Finite Group Databasebeta

← #11909721

Subgroups — #11909721

Order 1152.

3312 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
1C32×#548C211yesmaximal, characteristic, complemented
2#10596305C211yesmaximal, characteristic, complemented
3#10596298C211yesmaximal, characteristic, complemented
4#10596288C211yesmaximal, characteristic, complemented
5C2×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#10596153C213yesmaximal, complemented
7#10596152C213yesmaximal, complemented
8#10596114C213yesmaximal, complemented
9#10596106C213yesmaximal, complemented
10#10596082C213yesmaximal, complemented
11#10596067C213yesmaximal, complemented
12#10596045C211yesmaximal, characteristic, complemented
13Q8×#634C211yesmaximal, characteristic, complemented
14#10596026C213yesmaximal, complemented
15#10596009C213yesmaximal, complemented
16#86722—34nomaximal
17C2×C32×#41C2211yescharacteristic
18C22×C32×Q8C2211yescharacteristic
19C2×#3832C2211yescharacteristic
20C2×#3830C2216yes—
21C32×#124C2213yes—
22C32×C22⋊Q8C2213yes—
23C32×#121C2213yes—
24C32×C4×Q8C2211yescharacteristic
25C32×C4×D4C2213yes—
26#64156C2213yes—
27#64155C2213yes—
28Q8×C3⋊Dic3C2213yes—
29Q8×C3⋊Dic3C2213yes—
30Q8×C3⋊Dic3C2211yescharacteristic
31Q8×C3⋊Dic3C2211yescharacteristic
32Q8×C3⋊Dic3C2211yescharacteristic
33#64149C2213yes—
34#64148C2213yes—
35#64147C2213yes—
36#64144C2213yes—
37D4×C3⋊Dic3C2213yes—
38D4×C3⋊Dic3C2213yes—
39#64139C2213yes—
40#64139C2213yes—
41C4×#638C2213yes—
42C4×#638C2213yes—
43#64135C2213yes—
44C2×C4⋊(C3⋊Dic3)C2213yes—
45C2×C4⋊(C3⋊Dic3)C2213yes—
46C2×C4×C3⋊Dic3C2213yes—
47#64107C2213yes—
48#64106C2216yes—
49#64102C2213yes—
50C4⋊C4×C3⋊D3C2213yes—
51#64098C2213yes—
52#64097C2213yes—
53#64097C2213yes—
54#64095C2216yes—
55#64095C2213yes—
56#64094C2216yes—
57#64094C2213yes—
58#64092C2216yes—
59#64090C2216yes—
60#64088C2213yes—
61#64087C2216yes—
62#64087C2211yescharacteristic
63C4×C4⋊(C3⋊D3)C2213yes—
64#64081C2213yes—
65#64078C2213yes—
66C4×#634C2213yes—
67C3×#548D314yescomplemented
68#5834—34no—
69#5827—34no—
70#5817—34no—
71C2×Q8×Dic3—34no—
72#5682—312no—
73#5681—312no—
74#5643—312no—
75#5635—312no—
76#5611—312no—
77#5596—312no—
78#5574—34no—
79Q8×Dic6—34no—
80#5555—312no—
81#5538—312no—
82C32×#41—24no—
83C2×C32×Q8—24no—
84C2×C32×Q8C2313yes—
85C2×C32×Q8C2311yescharacteristic
86C2×C32×D4C2313yes—
87C2×C32×D4C2313yes—
88C22×C32×C4C2313yes—
89C22×C32×C4C2313yes—
90C2×#638C2313yes—
91C2×#638C2313yes—
92C2×#638C2316yes—
93C22×C3⋊Dic3C2311yescharacteristic
94C22×C3⋊Dic3C2313yes—
95C22×C3⋊Dic3C2316yes—
96#3832—24no—
97#3830—412no—
98C2×C4⋊(C3⋊D3)C2313yes—
99C2×C4×C3⋊D3C2313yes—
100C2×C4×C3⋊D3C2316yes—
101C2×#634C2311yescharacteristic
102C2×#634C2316yes—
103C32×C4⋊C4C2313yes—
104C32×C4⋊C4C2313yes—
105C32×C22⋊C4C2316yes—
106C32×C22⋊C4C2311yescharacteristic
107C32×C22⋊C4C2313yes—
108C32×C42C2313yes—
109#3757C2313yes—
110#3757C2311yescharacteristic
111#3757C2313yes—
112#3757C2313yes—
113#3752C2313yes—
114#3752C2313yes—
115#3752C2316yes—
116#3752C2313yes—
117#3752C2313yes—
118C4⋊(C3⋊Dic3)C2313yes—
119C4⋊(C3⋊Dic3)C2313yes—
120C4⋊(C3⋊Dic3)C2313yes—
121C4⋊(C3⋊Dic3)C2313yes—
122C4⋊(C3⋊Dic3)C2316yes—
123C4⋊(C3⋊Dic3)C2311yescharacteristic
124C4⋊(C3⋊Dic3)C2313yes—
125#3750C2313yes—
126#3750C2313yes—
127#3750C2311yescharacteristic
128#3750C2313yes—
129#3750C2313yes—
130#3750C2311yescharacteristic
131C4×C3⋊Dic3C2313yes—
132C4×C3⋊Dic3C2313yes—
133C4×C3⋊Dic3C2313yes—
134C4×C3⋊Dic3C2313yes—
135C4×C3⋊Dic3C2313yes—
136#3522—91noSylow
137C2×C3×#41C2×D314yes—
138C22×C3×Q8C2×D314yes—
139C2×#239—34no—
140C2×#237—324no—
141C3×#124C2×D3112yes—
142C3×C22⋊Q8C2×D3112yes—
143C3×#121C2×D3112yes—
144C3×C4×Q8C2×D314yes—
145C3×C4×D4C2×D3112yes—
146#947—312no—
147#946—312no—
148Q8×Dic3—312no—
149Q8×Dic3—312no—
150Q8×Dic3—34no—
151Q8×Dic3—34no—
152Q8×Dic3—34no—
153#940—312no—
154#939—312no—
155#938—312no—
156#935—312no—
157D4×Dic3—312no—
158D4×Dic3—312no—
159#930—312no—
160#930—312no—
161C4×#67—312no—
162C4×#67—312no—
163#926—312no—
164C2×C4⋊Dic3—312no—
165C2×C4⋊Dic3—312no—
166C2×C4×Dic3—312no—
167#898—312no—
168#897—324no—
169#893—312no—
170D3×C4⋊C4—312no—
171#889—312no—
172#888—312no—
173#888—312no—
174#886—324no—
175#886—312no—
176#885—324no—
177#885—312no—
178#883—324no—
179#67⋊C4—324no—
180#879—312no—
181#878—324no—
182#878—34no—
183C4×D12—312no—
184#872—312no—
185#869—312no—
186C4×Dic6—312no—
187C23×C32C2413yes—
188C23×C32C2411yescharacteristic
189C22×C3⋊D3C2413yes—
190C32×Q8—26no—
191C32×Q8#4114yescomplemented
192C32×D4—26no—
193C32×D4—26no—
194C2×C32×C4—26no—
195C2×C32×C4—26no—
196C2×C32×C4C2413yes—
197C2×C32×C4C2413yes—
198C2×C32×C4C2411yescharacteristic
199C2×C32×C4C2411yescharacteristic
200C2×C32×C4C2413yes—
201#638—26no—
202#638—26no—
203#638—46no—
204C2×C3⋊Dic3—26no—
205C2×C3⋊Dic3—22no—
206C2×C3⋊Dic3—46no—
207C2×C3⋊Dic3C2411yescharacteristic
208C2×C3⋊Dic3C2411yescharacteristic
209C2×C3⋊Dic3C2413yes—
210C2×C3⋊Dic3C2413yes—
211C2×C3⋊Dic3C2413yes—
212C2×C3⋊Dic3C2411yescharacteristic
213C2×C3⋊Dic3C2411yescharacteristic
214C4⋊(C3⋊D3)—26no—
215C4×C3⋊D3—26no—
216C4×C3⋊D3—46no—
217#634—22no—
218#634—46no—
219#559—93no—
220Q82—91no—
221#554—93no—
222#550—91no—
223#548—91no—
224#548C3⋊D311yescharacteristic, 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—216no—
235C2×C3×Q8—216no—
236C2×C3×Q8C22×D3112yes—
237C2×C3×Q8C22×D314yes—
238C2×C3×D4C22×D3112yes—
239C2×C3×D4C22×D3112yes—
240C22×C3×C4C22×D3112yes—
241C22×C3×C4C22×D3112yes—
242C2×#67—312no—
243C2×#67—312no—
244C2×#67—324no—
245C22×Dic3—34no—
246C22×Dic3—312no—
247C22×Dic3—324no—
248#239—616no—
249#237—1248no—
250C2×D12—312no—
251C2×C4×D3—312no—
252C2×C4×D3—324no—
253C2×Dic6—34no—
254C2×Dic6—324no—
255C3×C4⋊C4C22×D3112yes—
256C3×C4⋊C4C22×D3112yes—
257C3×C22⋊C4C22×D3124yes—
258C3×C22⋊C4C22×D314yes—
259C3×C22⋊C4C22×D3112yes—
260C3×C42C22×D3112yes—
261#217—312no—
262#217—34no—
263#217—312no—
264#217—312no—
265(C2×D3)⋊C4—312no—
266(C2×D3)⋊C4—312no—
267(C2×D3)⋊C4—324no—
268(C2×D3)⋊C4—312no—
269(C2×D3)⋊C4—312no—
270C4⋊Dic3—312no—
271C4⋊Dic3—312no—
272C4⋊Dic3—312no—
273C4⋊Dic3—312no—
274C4⋊Dic3—324no—
275C4⋊Dic3—34no—
276C4⋊Dic3—312no—
277Dic3⋊C4—312no—
278Dic3⋊C4—312no—
279Dic3⋊C4—34no—
280Dic3⋊C4—312no—
281Dic3⋊C4—312no—
282Dic3⋊C4—34no—
283C4×Dic3—312no—
284C4×Dic3—312no—
285C4×Dic3—312no—
286C4×Dic3—312no—
287C4×Dic3—312no—
288C22×C32—21no—
289C22×C32—21no—
290C22×C32—23no—
291C22×C32—23no—
292C22×C32—23no—
293C22×C32H(5,𝔽2)12yes—
294C22×C32C2511yescharacteristic
295C2×C3⋊D3—23no—
296C2×C3⋊D3—23no—
297C2×C3⋊D3—23no—
298C32×C4—21no—
299C32×C4—23no—
300C32×C4—23no—
301C32×C4—21no—
302C32×C4C2×#4116yes—
303C3⋊Dic3—21no—
304C3⋊Dic3—23no—
305C3⋊Dic3—23no—
306C3⋊Dic3—23no—
307C3⋊Dic3—21no—
308C3⋊Dic3—21no—
309C3⋊Dic32−1+412yes—
310C2×#41—91no—
311C2×#41—96no—
312C2×#41C2×C3⋊D311yescharacteristic
313C22×Q8C2×C3⋊D311yescharacteristic
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×C3⋊D313yes—
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×C3⋊D313yes—
334#121—93no—
335#121—93no—
336#121—93no—
337#121—93no—
338#121C2×C3⋊D313yes—
339C4×Q8—91no—
340C4×Q8—93no—
341C4×Q8—91no—
342C4×Q8—93no—
343C4×Q8—91no—
344C4×Q8—93no—
345C4×Q8C2×C3⋊D311yescharacteristic
346C4×D4—93no—
347C4×D4—93no—
348C4×D4—93no—
349C4×D4—93no—
350C4×D4—93no—
351C4×D4—96no—
352C4×D4—93no—
353C4×D4C2×C3⋊D313yes—
354#117—93no—
355#117—93no—
356C2×C4⋊C4—93no—
357C2×C4⋊C4—93no—
358C2×C4⋊C4—93no—
359C2×C42—93no—
360C23×C3C23×D3112yes—
361C23×C3C23×D314yes—
362C22×D3—312no—
363C3×Q8—224no—
364C3×Q8#237116yescomplemented
365C3×D4—224no—
366C3×D4—224no—
367C2×C3×C4—224no—
368C2×C3×C4—224no—
369C2×C3×C4C23×D3112yes—
370C2×C3×C4C23×D3112yes—
371C2×C3×C4C23×D314yes—
372C2×C3×C4C23×D314yes—
373C2×C3×C4C23×D3112yes—
374#67—624no—
375#67—624no—
376#67—1224no—
377C2×Dic3—34no—
378C2×Dic3—34no—
379C2×Dic3—34no—
380C2×Dic3—312no—
381C2×Dic3—312no—
382C2×Dic3—312no—
383C2×Dic3—34no—
384C2×Dic3—68no—
385C2×Dic3—624no—
386C2×Dic3—1224no—
387D12—624no—
388C4×D3—624no—
389C4×D3—1224no—
390Dic6—68no—
391Dic6—1224no—
392C2×C32—22no—
393C2×C32—43no—
394C2×C32C22×#4111yescharacteristic
395C2×C32C2×H(5,𝔽2)11yescharacteristic
396C2×C32C2×2−1+411yescharacteristic
397C3⋊D3—43no—
398#41—24no—
399#41—184no—
400#41—3612no—
401C2×Q8—24no—
402C2×Q8—91no—
403C2×Q8—96no—
404C2×Q8C22×C3⋊D313yes—
405C2×Q8C22×C3⋊D311yescharacteristic
406C2×D4—93no—
407C2×D4—93no—
408C2×D4—96no—
409C2×D4—93no—
410C2×D4C22×C3⋊D313yes—
411C2×D4C22×C3⋊D313yes—
412C22×C4—91no—
413C22×C4—93no—
414C22×C4—96no—
415C22×C4—93no—
416C22×C4—96no—
417C22×C4C22×C3⋊D313yes—
418C22×C4C22×C3⋊D313yes—
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×C3⋊D313yes—
433C4⋊C4C22×C3⋊D313yes—
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×C3⋊D316yes—
444C22⋊C4C22×C3⋊D311yescharacteristic
445C22⋊C4C22×C3⋊D313yes—
446C42—93no—
447C42—93no—
448C42—93no—
449C42—93no—
450C42—93no—
451C42C22×C3⋊D313yes—
452C22×C3—24no—
453C22×C3—24no—
454C22×C3—212no—
455C22×C3—212no—
456C22×C3—212no—
457C22×C3#100918yes—
458C22×C3C24×D314yes—
459C2×D3—612no—
460C2×D3—612no—
461C2×D3—612no—
462C3×C4—24no—
463C3×C4—212no—
464C3×C4—212no—
465C3×C4—24no—
466C3×C4C2×#237124yes—
467Dic3—38no—
468Dic3—64no—
469Dic3—612no—
470Dic3—612no—
471Dic3—612no—
472Dic3—64no—
473Dic3—64no—
474C32#352211yescharacteristic, complemented, Sylow
475C23—93no—
476C23C23×C3⋊D313yes—
477C23C23×C3⋊D311yescharacteristic
478Q8—26no—
479Q8—182no—
480Q8—366no—
481Q8#383014yescomplemented
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×C3⋊D313yes—
503C2×C4C23×C3⋊D311yescharacteristic
504C2×C4C23×C3⋊D313yes—
505C2×C4C23×C3⋊D311yescharacteristic
506C2×C4C23×C3⋊D313yes—
507C2×C3—28no—
508C2×C3—412no—
509C2×C3C22×#23714yes—
510C2×C3C2×#100914yes—
511C2×C3D3×2−1+414yes—
512D3—1212no—
513C22—21no—
514C22—21no—
515C22—23no—
516C22—23no—
517C22—23no—
518C22—183no—
519C22—183no—
520C22—183no—
521C22#6436612yes—
522C22C24×C3⋊D311yescharacteristic
523C4—21no—
524C4—23no—
525C4—23no—
526C4—21no—
527C4—92no—
528C4—181no—
529C4—183no—
530C4—183no—
531C4—183no—
532C4—181no—
533C4—181no—
534C4C2×#383016yes—
535C3#8672214yescomplemented
536C2—22no—
537C2—43no—
538C2—363no—
539C2C22×#383011yescharacteristic
540C2C2×#6436611yescharacteristic
541C2C3⋊D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×C3⋊D3
commutator, socle, cosocleC22×C32C25
FittingC32×#548C2
hypercenter#548C3⋊D3
hypocenterC32#3522
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C32×#548C2semidirect
2. #10596305C2semidirect
3. #10596298C2semidirect
4. #10596288C2semidirect
5. C2×Q8×C3⋊Dic3C2semidirect
6. #10596153C2semidirect
7. #10596152C2semidirect
8. #10596114C2semidirect
9. #10596106C2semidirect
10. #10596082C2semidirect
11. #10596067C2semidirect
12. #10596045C2semidirect
13. Q8×#634C2semidirect
14. #10596026C2semidirect
15. #10596009C2semidirect
67. C3×#548D3semidirect
191. C32×Q8#41semidirect
224. #548C3⋊D3semidirect
364. C3×Q8#237semidirect
474. C32#3522semidirect
481. Q8#3830semidirect
535. C3#86722semidirect

Maximal subgroups

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

subgroupindex
C32×#5482
#105963052
#105962982
#105962882
C2×Q8×C3⋊Dic32
#105961532
#105961522
#105961142
#105961062
#105960822
#105960672
#105960452
Q8×#6342
#105960262
#105960092
#867223