Finite Group Databasebeta

← #11792733

Subgroups — #11792733

Order 1152.

853 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64118C211yesmaximal, characteristic, complemented
2C2×#63830C211yesmaximal, characteristic
3C22×#3716C211yesmaximal, characteristic, complemented
4C2×C3×#823C211yesmaximal, characteristic, complemented
5C2×#63561C211yesmaximal, characteristic, complemented
6#10591073C211yesmaximal, characteristic, complemented
7C3×#4558C211yesmaximal, characteristic, complemented
8#70091—31nomaximal
9#68943—31nomaximal
10C2×#3828C2211yescharacteristic
11C22×#634C2211yescharacteristic
12C22×C3×D12C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14#64118—22no—
15#64118—22no—
16C2×#3746—21no—
17C2×#3744—21no—
18C2×C3×#208C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21#63830—22no—
22#63830—22no—
23C2×#3716—22no—
24C2×#3716—21no—
25C2×#3716—22no—
26C2×#3716—22no—
27C2×#3714—21no—
28C3×#823—22no—
29#63561—22no—
30#63557—22no—
31C22×#214—31no—
32C2×#932—31no—
33C2×#909—31no—
34C2×#909—31no—
35C2×#909—61no—
36C2×#908—31no—
37C22×#204—31no—
38C2×C3×#100—31no—
39C2×#824—31no—
40C2×#824—31no—
41C2×#823D311yescharacteristic, complemented
42#5129—31no—
43#5110—31no—
44C3×#341—31no—
45#4558D311yescharacteristic, complemented
46C23×C3×D3—21no—
47C22×C32×C4C2311yescharacteristic
48C2×#638—21no—
49C22×C3⋊Dic3—21no—
50#3828—21no—
51#3828—21no—
52#3828—21no—
53#3828—21no—
54C2×C4⋊(C3⋊D3)D411yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×#634—21no—
57C2×#634—21no—
58C2×#634—21no—
59C2×#634—21no—
60C2×#634—21no—
61C2×#634D412yes—
62C2×#634D411yescharacteristic
63C2×C3×D12—21no—
64C2×C3×D12—21no—
65C2×C3×D12—21no—
66C2×C3×D12—21no—
67C2×C3×D12—21no—
68C2×C3×D12D412yes—
69C32×M16—21no—
70C32×M16D412yes—
71C2×C32×C8—21no—
72#3746—24no—
73#3744—24no—
74C3×#208—21no—
75C3×#208—21no—
76C3×#208—21no—
77C3×#208D412yes—
78C2×C3×C3⋊C8—21no—
79C2×C3×C3⋊C8—21no—
80C2×C3×C3⋊C8—21no—
81C2×C3×C3⋊C8—21no—
82C2×C3×C3⋊C8D412yes—
83#3716—28no—
84#3716—24no—
85#3714—24no—
86#2019—91noSylow
87C22×C3×D4—31no—
88C2×#235—31no—
89C2×#235—31no—
90C2×#235—61no—
91C22×D12C2×D311yescharacteristic
92C22×Dic6—31no—
93C22×Dic6—31no—
94C22×Dic6—61no—
95C2×C3×M16—21no—
96C2×C3×M16—31no—
97C2×C3×M16—31no—
98C2×C3×M16C2×D311yescharacteristic
99C2×C3×M16C2×D311yescharacteristic
100C22×C3×C8—31no—
101C2×#214—61no—
102C2×#214—62no—
103C2×#214—62no—
104C2×#214—62no—
105#932—62no—
106#932—62no—
107C2×#213—61no—
108C2×#208C2×D311yescharacteristic
109C2×#208C2×D311yescharacteristic
110C22×C3⋊C8C2×D311yescharacteristic
111#909—62no—
112#909—62no—
113#909—64no—
114#909—64no—
115#909—62no—
116#909—62no—
117#908—62no—
118#908—62no—
119C2×Dic12—61no—
120C2×Dic12—61no—
121C2×Dic12—62no—
122C2×D24—61no—
123C2×#204—61no—
124C2×#204—61no—
125C2×#204—62no—
126C2×#204—62no—
127C2×#204—62no—
128C2×#204—61no—
129C2×#204—62no—
130C3×#100—62no—
131#824—62no—
132#824—62no—
133#823—22no—
134(C2×D3)⋊C8—62no—
135(C2×D3)⋊C8—62no—
136C23×C32#4111yescharacteristic
137C22×C3⋊D3—21no—
138C22×C3×D3—22no—
139C22×C3×D3—41no—
140C22×C3×D3—41no—
141C22×C3×D3—82no—
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148#638—41no—
149#638—41no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3—41no—
153C2×C3⋊Dic3—41no—
154C4⋊(C3⋊D3)—21no—
155C4⋊(C3⋊D3)—21no—
156C4×C3⋊D3—41no—
157C4×C3⋊D3—41no—
158#634—22no—
159#634—21no—
160#634—22no—
161#634—21no—
162#634—41no—
163#634—41no—
164C3×D12—22no—
165C3×D12—22no—
166C3×D12—41no—
167C3×D12—41no—
168C32×C8—22no—
169C3×C3⋊C8—22no—
170C3×C3⋊C8—22no—
171C3×C3⋊C8—41no—
172C2×#137—91no—
173C2×AGL(1,ℤ/8)—91no—
174C22×SD16—91no—
175C2×#101—91no—
176C2×#100—31no—
177#408—91no—
178#341—31no—
179C24×C3—61no—
180C23×D3—21no—
181C2×C3×D4—32no—
182C2×C3×D4—61no—
183C2×C3×D4—61no—
184C2×C3×D4—61no—
185C2×C3×D4—61no—
186C2×C3×D4—61no—
187C22×C3×C4—21no—
188C22×C3×C4C22×D311yescharacteristic
189C22×C3×C4C22×D311yescharacteristic
190C2×#67—61no—
191C2×#67—61no—
192C2×#67—121no—
193C22×Dic3—61no—
194C22×Dic3—61no—
195C22×Dic3—121no—
196#235—62no—
197#235—62no—
198#235—61no—
199#235—61no—
200#235—61no—
201#235—61no—
202#235—61no—
203#235—61no—
204#235—61no—
205#235—61no—
206#235—121no—
207#235—121no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—31no—
214C2×D12—31no—
215C2×D12—61no—
216C2×D12D1212yes—
217C2×C4×D3—61no—
218C2×C4×D3—61no—
219C2×C4×D3—121no—
220C2×Dic6—31no—
221C2×Dic6—32no—
222C2×Dic6—31no—
223C2×Dic6—32no—
224C2×Dic6—61no—
225C2×Dic6—62no—
226C2×Dic6—62no—
227C2×Dic6—62no—
228C2×Dic6—61no—
229C2×Dic6—61no—
230C2×Dic6—61no—
231C2×Dic6—61no—
232C2×Dic6—61no—
233C2×Dic6—61no—
234C2×Dic6—61no—
235C2×Dic6—61no—
236C2×Dic6—61no—
237C2×Dic6—61no—
238C2×Dic6—121no—
239C2×Dic6—121no—
240C2×Dic6—121no—
241C3×M16—22no—
242C3×M16—21no—
243C3×M16—21no—
244C3×M16—22no—
245C3×M16—32no—
246C3×M16—61no—
247C3×M16—61no—
248C3×M16—61no—
249C3×M16D1212yes—
250C3×M16#6712yes—
251C2×C3×C8—22no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—32no—
255C2×C3×C8—61no—
256C2×C3×C8—61no—
257C2×C3×C8—61no—
258C2×C3×C8—61no—
259#214—64no—
260#214—68no—
261#213—64no—
262#208—21no—
263#208—21no—
264#208—21no—
265#208D1212yes—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8D1212yes—
271Dic12—68no—
272Dic12—64no—
273Dic12—64no—
274D24—64no—
275#204—68no—
276#204—64no—
277#204—64no—
278#204—68no—
279#204—64no—
280C22×C32—22no—
281C22×C32#12111yescharacteristic
282C22×C32#12111yescharacteristic
283C22×C32#12111yescharacteristic
284C2×C3⋊D3—41no—
285C2×C3⋊D3—42no—
286C2×C3×D3—42no—
287C2×C3×D3—82no—
288C2×C3×D3—82no—
289C2×C3×D3—82no—
290C32×C4#12011yescharacteristic
291C32×C4#12011yescharacteristic
292C32×C4#12011yescharacteristic
293C32×C4#12711yescharacteristic
294C3⋊Dic3—42no—
295C3⋊Dic3—41no—
296C2×#41—91no—
297C22×Q8—91no—
298C22×D4—31no—
299#137—182no—
300#137—182no—
301AGL(1,ℤ/8)—182no—
302AGL(1,ℤ/8)—182no—
303C2×Q16—181no—
304C2×SD16—181no—
305C2×SD16—182no—
306C2×SD16—182no—
307C2×SD16—182no—
308C2×SD16—181no—
309C2×D8—181no—
310C2×M16—31no—
311C2×M16—31no—
312C2×M16D3211yescharacteristic
313C22×C8—31no—
314#101—182no—
315#100—62no—
316C22⋊C8—182no—
317C23×C3—21no—
318C23×C3—62no—
319C23×C3—121no—
320C23×C3—121no—
321C23×C3—242no—
322C23×C3#23511yescharacteristic
323C23×C3#23911yescharacteristic
324C22×D3—22no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—61no—
328C22×D3—61no—
329C22×D3—82no—
330C22×D3—121no—
331C3×D4—62no—
332C3×D4—62no—
333C3×D4—121no—
334C3×D4—121no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4C2×D1211yescharacteristic
342C2×C3×C4C2×D1211yescharacteristic
343C2×C3×C4C2×D1211yescharacteristic
344C2×C3×C4C2×D1211yescharacteristic
345C2×C3×C4D3×D411yescharacteristic
346C2×C3×C4D3×D411yescharacteristic
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4D3×D411yescharacteristic
350C2×C3×C4D3×D411yescharacteristic
351C2×C3×C4D3×D411yescharacteristic
352C2×C3×C4C2×#6711yescharacteristic
353#67—122no—
354#67—122no—
355#67—121no—
356#67—121no—
357#67—121no—
358#67—121no—
359C2×Dic3—62no—
360C2×Dic3—61no—
361C2×Dic3—62no—
362C2×Dic3—61no—
363C2×Dic3—122no—
364C2×Dic3—121no—
365C2×Dic3—122no—
366C2×Dic3—122no—
367C2×Dic3—121no—
368C2×Dic3—121no—
369C2×Dic3—121no—
370C2×Dic3—121no—
371D12—22no—
372D12—22no—
373D12—41no—
374D12—41no—
375D12—62no—
376D12—61no—
377D12—61no—
378D12—61no—
379D12—61no—
380D12—121no—
381C4×D3—122no—
382C4×D3—122no—
383C4×D3—121no—
384C4×D3—121no—
385C4×D3—121no—
386C4×D3—121no—
387Dic6—64no—
388Dic6—62no—
389Dic6—62no—
390Dic6—61no—
391Dic6—62no—
392Dic6—61no—
393Dic6—62no—
394Dic6—61no—
395Dic6—62no—
396Dic6—61no—
397Dic6—122no—
398Dic6—121no—
399Dic6—122no—
400Dic6—122no—
401Dic6—121no—
402Dic6—121no—
403Dic6—121no—
404Dic6—121no—
405C3×C8—24no—
406C3×C8—22no—
407C3×C8—22no—
408C3×C8—62no—
409C3×C8—62no—
410C3×C8—121no—
411C3⋊C8—22no—
412C3⋊C8—22no—
413C3⋊C8—41no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32#39211yescharacteristic
417C2×C32#47112yes—
418C3⋊D3—81no—
419C3×D3—82no—
420C24—61no—
421#41—181no—
422#41—181no—
423#41—181no—
424#41—181no—
425C2×Q8—91no—
426C2×Q8—92no—
427C2×Q8—181no—
428C2×Q8—181no—
429C2×Q8—181no—
430C2×Q8—181no—
431C2×Q8—181no—
432C2×D4—32no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—91no—
439C2×D4—181no—
440C22×C4—181no—
441C22×C4—181no—
442C22×C4C2×D3211yescharacteristic
443Q16—184no—
444SD16—184no—
445SD16—188no—
446SD16—184no—
447D8—184no—
448M16—21no—
449M16—32no—
450M16—61no—
451M16—61no—
452M16—61no—
453M16#62612yes—
454C2×C8—21no—
455C2×C8—32no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—42no—
466C22×C3—122no—
467C22×C3—242no—
468C22×C3—242no—
469C22×C3—242no—
470C22×C3#88411yescharacteristic
471C22×C3#88411yescharacteristic
472C22×C3#89511yescharacteristic
473C22×C3#89511yescharacteristic
474C22×C3#89511yescharacteristic
475C22×C3#93011yescharacteristic
476C2×D3—42no—
477C2×D3—82no—
478C2×D3—82no—
479C2×D3—82no—
480C2×D3—122no—
481C2×D3—122no—
482C2×D3—121no—
483C2×D3—121no—
484C2×D3—122no—
485C2×D3—242no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4#87411yescharacteristic
491C3×C4#88211yescharacteristic
492C3×C4#88211yescharacteristic
493C3×C4#88211yescharacteristic
494C3×C4#88211yescharacteristic
495C3×C4#88211yescharacteristic
496C3×C4#93711yescharacteristic
497C3×C4#94011yescharacteristic
498Dic3—124no—
499Dic3—122no—
500Dic3—122no—
501Dic3—121no—
502Dic3—122no—
503Dic3—121no—
504C32#201911yescharacteristic, complemented, Sylow
505C23—62no—
506C23—121no—
507C23—121no—
508C23—181no—
509C23—242no—
510C23#379911yescharacteristic
511Q8—182no—
512Q8—181no—
513Q8—182no—
514Q8—181no—
515Q8—361no—
516Q8—361no—
517D4—62no—
518D4—62no—
519D4—121no—
520D4—121no—
521D4—181no—
522D4—181no—
523D4—361no—
524D4—361no—
525C2×C4—182no—
526C2×C4—181no—
527C2×C4—361no—
528C2×C4—361no—
529C2×C4—361no—
530C2×C4—361no—
531C2×C4D3×D1211yescharacteristic
532C2×C4D3×D1211yescharacteristic
533C2×C4D3×D1211yescharacteristic
534C2×C4C2×#62611yescharacteristic
535C2×C4#381111yescharacteristic
536C2×C4#381111yescharacteristic
537C8—22no—
538C8—62no—
539C8—62no—
540C8—121no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—22no—
546C2×C3—21no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—242no—
550C2×C3#467111yescharacteristic
551C2×C3#474812yes—
552C2×C3#489812yes—
553C2×C3#496011yescharacteristic
554D3—82no—
555D3—242no—
556D3—241no—
557D3—241no—
558C22—22no—
559C22—122no—
560C22—242no—
561C22—242no—
562C22—242no—
563C22—362no—
564C22—361no—
565C22#6390911yescharacteristic
566C22#6391011yescharacteristic
567C22#6391011yescharacteristic
568C4—361no—
569C4—362no—
570C4#6391111yescharacteristic
571C4#6392311yescharacteristic
572C4#6392311yescharacteristic
573C4#6398111yescharacteristic
574C3—21no—
575C3#6894311yescharacteristic, complemented
576C3#7009111yescharacteristic, complemented
577C2—21no—
578C2—21no—
579C2—242no—
580C2—721no—
581C2#1059098912yes—
582C2#1059139511yescharacteristic
subgroupG/H
centerC22#63910
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#121
prime radicalC2×#63830C2
hypercenterC2×M16D32
hypocenterC32#2019
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×#64118C2semidirect
3. C22×#3716C2semidirect
4. C2×C3×#823C2semidirect
5. C2×#63561C2semidirect
6. #10591073C2semidirect
7. C3×#4558C2semidirect
41. C2×#823D3semidirect
45. #4558D3semidirect
504. C32#2019semidirect
575. C3#68943semidirect
576. C3#70091semidirect

Maximal subgroups

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

subgroupindex
C2×#641182
C2×#638302
C22×#37162
C2×C3×#8232
C2×#635612
#105910732
C3×#45582
#700913
#689433