Finite Group Databasebeta

← #11865222

Subgroups — #11865222

Order 1152.

1543 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic
2#10596296C211yesmaximal, characteristic, complemented
3C2×#63831C212yesmaximal, complemented
4C3×#5237C211yesmaximal, characteristic, complemented
5#10591649C214yesmaximal, complemented
6#10591513C214yesmaximal, complemented
7#10591110C212yesmaximal, complemented
8#84123—31nomaximal
9#78718—31nomaximal
10C2×C32×#41C2211yescharacteristic
11C22×#634C2211yescharacteristic, complemented
12C3×#1010—22no—
13C3×#1009—22no—
14C3×D3×#41—24no—
15C3×#1007C2212yes—
16C3×#1004—22no—
17C3×#1004C2212yes—
18C2×C3×#235—22no—
19C2×C3×#235C2212yes—
20#64153C2212yescomplemented
21#64145C2212yescomplemented
22#64144—22no—
23#64135C2211yescharacteristic, complemented
24#64133—22no—
25C2×C3×#208C2212yescomplemented
26C2×#3719—22no—
27#63831—24no—
28#63831C2218yescomplemented
29C2×#3716—22no—
30C3×#836C2212yescomplemented
31C3×#833C2212yescomplemented
32#63568C2218yescomplemented
33C3×#585—31no—
34#5964D311yescharacteristic
35#5825—31no—
36#5825—31no—
37#5825—61no—
38C2×#951—32no—
39C2×#909—32no—
40C3×#413—31no—
41#5246—34no—
42#5242—34no—
43#5237D311yescharacteristic, complemented
44#5236—32no—
45#5139—32no—
46#4752—34no—
47#4746—34no—
48C32×#41—22no—
49C32×#41—22no—
50C2×C32×Q8C2311yescharacteristic
51C2×C32×D4D412yes—
52C2×C32×D4C2311yescharacteristic
53C22×C32×C4D412yes—
54C22×C32×C4C2311yescharacteristic
55C22×C3⋊Dic3—21no—
56C2×#634—22no—
57C2×#634—22no—
58C2×#634—21no—
59C2×#634C2312yes—
60C2×C3×#67—24no—
61C2×C3×#67—22no—
62C3×#239—24no—
63C3×D3×Q8—24no—
64C3×#237—28no—
65C3×#237—24no—
66C3×D3×D4—28no—
67C3×D3×D4—24no—
68C3×#235—24no—
69C3×#235—24no—
70C3×#235—22no—
71C3×#235—24no—
72C3×#235—22no—
73C3×#235D414yes—
74C3×#235C2314yes—
75C2×C3×D12—22no—
76C2×C3×D12D412yes—
77C2×C3×C4×D3—24no—
78C2×C3×C4×D3—22no—
79C2×C3×Dic6—22no—
80C2×C3×Dic6D412yes—
81#3757—24no—
82#3757—21no—
83C4⋊(C3⋊Dic3)—21no—
84#3750—24no—
85C4×C3⋊Dic3C2312yes—
86C3×#208—22no—
87C3×#208C2314yes—
88C2×C3×C3⋊C8—22no—
89#3719—28no—
90#3716—28no—
91#3022—91noSylow
92C3×2−1+4—32no—
93C3×2−1+4—62no—
94C3×H(5,𝔽2)—32no—
95C3×H(5,𝔽2)—62no—
96C3×H(5,𝔽2)—62no—
97C2×C3×#41—21no—
98C2×C3×#41—32no—
99C2×C3×#41—64no—
100C2×C3×#41—62no—
101C2×C3×#41C2×D311yescharacteristic
102C2×C3×#41C2×D311yescharacteristic
103#1010—22no—
104#1009—22no—
105D3×#41—24no—
106#1007C2×D312yes—
107#1004—22no—
108#1004C2×D312yes—
109C2×#235—22no—
110C2×#235C2×D312yes—
111C22×Dic6—31no—
112C22×Dic6—31no—
113C22×Dic6—61no—
114C2×C3×M16—32no—
115#951—38no—
116#951—64no—
117#944—32no—
118#944—32no—
119#944—62no—
120C2×#216—62no—
121#936—32no—
122#936—32no—
123#936—62no—
124#935—64no—
125#935—62no—
126#935—62no—
127C2×#214—62no—
128#926—31no—
129#926—31no—
130#926—61no—
131#924—64no—
132#924—62no—
133#924—62no—
134C2×#208C2×D312yescomplemented
135#909—38no—
136#909—64no—
137C2×Dic12—62no—
138C2×#204—62no—
139C3×#101—32no—
140C3×#100—32no—
141#837—38no—
142#836C2×D312yescomplemented
143#833C2×D312yescomplemented
144#825—38no—
145C23×C32C2×D412yes—
146C23×C32C2×D411yescharacteristic
147C22×C3×D3—24no—
148C22×C3×D3—22no—
149C32×Q8—22no—
150C32×D4—24no—
151C32×D4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4C2×D412yes—
155C2×C32×C4C2×D411yescharacteristic
156C2×C32×C4C2×D412yes—
157C2×C32×C4C2×D412yes—
158C2×C32×C4C2411yescharacteristic
159C2×C3⋊Dic3—22no—
160C2×C3⋊Dic3—22no—
161C2×C3⋊Dic3—42no—
162#634—24no—
163#634—41no—
164#634—41no—
165C3×#67—28no—
166C3×#67—24no—
167C3×#67—24no—
168C2×C3×Dic3—24no—
169C2×C3×Dic3—22no—
170C3×D12—24no—
171C3×D12—22no—
172C3×D12C2×D414yes—
173C3×C4×D3—28no—
174C3×C4×D3—24no—
175C3×C4×D3—24no—
176C3×Dic6—24no—
177C3×Dic6—22no—
178C3×Dic6C2×D414yes—
179C3×C3⋊C8—24no—
180#585—31no—
181C2×#137—92no—
182#536—91no—
183#456—94no—
184#455—94no—
185#421—92no—
186#413—31no—
187C3×#41—22no—
188C3×#41—22no—
189C3×#41—22no—
190C3×#41—22no—
191C3×#41—34no—
192C3×#41—34no—
193C3×#41—42no—
194C3×#41—42no—
195C3×#41—64no—
196C3×#41—64no—
197C3×#41—64no—
198C3×#41—68no—
199C3×#41—64no—
200C3×#41—62no—
201C3×#41—64no—
202C3×#41—62no—
203C2×C3×Q8—21no—
204C2×C3×Q8—32no—
205C2×C3×Q8—62no—
206C2×C3×Q8—64no—
207C2×C3×Q8C22×D311yescharacteristic
208C2×C3×Q8C22×D311yescharacteristic
209C2×C3×D4—22no—
210C2×C3×D4—21no—
211C2×C3×D4—32no—
212C2×C3×D4—62no—
213C2×C3×D4—64no—
214C2×C3×D4—68no—
215C2×C3×D4—62no—
216C2×C3×D4—64no—
217C2×C3×D4D1212yes—
218C2×C3×D4#6712yes—
219C2×C3×D4C22×D311yescharacteristic
220C2×C3×D4C22×D311yescharacteristic
221C22×C3×C4—22no—
222C22×C3×C4—21no—
223C22×C3×C4—64no—
224C22×C3×C4—62no—
225C22×C3×C4D1212yes—
226C22×C3×C4#6712yes—
227C22×C3×C4C22×D311yescharacteristic
228C22×C3×C4C22×D311yescharacteristic
229C2×#67—24no—
230C2×#67—22no—
231C22×Dic3—62no—
232C22×Dic3—61no—
233C22×Dic3—61no—
234#239—24no—
235D3×Q8—24no—
236#237—28no—
237#237—24no—
238D3×D4—28no—
239D3×D4—24no—
240#235—24no—
241#235—24no—
242#235—22no—
243#235—24no—
244#235—22no—
245#235#6714yes—
246#235C22×D314yes—
247C2×D12—22no—
248C2×D12#6712yes—
249C2×C4×D3—24no—
250C2×C4×D3—22no—
251C2×Dic6—22no—
252C2×Dic6—32no—
253C2×Dic6—32no—
254C2×Dic6—62no—
255C2×Dic6—62no—
256C2×Dic6—62no—
257C2×Dic6—62no—
258C2×Dic6—62no—
259C2×Dic6—62no—
260C2×Dic6—61no—
261C2×Dic6—61no—
262C2×Dic6—122no—
263C2×Dic6—122no—
264C2×Dic6#6712yes—
265C3×M16—34no—
266C3×M16—62no—
267C2×C3×C8—62no—
268#217—68no—
269#217—62no—
270#217—64no—
271#217—61no—
272#217—64no—
273#217—61no—
274#216—68no—
275#214—68no—
276C4⋊Dic3—62no—
277C4⋊Dic3—61no—
278C4⋊Dic3—61no—
279Dic3⋊C4—68no—
280Dic3⋊C4—64no—
281Dic3⋊C4—64no—
282C4×Dic3—32no—
283C4×Dic3—32no—
284C4×Dic3—62no—
285#208—22no—
286#208C22×D314yes—
287C2×C3⋊C8—22no—
288Dic12—68no—
289#204—68no—
290C22×C32—22no—
291C22×C32—41no—
292C22×C32—42no—
293C22×C32#12012yes—
294C22×C32C22×D411yescharacteristic
295C2×C3×D3—24no—
296C2×C3×D3—82no—
297C2×C3×D3—84no—
298C32×C4—22no—
299C32×C4#12011yescharacteristic
300C32×C4#12011yescharacteristic
301C32×C4C22×D412yes—
302C3⋊Dic3—42no—
303C3⋊Dic3—42no—
304C3×Dic3—24no—
3052−1+4—32no—
3062−1+4—62no—
307H(5,𝔽2)—32no—
308H(5,𝔽2)—62no—
309H(5,𝔽2)—62no—
310C2×#41—32no—
311C2×#41—64no—
312C2×#41—62no—
313C2×#41D3211yescharacteristic
314C22×Q8—91no—
315#137—98no—
316#137—184no—
317C2×Q16—182no—
318C2×SD16—182no—
319C2×M16—32no—
320C4⋊Q8—92no—
321#124—92no—
322#123—182no—
323C22⋊Q8—182no—
324#117—91no—
325#104—98no—
326#101—32no—
327#100—32no—
328C23×C3—21no—
329C23×C3—22no—
330C23×C3—64no—
331C23×C3—62no—
332C23×C3C2×D1212yes—
333C23×C3C2×D1211yescharacteristic
334C23×C3C2×#6712yes—
335C23×C3C2×#6711yescharacteristic
336C22×D3—24no—
337C22×D3—22no—
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—34no—
341C3×Q8—42no—
342C3×Q8—64no—
343C3×Q8—62no—
344C3×D4—24no—
345C3×D4—24no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—34no—
349C3×D4—42no—
350C3×D4—44no—
351C3×D4—64no—
352C3×D4—68no—
353C3×D4—64no—
354C3×D4—64no—
355C3×D4—62no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—68no—
368C2×C3×C4—64no—
369C2×C3×C4—62no—
370C2×C3×C4—64no—
371C2×C3×C4—64no—
372C2×C3×C4C2×D1212yes—
373C2×C3×C4C2×D1211yescharacteristic
374C2×C3×C4D3×D412yes—
375C2×C3×C4D3×D412yes—
376C2×C3×C4C2×#6712yes—
377C2×C3×C4C2×#6711yescharacteristic
378C2×C3×C4C2×#6712yes—
379C2×C3×C4C2×#6712yes—
380C2×C3×C4C23×D311yescharacteristic
381C2×C3×C4C23×D311yescharacteristic
382#67—28no—
383#67—24no—
384#67—24no—
385C2×Dic3—24no—
386C2×Dic3—22no—
387C2×Dic3—64no—
388C2×Dic3—64no—
389C2×Dic3—62no—
390C2×Dic3—62no—
391C2×Dic3—62no—
392C2×Dic3—62no—
393C2×Dic3—124no—
394C2×Dic3—122no—
395C2×Dic3—122no—
396D12—24no—
397D12—22no—
398D12C2×#6714yes—
399C4×D3—28no—
400C4×D3—24no—
401C4×D3—24no—
402Dic6—24no—
403Dic6—22no—
404Dic6—64no—
405Dic6—64no—
406Dic6—124no—
407Dic6—122no—
408Dic6—122no—
409Dic6—121no—
410Dic6—121no—
411Dic6—121no—
412Dic6—121no—
413Dic6C2×#6714yes—
414C3×C8—64no—
415C3⋊C8—24no—
416C2×C32—21no—
417C2×C32—22no—
418C2×C32—42no—
419C2×C32C2×#12011yescharacteristic
420C3×D3—44no—
421#41—22no—
422#41—22no—
423#41—34no—
424#41—34no—
425#41—64no—
426#41—64no—
427#41—64no—
428#41—68no—
429#41—64no—
430#41—62no—
431#41—64no—
432#41—62no—
433C2×Q8—32no—
434C2×Q8—62no—
435C2×Q8—64no—
436C2×Q8—92no—
437C2×Q8—181no—
438C2×Q8—182no—
439C2×Q8—182no—
440C2×Q8C2×D3211yescharacteristic
441C2×D4—32no—
442C2×D4—62no—
443C2×D4—64no—
444C2×D4—68no—
445C2×D4—62no—
446C2×D4—64no—
447C2×D4#62612yes—
448C2×D4C2×D3211yescharacteristic
449C22×C4—64no—
450C22×C4—62no—
451C22×C4—181no—
452C22×C4#62612yes—
453C22×C4C2×D3211yescharacteristic
454Q16—188no—
455SD16—188no—
456M16—34no—
457M16—62no—
458C2×C8—62no—
459C4⋊C4—181no—
460C4⋊C4—184no—
461C22⋊C4—181no—
462C22⋊C4—184no—
463C42—92no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—21no—
467C22×C3—22no—
468C22×C3—42no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—44no—
475C22×C3—64no—
476C22×C3—242no—
477C22×C3—244no—
478C22×C3#88212yes—
479C22×C3#95312yes—
480C22×C3C22×D1211yescharacteristic
481C22×C3C22×#6711yescharacteristic
482C2×D3—24no—
483C2×D3—82no—
484C2×D3—84no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—42no—
491C3×C4—64no—
492C3×C4#88211yescharacteristic
493C3×C4#88211yescharacteristic
494C3×C4#95311yescharacteristic
495C3×C4#95311yescharacteristic
496C3×C4C2×D3×D412yes—
497C3×C4C22×#6712yes—
498Dic3—24no—
499Dic3—124no—
500Dic3—124no—
501Dic3—122no—
502Dic3—122no—
503Dic3—122no—
504Dic3—122no—
505C32#302211yescharacteristic, complemented, Sylow
506C23—64no—
507C23—62no—
508C23C2×#62612yes—
509C23C2×#62611yescharacteristic
510Q8—22no—
511Q8—34no—
512Q8—64no—
513Q8—62no—
514Q8—184no—
515Q8—361no—
516Q8—361no—
517D4—24no—
518D4—22no—
519D4—34no—
520D4—64no—
521D4—68no—
522D4—64no—
523D4—64no—
524D4—62no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—68no—
528C2×C4—64no—
529C2×C4—62no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—182no—
533C2×C4—182no—
534C2×C4—362no—
535C2×C4C2×#62612yes—
536C2×C4C2×#62611yescharacteristic
537C2×C4D3×#6712yes—
538C2×C4D3×#6712yes—
539C2×C4C22×D3211yescharacteristic
540C8—64no—
541C2×C3—22no—
542C2×C3—21no—
543C2×C3—22no—
544C2×C3—21no—
545C2×C3—22no—
546C2×C3—21no—
547C2×C3—42no—
548C2×C3—42no—
549C2×C3—42no—
550C2×C3—82no—
551C2×C3—124no—
552C2×C3C2×#88211yescharacteristic
553C2×C3C2×#95311yescharacteristic
554D3—44no—
555C22—22no—
556C22—41no—
557C22—42no—
558C22—64no—
559C22—242no—
560C22—244no—
561C22#6397712yes—
562C22C22×#62611yescharacteristic
563C4—22no—
564C4—64no—
565C4—362no—
566C4—362no—
567C4#6397711yescharacteristic
568C4#6397711yescharacteristic
569C4C2×D3×#6712yes—
570C3—21no—
571C3#7871811yescharacteristic, complemented
572C3#8412311yescharacteristic, complemented
573C2—21no—
574C2—22no—
575C2—42no—
576C2—124no—
577C2C2×#6397711yescharacteristic
subgroupG/H
centerC2C2×#63977
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×#41C22
socleC2×C32C2×#120
prime radicalC3×#5964C2
hypercenterC2×#41D32
hypocenterC32#3022
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. #10596296C2semidirect
3. C2×#63831C2semidirect
4. C3×#5237C2semidirect
5. #10591649C2semidirect
6. #10591513C2semidirect
7. #10591110C2semidirect
11. C22×#634C22semidirect
20. #64153C22semidirect
21. #64145C22semidirect
23. #64135C22semidirect
25. C2×C3×#208C22semidirect
28. #63831C22semidirect
30. C3×#836C22semidirect
31. C3×#833C22semidirect
32. #63568C22semidirect
43. #5237D3semidirect
134. C2×#208C2×D3semidirect
142. #836C2×D3semidirect
143. #833C2×D3semidirect
505. C32#3022semidirect
571. C3#78718semidirect
572. C3#84123semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
#105962962
C2×#638312
C3×#52372
#105916492
#105915132
#105911102
#841233
#787183