Finite Group Databasebeta

← #11887063

Subgroups — #11887063

Order 1152.

1058 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5717C211yesmaximal, characteristic, complemented
2#10593896C211yesmaximal, characteristic, complemented
3#10592771C211yesmaximal, characteristic, complemented
4C3×#5070C211yesmaximal, characteristic, complemented
5#10591449C212yesmaximal, complemented
6#10591448C212yesmaximal, complemented
7#10591413C212yesmaximal, complemented
8D3×#832C212yesmaximal, complemented
9C4×#3714C211yesmaximal, characteristic, complemented
10C3⋊C8×D12C211yesmaximal, characteristic, complemented
11#10590269C211yesmaximal, characteristic, complemented
12#83482—31nomaximal
13#81576—31nomaximal
14C2×C3×D3×D4—21no—
15C2×C3×D3×D4C2212yes—
16C2×#3796—21no—
17C32×#127C2211yescharacteristic
18C2×#3753C2212yescomplemented
19C4×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
20C3×#938—21no—
21C3×#938C2212yes—
22C3×#937—22no—
23C3×C4×D12C2211yescharacteristic
24C3×C4×D12C2211yescharacteristic
25C3×#869C2211yescharacteristic, complemented
26#63909C2212yescomplemented
27#63890—22no—
28D3×C4⋊Dic3C2212yescomplemented
29#63871—21no—
30C2×#3714C2211yescharacteristic
31C2×D3×C3⋊C8C2212yescomplemented
32#63659C2212yescomplemented
33C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
34C3×#832C2212yescomplemented
35C3×#832C2212yescomplemented
36C3×C4×C3⋊C8C2211yescharacteristic, complemented
37#63576C2212yescomplemented
38#63575C2211yescharacteristic, complemented
39#63564C2211yescharacteristic, complemented
40#63563C2211yescharacteristic
41#63562C2212yes—
42(C2×D3)⋊(C3⋊C8)C2212yescomplemented
43C3×D42—31no—
44#5726—31no—
45#5717D311yescharacteristic, complemented
46#5555—31no—
47C3×#486—31no—
48#5217—32no—
49C2×#832—32no—
50#5087—31no—
51#5072—31no—
52#5072—31no—
53#5072—61no—
54#5070D311yescharacteristic, complemented
55#5038—32no—
56#5033—32no—
57C4×#213—31no—
58D4×C3⋊C8—31no—
59#5007—31no—
60#4775—32no—
61#4774—32no—
62#4773—32no—
63D3×#102—32no—
64C4×D24—31no—
65C8×D12—31no—
66C23×C3×D3—22no—
67C2×C32×D4—21no—
68C2×C32×D4—21no—
69C2×C32×D4C2312yes—
70C2×C4⋊(C3⋊D3)C2311yescharacteristic
71C2×C4×C3⋊D3—21no—
72C2×C3×#67—22no—
73C2×C3×#67—22no—
74C3×D3×D4—24no—
75C3×D3×D4—42no—
76C3×D3×D4—44no—
77C2×C3×D12D411yescharacteristic
78C2×C3×D12C2311yescharacteristic
79C2×C3×C4×D3—21no—
80C2×C3×C4×D3C2312yes—
81C2×C3×Dic6—21no—
82C2×#626—22no—
83C2×D3×Dic3—22no—
84#3796—44no—
85C32×C42C2311yescharacteristic
86#3753—42no—
87#3752—21no—
88C4⋊(C3⋊Dic3)C2311yescharacteristic
89C2×C3⋊(C3⋊C8)C2312yes—
90C3×#217—22no—
91C3×(C2×D3)⋊C4—21no—
92C3×(C2×D3)⋊C4D412yes—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3D411yescharacteristic
95C3×C4⋊Dic3C2311yescharacteristic
96C3×C4⋊Dic3C2312yes—
97C2×C3×C3⋊C8C2311yescharacteristic
98C2×C3×C3⋊C8C2311yescharacteristic
99Dic3⋊Dic3—22no—
100#3722—22no—
101#3714—22no—
102D3×C3⋊C8—42no—
103#3294—91noSylow
104C22×C3×D4—32no—
105C22×C3×D4—61no—
106C2×#239—61no—
107C2×#237—61no—
108C2×D3×D4—21no—
109C2×D3×D4C2×D312yes—
110C3×C4⋊Q8—31no—
111C3×#127—21no—
112C3×#127C2×D311yescharacteristic
113C3×#127C2×D311yescharacteristic
114C3×#121—32no—
115C3×#121—61no—
116C3×#120—62no—
117C3×C4×D4—31no—
118C3×C4×D4—31no—
119#946—61no—
120#938—21no—
121#938C2×D312yes—
122#937—22no—
123C2×#213—32no—
124C2×#213—32no—
125C2×#213—31no—
126C2×#213—62no—
127#930—32no—
128C2×C4⋊Dic3—32no—
129C22×C3⋊C8—32no—
130C2×D24—31no—
131C2×D3×C8—32no—
132#897—61no—
133#895—32no—
134#894—62no—
135D3×C4⋊C4—32no—
136#886—62no—
137C4×D12—31no—
138C4×D12—31no—
139C4×D12—61no—
140C4×D12C2×D311yescharacteristic
141C4×D12C2×D311yescharacteristic
142#869C2×D311yescharacteristic, complemented
143C3×#102—32no—
144C3×#102—32no—
145C3×C4×C8—31no—
146#832—32no—
147#832—32no—
148#832—32no—
149#832—31no—
150#832—62no—
151#832C2×D312yescomplemented
152#832C2×D312yescomplemented
153#830—32no—
154#821—31no—
155#821—31no—
156(C2×D3)⋊C8—32no—
157#818—31no—
158#809—32no—
159#809—31no—
160#807—32no—
161#807—32no—
162#807—31no—
163C4⋊(C3⋊C8)—31no—
164C4⋊(C3⋊C8)—31no—
165C4⋊(C3⋊C8)—61no—
166C4×C3⋊C8C2×D311yescharacteristic, complemented
167C23×C32—22no—
168C22×C3⋊D3—21no—
169C22×C3×D3—21no—
170C22×C3×D3—44no—
171C22×C3×D3—44no—
172C22×C3×D3—82no—
173C22×C3×D3C2×D412yes—
174C32×D4—22no—
175C32×D4—22no—
176C32×D4—42no—
177C32×D4—81no—
178C2×C32×C4C2×D411yescharacteristic
179C2×C32×C4C2×D411yescharacteristic
180C2×C32×C4C2411yescharacteristic
181C2×C3⋊Dic3—21no—
182C4⋊(C3⋊D3)—21no—
183C4⋊(C3⋊D3)—21no—
184C4×C3⋊D3—42no—
185C3×#67—44no—
186C3×#67—44no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3C2×D412yes—
190C3×D12—21no—
191C3×D12D814yescomplemented
192C3×D12#4112yescomplemented
193C3×C4×D3—22no—
194C3×C4×D3—22no—
195C3×C4×D3—42no—
196C3×Dic6—42no—
197#626—44no—
198D3×Dic3—44no—
199C3⋊(C3⋊C8)—22no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202#547—91no—
203D42—31no—
204#486—31no—
205#480—92no—
206#474—91no—
207#466—92no—
208#459—91no—
209#447—92no—
210C4×D8—91no—
211C8×D4—91no—
212C2×#102—92no—
213C24×C3—62no—
214C23×D3—22no—
215C2×C3×Q8—61no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—22no—
221C2×C3×D4—31no—
222C2×C3×D4—31no—
223C2×C3×D4—41no—
224C2×C3×D4—41no—
225C2×C3×D4—62no—
226C2×C3×D4—64no—
227C2×C3×D4—62no—
228C2×C3×D4—122no—
229C2×C3×D4—124no—
230C2×C3×D4C22×D312yes—
231C2×C3×D4C22×D312yes—
232C22×C3×C4—32no—
233C22×C3×C4—61no—
234C2×#67—22no—
235C2×#67—22no—
236C2×#67—62no—
237C22×Dic3—62no—
238#239—124no—
239#237—124no—
240D3×D4—24no—
241D3×D4—42no—
242D3×D4—44no—
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—61no—
246C2×D12—62no—
247C2×D12#6711yescharacteristic
248C2×D12C22×D311yescharacteristic
249C2×C4×D3—21no—
250C2×C4×D3—62no—
251C2×C4×D3—61no—
252C2×C4×D3—61no—
253C2×C4×D3—121no—
254C2×C4×D3C22×D312yes—
255C2×Dic6—21no—
256C2×C3×C8—31no—
257C2×C3×C8—31no—
258C3×C4⋊C4—31no—
259C3×C4⋊C4—31no—
260C3×C4⋊C4—32no—
261C3×C4⋊C4—61no—
262C3×C22⋊C4—32no—
263C3×C22⋊C4—62no—
264C3×C22⋊C4—61no—
265C3×C42—21no—
266C3×C42C22×D311yescharacteristic
267C3×C42C22×D311yescharacteristic
268#217—22no—
269#213—62no—
270#213—122no—
271#213—122no—
272#213—124no—
273(C2×D3)⋊C4—21no—
274(C2×D3)⋊C4—62no—
275(C2×D3)⋊C4—61no—
276(C2×D3)⋊C4—61no—
277(C2×D3)⋊C4—62no—
278(C2×D3)⋊C4—121no—
279(C2×D3)⋊C4#6712yes—
280C4⋊Dic3—21no—
281C4⋊Dic3—31no—
282C4⋊Dic3—31no—
283C4⋊Dic3—61no—
284C4⋊Dic3—62no—
285C4⋊Dic3#6711yescharacteristic
286C4⋊Dic3C22×D311yescharacteristic
287C4⋊Dic3C22×D312yes—
288Dic3⋊C4—62no—
289C2×C3⋊C8—32no—
290C2×C3⋊C8—32no—
291C2×C3⋊C8—62no—
292C2×C3⋊C8—122no—
293C2×C3⋊C8C22×D311yescharacteristic
294C2×C3⋊C8C22×D311yescharacteristic
295D24—62no—
296D3×C8—122no—
297C22×C32—42no—
298C22×C32—42no—
299C22×C32—42no—
300C22×C32C22×D411yescharacteristic
301C2×C3⋊D3—41no—
302C2×C3⋊D3—41no—
303C2×C3⋊D3—41no—
304C2×C3×D3—22no—
305C2×C3×D3—22no—
306C2×C3×D3—22no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—88no—
310C2×C3×D3C2×D814yes—
311C32×C4—21no—
312C32×C4C2×D812yes—
313C32×C4C2×#4111yescharacteristic
314C32×C4H(5,𝔽2)11yescharacteristic
315C3⋊Dic3—41no—
316C3×Dic3—22no—
317C3×Dic3—42no—
318C3×Dic3—41no—
319C2×#41—181no—
320C22×D4—32no—
321C22×D4—61no—
322C2×D8—91no—
323C2×D8—92no—
324C22×C8—92no—
325C4⋊Q8—31no—
326#127D3211yescharacteristic
327#123—182no—
328C22⋊Q8—181no—
329#121—32no—
330#121—61no—
331#121—92no—
332#120—62no—
333C4×D4—31no—
334C4×D4—31no—
335C4×D4—91no—
336C2×C4⋊C4—92no—
337#107—92no—
338#107—91no—
339C4⋊C8—91no—
340#102—32no—
341#102—32no—
342#102—91no—
343#102—92no—
344#102—91no—
345#102—92no—
346C22⋊C8—92no—
347C4×C8—31no—
348C23×C3—22no—
349C23×C3—22no—
350C23×C3—32no—
351C23×C3—42no—
352C23×C3—61no—
353C23×C3—124no—
354C23×C3—124no—
355C23×C3—242no—
356C22×D3—21no—
357C22×D3—44no—
358C22×D3—44no—
359C22×D3—61no—
360C22×D3—61no—
361C22×D3—82no—
362C22×D3—121no—
363C22×D3C2×#6712yes—
364C3×Q8—122no—
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—34no—
370C3×D4—32no—
371C3×D4—42no—
372C3×D4—42no—
373C3×D4—42no—
374C3×D4—42no—
375C3×D4—61no—
376C3×D4—82no—
377C3×D4—81no—
378C3×D4—81no—
379C3×D4—82no—
380C3×D4—124no—
381C3×D4—124no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—32no—
386C2×C3×C4—62no—
387C2×C3×C4—62no—
388C2×C3×C4—61no—
389C2×C3×C4—62no—
390C2×C3×C4—122no—
391C2×C3×C4D3×D411yescharacteristic
392C2×C3×C4D3×D411yescharacteristic
393C2×C3×C4C2×#6711yescharacteristic
394C2×C3×C4C2×#6711yescharacteristic
395C2×C3×C4C23×D311yescharacteristic
396C2×C3×C4C23×D311yescharacteristic
397#67—44no—
398#67—44no—
399#67—124no—
400C2×Dic3—22no—
401C2×Dic3—21no—
402C2×Dic3—61no—
403C2×Dic3—61no—
404C2×Dic3—121no—
405C2×Dic3—124no—
406C2×Dic3C2×#6712yes—
407D12—21no—
408D12—62no—
409D12—61no—
410D12—61no—
411D12—61no—
412D12—61no—
413D12—121no—
414D12—124no—
415D12#21314yescomplemented
416D12#23712yescomplemented
417C4×D3—22no—
418C4×D3—22no—
419C4×D3—42no—
420C4×D3—124no—
421C4×D3—122no—
422C4×D3—122no—
423C4×D3—242no—
424Dic6—42no—
425C3×C8—61no—
426C3×C8—61no—
427C3⋊C8—21no—
428C3⋊C8—21no—
429C3⋊C8—62no—
430C3⋊C8—62no—
431C3⋊C8—122no—
432C2×C32—82no—
433C2×C32#54611yescharacteristic
434C2×C32C22×D811yescharacteristic
435C2×C32#57711yescharacteristic
436C3⋊D3—81no—
437C3×D3—24no—
438C3×D3—42no—
439C24—62no—
440#41—364no—
441C2×Q8—61no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—31no—
445C2×D4—31no—
446C2×D4—62no—
447C2×D4—64no—
448C2×D4—62no—
449C2×D4—91no—
450C2×D4—122no—
451C2×D4—124no—
452C2×D4—182no—
453C2×D4C2×D3212yes—
454C22×C4—32no—
455C22×C4—61no—
456C22×C4—182no—
457C22×C4—181no—
458D8—182no—
459D8—362no—
460C2×C8—31no—
461C2×C8—31no—
462C2×C8—92no—
463C2×C8—362no—
464C4⋊C4—31no—
465C4⋊C4—32no—
466C4⋊C4—31no—
467C4⋊C4—61no—
468C4⋊C4—91no—
469C4⋊C4—182no—
470C22⋊C4—32no—
471C22⋊C4—62no—
472C22⋊C4—61no—
473C22⋊C4—182no—
474C22⋊C4—181no—
475C42C2×D3211yescharacteristic
476C22×C3—21no—
477C22×C3—34no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—62no—
485C22×C3—62no—
486C22×C3—62no—
487C22×C3—82no—
488C22×C3—82no—
489C22×C3—82no—
490C22×C3—121no—
491C22×C3—121no—
492C22×C3—248no—
493C22×C3C2×D3×D411yescharacteristic
494C22×C3C22×#6711yescharacteristic
495C2×D3—22no—
496C2×D3—22no—
497C2×D3—22no—
498C2×D3—41no—
499C2×D3—41no—
500C2×D3—88no—
501C2×D3—122no—
502C2×D3—121no—
503C2×D3—121no—
504C2×D3—121no—
505C2×D3—121no—
506C2×D3—121no—
507C2×D3—121no—
508C2×D3—241no—
509C2×D3—241no—
510C2×D3C2×#21314yes—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—22no—
516C3×C4—41no—
517C3×C4—62no—
518C3×C4—121no—
519C3×C4—122no—
520C3×C4D3×D812yes—
521C3×C4C2×#21312yes—
522C3×C4C2×#23711yescharacteristic
523C3×C4#100411yescharacteristic
524C3×C4D3×#4111yescharacteristic
525C3×C4#100911yescharacteristic
526Dic3—22no—
527Dic3—42no—
528Dic3—41no—
529Dic3—121no—
530Dic3—121no—
531Dic3—241no—
532C32#329411yescharacteristic, complemented, Sylow
533C23—22no—
534C23—32no—
535C23—61no—
536C23—124no—
537C23—124no—
538C23—181no—
539C23—242no—
540Q8—122no—
541D4—22no—
542D4—22no—
543D4—32no—
544D4—34no—
545D4—42no—
546D4—61no—
547D4—81no—
548D4—124no—
549D4—124no—
550D4—181no—
551D4—181no—
552D4—364no—
553C2×C4—32no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—61no—
557C2×C4—62no—
558C2×C4—122no—
559C2×C4—181no—
560C2×C4—364no—
561C2×C4—362no—
562C2×C4D3×#6711yescharacteristic
563C2×C4D3×#6711yescharacteristic
564C2×C4C22×D3211yescharacteristic
565C8—61no—
566C8—61no—
567C8—182no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—64no—
572C2×C3—82no—
573C2×C3—82no—
574C2×C3—122no—
575C2×C3—162no—
576C2×C3#567611yescharacteristic
577C2×C3C2×D3×D811yescharacteristic
578C2×C3#576211yescharacteristic
579C2×C3C22×#21311yescharacteristic
580C2×C3#580611yescharacteristic
581C2×C3#583611yescharacteristic
582D3—24no—
583D3—42no—
584D3—242no—
585D3—241no—
586D3—241no—
587C22—34no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—62no—
592C22—62no—
593C22—62no—
594C22—121no—
595C22—121no—
596C22—248no—
597C22—361no—
598C22—361no—
599C22—361no—
600C22C2×D3×#6711yescharacteristic
601C4—21no—
602C4—62no—
603C4—121no—
604C4—122no—
605C4—361no—
606C4D3×#21312yes—
607C4D3×#23711yescharacteristic
608C4#6431411yescharacteristic
609C3—21no—
610C3#8157611yescharacteristic, complemented
611C3#8348211yescharacteristic, complemented
612C2—64no—
613C2—82no—
614C2—122no—
615C2—721no—
616C2C2×D3×#21311yescharacteristic
617C2#1059510511yescharacteristic
618C2#1059522911yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#127C22
socleC22×C32C22×D4
hypercenter#127D32
hypocenterC32#3294
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C3×#5717C2semidirect
2. #10593896C2semidirect
3. #10592771C2semidirect
4. C3×#5070C2semidirect
5. #10591449C2semidirect
6. #10591448C2semidirect
7. #10591413C2semidirect
8. D3×#832C2semidirect
9. C4×#3714C2semidirect
10. C3⋊C8×D12C2semidirect
11. #10590269C2semidirect
18. C2×#3753C22semidirect
19. C4×C4⋊(C3⋊D3)C22semidirect
25. C3×#869C22semidirect
26. #63909C22semidirect
28. D3×C4⋊Dic3C22semidirect
31. C2×D3×C3⋊C8C22semidirect
32. #63659C22semidirect
33. C4⋊(C3⋊(C3⋊C8))C22semidirect
34. C3×#832C22semidirect
35. C3×#832C22semidirect
36. C3×C4×C3⋊C8C22semidirect
37. #63576C22semidirect
38. #63575C22semidirect
39. #63564C22semidirect
42. (C2×D3)⋊(C3⋊C8)C22semidirect
45. #5717D3semidirect
54. #5070D3semidirect
142. #869C2×D3semidirect
151. #832C2×D3semidirect
152. #832C2×D3semidirect
166. C4×C3⋊C8C2×D3semidirect
191. C3×D12D8semidirect
192. C3×D12#41semidirect
415. D12#213semidirect
416. D12#237semidirect
532. C32#3294semidirect
610. C3#81576semidirect
611. C3#83482semidirect

Maximal subgroups

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

subgroupindex
C3×#57172
#105938962
#105927712
C3×#50702
#105914492
#105914482
#105914132
D3×#8322
C4×#37142
C3⋊C8×D122
#105902692
#834823
#815763