
24C9×#124C2212yes—
25C9×#124C2212yes—
26C9×#123C2214yes—
27C4×Q8×C9C2211yescharacteristic
28C4×Q8×C9C2211yescharacteristic
29C4×D4×C9C2212yes—
30C4×D4×C9C2212yes—
31#63509C2211yescharacteristic
32#63508C2212yes—
33#63508C2211yescharacteristic
34Q8×Dic9C2212yes—
35Q8×Dic9C2211yescharacteristic
36#63502C2212yes—
37#63501C2214yes—
38#63500C2212yes—
39#63500C2212yes—
40#63500C2212yes—
41#63498C2212yes—
42D4×Dic9C2212yes—
43D4×Dic9C2212yes—
44D4×Dic9C2212yes—
45C4×#611C2214yes—
46#63486C2214yes—
47C2×Dic9⋊C4C2214yes—
48#63460C2212yes—
49#63458C2214yes—
50#63457C2211yescharacteristic
51#63456C2212yes—
52C4⋊C4×D9C2212yes—
53C4⋊C4×D9C2212yes—
54C4⋊C4×D9C2211yescharacteristic
55#63452C2211yescharacteristic
56#63450C2214yes—
57#63450C2212yes—
58#63449C2212yes—
59#63449C2211yescharacteristic
60#63448C2214yes—
61#63447C2214yes—
62#63447C2214yes—
63#63446C2214yes—
64#63445C2214yes—
65#63445C2214yes—
66#611⋊C4C2214yes—
67#611⋊C4C2214yes—
68#63441C2214yes—
69#63440C2214yes—
70#63440C2214yes—
71#63439C2214yes—
72C4×D36C2212yes—
73C4×D36C2212yes—
74C4×D36C2211yescharacteristic
75#63434C2212yes—
76C42×D9C2212yes—
77C42×D9C2211yescharacteristic
78#63432C2212yes—
79#63431C2212yes—
80#63431C2211yescharacteristic
81C4×Dic18C2212yes—
82C4×Dic18C2212yes—
83C4×Dic18C2211yescharacteristic
84C3×#559D311yescharacteristic
85#5718—31no—
86#5716—31no—
87#5682—32no—
88#5681—32no—
89#5680—32no—
90#5677—32no—
91#5662—34no—
92#5655—34no—
93#5580—31no—
94#5579—31no—
95C4×D3×Q8—31no—
96Q8×Dic6—31no—
97#5551—32no—
98#5548—32no—
99#5546—32no—
100#5539—32no—
101C2×Q8×C9C2311yescharacteristic
102C2×Q8×C9C2311yescharacteristic
103C2×D4×C9C2312yes—
104C2×D4×C9C2312yes—
105C2×D4×C9C2312yes—
106C22×C4×C9C2314yes—
107C2×#611C2314yes—
108C2×#611C2314yes—
109C2×#611C2314yes—
110C22×Dic9C2314yes—
111C22×Dic9C2314yes—
112Q8×D9—24no—
113#3699—28no—
114#3699—44no—
115#3697—48no—
116C2×D36C2311yescharacteristic
117C2×D36C2312yes—
118C2×C4×D9C2311yescharacteristic
119C2×C4×D9C2311yescharacteristic
120C2×C4×D9C2312yes—
121C2×C4×D9C2312yes—
122C2×C4×D9C2312yes—
123C2×C4×D9C2311yescharacteristic
124C2×C4×D9C2314yes—
125C2×Dic18C2312yes—
126C2×Dic18C2312yes—
127C2×Dic18C2314yes—
128C2×Dic18C2311yescharacteristic
129C9×C4⋊C4C2311yescharacteristic
130C9×C4⋊C4C2311yescharacteristic
131C9×C4⋊C4C2312yes—
132C9×C4⋊C4C2312yes—
133C9×C22⋊C4C2314yes—
134C9×C22⋊C4C2314yes—
135C9×C22⋊C4C2314yes—
136C42×C9C2312yes—
137C42×C9C2312yes—
138C42×C9C2311yescharacteristic
139#3676C2314yes—
140#3676C2314yes—
141#3676C2314yes—
142(C2×D9)⋊C4C2312yes—
143(C2×D9)⋊C4C2312yes—
144(C2×D9)⋊C4C2312yes—
145(C2×D9)⋊C4C2314yes—
146(C2×D9)⋊C4C2312yes—
147(C2×D9)⋊C4C2312yes—
148C4⋊Dic9C2311yescharacteristic
149C4⋊Dic9C2312yes—
150C4⋊Dic9C2312yes—
151C4⋊Dic9C2311yescharacteristic
152C4⋊Dic9C2312yes—
153C4⋊Dic9C2314yes—
154C4⋊Dic9C2312yes—
155Dic9⋊C4C2314yes—
156Dic9⋊C4C2312yes—
157Dic9⋊C4C2314yes—
158Dic9⋊C4C2312yes—
159Dic9⋊C4C2312yes—
160Dic9⋊C4C2312yes—
161Dic9⋊C4C2314yes—
162Dic9⋊C4C2312yes—
163C4×Dic9C2312yes—
164C4×Dic9C2311yescharacteristic
165C4×Dic9C2311yescharacteristic
166C4×Dic9C2312yes—
167C4×Dic9C2311yescharacteristic
168C4×Dic9C2314yes—
169C4×Dic9C2312yes—
170#3522—91noSylow
171C2×D3×Q8—31no—
172C2×#237—32no—
173C2×#237—32no—
174C2×#235—34no—
175C3×#127C2×D311yescharacteristic
176C3×#124C2×D312yes—
177C3×#124C2×D312yes—
178C3×#123C2×D314yes—
179C3×C4×Q8C2×D311yescharacteristic
180C3×C4×Q8C2×D311yescharacteristic
181C3×C4×D4C2×D312yes—
182C3×C4×D4C2×D312yes—
183#947—31no—
184#946—32no—
185#946—31no—
186Q8×Dic3—32no—
187Q8×Dic3—31no—
188#940—32no—
189#939—34no—
190#938—32no—
191#938—32no—
192#938—32no—
193#936—32no—
194D4×Dic3—32no—
195D4×Dic3—32no—
196D4×Dic3—32no—
197C4×#67—34no—
198#924—34no—
199C2×Dic3⋊C4—34no—
200#898—32no—
201#896—34no—
202#895—31no—
203#894—32no—
204D3×C4⋊C4—32no—
205D3×C4⋊C4—32no—
206D3×C4⋊C4—31no—
207#890—31no—
208#888—34no—
209#888—32no—
210#887—32no—
211#887—31no—
212#886—34no—
213#885—34no—
214#885—34no—
215#884—34no—
216#883—34no—
217#883—34no—
218#67⋊C4—34no—
219#67⋊C4—34no—
220#879—34no—
221#878—34no—
222#878—34no—
223#877—34no—
224C4×D12—32no—
225C4×D12—32no—
226C4×D12—31no—
227#872—32no—
228C42×D3—32no—
229C42×D3—31no—
230#870—32no—
231#869—32no—
232#869—31no—
233C4×Dic6—32no—
234C4×Dic6—32no—
235C4×Dic6—31no—
236C23×C9C2414yes—
237C22×D9C2412yes—
238C22×D9C2411yescharacteristic
239Q8×C9—22no—
240Q8×C9—22no—
241D4×C9—24no—
242D4×C9—24no—
243D4×C9—42no—
244C2×C4×C9—44no—
245C2×C4×C9C2412yes—
246C2×C4×C9C2412yes—
247C2×C4×C9C2412yes—
248C2×C4×C9C2411yescharacteristic
249C2×C4×C9C2412yes—
250C2×C4×C9C2411yescharacteristic
251C2×C4×C9C2411yescharacteristic
252#611—28no—
253#611—28no—
254#611—44no—
255C2×Dic9—28no—
256C2×Dic9—44no—
257C2×Dic9C2411yescharacteristic
258C2×Dic9C2411yescharacteristic
259C2×Dic9C2412yes—
260C2×Dic9C2411yescharacteristic
261C2×Dic9C2414yes—
262C2×Dic9C2412yes—
263C2×Dic9C2412yes—
264D36—24no—
265D36—22no—
266C4×D9—24no—
267C4×D9—22no—
268C4×D9—24no—
269C4×D9—22no—
270C4×D9—22no—
271C4×D9—24no—
272C4×D9—44no—
273Dic18—24no—
274Dic18—44no—
275Dic18—42no—
276Dic18#4114yescomplemented
277#559—92no—
278#559D911yescharacteristic, complemented
279Q82—91no—
280#554—91no—
281#554—92no—
282#550—91no—
283#548—92no—
284#547—94no—
285#547—92no—
286#542—92no—
287#542—94no—
288#532—91no—
289#532—92no—
290#529—92no—
291#529—91no—
292#520—92no—
293C2×C4×Q8—91no—
294C2×C3×Q8C22×D311yescharacteristic
295C2×C3×Q8C22×D311yescharacteristic
296C2×C3×D4C22×D312yes—
297C2×C3×D4C22×D312yes—
298C2×C3×D4C22×D312yes—
299C22×C3×C4C22×D314yes—
300C2×#67—34no—
301C2×#67—34no—
302C2×#67—34no—
303C22×Dic3—34no—
304C22×Dic3—34no—
305D3×Q8—64no—
306#237—68no—
307#237—124no—
308#235—128no—
309C2×D12—31no—
310C2×D12—32no—
311C2×C4×D3—31no—
312C2×C4×D3—31no—
313C2×C4×D3—32no—
314C2×C4×D3—32no—
315C2×C4×D3—31no—
316C2×C4×D3—32no—
317C2×C4×D3—34no—
318C2×Dic6—31no—
319C2×Dic6—32no—
320C2×Dic6—34no—
321C2×Dic6—32no—
322C3×C4⋊C4C22×D311yescharacteristic
323C3×C4⋊C4C22×D312yes—
324C3×C4⋊C4C22×D311yescharacteristic
325C3×C4⋊C4C22×D312yes—
326C3×C22⋊C4C22×D314yes—
327C3×C22⋊C4C22×D314yes—
328C3×C22⋊C4C22×D314yes—
329C3×C42C22×D311yescharacteristic
330C3×C42C22×D312yes—
331C3×C42C22×D312yes—
332#217—34no—
333#217—34no—
334#217—34no—
335(C2×D3)⋊C4—32no—
336(C2×D3)⋊C4—32no—
337(C2×D3)⋊C4—32no—
338(C2×D3)⋊C4—32no—
339(C2×D3)⋊C4—34no—
340(C2×D3)⋊C4—32no—
341C4⋊Dic3—32no—
342C4⋊Dic3—31no—
343C4⋊Dic3—32no—
344C4⋊Dic3—32no—
345C4⋊Dic3—34no—
346C4⋊Dic3—31no—
347C4⋊Dic3—32no—
348Dic3⋊C4—34no—
349Dic3⋊C4—32no—
350Dic3⋊C4—32no—
351Dic3⋊C4—32no—
352Dic3⋊C4—34no—
353Dic3⋊C4—34no—
354Dic3⋊C4—32no—
355Dic3⋊C4—32no—
356C4×Dic3—32no—
357C4×Dic3—31no—
358C4×Dic3—34no—
359C4×Dic3—31no—
360C4×Dic3—32no—
361C4×Dic3—32no—
362C4×Dic3—31no—
363C22×C9—24no—
364C22×C9—24no—
365C22×C9—24no—
366C22×C9C2511yescharacteristic
367C2×D9—21no—
368C2×D9—21no—
369C2×D9—22no—
370C2×D9—22no—
371C2×D9—22no—
372C2×D9H(5,𝔽2)12yes—
373C4×C9—22no—
374C4×C9—21no—
375C4×C9—22no—
376C4×C9—22no—
377C4×C9—22no—
378C4×C9C2×#4112yes—
379C4×C92−1+412yes—
380Dic9—21no—
381Dic9—22no—
382Dic9—22no—
383Dic9—24no—
384Dic9—21no—
385Dic9—21no—
386Dic9C2×#4114yes—
387C2×#41—92no—
388C2×#41—92no—
389C2×#41—94no—
390C22×Q8—91no—
391C4⋊Q8—92no—
392C4⋊Q8—94no—
393C4⋊Q8—91no—
394C4⋊Q8—92no—
395#127—92no—
396#127C2×D911yescharacteristic
397#126—92no—
398#126—94no—
399#125—91no—
400#125—92no—
401#124—91no—
402#124—92no—
403#124—94no—
404#124—94no—
405#124C2×D912yes—
406#124C2×D912yes—
407#123—92no—
408#123—94no—
409#123—94no—
410#123—94no—
411#123C2×D914yes—
412C22⋊Q8—91no—
413C22⋊Q8—92no—
414C22⋊Q8—94no—
415C22⋊Q8—94no—
416C22⋊Q8—94no—
417C22⋊Q8—94no—
418#121—92no—
419#121—92no—
420#121—92no—
421#121—94no—
422#121—91no—
423#121—94no—
424C4×Q8—91no—
425C4×Q8—92no—
426C4×Q8—91no—
427C4×Q8—92no—
428C4×Q8—91no—
429C4×Q8—92no—
430C4×Q8—92no—
431C4×Q8C2×D911yescharacteristic
432C4×Q8C2×D911yescharacteristic
433C4×D4—92no—
434C4×D4—92no—
435C4×D4—92no—
436C4×D4—94no—
437C4×D4—94no—
438C4×D4—94no—
439C4×D4—92no—
440C4×D4—91no—
441C4×D4—92no—
442C4×D4C2×D912yes—
443C4×D4C2×D912yes—
444#117—94no—
445#117—92no—
446C2×C4⋊C4—94no—
447C2×C4⋊C4—92no—
448C2×C4⋊C4—91no—
449C2×C4⋊C4—92no—
450C2×C42—91no—
451C2×C42—92no—
452C23×C3C23×D314yes—
453C22×D3—31no—
454C22×D3—32no—
455C3×Q8—22no—
456C3×Q8—22no—
457C3×D4—24no—
458C3×D4—24no—
459C3×D4—42no—
460C2×C3×C4—44no—
461C2×C3×C4C23×D312yes—
462C2×C3×C4C23×D312yes—
463C2×C3×C4C23×D312yes—
464C2×C3×C4C23×D311yescharacteristic
465C2×C3×C4C23×D312yes—
466C2×C3×C4C23×D311yescharacteristic
467C2×C3×C4C23×D311yescharacteristic
468#67—68no—
469#67—68no—
470#67—124no—
471C2×Dic3—32no—
472C2×Dic3—31no—
473C2×Dic3—31no—
474C2×Dic3—32no—
475C2×Dic3—31no—
476C2×Dic3—34no—
477C2×Dic3—32no—
478C2×Dic3—68no—
479C2×Dic3—124no—
480D12—64no—
481D12—62no—
482C4×D3—64no—
483C4×D3—64no—
484C4×D3—62no—
485C4×D3—64no—
486C4×D3—62no—
487C4×D3—62no—
488C4×D3—124no—
489Dic6—34no—
490Dic6—64no—
491Dic6—124no—
492Dic6—122no—
493C2×C9—44no—
494C2×C9C22×#4111yescharacteristic
495C2×C9C2×H(5,𝔽2)11yescharacteristic
496C2×C9C2×2−1+411yescharacteristic
497D9—22no—
498D9—42no—
499#41—188no—
500#41—368no—
501#41—364no—
502C2×Q8—91no—
503C2×Q8—92no—
504C2×Q8—94no—
505C2×Q8—92no—
506C2×Q8—184no—
507C2×Q8C22×D911yescharacteristic
508C2×Q8C22×D911yescharacteristic
509C2×D4—94no—
510C2×D4—94no—
511C2×D4—94no—
512C2×D4—91no—
513C2×D4—92no—
514C2×D4C22×D912yes—
515C2×D4C22×D912yes—
516C2×D4C22×D912yes—
517C22×C4—94no—
518C22×C4—94no—
519C22×C4—91no—
520C22×C4—91no—
521C22×C4—92no—
522C22×C4—92no—
523C22×C4—91no—
524C22×C4—92no—
525C22×C4—94no—
526C22×C4C22×D914yes—
527C4⋊C4—92no—
528C4⋊C4—92no—
529C4⋊C4—94no—
530C4⋊C4—94no—
531C4⋊C4—92no—
532C4⋊C4—92no—
533C4⋊C4—91no—
534C4⋊C4—92no—
535C4⋊C4—92no—
536C4⋊C4—92no—
537C4⋊C4—91no—
538C4⋊C4—92no—
539C4⋊C4—94no—
540C4⋊C4—92no—
541C4⋊C4—94no—
542C4⋊C4C22×D911yescharacteristic
543C4⋊C4C22×D912yes—
544C4⋊C4C22×D911yescharacteristic
545C4⋊C4C22×D912yes—
546C22⋊C4—94no—
547C22⋊C4—94no—
548C22⋊C4—94no—
549C22⋊C4—92no—
550C22⋊C4—92no—
551C22⋊C4—92no—
552C22⋊C4—92no—
553C22⋊C4—94no—
554C22⋊C4—92no—
555C22⋊C4C22×D914yes—
556C22⋊C4C22×D914yes—
557C22⋊C4C22×D914yes—
558C42—92no—
559C42—91no—
560C42—91no—
561C42—92no—
562C42—91no—
563C42—92no—
564C42—94no—
565C42C22×D912yes—
566C42C22×D911yescharacteristic
567C42C22×D912yes—
568C22×C3—24no—
569C22×C3—24no—
570C22×C3—24no—
571C22×C3C24×D311yescharacteristic
572C2×D3—32no—
573C2×D3—61no—
574C2×D3—61no—
575C2×D3—62no—
576C2×D3—62no—
577C2×D3—62no—
578C3×C4—22no—
579C3×C4—22no—
580C3×C4—21no—
581C3×C4—22no—
582C3×C4—22no—
583C3×C4D3×#4112yes—
584C3×C4#101012yes—
585Dic3—34no—
586Dic3—61no—
587Dic3—62no—
588Dic3—61no—
589Dic3—62no—
590Dic3—64no—
591Dic3—61no—
592C9#352211yescharacteristic, complemented, Sylow
593C23—91no—
594C23—92no—
595C23C23×D914yes—
596Q8—22no—
597Q8—22no—
598Q8—94no—
599Q8—184no—
600Q8—364no—
601Q8—362no—
602D4—24no—
603D4—24no—
604D4—42no—
605D4—184no—
606D4—188no—
607D4—188no—
608D4—182no—
609D4—364no—
610C2×C4—44no—
611C2×C4—92no—
612C2×C4—91no—
613C2×C4—91no—
614C2×C4—91no—
615C2×C4—92no—
616C2×C4—94no—
617C2×C4—92no—
618C2×C4—184no—
619C2×C4—184no—
620C2×C4—182no—
621C2×C4—188no—
622C2×C4—182no—
623C2×C4—184no—
624C2×C4—182no—
625C2×C4—364no—
626C2×C4—364no—
627C2×C4C23×D912yes—
628C2×C4C23×D912yes—
629C2×C4C23×D911yescharacteristic
630C2×C4C23×D912yes—
631C2×C4C23×D912yes—
632C2×C4C23×D911yescharacteristic
633C2×C4C23×D911yescharacteristic
634C2×C3—44no—
635C2×C3C2×D3×#4111yescharacteristic
636C2×C3C2×#101011yescharacteristic
637C2×C3D3×H(5,𝔽2)11yescharacteristic
638D3—62no—
639D3—122no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—92no—
644C22—181no—
645C22—181no—
646C22—182no—
647C22—182no—
648C22—182no—
649C22C24×D911yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—21no—
655C4—94no—
656C4—182no—
657C4—181no—
658C4—181no—
659C4—181no—
660C4—182no—
661C4—184no—
662C4#41×D912yes—
663C4#6371612yes—
664C3#8705511yescharacteristic
665C2—44no—
666C2—182no—
667C2—362no—
668C2C2×#41×D911yescharacteristic
669C2C2×#6371611yescharacteristic
670C2D9×H(5,𝔽2)11yescharacteristic
subgroupG/H
centerC22C24×D9
commutator, cosocleC22×C9C25
Frattini, socleC22×C3C24×D3
FittingC9×#559C2
hypercenter#559D9
hypocenterC9#3522
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #559

Complements

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

NH ≅ G/Ntype
1. C9×#559C2semidirect
2. #10589820C2semidirect
3. #10589818C2semidirect
4. #10589784C2semidirect
5. #10589783C2semidirect
6. #10589782C2semidirect
7. #10589779C2semidirect
8. #10589764C2semidirect
9. #10589757C2semidirect
10. #10589682C2semidirect
11. #10589681C2semidirect
12. C4×Q8×D9C2semidirect
13. Q8×Dic18C2semidirect
14. #10589653C2semidirect
15. #10589650C2semidirect
16. #10589648C2semidirect
17. #10589641C2semidirect
276. Dic18#41semidirect
278. #559D9semidirect
592. C9#3522semidirect

Maximal subgroups

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

subgroupindex
C9×#5592
#105898202
#105898182
#105897842
#105897832
#105897822
#105897792
#105897642
#105897572
#105896822
#105896812
C4×Q8×D92
Q8×Dic182
#105896532
#105896502
#105896482
#105896412
#870553