Finite Group Databasebeta

← #11906322

Subgroups — #11906322

Order 1152.

1590 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#544C211yesmaximal, characteristic, complemented
2#10589831C212yesmaximal, complemented
3#10589829C212yesmaximal, complemented
4D9×C4⋊Q8C212yesmaximal, complemented
5#10589824C212yesmaximal, complemented
6#10589792C212yesmaximal, complemented
7#10589791C212yesmaximal, complemented
8#10589790C212yesmaximal, complemented
9#10589787C212yesmaximal, complemented
10#10589738C214yesmaximal, complemented
11#10589734C214yesmaximal, complemented
12#10589625C211yesmaximal, characteristic
13#10589624C211yesmaximal, characteristic
14#10589621C211yesmaximal, characteristic, complemented
15#10589619C211yesmaximal, characteristic, complemented
16#10589606C211yesmaximal, characteristic
17C2×#63450C211yesmaximal, characteristic, complemented
18#86586—31nomaximal
19C2×Q8×D9C2214yes—
20C2×#3697C2211yescharacteristic
21C22×Dic18C2212yes—
22C9×C4⋊Q8C2212yes—
23C9×C4⋊Q8C2212yes—
24C9×#125C2212yes—
25C9×#125C2212yes—
26C9×C22⋊Q8C2214yes—
27C9×#117C2211yescharacteristic
28C9×#117C2211yescharacteristic
29C2×C9×C4⋊C4C2211yescharacteristic
30#63508C2214yes—
31#63506C2214yes—
32#63506C2214yes—
33C4×#611C2214yes—
34#63488C2211yescharacteristic
35C2×C4⋊Dic9C2211yescharacteristic
36C2×Dic9⋊C4C2212yes—
37C2×C4×Dic9C2211yescharacteristic
38#63459C2214yes—
39#63458C2214yes—
40#63458C2214yes—
41#63458C2214yes—
42#63458C2214yes—
43#63455C2211yescharacteristic
44#63455C2212yes—
45#63455C2211yescharacteristic
46#63454C2212yes—
47#63454C2212yes—
48C4⋊C4×D9C2214yes—
49C4⋊C4×D9C2212yes—
50C4⋊C4×D9C2212yes—
51#63452C2212yes—
52#63452C2211yescharacteristic
53#63452C2211yescharacteristic
54#63451C2214yes—
55#63451C2214yes—
56#63450C2214yes—
57#63450C2214yes—
58#63450C2212yes—
59#63450C2211yescharacteristic
60#63450C2212yes—
61#63450C2212yes—
62#63450C2212yes—
63#63450C2212yes—
64#63450C2211yescharacteristic
65#63449C2214yes—
66#63449C2214yes—
67#63449C2211yescharacteristic
68#63449C2212yes—
69#63449C2211yescharacteristic
70#611⋊C4C2212yes—
71#611⋊C4C2214yes—
72#611⋊C4C2212yes—
73#63440C2212yes—
74#63440C2214yes—
75#63440C2212yes—
76C4×D36C2212yes—
77#63434C2212yes—
78C42×D9C2212yes—
79#63432C2212yes—
80#63431C2212yes—
81C4×Dic18C2212yes—
82C4×Dic18C2212yes—
83C4×Dic18C2212yes—
84C3×#544D311yescharacteristic
85#5729—32no—
86#5727—32no—
87D3×C4⋊Q8—32no—
88#5722—32no—
89#5690—32no—
90#5689—32no—
91#5688—32no—
92#5685—32no—
93#5636—34no—
94#5632—34no—
95#5523—31no—
96#5522—31no—
97#5519—31no—
98#5517—31no—
99#5504—31no—
100C2×#888—31no—
101C2×Q8×C9C2314yes—
102C22×C4×C9C2312yes—
103C22×C4×C9C2311yescharacteristic
104C2×#611C2312yes—
105C22×Dic9C2311yescharacteristic
106C22×Dic9C2311yescharacteristic
107C22×Dic9C2312yes—
108Q8×D9—48no—
109#3697Q818yescomplemented
110C2×D36C2311yescharacteristic
111C2×C4×D9C2312yes—
112C2×C4×D9C2312yes—
113C2×C4×D9C2314yes—
114C2×C4×D9C2314yes—
115C2×C4×D9C2312yes—
116C2×Dic18—44no—
117C2×Dic18C2312yes—
118C2×Dic18C2312yes—
119C2×Dic18C2314yes—
120C2×Dic18C2314yes—
121C2×Dic18C2314yes—
122C2×Dic18C2311yescharacteristic
123C9×C4⋊C4C2312yes—
124C9×C4⋊C4C2312yes—
125C9×C4⋊C4C2311yescharacteristic
126C9×C4⋊C4C2311yescharacteristic
127C9×C4⋊C4C2311yescharacteristic
128C9×C4⋊C4C2311yescharacteristic
129C9×C4⋊C4C2314yes—
130C9×C4⋊C4C2314yes—
131C9×C4⋊C4C2314yes—
132C9×C22⋊C4C2312yes—
133C9×C22⋊C4C2312yes—
134C42×C9C2312yes—
135C42×C9C2312yes—
136#3676C2312yes—
137(C2×D9)⋊C4C2314yes—
138(C2×D9)⋊C4C2312yes—
139(C2×D9)⋊C4C2312yes—
140(C2×D9)⋊C4C2314yes—
141C4⋊Dic9C2311yescharacteristic
142C4⋊Dic9C2311yescharacteristic
143C4⋊Dic9C2312yes—
144C4⋊Dic9C2312yes—
145C4⋊Dic9C2312yes—
146C4⋊Dic9C2312yes—
147C4⋊Dic9C2314yes—
148Dic9⋊C4C2314yes—
149Dic9⋊C4C2312yes—
150Dic9⋊C4C2312yes—
151Dic9⋊C4C2314yes—
152Dic9⋊C4C2312yes—
153Dic9⋊C4C2314yes—
154Dic9⋊C4C2314yes—
155Dic9⋊C4C2314yes—
156Dic9⋊C4C2312yes—
157Dic9⋊C4C2312yes—
158Dic9⋊C4C2314yes—
159Dic9⋊C4C2312yes—
160C4×Dic9C2312yes—
161C4×Dic9C2314yes—
162C4×Dic9C2312yes—
163C4×Dic9C2311yescharacteristic
164C4×Dic9C2311yescharacteristic
165C4×Dic9C2314yes—
166C4×Dic9C2311yescharacteristic
167C4×Dic9C2312yes—
168C4×Dic9C2311yescharacteristic
169#3503—91noSylow
170C2×D3×Q8—34no—
171C2×#235—31no—
172C22×Dic6—32no—
173C3×C4⋊Q8C2×D312yes—
174C3×C4⋊Q8C2×D312yes—
175C3×#125C2×D312yes—
176C3×#125C2×D312yes—
177C3×C22⋊Q8C2×D314yes—
178C3×#117C2×D311yescharacteristic
179C3×#117C2×D311yescharacteristic
180C2×C3×C4⋊C4C2×D311yescharacteristic
181#946—34no—
182#944—34no—
183#944—34no—
184C4×#67—34no—
185#926—31no—
186C2×C4⋊Dic3—31no—
187C2×Dic3⋊C4—32no—
188C2×C4×Dic3—31no—
189#897—34no—
190#896—34no—
191#896—34no—
192#896—34no—
193#896—34no—
194#893—31no—
195#893—32no—
196#893—31no—
197#892—32no—
198#892—32no—
199D3×C4⋊C4—34no—
200D3×C4⋊C4—32no—
201D3×C4⋊C4—32no—
202#890—32no—
203#890—31no—
204#890—31no—
205#889—34no—
206#889—34no—
207#888—34no—
208#888—34no—
209#888—32no—
210#888—31no—
211#888—32no—
212#888—32no—
213#888—32no—
214#888—32no—
215#888—31no—
216#887—34no—
217#887—34no—
218#887—31no—
219#887—32no—
220#887—31no—
221#67⋊C4—32no—
222#67⋊C4—34no—
223#67⋊C4—32no—
224#878—32no—
225#878—34no—
226#878—32no—
227C4×D12—32no—
228#872—32no—
229C42×D3—32no—
230#870—32no—
231#869—32no—
232C4×Dic6—32no—
233C4×Dic6—32no—
234C4×Dic6—32no—
235C23×C9C2411yescharacteristic
236C22×D9C2412yes—
237Q8×C9—44no—
238C2×C4×C9—42no—
239C2×C4×C9C2×Q814yescomplemented
240C2×C4×C9C2412yes—
241C2×C4×C9C2412yes—
242C2×C4×C9C2412yes—
243C2×C4×C9C2414yes—
244C2×C4×C9C2412yes—
245C2×C4×C9C2411yescharacteristic
246C2×C4×C9C2411yescharacteristic
247#611C2×Q818yescomplemented
248C2×Dic9—22no—
249C2×Dic9—22no—
250C2×Dic9—42no—
251C2×Dic9C2412yes—
252C2×Dic9C2411yescharacteristic
253C2×Dic9C2411yescharacteristic
254C2×Dic9C2414yes—
255C2×Dic9C2412yes—
256C2×Dic9C2412yes—
257C2×Dic9C2411yescharacteristic
258C2×Dic9C2411yescharacteristic
259D36C2×Q814yescomplemented
260C4×D9—24no—
261C4×D9—24no—
262C4×D9—44no—
263C4×D9—44no—
264C4×D9C2×Q818yescomplemented
265Dic18—28no—
266Dic18—24no—
267Dic18—24no—
268Dic18—44no—
269Dic18—44no—
270Dic18C2×Q814yescomplemented
271Q82—92no—
272#557—92no—
273#554—92no—
274#554—94no—
275D4×Q8—92no—
276D4×Q8—94no—
277#544—92no—
278#544—92no—
279#544—91no—
280#544D911yescharacteristic, complemented
281#533—92no—
282#533—91no—
283C2×C4⋊Q8—92no—
284C2×C4⋊Q8—91no—
285#520—91no—
286#520—91no—
287C4×#41—91no—
288C2×C3×Q8C22×D314yes—
289C22×C3×C4C22×D312yes—
290C22×C3×C4C22×D311yescharacteristic
291C2×#67—32no—
292C22×Dic3—31no—
293C22×Dic3—31no—
294C22×Dic3—32no—
295D3×Q8—128no—
296#235—38no—
297C2×D12—31no—
298C2×C4×D3—32no—
299C2×C4×D3—32no—
300C2×C4×D3—32no—
301C2×C4×D3—34no—
302C2×C4×D3—34no—
303C2×Dic6—31no—
304C2×Dic6—32no—
305C2×Dic6—32no—
306C2×Dic6—34no—
307C2×Dic6—34no—
308C2×Dic6—34no—
309C2×Dic6—124no—
310C3×C4⋊C4C22×D312yes—
311C3×C4⋊C4C22×D311yescharacteristic
312C3×C4⋊C4C22×D314yes—
313C3×C4⋊C4C22×D314yes—
314C3×C4⋊C4C22×D311yescharacteristic
315C3×C4⋊C4C22×D312yes—
316C3×C4⋊C4C22×D314yes—
317C3×C4⋊C4C22×D311yescharacteristic
318C3×C4⋊C4C22×D311yescharacteristic
319C3×C22⋊C4C22×D312yes—
320C3×C22⋊C4C22×D312yes—
321C3×C42C22×D312yes—
322C3×C42C22×D312yes—
323#217—32no—
324(C2×D3)⋊C4—32no—
325(C2×D3)⋊C4—32no—
326(C2×D3)⋊C4—34no—
327(C2×D3)⋊C4—34no—
328C4⋊Dic3—32no—
329C4⋊Dic3—32no—
330C4⋊Dic3—32no—
331C4⋊Dic3—31no—
332C4⋊Dic3—32no—
333C4⋊Dic3—34no—
334C4⋊Dic3—31no—
335Dic3⋊C4—32no—
336Dic3⋊C4—32no—
337Dic3⋊C4—32no—
338Dic3⋊C4—32no—
339Dic3⋊C4—34no—
340Dic3⋊C4—34no—
341Dic3⋊C4—34no—
342Dic3⋊C4—32no—
343Dic3⋊C4—32no—
344Dic3⋊C4—34no—
345Dic3⋊C4—34no—
346Dic3⋊C4—34no—
347C4×Dic3—31no—
348C4×Dic3—31no—
349C4×Dic3—32no—
350C4×Dic3—34no—
351C4×Dic3—31no—
352C4×Dic3—32no—
353C4×Dic3—32no—
354C4×Dic3—31no—
355C4×Dic3—34no—
356C22×C9—21no—
357C22×C9—21no—
358C22×C9C22×Q812yes—
359C22×C9C2511yescharacteristic
360C2×D9—22no—
361C2×D9—22no—
362C2×D9C22×Q814yes—
363C4×C9—24no—
364C4×C9—22no—
365C4×C9—22no—
366C4×C9—22no—
367C4×C9—22no—
368C4×C9C22×Q812yes—
369C4×C9C22×Q812yes—
370Dic9—22no—
371Dic9—24no—
372Dic9—21no—
373Dic9—22no—
374Dic9—21no—
375Dic9C22×Q814yes—
376Dic9H(5,𝔽2)12yes—
377Dic92−1+412yes—
378C2×#41—91no—
379C22×Q8—94no—
380C22×Q8—92no—
381C4⋊Q8—94no—
382C4⋊Q8—94no—
383C4⋊Q8—91no—
384C4⋊Q8—92no—
385C4⋊Q8—92no—
386C4⋊Q8—94no—
387C4⋊Q8—92no—
388C4⋊Q8—92no—
389C4⋊Q8—91no—
390C4⋊Q8—94no—
391C4⋊Q8—92no—
392C4⋊Q8—92no—
393C4⋊Q8C2×D912yes—
394C4⋊Q8C2×D912yes—
395#125—91no—
396#125—91no—
397#125—94no—
398#125—94no—
399#125—92no—
400#125—92no—
401#125C2×D912yes—
402#125C2×D912yes—
403C22⋊Q8—94no—
404C22⋊Q8—94no—
405C22⋊Q8—94no—
406C22⋊Q8—94no—
407C22⋊Q8—94no—
408C22⋊Q8—94no—
409C22⋊Q8—92no—
410C22⋊Q8—92no—
411C22⋊Q8—94no—
412C22⋊Q8C2×D914yes—
413C4×Q8—91no—
414C4×Q8—91no—
415C4×Q8—92no—
416C4×Q8—94no—
417C4×Q8—94no—
418C4×Q8—92no—
419C4×Q8—92no—
420C4×Q8—92no—
421C4×D4—94no—
422C4×D4—91no—
423C4×D4—91no—
424C4×D4—92no—
425C4×D4—92no—
426C4×D4—92no—
427C4×D4—94no—
428C4×D4—92no—
429#117—91no—
430#117—92no—
431#117—92no—
432#117—92no—
433#117C2×D911yescharacteristic
434#117C2×D911yescharacteristic
435C2×C4⋊C4—91no—
436C2×C4⋊C4—92no—
437C2×C4⋊C4—92no—
438C2×C4⋊C4—92no—
439C2×C4⋊C4—94no—
440C2×C4⋊C4C2×D911yescharacteristic
441C2×C42—91no—
442C2×C42—92no—
443C23×C3C23×D311yescharacteristic
444C22×D3—32no—
445C3×Q8—44no—
446C2×C3×C4—42no—
447C2×C3×C4D3×Q814yes—
448C2×C3×C4C23×D312yes—
449C2×C3×C4C23×D312yes—
450C2×C3×C4C23×D312yes—
451C2×C3×C4C23×D314yes—
452C2×C3×C4C23×D312yes—
453C2×C3×C4C23×D311yescharacteristic
454C2×C3×C4C23×D311yescharacteristic
455#67—38no—
456C2×Dic3—32no—
457C2×Dic3—31no—
458C2×Dic3—31no—
459C2×Dic3—32no—
460C2×Dic3—31no—
461C2×Dic3—31no—
462C2×Dic3—34no—
463C2×Dic3—32no—
464C2×Dic3—62no—
465C2×Dic3—62no—
466C2×Dic3—122no—
467D12—34no—
468C4×D3—38no—
469C4×D3—64no—
470C4×D3—64no—
471C4×D3—124no—
472C4×D3—124no—
473Dic6—34no—
474Dic6—64no—
475Dic6—68no—
476Dic6—64no—
477Dic6—124no—
478Dic6—124no—
479C2×C9—22no—
480C2×C9C23×Q811yescharacteristic
481C2×C9C2×H(5,𝔽2)11yescharacteristic
482C2×C9C2×2−1+411yescharacteristic
483D9—24no—
484#41—98no—
485C2×Q8—91no—
486C2×Q8—92no—
487C2×Q8—94no—
488C2×Q8—92no—
489C2×Q8—94no—
490C2×Q8—94no—
491C2×Q8—364no—
492C2×Q8—368no—
493C2×Q8C22×D914yes—
494C2×D4—92no—
495C2×D4—91no—
496C22×C4—92no—
497C22×C4—91no—
498C22×C4—91no—
499C22×C4—92no—
500C22×C4—92no—
501C22×C4—92no—
502C22×C4—94no—
503C22×C4—94no—
504C22×C4C22×D912yes—
505C22×C4C22×D911yescharacteristic
506C4⋊C4—92no—
507C4⋊C4—91no—
508C4⋊C4—94no—
509C4⋊C4—92no—
510C4⋊C4—94no—
511C4⋊C4—94no—
512C4⋊C4—92no—
513C4⋊C4—94no—
514C4⋊C4—94no—
515C4⋊C4—91no—
516C4⋊C4—92no—
517C4⋊C4—92no—
518C4⋊C4—92no—
519C4⋊C4—94no—
520C4⋊C4—92no—
521C4⋊C4—92no—
522C4⋊C4—92no—
523C4⋊C4—94no—
524C4⋊C4—92no—
525C4⋊C4C22×D911yescharacteristic
526C4⋊C4C22×D912yes—
527C4⋊C4C22×D914yes—
528C4⋊C4C22×D912yes—
529C4⋊C4C22×D911yescharacteristic
530C4⋊C4C22×D911yescharacteristic
531C4⋊C4C22×D911yescharacteristic
532C4⋊C4C22×D914yes—
533C4⋊C4C22×D914yes—
534C22⋊C4—92no—
535C22⋊C4—94no—
536C22⋊C4—92no—
537C22⋊C4—92no—
538C22⋊C4—94no—
539C22⋊C4C22×D912yes—
540C22⋊C4C22×D912yes—
541C42—91no—
542C42—91no—
543C42—92no—
544C42—92no—
545C42—91no—
546C42—91no—
547C42—94no—
548C42—94no—
549C42—92no—
550C42C22×D912yes—
551C42C22×D912yes—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3C2×D3×Q812yes—
555C22×C3C24×D311yescharacteristic
556C2×D3—34no—
557C2×D3—62no—
558C2×D3—62no—
559C3×C4—24no—
560C3×C4—22no—
561C3×C4—22no—
562C3×C4—22no—
563C3×C4—22no—
564C3×C4C2×D3×Q812yes—
565C3×C4C2×D3×Q812yes—
566Dic3—34no—
567Dic3—32no—
568Dic3—32no—
569Dic3—62no—
570Dic3—64no—
571Dic3—61no—
572Dic3—62no—
573Dic3—61no—
574C9#350311yescharacteristic, complemented, Sylow
575C23—92no—
576C23C23×D911yescharacteristic
577Q8—44no—
578Q8—94no—
579Q8—188no—
580Q8—184no—
581Q8—184no—
582Q8—364no—
583Q8—364no—
584D4—98no—
585D4—94no—
586C2×C4—42no—
587C2×C4—92no—
588C2×C4—91no—
589C2×C4—91no—
590C2×C4—98no—
591C2×C4—92no—
592C2×C4—91no—
593C2×C4—91no—
594C2×C4—92no—
595C2×C4—94no—
596C2×C4—184no—
597C2×C4—184no—
598C2×C4—182no—
599C2×C4—182no—
600C2×C4—364no—
601C2×C4—364no—
602C2×C4—362no—
603C2×C4Q8×D914yescomplemented
604C2×C4C23×D912yes—
605C2×C4C23×D912yes—
606C2×C4C23×D912yes—
607C2×C4C23×D914yes—
608C2×C4C23×D912yes—
609C2×C4C23×D911yescharacteristic
610C2×C4C23×D911yescharacteristic
611C2×C3—22no—
612C2×C3C22×D3×Q811yescharacteristic
613C2×C3D3×H(5,𝔽2)11yescharacteristic
614C2×C3D3×2−1+411yescharacteristic
615D3—64no—
616C22—21no—
617C22—21no—
618C22—94no—
619C22—182no—
620C22—182no—
621C22C2×Q8×D912yes—
622C22C24×D911yescharacteristic
623C4—24no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—94no—
629C4—92no—
630C4—92no—
631C4—181no—
632C4—182no—
633C4—184no—
634C4—181no—
635C4—182no—
636C4C2×Q8×D912yes—
637C4C2×Q8×D912yes—
638C3#8658611yescharacteristic
639C2—22no—
640C2—184no—
641C2C22×Q8×D911yescharacteristic
642C2D9×H(5,𝔽2)11yescharacteristic
643C2D9×2−1+411yescharacteristic
subgroupG/H
centerC22C24×D9
commutator, cosocleC22×C9C25
Frattini, socleC22×C3C24×D3
FittingC9×#544C2
prime radicalC2×#3697C22
hypercenter#544D9
hypocenterC9#3503
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #544

Complements

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

NH ≅ G/Ntype
1. C9×#544C2semidirect
2. #10589831C2semidirect
3. #10589829C2semidirect
4. D9×C4⋊Q8C2semidirect
5. #10589824C2semidirect
6. #10589792C2semidirect
7. #10589791C2semidirect
8. #10589790C2semidirect
9. #10589787C2semidirect
10. #10589738C2semidirect
11. #10589734C2semidirect
14. #10589621C2semidirect
15. #10589619C2semidirect
17. C2×#63450C2semidirect
109. #3697Q8semidirect
239. C2×C4×C9C2×Q8semidirect
247. #611C2×Q8semidirect
259. D36C2×Q8semidirect
264. C4×D9C2×Q8semidirect
270. Dic18C2×Q8semidirect
280. #544D9semidirect
574. C9#3503semidirect
603. C2×C4Q8×D9semidirect

Maximal subgroups

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

subgroupindex
C9×#5442
#105898312
#105898292
D9×C4⋊Q82
#105898242
#105897922
#105897912
#105897902
#105897872
#105897382
#105897342
#105896252
#105896242
#105896212
#105896192
#105896062
C2×#634502
#865863