Finite Group Databasebeta

← #11909720

Subgroups — #11909720

Order 1152.

1656 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#548C211yesmaximal, characteristic, complemented
2#10589936C211yesmaximal, characteristic, complemented
3#10589929C211yesmaximal, characteristic, complemented
4#10589919C211yesmaximal, characteristic, complemented
5C2×Q8×Dic9C211yesmaximal, characteristic, complemented
6#10589784C213yesmaximal, complemented
7#10589783C213yesmaximal, complemented
8#10589745C213yesmaximal, complemented
9#10589737C213yesmaximal, complemented
10#10589713C213yesmaximal, complemented
11#10589698C213yesmaximal, complemented
12#10589676C211yesmaximal, characteristic, complemented
13Q8×Dic18C211yesmaximal, characteristic, complemented
14#10589657C213yesmaximal, complemented
15#10589640C213yesmaximal, complemented
16#86722—31nomaximal
17C2×C9×#41C2211yescharacteristic
18C22×Q8×C9C2211yescharacteristic
19C2×#3701C2211yescharacteristic
20C2×#3699C2216yes—
21C9×#124C2213yes—
22C9×C22⋊Q8C2213yes—
23C9×#121C2213yes—
24C4×Q8×C9C2211yescharacteristic
25C4×D4×C9C2213yes—
26#63509C2213yes—
27#63508C2213yes—
28Q8×Dic9C2213yes—
29Q8×Dic9C2213yes—
30Q8×Dic9C2211yescharacteristic
31Q8×Dic9C2211yescharacteristic
32Q8×Dic9C2211yescharacteristic
33#63502C2213yes—
34#63501C2213yes—
35#63500C2213yes—
36#63497C2213yes—
37D4×Dic9C2213yes—
38D4×Dic9C2213yes—
39#63492C2213yes—
40#63492C2213yes—
41C4×#611C2213yes—
42C4×#611C2213yes—
43#63488C2213yes—
44C2×C4⋊Dic9C2213yes—
45C2×C4⋊Dic9C2213yes—
46C2×C4×Dic9C2213yes—
47#63460C2213yes—
48#63459C2216yes—
49#63455C2213yes—
50C4⋊C4×D9C2213yes—
51#63451C2213yes—
52#63450C2213yes—
53#63450C2213yes—
54#63448C2216yes—
55#63448C2213yes—
56#63447C2216yes—
57#63447C2213yes—
58#63445C2216yes—
59#611⋊C4C2216yes—
60#63441C2213yes—
61#63440C2216yes—
62#63440C2211yescharacteristic
63C4×D36C2213yes—
64#63434C2213yes—
65#63431C2213yes—
66C4×Dic18C2213yes—
67C3×#548D311yescharacteristic
68#5834—31no—
69#5827—31no—
70#5817—31no—
71C2×Q8×Dic3—31no—
72#5682—33no—
73#5681—33no—
74#5643—33no—
75#5635—33no—
76#5611—33no—
77#5596—33no—
78#5574—31no—
79Q8×Dic6—31no—
80#5555—33no—
81#5538—33no—
82C9×#41—24no—
83C2×Q8×C9—24no—
84C2×Q8×C9C2313yes—
85C2×Q8×C9C2311yescharacteristic
86C2×D4×C9C2313yes—
87C2×D4×C9C2313yes—
88C22×C4×C9C2313yes—
89C22×C4×C9C2313yes—
90C2×#611C2313yes—
91C2×#611C2313yes—
92C2×#611C2316yes—
93C22×Dic9C2313yes—
94C22×Dic9C2311yescharacteristic
95C22×Dic9C2316yes—
96#3701—24no—
97#3699—412no—
98C2×D36C2313yes—
99C2×C4×D9C2313yes—
100C2×C4×D9C2316yes—
101C2×Dic18C2311yescharacteristic
102C2×Dic18C2316yes—
103C9×C4⋊C4C2313yes—
104C9×C4⋊C4C2313yes—
105C9×C22⋊C4C2311yescharacteristic
106C9×C22⋊C4C2313yes—
107C9×C22⋊C4C2316yes—
108C42×C9C2313yes—
109#3676C2313yes—
110#3676C2313yes—
111#3676C2311yescharacteristic
112#3676C2313yes—
113(C2×D9)⋊C4C2313yes—
114(C2×D9)⋊C4C2313yes—
115(C2×D9)⋊C4C2316yes—
116(C2×D9)⋊C4C2313yes—
117(C2×D9)⋊C4C2313yes—
118C4⋊Dic9C2311yescharacteristic
119C4⋊Dic9C2313yes—
120C4⋊Dic9C2313yes—
121C4⋊Dic9C2313yes—
122C4⋊Dic9C2313yes—
123C4⋊Dic9C2316yes—
124C4⋊Dic9C2313yes—
125Dic9⋊C4C2311yescharacteristic
126Dic9⋊C4C2313yes—
127Dic9⋊C4C2313yes—
128Dic9⋊C4C2313yes—
129Dic9⋊C4C2313yes—
130Dic9⋊C4C2311yescharacteristic
131C4×Dic9C2313yes—
132C4×Dic9C2313yes—
133C4×Dic9C2313yes—
134C4×Dic9C2313yes—
135C4×Dic9C2313yes—
136#3522—91noSylow
137C2×C3×#41C2×D311yescharacteristic
138C22×C3×Q8C2×D311yescharacteristic
139C2×#239—31no—
140C2×#237—36no—
141C3×#124C2×D313yes—
142C3×C22⋊Q8C2×D313yes—
143C3×#121C2×D313yes—
144C3×C4×Q8C2×D311yescharacteristic
145C3×C4×D4C2×D313yes—
146#947—33no—
147#946—33no—
148Q8×Dic3—33no—
149Q8×Dic3—33no—
150Q8×Dic3—31no—
151Q8×Dic3—31no—
152Q8×Dic3—31no—
153#940—33no—
154#939—33no—
155#938—33no—
156#935—33no—
157D4×Dic3—33no—
158D4×Dic3—33no—
159#930—33no—
160#930—33no—
161C4×#67—33no—
162C4×#67—33no—
163#926—33no—
164C2×C4⋊Dic3—33no—
165C2×C4⋊Dic3—33no—
166C2×C4×Dic3—33no—
167#898—33no—
168#897—36no—
169#893—33no—
170D3×C4⋊C4—33no—
171#889—33no—
172#888—33no—
173#888—33no—
174#886—36no—
175#886—33no—
176#885—36no—
177#885—33no—
178#883—36no—
179#67⋊C4—36no—
180#879—33no—
181#878—36no—
182#878—31no—
183C4×D12—33no—
184#872—33no—
185#869—33no—
186C4×Dic6—33no—
187C23×C9C2413yes—
188C23×C9C2411yescharacteristic
189C22×D9C2413yes—
190Q8×C9—26no—
191Q8×C9#4114yescomplemented
192D4×C9—26no—
193D4×C9—26no—
194C2×C4×C9—26no—
195C2×C4×C9—26no—
196C2×C4×C9C2413yes—
197C2×C4×C9C2413yes—
198C2×C4×C9C2411yescharacteristic
199C2×C4×C9C2411yescharacteristic
200C2×C4×C9C2413yes—
201#611—26no—
202#611—26no—
203#611—46no—
204C2×Dic9—26no—
205C2×Dic9—22no—
206C2×Dic9—46no—
207C2×Dic9C2411yescharacteristic
208C2×Dic9C2411yescharacteristic
209C2×Dic9C2413yes—
210C2×Dic9C2413yes—
211C2×Dic9C2413yes—
212C2×Dic9C2411yescharacteristic
213C2×Dic9C2411yescharacteristic
214D36—26no—
215C4×D9—26no—
216C4×D9—46no—
217Dic18—22no—
218Dic18—46no—
219#559—93no—
220Q82—91no—
221#554—93no—
222#550—91no—
223#548—91no—
224#548D911yescharacteristic, complemented
225#547—93no—
226#547—93no—
227#542—93no—
228#542—93no—
229#532—93no—
230#529—93no—
231#520—91no—
232#520—91no—
233C2×C4×Q8—91no—
234C3×#41—24no—
235C2×C3×Q8—24no—
236C2×C3×Q8C22×D313yes—
237C2×C3×Q8C22×D311yescharacteristic
238C2×C3×D4C22×D313yes—
239C2×C3×D4C22×D313yes—
240C22×C3×C4C22×D313yes—
241C22×C3×C4C22×D313yes—
242C2×#67—33no—
243C2×#67—33no—
244C2×#67—36no—
245C22×Dic3—31no—
246C22×Dic3—33no—
247C22×Dic3—36no—
248#239—64no—
249#237—1212no—
250C2×D12—33no—
251C2×C4×D3—33no—
252C2×C4×D3—36no—
253C2×Dic6—31no—
254C2×Dic6—36no—
255C3×C4⋊C4C22×D313yes—
256C3×C4⋊C4C22×D313yes—
257C3×C22⋊C4C22×D316yes—
258C3×C22⋊C4C22×D311yescharacteristic
259C3×C22⋊C4C22×D313yes—
260C3×C42C22×D313yes—
261#217—33no—
262#217—31no—
263#217—33no—
264#217—33no—
265(C2×D3)⋊C4—33no—
266(C2×D3)⋊C4—33no—
267(C2×D3)⋊C4—36no—
268(C2×D3)⋊C4—33no—
269(C2×D3)⋊C4—33no—
270C4⋊Dic3—33no—
271C4⋊Dic3—33no—
272C4⋊Dic3—33no—
273C4⋊Dic3—33no—
274C4⋊Dic3—36no—
275C4⋊Dic3—31no—
276C4⋊Dic3—33no—
277Dic3⋊C4—33no—
278Dic3⋊C4—33no—
279Dic3⋊C4—31no—
280Dic3⋊C4—33no—
281Dic3⋊C4—33no—
282Dic3⋊C4—31no—
283C4×Dic3—33no—
284C4×Dic3—33no—
285C4×Dic3—33no—
286C4×Dic3—33no—
287C4×Dic3—33no—
288C22×C9—21no—
289C22×C9—21no—
290C22×C9—23no—
291C22×C9—23no—
292C22×C9—23no—
293C22×C9H(5,𝔽2)12yes—
294C22×C9C2511yescharacteristic
295C2×D9—23no—
296C2×D9—23no—
297C2×D9—23no—
298C4×C9—21no—
299C4×C9—23no—
300C4×C9—23no—
301C4×C9—21no—
302C4×C9C2×#4116yes—
303Dic9—21no—
304Dic9—23no—
305Dic9—23no—
306Dic9—23no—
307Dic9—21no—
308Dic9—21no—
309Dic92−1+412yes—
310C2×#41—91no—
311C2×#41—96no—
312C2×#41C2×D911yescharacteristic
313C22×Q8C2×D911yescharacteristic
314C4⋊Q8—93no—
315C4⋊Q8—93no—
316C4⋊Q8—93no—
317#127—93no—
318#126—93no—
319#126—93no—
320#125—93no—
321#124—93no—
322#124—93no—
323#124—96no—
324#124C2×D913yes—
325#123—93no—
326#123—93no—
327#123—96no—
328#123—96no—
329C22⋊Q8—93no—
330C22⋊Q8—96no—
331C22⋊Q8—91no—
332C22⋊Q8—96no—
333C22⋊Q8C2×D913yes—
334#121—93no—
335#121—93no—
336#121—93no—
337#121—93no—
338#121C2×D913yes—
339C4×Q8—91no—
340C4×Q8—93no—
341C4×Q8—91no—
342C4×Q8—93no—
343C4×Q8—91no—
344C4×Q8—93no—
345C4×Q8C2×D911yescharacteristic
346C4×D4—93no—
347C4×D4—93no—
348C4×D4—93no—
349C4×D4—93no—
350C4×D4—93no—
351C4×D4—96no—
352C4×D4—93no—
353C4×D4C2×D913yes—
354#117—93no—
355#117—93no—
356C2×C4⋊C4—93no—
357C2×C4⋊C4—93no—
358C2×C4⋊C4—93no—
359C2×C42—93no—
360C23×C3C23×D313yes—
361C23×C3C23×D311yescharacteristic
362C22×D3—33no—
363C3×Q8—26no—
364C3×Q8#23714yes—
365C3×D4—26no—
366C3×D4—26no—
367C2×C3×C4—26no—
368C2×C3×C4—26no—
369C2×C3×C4C23×D313yes—
370C2×C3×C4C23×D313yes—
371C2×C3×C4C23×D311yescharacteristic
372C2×C3×C4C23×D311yescharacteristic
373C2×C3×C4C23×D313yes—
374#67—66no—
375#67—66no—
376#67—126no—
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—31no—
380C2×Dic3—33no—
381C2×Dic3—33no—
382C2×Dic3—33no—
383C2×Dic3—31no—
384C2×Dic3—62no—
385C2×Dic3—66no—
386C2×Dic3—126no—
387D12—66no—
388C4×D3—66no—
389C4×D3—126no—
390Dic6—62no—
391Dic6—126no—
392C2×C9—22no—
393C2×C9—43no—
394C2×C9C22×#4111yescharacteristic
395C2×C9C2×H(5,𝔽2)11yescharacteristic
396C2×C9C2×2−1+411yescharacteristic
397D9—43no—
398#41—24no—
399#41—184no—
400#41—3612no—
401C2×Q8—24no—
402C2×Q8—91no—
403C2×Q8—96no—
404C2×Q8C22×D913yes—
405C2×Q8C22×D911yescharacteristic
406C2×D4—93no—
407C2×D4—93no—
408C2×D4—96no—
409C2×D4—93no—
410C2×D4C22×D913yes—
411C2×D4C22×D913yes—
412C22×C4—91no—
413C22×C4—93no—
414C22×C4—96no—
415C22×C4—93no—
416C22×C4—96no—
417C22×C4C22×D913yes—
418C22×C4C22×D913yes—
419C4⋊C4—93no—
420C4⋊C4—93no—
421C4⋊C4—93no—
422C4⋊C4—93no—
423C4⋊C4—93no—
424C4⋊C4—96no—
425C4⋊C4—91no—
426C4⋊C4—91no—
427C4⋊C4—91no—
428C4⋊C4—93no—
429C4⋊C4—93no—
430C4⋊C4—93no—
431C4⋊C4—93no—
432C4⋊C4C22×D913yes—
433C4⋊C4C22×D913yes—
434C22⋊C4—93no—
435C22⋊C4—93no—
436C22⋊C4—91no—
437C22⋊C4—93no—
438C22⋊C4—93no—
439C22⋊C4—93no—
440C22⋊C4—93no—
441C22⋊C4—96no—
442C22⋊C4—93no—
443C22⋊C4C22×D916yes—
444C22⋊C4C22×D911yescharacteristic
445C22⋊C4C22×D913yes—
446C42—93no—
447C42—93no—
448C42—93no—
449C42—93no—
450C42—93no—
451C42C22×D913yes—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—23no—
455C22×C3—23no—
456C22×C3—23no—
457C22×C3#100912yes—
458C22×C3C24×D311yescharacteristic
459C2×D3—63no—
460C2×D3—63no—
461C2×D3—63no—
462C3×C4—21no—
463C3×C4—23no—
464C3×C4—23no—
465C3×C4—21no—
466C3×C4C2×#23716yes—
467Dic3—32no—
468Dic3—61no—
469Dic3—63no—
470Dic3—63no—
471Dic3—63no—
472Dic3—61no—
473Dic3—61no—
474C9#352211yescharacteristic, complemented, Sylow
475C23—93no—
476C23C23×D913yes—
477C23C23×D911yescharacteristic
478Q8—26no—
479Q8—182no—
480Q8—366no—
481Q8#369914yescomplemented
482D4—26no—
483D4—26no—
484D4—186no—
485D4—186no—
486D4—186no—
487D4—366no—
488C2×C4—26no—
489C2×C4—26no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—91no—
493C2×C4—93no—
494C2×C4—93no—
495C2×C4—93no—
496C2×C4—91no—
497C2×C4—182no—
498C2×C4—186no—
499C2×C4—186no—
500C2×C4—366no—
501C2×C4—366no—
502C2×C4C23×D913yes—
503C2×C4C23×D911yescharacteristic
504C2×C4C23×D913yes—
505C2×C4C23×D911yescharacteristic
506C2×C4C23×D913yes—
507C2×C3—22no—
508C2×C3—43no—
509C2×C3C22×#23711yescharacteristic
510C2×C3C2×#100911yescharacteristic
511C2×C3D3×2−1+411yescharacteristic
512D3—123no—
513C22—21no—
514C22—21no—
515C22—23no—
516C22—23no—
517C22—23no—
518C22—183no—
519C22—183no—
520C22—183no—
521C22#6371512yes—
522C22C24×D911yescharacteristic
523C4—21no—
524C4—23no—
525C4—23no—
526C4—21no—
527C4—92no—
528C4—181no—
529C4—183no—
530C4—183no—
531C4—183no—
532C4—181no—
533C4—181no—
534C4C2×#369916yes—
535C3#8672211yescharacteristic
536C2—22no—
537C2—43no—
538C2—363no—
539C2C22×#369911yescharacteristic
540C2C2×#6371511yescharacteristic
541C2D9×2−1+411yescharacteristic
subgroupG/H
centerC22C24×D9
commutator, cosocleC22×C9C25
Frattini, socleC22×C3C24×D3
FittingC9×#548C2
hypercenter#548D9
hypocenterC9#3522
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C9×#548C2semidirect
2. #10589936C2semidirect
3. #10589929C2semidirect
4. #10589919C2semidirect
5. C2×Q8×Dic9C2semidirect
6. #10589784C2semidirect
7. #10589783C2semidirect
8. #10589745C2semidirect
9. #10589737C2semidirect
10. #10589713C2semidirect
11. #10589698C2semidirect
12. #10589676C2semidirect
13. Q8×Dic18C2semidirect
14. #10589657C2semidirect
15. #10589640C2semidirect
191. Q8×C9#41semidirect
224. #548D9semidirect
474. C9#3522semidirect
481. Q8#3699semidirect

Maximal subgroups

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

subgroupindex
C9×#5482
#105899362
#105899292
#105899192
C2×Q8×Dic92
#105897842
#105897832
#105897452
#105897372
#105897132
#105896982
#105896762
Q8×Dic182
#105896572
#105896402
#867223