Finite Group Databasebeta

← #11887055

Subgroups — #11887055

Order 1152.

1055 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5551C211yesmaximal, characteristic, complemented
2#10594004C211yesmaximal, characteristic, complemented
3C4×#3753C211yesmaximal, characteristic, complemented
4C3×#5015C211yesmaximal, characteristic, complemented
5#10591515C212yesmaximal, complemented
6#10591498C212yesmaximal, complemented
7D3×#809C212yesmaximal, complemented
8#10590498C212yesmaximal, complemented
9#10590329C211yesmaximal, characteristic, complemented
10#10590313C211yesmaximal, characteristic, complemented
11#10590277C211yesmaximal, characteristic, complemented
12#83525—31nomaximal
13#80352—31nomaximal
14C2×C3×#235—21no—
15C2×#3802—21no—
16C2×D3×D12C2212yes—
17C32×C4×D4C2211yescharacteristic, complemented
18C2×#3753C2211yescharacteristic
19C4×C4⋊(C3⋊D3)C2211yescharacteristic
20C3×#938C2212yescomplemented
21C2×C3×#213C2212yescomplemented
22C3×#924—21no—
23C3×D3×C4⋊C4C2212yescomplemented
24C3×#883—22no—
25C3×#874C2211yescharacteristic
26C3×C4×D12C2211yescharacteristic
27C3×#869C2211yescharacteristic, complemented
28#63922—22no—
29#63916—21no—
30#63908C2212yes—
31C2×D3×C3⋊C8C2212yescomplemented
32#63659C2211yescharacteristic, complemented
33#63636C2211yescharacteristic
34#63634C2211yescharacteristic, complemented
35C4×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
36C3×#809C2212yescomplemented
37C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
38#63575C2212yescomplemented
39#63564C2212yes—
40#63562C2212yescomplemented
41#63562C2212yescomplemented
42(C2×D3)⋊(C3⋊C8)C2212yescomplemented
43C3×#547—31no—
44#5717—31no—
45#5551D311yescharacteristic, complemented
46D4×D12—31no—
47C3×#459—31no—
48#5157—32no—
49#5156—32no—
50#5088—31no—
51#5087—31no—
52#5070—31no—
53#5038—32no—
54#5036—32no—
55#5015D311yescharacteristic, complemented
56C4×#213—31no—
57C4×#213—31no—
58C4×#213—61no—
59D4×C3⋊C8—31no—
60#4967—32no—
61C2×#809—32no—
62#4846—31no—
63#4834—31no—
64#4774—32no—
65D3×#102—32no—
66C22×D32—22no—
67C2×C32×D4C2311yescharacteristic
68C22×C32×C4—21no—
69C2×C4⋊(C3⋊D3)C2311yescharacteristic
70C2×C4×C3⋊D3—21no—
71C2×C3×#67—22no—
72C3×#235—44no—
73C2×C3×D12—21no—
74C2×C3×D12—21no—
75C2×C3×D12D411yescharacteristic
76C2×C3×D12C2312yes—
77C2×C3×C4×D3—22no—
78C2×C3×C4×D3C2312yes—
79C2×C3×Dic6—21no—
80C2×#626—22no—
81C2×#625—22no—
82#3802—44no—
83D3×D12—24no—
84D3×D12—42no—
85C32×C4⋊C4C2311yescharacteristic
86C32×C22⋊C4—21no—
87C32×C42C2311yescharacteristic
88#3753—22no—
89#3752—21no—
90C4⋊(C3⋊Dic3)C2311yescharacteristic
91C2×C3⋊(C3⋊C8)C2311yescharacteristic
92C2×C3⋊(C3⋊C8)C2311yescharacteristic
93C3×#217—22no—
94C3×#213—42no—
95C3×(C2×D3)⋊C4D412yes—
96C3×C4⋊Dic3—21no—
97C3×C4⋊Dic3D411yescharacteristic
98C3×C4⋊Dic3C2312yes—
99C3×Dic3⋊C4—22no—
100C2×C3×C3⋊C8C2312yes—
101(C2×D3)⋊Dic3—22no—
102D3×C3⋊C8—42no—
103#3294—91noSylow
104C2×C3×#41—61no—
105C2×D3×D4—32no—
106C2×D3×D4—61no—
107C2×D3×D4—61no—
108C2×#235—21no—
109C22×D12—32no—
110C2×C3×D8—32no—
111C3×C4⋊Q8—31no—
112C3×#127—31no—
113C3×#123—62no—
114C3×C22⋊Q8—61no—
115C3×#121—32no—
116C3×C4×D4—21no—
117C3×C4×D4—31no—
118C3×C4×D4C2×D311yescharacteristic, complemented
119C3×C4×D4C2×D311yescharacteristic, complemented
120C2×C3×C4⋊C4—32no—
121#938—32no—
122#938—61no—
123#938C2×D312yescomplemented
124#937—62no—
125C2×#213—31no—
126C2×#213—31no—
127C2×#213—61no—
128C2×#213C2×D312yescomplemented
129#930—32no—
130#924—21no—
131C22×C3⋊C8—32no—
132C2×D3×C8—32no—
133#895—61no—
134D3×C4⋊C4C2×D312yescomplemented
135#883—22no—
136#882—62no—
137#874C2×D311yescharacteristic
138C4×D12—31no—
139C4×D12—31no—
140C4×D12—61no—
141C4×D12C2×D311yescharacteristic
142#869C2×D311yescharacteristic, complemented
143C3×C4⋊C8—31no—
144C3×#102—32no—
145#832—31no—
146#832—31no—
147#832—32no—
148#832—32no—
149#832—61no—
150#830—32no—
151#821—32no—
152(C2×D3)⋊C8—32no—
153#818—32no—
154#809—32no—
155#809—32no—
156#809—31no—
157#809—31no—
158#809—32no—
159#809—61no—
160#809C2×D312yescomplemented
161#807—31no—
162#807—31no—
163#807—32no—
164#807—61no—
165C4⋊(C3⋊C8)C2×D311yescharacteristic, complemented
166C4×C3⋊C8—31no—
167C4×C3⋊C8—31no—
168C4×C3⋊C8—61no—
169C23×C32—21no—
170C22×C3⋊D3—21no—
171C22×C3×D3—22no—
172C22×C3×D3C2×D412yes—
173C2×D32—44no—
174C2×D32—44no—
175C2×D32—82no—
176C32×D4—21no—
177C32×D4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—42no—
180C2×C32×C4C2×D411yescharacteristic
181C2×C32×C4C2×D411yescharacteristic
182C2×C32×C4C2411yescharacteristic
183C2×C3⋊Dic3—21no—
184C4⋊(C3⋊D3)—21no—
185C4⋊(C3⋊D3)#4112yescomplemented
186C4×C3⋊D3—42no—
187C3×#67—44no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3C2×D412yes—
190C3×D12—22no—
191C3×D12—22no—
192C3×D12—42no—
193C3×D12—81no—
194C3×D12D814yescomplemented
195C3×C4×D3—22no—
196C3×C4×D3—22no—
197C3×C4×D3—44no—
198C3×Dic6—42no—
199#626—44no—
200#625—44no—
201C3⋊(C3⋊C8)—21no—
202C3⋊(C3⋊C8)—21no—
203C3×C3⋊C8—22no—
204#547—31no—
205D42—91no—
206#486—91no—
207#480—92no—
208#474—91no—
209#466—92no—
210#459—31no—
211#447—92no—
212C4×D8—91no—
213C8×D4—91no—
214C2×#102—92no—
215C23×D3—62no—
216C23×D3—62no—
217C3×#41—124no—
218C2×C3×Q8—61no—
219C2×C3×D4—21no—
220C2×C3×D4—31no—
221C2×C3×D4—32no—
222C2×C3×D4—62no—
223C2×C3×D4—61no—
224C2×C3×D4—61no—
225C2×C3×D4C22×D311yescharacteristic
226C2×C3×D4C22×D311yescharacteristic
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—32no—
230C22×C3×C4—41no—
231C22×C3×C4—62no—
232C2×#67—22no—
233C2×#67—62no—
234C2×#67—62no—
235C2×#67—62no—
236D3×D4—64no—
237D3×D4—122no—
238D3×D4—124no—
239D3×D4—124no—
240#235—44no—
241C2×D12—21no—
242C2×D12—21no—
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—61no—
246C2×D12—64no—
247C2×D12—62no—
248C2×D12—122no—
249C2×D12#6711yescharacteristic
250C2×D12C22×D312yes—
251C2×C4×D3—22no—
252C2×C4×D3—61no—
253C2×C4×D3—61no—
254C2×C4×D3—121no—
255C2×C4×D3C22×D312yes—
256C2×Dic6—21no—
257C3×D8—122no—
258C2×C3×C8—32no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—31no—
261C3×C4⋊C4—32no—
262C3×C4⋊C4—62no—
263C3×C4⋊C4—61no—
264C3×C4⋊C4C22×D311yescharacteristic
265C3×C4⋊C4C22×D311yescharacteristic
266C3×C22⋊C4—21no—
267C3×C22⋊C4—21no—
268C3×C22⋊C4—32no—
269C3×C22⋊C4—41no—
270C3×C22⋊C4—62no—
271C3×C42—21no—
272C3×C42C22×D311yescharacteristic
273C3×C42C22×D311yescharacteristic
274#217—22no—
275#217—62no—
276#213—42no—
277#213—62no—
278#213—62no—
279#213—64no—
280(C2×D3)⋊C4—62no—
281(C2×D3)⋊C4—61no—
282(C2×D3)⋊C4—61no—
283(C2×D3)⋊C4—121no—
284(C2×D3)⋊C4#6712yes—
285C4⋊Dic3—21no—
286C4⋊Dic3—31no—
287C4⋊Dic3—31no—
288C4⋊Dic3—61no—
289C4⋊Dic3#6711yescharacteristic
290C4⋊Dic3C22×D312yes—
291Dic3⋊C4—22no—
292C2×C3⋊C8—31no—
293C2×C3⋊C8—31no—
294C2×C3⋊C8—31no—
295C2×C3⋊C8—31no—
296C2×C3⋊C8—61no—
297C2×C3⋊C8—61no—
298C2×C3⋊C8—122no—
299C2×C3⋊C8C22×D312yes—
300D3×C8—122no—
301C22×C32—41no—
302C22×C32—41no—
303C22×C32—41no—
304C22×C32C22×D411yescharacteristic
305C2×C3⋊D3—22no—
306C2×C3⋊D3—41no—
307C2×C3⋊D3—41no—
308C2×C3×D3—22no—
309C2×C3×D3—22no—
310C2×C3×D3—42no—
311C2×C3×D3—42no—
312C2×C3×D3—42no—
313C2×C3×D3C2×D814yes—
314D32—88no—
315C32×C4—21no—
316C32×C4—41no—
317C32×C4C2×D812yes—
318C32×C4C2×#4111yescharacteristic
319C32×C4H(5,𝔽2)11yescharacteristic
320C3⋊Dic3—41no—
321C3×Dic3—22no—
322C3×Dic3—42no—
323C2×#41—61no—
324C22×D4—92no—
325C22×D4—181no—
326C2×D8—32no—
327C2×D8—91no—
328C22×C8—92no—
329C4⋊Q8—31no—
330#127—31no—
331#123—62no—
332C22⋊Q8—61no—
333#121—32no—
334#121—92no—
335#121—181no—
336#120—182no—
337C4×D4—31no—
338C4×D4—91no—
339C4×D4D3211yescharacteristic, complemented
340C2×C4⋊C4—32no—
341#107—92no—
342#107—91no—
343C4⋊C8—31no—
344#102—32no—
345#102—92no—
346#102—92no—
347#102—92no—
348#102—91no—
349#102—91no—
350C22⋊C8—92no—
351C4×C8—91no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—32no—
355C23×C3—41no—
356C23×C3—62no—
357C22×D3—22no—
358C22×D3—61no—
359C22×D3—61no—
360C22×D3—121no—
361C22×D3—124no—
362C22×D3—124no—
363C22×D3—124no—
364C22×D3—124no—
365C22×D3—242no—
366C22×D3—242no—
367C22×D3C2×#6712yes—
368C3×Q8—122no—
369C3×D4—22no—
370C3×D4—21no—
371C3×D4—21no—
372C3×D4—21no—
373C3×D4—21no—
374C3×D4—34no—
375C3×D4—41no—
376C3×D4—62no—
377C3×D4—62no—
378C3×D4—124no—
379C3×D4—122no—
380C3×D4—241no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—32no—
387C2×C3×C4—41no—
388C2×C3×C4—42no—
389C2×C3×C4—42no—
390C2×C3×C4—62no—
391C2×C3×C4—62no—
392C2×C3×C4—62no—
393C2×C3×C4—82no—
394C2×C3×C4—124no—
395C2×C3×C4D3×D411yescharacteristic
396C2×C3×C4D3×D411yescharacteristic
397C2×C3×C4C2×#6711yescharacteristic
398C2×C3×C4C2×#6711yescharacteristic
399C2×C3×C4C23×D311yescharacteristic
400C2×C3×C4C23×D311yescharacteristic
401#67—44no—
402#67—124no—
403#67—124no—
404#67—124no—
405C2×Dic3—22no—
406C2×Dic3—61no—
407C2×Dic3—61no—
408C2×Dic3—121no—
409C2×Dic3C2×#6712yes—
410D12—22no—
411D12—22no—
412D12—32no—
413D12—32no—
414D12—42no—
415D12—62no—
416D12—61no—
417D12—61no—
418D12—81no—
419D12—121no—
420D12—124no—
421D12#21314yescomplemented
422C4×D3—22no—
423C4×D3—22no—
424C4×D3—44no—
425C4×D3—122no—
426C4×D3—122no—
427C4×D3—242no—
428Dic6—42no—
429C3×C8—62no—
430C3⋊C8—22no—
431C3⋊C8—62no—
432C3⋊C8—61no—
433C3⋊C8—61no—
434C3⋊C8—61no—
435C3⋊C8—61no—
436C3⋊C8—121no—
437C2×C32—81no—
438C2×C32#54611yescharacteristic
439C2×C32C22×D811yescharacteristic
440C2×C32#57711yescharacteristic
441C3⋊D3—42no—
442C3×D3—24no—
443C3×D3—82no—
444C24—182no—
445#41—124no—
446C2×Q8—61no—
447C2×D4—32no—
448C2×D4—31no—
449C2×D4—62no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—91no—
453C2×D4—182no—
454C2×D4—184no—
455C2×D4—182no—
456C2×D4—362no—
457C2×D4—364no—
458C2×D4C2×D3211yescharacteristic
459C22×C4—21no—
460C22×C4—32no—
461C22×C4—62no—
462C22×C4—181no—
463D8—122no—
464D8—182no—
465C2×C8—32no—
466C2×C8—91no—
467C2×C8—91no—
468C2×C8—362no—
469C4⋊C4—31no—
470C4⋊C4—32no—
471C4⋊C4—62no—
472C4⋊C4—61no—
473C4⋊C4—91no—
474C4⋊C4C2×D3211yescharacteristic
475C22⋊C4—21no—
476C22⋊C4—32no—
477C22⋊C4—62no—
478C22⋊C4—182no—
479C22⋊C4—181no—
480C42C2×D3211yescharacteristic
481C22×C3—21no—
482C22×C3—34no—
483C22×C3—42no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—62no—
491C22×C3—62no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3—122no—
495C22×C3—122no—
496C22×C3—122no—
497C22×C3C2×D3×D411yescharacteristic
498C22×C3C22×#6711yescharacteristic
499C2×D3—22no—
500C2×D3—22no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—62no—
505C2×D3—62no—
506C2×D3—122no—
507C2×D3—121no—
508C2×D3—121no—
509C2×D3—121no—
510C2×D3—121no—
511C2×D3—241no—
512C2×D3—241no—
513C2×D3—248no—
514C2×D3—248no—
515C2×D3C2×#21314yes—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—22no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4—62no—
525C3×C4—81no—
526C3×C4—122no—
527C3×C4D3×D812yes—
528C3×C4C2×#21312yes—
529C3×C4#100411yescharacteristic
530C3×C4D3×#4111yescharacteristic
531C3×C4D3×#4111yescharacteristic
532C3×C4#100911yescharacteristic
533Dic3—22no—
534Dic3—42no—
535Dic3—121no—
536Dic3—121no—
537Dic3—241no—
538C32#329411yescharacteristic, complemented, Sylow
539C23—21no—
540C23—32no—
541C23—62no—
542C23—181no—
543C23—364no—
544C23—364no—
545C23—722no—
546Q8—122no—
547D4—21no—
548D4—21no—
549D4—34no—
550D4—62no—
551D4—62no—
552D4—92no—
553D4—124no—
554D4—122no—
555D4—181no—
556D4—241no—
557D4—364no—
558D4—364no—
559C2×C4—21no—
560C2×C4—32no—
561C2×C4—42no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—124no—
566C2×C4—181no—
567C2×C4—362no—
568C2×C4D3×#6711yescharacteristic
569C2×C4D3×#6711yescharacteristic
570C2×C4C22×D3211yescharacteristic
571C8—62no—
572C8—181no—
573C8—181no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—64no—
578C2×C3—82no—
579C2×C3—81no—
580C2×C3—81no—
581C2×C3—242no—
582C2×C3#561211yescharacteristic
583C2×C3C2×D3×D811yescharacteristic
584C2×C3#577511yescharacteristic
585C2×C3C22×#21311yescharacteristic
586C2×C3#582911yescharacteristic
587C2×C3#583711yescharacteristic
588D3—24no—
589D3—82no—
590D3—122no—
591D3—122no—
592D3—242no—
593C22—34no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—62no—
598C22—62no—
599C22—122no—
600C22—122no—
601C22—122no—
602C22—182no—
603C22—361no—
604C22—361no—
605C22—728no—
606C22C2×D3×#6711yescharacteristic
607C4—21no—
608C4—41no—
609C4—62no—
610C4—122no—
611C4—361no—
612C4D3×#21312yes—
613C4#6430611yescharacteristic
614C4#6430711yescharacteristic
615C3—21no—
616C3#8035211yescharacteristic, complemented
617C3#8352511yescharacteristic, complemented
618C2—64no—
619C2—81no—
620C2—242no—
621C2—362no—
622C2#1059507011yescharacteristic
623C2C2×D3×#21311yescharacteristic
624C2#1059545511yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×D4C22
socleC22×C32C22×D4
hypercenterC4×D4D32
hypocenterC32#3294
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×#5551C2semidirect
2. #10594004C2semidirect
3. C4×#3753C2semidirect
4. C3×#5015C2semidirect
5. #10591515C2semidirect
6. #10591498C2semidirect
7. D3×#809C2semidirect
8. #10590498C2semidirect
9. #10590329C2semidirect
10. #10590313C2semidirect
11. #10590277C2semidirect
17. C32×C4×D4C22semidirect
20. C3×#938C22semidirect
21. C2×C3×#213C22semidirect
23. C3×D3×C4⋊C4C22semidirect
27. C3×#869C22semidirect
31. C2×D3×C3⋊C8C22semidirect
32. #63659C22semidirect
34. #63634C22semidirect
35. C4×C3⋊(C3⋊C8)C22semidirect
36. C3×#809C22semidirect
37. C3×C4⋊(C3⋊C8)C22semidirect
38. #63575C22semidirect
40. #63562C22semidirect
41. #63562C22semidirect
42. (C2×D3)⋊(C3⋊C8)C22semidirect
45. #5551D3semidirect
55. #5015D3semidirect
118. C3×C4×D4C2×D3semidirect
119. C3×C4×D4C2×D3semidirect
123. #938C2×D3semidirect
128. C2×#213C2×D3semidirect
134. D3×C4⋊C4C2×D3semidirect
142. #869C2×D3semidirect
160. #809C2×D3semidirect
165. C4⋊(C3⋊C8)C2×D3semidirect
185. C4⋊(C3⋊D3)#41semidirect
194. C3×D12D8semidirect
339. C4×D4D32semidirect
421. D12#213semidirect
538. C32#3294semidirect
616. C3#80352semidirect
617. C3#83525semidirect

Maximal subgroups

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

subgroupindex
C3×#55512
#105940042
C4×#37532
C3×#50152
#105915152
#105914982
D3×#8092
#105904982
#105903292
#105903132
#105902772
#835253
#803523