Finite Group Databasebeta

← #11909851

Subgroups — #11909851

Order 1152.

1488 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#519C211yesmaximal, characteristic, complemented
2Q8×#611C211yesmaximal, characteristic, complemented
3Q8×#611C211yesmaximal, characteristic, complemented
4#10589685C213yesmaximal, complemented
5#10589683C213yesmaximal, complemented
6#10589680C213yesmaximal, complemented
7Q8×D36C211yesmaximal, characteristic, complemented
8#10589674C211yesmaximal, characteristic, complemented
9C4×Q8×D9C211yesmaximal, characteristic, complemented
10#10589671C211yesmaximal, characteristic, complemented
11#10589669C213yesmaximal, complemented
12#10589635C213yesmaximal, complemented
13#10589631C213yesmaximal, complemented
14#10589630C213yesmaximal
15#10589620C213yesmaximal, complemented
16#85884—31nomaximal
17C22×Q8×C9C2211yescharacteristic
18C2×Q8×D9C2211yescharacteristic
19C2×Q8×D9C2211yescharacteristic
20C4×Q8×C9C2213yes—
21C4×Q8×C9C2211yescharacteristic
22C4×Q8×C9C2211yescharacteristic
23C4×Q8×C9C2213yes—
24C9×#117C2213yes—
25C9×#117C2213yes—
26#63508C2213yes—
27#63508C2213yes—
28Q8×Dic9C2211yescharacteristic
29Q8×Dic9C2211yescharacteristic
30#63506C2213yes—
31#63506C2213yes—
32C4×#611C2213yes—
33C4×#611C2213yes—
34#63486C2213yes—
35#63486C2213yes—
36#63460C2213yes—
37#63459C2213yes—
38#63459C2213yes—
39#63458C2213yes—
40#63456C2213yes—
41#63454C2213yes—
42C4⋊C4×D9C2213yes—
43#63452C2213yes—
44#63451C2216yes—
45#63451C2216yes—
46#63451C2213yes—
47#63450C2213yes—
48#63449C2213yes—
49#63449C2213yes—
50#63447C2213yes—
51#63446C2211yescharacteristic
52#63445C2213yes—
53#63441C2213yes—
54#63441C2216yes—
55#63438C2216yes—
56C4×D36C2213yes—
57#63434C2213yes—
58#63434C2216yes—
59C42×D9C2213yes—
60#63432C2213yes—
61#63432C2216yes—
62#63431C2213yes—
63C4×Dic18C2213yes—
64C4×Dic18C2213yes—
65C4×Dic18C2213yes—
66C4×Dic18C2216yes—
67C3×#519D311yescharacteristic
68Q8×#67—31no—
69Q8×#67—31no—
70#5583—33no—
71#5581—33no—
72#5578—33no—
73Q8×D12—31no—
74#5572—31no—
75C4×D3×Q8—31no—
76#5569—31no—
77#5567—33no—
78#5533—33no—
79#5529—33no—
80#5528—33no—
81#5518—33no—
82C2×Q8×C9—24no—
83C2×Q8×C9C2313yes—
84C2×Q8×C9C2311yescharacteristic
85C22×C4×C9C2313yes—
86C2×#611C2311yescharacteristic
87C2×#611C2311yescharacteristic
88Q8×D9—24no—
89Q8×D9—24no—
90C2×D36C2311yescharacteristic
91C2×C4×D9C2313yes—
92C2×C4×D9C2313yes—
93C2×C4×D9C2311yescharacteristic
94C2×C4×D9C2313yes—
95C2×Dic18C2313yes—
96C2×Dic18C2313yes—
97C2×Dic18C2313yes—
98C9×C4⋊C4C2313yes—
99C9×C4⋊C4C2313yes—
100C9×C4⋊C4C2313yes—
101C9×C4⋊C4C2313yes—
102C9×C22⋊C4C2311yescharacteristic
103C9×C22⋊C4C2313yes—
104C42×C9C2313yes—
105C42×C9C2313yes—
106C42×C9C2316yes—
107#3676C2313yes—
108#3676C2313yes—
109(C2×D9)⋊C4C2313yes—
110(C2×D9)⋊C4C2313yes—
111(C2×D9)⋊C4C2311yescharacteristic
112(C2×D9)⋊C4C2313yes—
113(C2×D9)⋊C4C2313yes—
114(C2×D9)⋊C4C2313yes—
115C4⋊Dic9C2313yes—
116C4⋊Dic9C2313yes—
117C4⋊Dic9C2316yes—
118C4⋊Dic9C2313yes—
119C4⋊Dic9C2313yes—
120Dic9⋊C4C2313yes—
121Dic9⋊C4C2313yes—
122Dic9⋊C4C2316yes—
123Dic9⋊C4C2316yes—
124Dic9⋊C4C2311yescharacteristic
125Dic9⋊C4C2313yes—
126Dic9⋊C4C2313yes—
127Dic9⋊C4C2316yes—
128Dic9⋊C4C2313yes—
129Dic9⋊C4C2316yes—
130C4×Dic9C2313yes—
131C4×Dic9C2313yes—
132C4×Dic9C2313yes—
133C4×Dic9C2313yes—
134C4×Dic9C2316yes—
135#3523—91noSylow
136C22×C3×Q8C2×D311yescharacteristic
137C2×D3×Q8—31no—
138C2×D3×Q8—31no—
139C3×C4×Q8C2×D313yes—
140C3×C4×Q8C2×D311yescharacteristic
141C3×C4×Q8C2×D311yescharacteristic
142C3×C4×Q8C2×D313yes—
143C3×#117C2×D313yes—
144C3×#117C2×D313yes—
145#946—33no—
146#946—33no—
147Q8×Dic3—31no—
148Q8×Dic3—31no—
149#944—33no—
150#944—33no—
151C4×#67—33no—
152C4×#67—33no—
153#924—33no—
154#924—33no—
155#898—33no—
156#897—33no—
157#897—33no—
158#896—33no—
159#894—33no—
160#892—33no—
161D3×C4⋊C4—33no—
162#890—33no—
163#889—36no—
164#889—36no—
165#889—33no—
166#888—33no—
167#887—33no—
168#887—33no—
169#885—33no—
170#884—31no—
171#883—33no—
172#879—33no—
173#879—36no—
174#876—36no—
175C4×D12—33no—
176#872—33no—
177#872—36no—
178C42×D3—33no—
179#870—33no—
180#870—36no—
181#869—33no—
182C4×Dic6—33no—
183C4×Dic6—33no—
184C4×Dic6—33no—
185C4×Dic6—36no—
186C23×C9C2411yescharacteristic
187C22×D9C2411yescharacteristic
188C22×D9C2411yescharacteristic
189Q8×C9—26no—
190Q8×C9#4114yes—
191C2×C4×C9—26no—
192C2×C4×C9C2411yescharacteristic
193C2×C4×C9C2413yes—
194C2×C4×C9C2413yes—
195C2×C4×C9C2411yescharacteristic
196C2×C4×C9C2413yes—
197C2×C4×C9C2413yes—
198#611—22no—
199#611—22no—
200C2×Dic9C2411yescharacteristic
201C2×Dic9C2413yes—
202C2×Dic9C2413yes—
203C2×Dic9C2413yes—
204C2×Dic9C2413yes—
205C2×Dic9C2411yescharacteristic
206D36—22no—
207C4×D9—26no—
208C4×D9—26no—
209C4×D9—26no—
210C4×D9—22no—
211Dic18—26no—
212Dic18—26no—
213Dic18—26no—
214#557—93no—
215#557—91no—
216#556—93no—
217#552—93no—
218#552—93no—
219D4×Q8—91no—
220D4×Q8—91no—
221D4×Q8—91no—
222#541—93no—
223#541—93no—
224#533—93no—
225#529—93no—
226#519—91no—
227#519D911yescharacteristic, complemented
228C2×C4×Q8—91no—
229C2×C3×Q8—24no—
230C2×C3×Q8C22×D313yes—
231C2×C3×Q8C22×D311yescharacteristic
232C22×C3×C4C22×D313yes—
233C2×#67—31no—
234C2×#67—31no—
235D3×Q8—64no—
236D3×Q8—64no—
237C2×D12—31no—
238C2×C4×D3—31no—
239C2×C4×D3—33no—
240C2×C4×D3—33no—
241C2×C4×D3—33no—
242C2×Dic6—33no—
243C2×Dic6—33no—
244C2×Dic6—33no—
245C3×C4⋊C4C22×D313yes—
246C3×C4⋊C4C22×D313yes—
247C3×C4⋊C4C22×D313yes—
248C3×C4⋊C4C22×D313yes—
249C3×C22⋊C4C22×D311yescharacteristic
250C3×C22⋊C4C22×D313yes—
251C3×C42C22×D313yes—
252C3×C42C22×D313yes—
253C3×C42C22×D316yes—
254#217—33no—
255#217—33no—
256(C2×D3)⋊C4—31no—
257(C2×D3)⋊C4—33no—
258(C2×D3)⋊C4—33no—
259(C2×D3)⋊C4—33no—
260(C2×D3)⋊C4—33no—
261(C2×D3)⋊C4—33no—
262C4⋊Dic3—36no—
263C4⋊Dic3—33no—
264C4⋊Dic3—33no—
265C4⋊Dic3—33no—
266C4⋊Dic3—33no—
267Dic3⋊C4—31no—
268Dic3⋊C4—33no—
269Dic3⋊C4—33no—
270Dic3⋊C4—33no—
271Dic3⋊C4—36no—
272Dic3⋊C4—36no—
273Dic3⋊C4—33no—
274Dic3⋊C4—33no—
275Dic3⋊C4—36no—
276Dic3⋊C4—36no—
277C4×Dic3—33no—
278C4×Dic3—33no—
279C4×Dic3—36no—
280C4×Dic3—33no—
281C4×Dic3—33no—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—21no—
285C22×C9C2511yescharacteristic
286C2×D9—21no—
287C2×D9—21no—
288C2×D9—21no—
289C2×D9—21no—
290C2×D9—21no—
291C2×D92−1+412yes—
292C4×C9—23no—
293C4×C9—23no—
294C4×C9—23no—
295C4×C9—21no—
296C4×C9—21no—
297C4×C9C2×#4116yes—
298Dic9—23no—
299Dic9—23no—
300Dic9—23no—
301Dic9—23no—
302Dic9—21no—
303Dic92−1+412yes—
304C22×Q8—91no—
305C22×Q8—91no—
306C22×Q8C2×D911yescharacteristic
307C4⋊Q8—93no—
308C4⋊Q8—93no—
309C4⋊Q8—93no—
310C4⋊Q8—93no—
311#126—93no—
312#126—93no—
313#126—96no—
314#126—96no—
315#125—93no—
316#125—96no—
317#125—96no—
318#125—93no—
319#125—96no—
320#125—93no—
321#124—93no—
322#123—93no—
323#123—93no—
324C22⋊Q8—93no—
325C22⋊Q8—93no—
326C22⋊Q8—93no—
327C22⋊Q8—93no—
328C22⋊Q8—93no—
329C22⋊Q8—93no—
330C22⋊Q8—93no—
331#121—91no—
332C4×Q8—91no—
333C4×Q8—91no—
334C4×Q8—93no—
335C4×Q8—93no—
336C4×Q8—93no—
337C4×Q8—93no—
338C4×Q8—93no—
339C4×Q8—96no—
340C4×Q8C2×D911yescharacteristic
341C4×Q8C2×D913yes—
342C4×Q8C2×D911yescharacteristic
343C4×Q8C2×D913yes—
344C4×D4—93no—
345C4×D4—93no—
346C4×D4—93no—
347#117—93no—
348#117—93no—
349#117—96no—
350#117C2×D913yes—
351#117C2×D913yes—
352C2×C4⋊C4—93no—
353C2×C42—93no—
354C23×C3C23×D311yescharacteristic
355C22×D3—31no—
356C22×D3—31no—
357C3×Q8—26no—
358C3×Q8#23514yes—
359C2×C3×C4—26no—
360C2×C3×C4C23×D311yescharacteristic
361C2×C3×C4C23×D313yes—
362C2×C3×C4C23×D313yes—
363C2×C3×C4C23×D311yescharacteristic
364C2×C3×C4C23×D313yes—
365C2×C3×C4C23×D313yes—
366#67—62no—
367#67—62no—
368C2×Dic3—31no—
369C2×Dic3—31no—
370C2×Dic3—33no—
371C2×Dic3—33no—
372C2×Dic3—33no—
373C2×Dic3—33no—
374D12—62no—
375C4×D3—62no—
376C4×D3—66no—
377C4×D3—66no—
378C4×D3—66no—
379Dic6—66no—
380Dic6—66no—
381Dic6—66no—
382C2×C9—41no—
383C2×C9C22×#4111yescharacteristic
384C2×C9C2×2−1+411yescharacteristic
385C2×C9C2×2−1+411yescharacteristic
386D9—22no—
387D9—41no—
388C2×Q8—24no—
389C2×Q8—93no—
390C2×Q8—93no—
391C2×Q8—93no—
392C2×Q8—184no—
393C2×Q8—184no—
394C2×Q8C22×D913yes—
395C2×Q8C22×D911yescharacteristic
396C2×D4—91no—
397C2×D4—91no—
398C2×D4—91no—
399C22×C4—91no—
400C22×C4—93no—
401C22×C4—93no—
402C22×C4—93no—
403C22×C4C22×D913yes—
404C4⋊C4—93no—
405C4⋊C4—96no—
406C4⋊C4—93no—
407C4⋊C4—93no—
408C4⋊C4—93no—
409C4⋊C4—96no—
410C4⋊C4—96no—
411C4⋊C4—93no—
412C4⋊C4—93no—
413C4⋊C4—93no—
414C4⋊C4—91no—
415C4⋊C4—96no—
416C4⋊C4—96no—
417C4⋊C4—93no—
418C4⋊C4—93no—
419C4⋊C4C22×D913yes—
420C4⋊C4C22×D913yes—
421C4⋊C4C22×D913yes—
422C4⋊C4C22×D913yes—
423C22⋊C4—93no—
424C22⋊C4—93no—
425C22⋊C4—91no—
426C22⋊C4—93no—
427C22⋊C4—93no—
428C22⋊C4—93no—
429C22⋊C4—93no—
430C22⋊C4—93no—
431C22⋊C4C22×D911yescharacteristic
432C22⋊C4C22×D913yes—
433C42—93no—
434C42—93no—
435C42—93no—
436C42—96no—
437C42—93no—
438C42C22×D913yes—
439C42C22×D916yes—
440C42C22×D913yes—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3C24×D311yescharacteristic
445C2×D3—32no—
446C2×D3—61no—
447C2×D3—61no—
448C2×D3—61no—
449C2×D3—61no—
450C2×D3—61no—
451C3×C4—23no—
452C3×C4—23no—
453C3×C4—21no—
454C3×C4—23no—
455C3×C4—21no—
456C3×C4C2×#23516yes—
457Dic3—32no—
458Dic3—63no—
459Dic3—63no—
460Dic3—63no—
461Dic3—63no—
462Dic3—61no—
463C9#352311yescharacteristic, complemented, Sylow
464C23—91no—
465C23—91no—
466C23C23×D911yescharacteristic
467Q8—26no—
468Q8—186no—
469Q8—186no—
470Q8—186no—
471Q8#369714yes—
472D4—182no—
473D4—182no—
474D4—182no—
475C2×C4—26no—
476C2×C4—91no—
477C2×C4—93no—
478C2×C4—91no—
479C2×C4—93no—
480C2×C4—93no—
481C2×C4—93no—
482C2×C4—182no—
483C2×C4—186no—
484C2×C4—186no—
485C2×C4—186no—
486C2×C4C23×D913yes—
487C2×C4C23×D911yescharacteristic
488C2×C4C23×D913yes—
489C2×C4C23×D911yescharacteristic
490C2×C4C23×D913yes—
491C2×C4C23×D913yes—
492C2×C3—41no—
493C2×C3C22×#23511yescharacteristic
494C2×C3D3×2−1+411yescharacteristic
495C2×C3D3×2−1+411yescharacteristic
496D3—62no—
497D3—121no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—92no—
502C22—181no—
503C22—181no—
504C22—181no—
505C22—181no—
506C22—181no—
507C22C24×D911yescharacteristic
508C4—23no—
509C4—21no—
510C4—23no—
511C4—21no—
512C4—23no—
513C4—92no—
514C4—183no—
515C4—183no—
516C4—183no—
517C4—181no—
518C4—183no—
519C4C2×#369716yes—
520C3#8588411yescharacteristic
521C2—41no—
522C2—182no—
523C2—361no—
524C2C22×#369711yescharacteristic
525C2D9×2−1+411yescharacteristic
526C2D9×2−1+411yescharacteristic
subgroupG/H
centerC22C24×D9
commutator, cosocleC22×C9C25
Frattini, socleC22×C3C24×D3
FittingC9×#519C2
prime radical#63446C22
hypercenter#519D9
hypocenterC9#3523
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. C9×#519C2semidirect
2. Q8×#611C2semidirect
3. Q8×#611C2semidirect
4. #10589685C2semidirect
5. #10589683C2semidirect
6. #10589680C2semidirect
7. Q8×D36C2semidirect
8. #10589674C2semidirect
9. C4×Q8×D9C2semidirect
10. #10589671C2semidirect
11. #10589669C2semidirect
12. #10589635C2semidirect
13. #10589631C2semidirect
15. #10589620C2semidirect
227. #519D9semidirect
463. C9#3523semidirect

Maximal subgroups

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

subgroupindex
C9×#5192
Q8×#6112
#105896852
#105896832
#105896802
Q8×D362
#105896742
C4×Q8×D92
#105896712
#105896692
#105896352
#105896312
#105896302
#105896202
#858843