Finite Group Databasebeta

← #11900181

Subgroups — #11900181

Order 1152.

3769 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64366C211yesmaximal, characteristic
2C3×#5827C211yesmaximal, characteristic, complemented
3C3×Dic3×#41C211yesmaximal, characteristic, complemented
4#10595386C212yesmaximal, complemented
5#10595353C212yesmaximal, complemented
6#10595223C216yesmaximal, complemented
7D4×#624C216yesmaximal, complemented
8#10594996C213yesmaximal, complemented
9#10594990C213yesmaximal, complemented
10#10594954C213yesmaximal, complemented
11C2×#63894C213yesmaximal, complemented
12#85886—31nomaximal
13C4×#1009—31nomaximal
14C2×C32×#41C2211yescharacteristic
15#64366C4116yescomplemented
16C2×#3832C2212yes—
17C2×D4×C3⋊D3C2216yes—
18C2×#3828C2213yes—
19C22×C4⋊(C3⋊D3)C2213yes—
20C22×#624C2216yescomplemented
21C3×Q8×Dic3C2211yescharacteristic, complemented
22C3×Q8×Dic3C2212yescomplemented
23C3×Q8×Dic3C2211yescharacteristic, complemented
24C3×D4×Dic3C2213yescomplemented
25C3×D4×Dic3C2216yescomplemented
26C3×D4×Dic3C2213yescomplemented
27C3×#926C2213yescomplemented
28C3×#926C2213yescomplemented
29C2×C3×C4⋊Dic3C2213yescomplemented
30C2×C3×C4×Dic3C2213yescomplemented
31#63974C22112yescomplemented
32C2×#3722C2216yescomplemented
33#63952C2216yescomplemented
34#63952C22112yescomplemented
35#63952C2216yescomplemented
36#63900C2216yescomplemented
37#63894C2216yescomplemented
38#63894C2216yescomplemented
39#63894C2216yescomplemented
40#63894C2216yescomplemented
41#63894C2213yescomplemented
42#63894C2216yescomplemented
43#63894C2213yescomplemented
44C4×#624C2216yescomplemented
45#63874C2216yescomplemented
46#63849C2216yescomplemented
47#63844C2211yescharacteristic, complemented
48#63844C2213yescomplemented
49C2×#1009—31no—
50C2×#1009—31no—
51C2×#1009—61no—
52C3×#520—31no—
53C3×C4×#41—31no—
54#5827D311yescharacteristic, complemented
55Dic3×#41D311yescharacteristic, complemented
56#5574—32no—
57#5574—32no—
58C4×#239—32no—
59C4×#239—32no—
60#5545—36no—
61#5545—36no—
62C4×D3×D4—36no—
63C4×D3×D4—36no—
64#5523—33no—
65#5522—33no—
66#5521—33no—
67#5521—33no—
68#5504—33no—
69C2×#893—33no—
70C4×#235—33no—
71C2×C4×D12—33no—
72C23×C3⋊D3C2316yes—
73C32×#41C2×C418yescomplemented
74C2×C32×Q8C2311yescharacteristic
75C2×C32×D4C2313yes—
76C22×C32×C4C2313yes—
77C2×#638C2316yes—
78#3832C2×C4116yescomplemented
79D4×C3⋊D3C2×C4148yescomplemented
80#3828C2×C4124yescomplemented
81C2×C4⋊(C3⋊D3)C2×C4124yescomplemented
82C2×C4⋊(C3⋊D3)C2313yes—
83C2×C4⋊(C3⋊D3)C2316yes—
84C2×C4×C3⋊D3C2316yes—
85C2×#634C2311yescharacteristic
86C22×C3×Dic3C2313yes—
87C22×C3×Dic3C2313yes—
88C22×C3×Dic3C2316yes—
89C2×#624—412no—
90C2×#624C2316yes—
91C2×#624C23112yes—
92C2×#624C2316yes—
93C3×#217C2316yes—
94C3×#217C2316yes—
95C3×C4⋊Dic3C2313yes—
96C3×C4⋊Dic3C2316yes—
97C3×C4⋊Dic3C2311yescharacteristic
98C3×C4⋊Dic3C2313yes—
99C3×C4⋊Dic3C2313yes—
100C3×C4×Dic3C2311yescharacteristic
101C3×C4×Dic3C2313yes—
102C3×C4×Dic3C2313yes—
103C3×C4×Dic3C2313yes—
104C3×C4×Dic3C2316yes—
105#3724C2312yes—
106#3724C2316yes—
107#3722C23112yes—
108#3722C2316yes—
109#3722C2316yes—
110Dic32C2316yes—
111Dic32C2312yes—
112C4×H(5,𝔽2)—91noSylow
113C2×C3×#41—21no—
114C2×C3×#41C2×D311yescharacteristic
115C2×C3×#41C2×D311yescharacteristic
116#1009—316no—
117#1009—316no—
118#1009—616no—
119C2×#239—32no—
120C2×#239—32no—
121C2×#239—62no—
122C2×D3×D4—36no—
123C2×D3×D4—36no—
124C2×D3×D4—66no—
125C2×#235—33no—
126C2×#235—33no—
127C2×#235—63no—
128C22×D12—33no—
129C22×D12—33no—
130C22×D12—63no—
131C22×C4×D3—36no—
132C22×C4×D3—36no—
133C3×C4×Q8—31no—
134C3×C4×Q8—32no—
135C3×C4×Q8—31no—
136C3×C4×D4—33no—
137C3×C4×D4—36no—
138C3×C4×D4—33no—
139C3×#117—33no—
140C3×#117—33no—
141C2×C3×C4⋊C4—33no—
142C2×C3×C42—33no—
143Q8×Dic3C2×D311yescharacteristic, complemented
144Q8×Dic3C2×D311yescharacteristic, complemented
145Q8×Dic3C2×D312yescomplemented
146D4×Dic3C2×D313yescomplemented
147D4×Dic3C2×D316yescomplemented
148D4×Dic3C2×D313yescomplemented
149C4×#67—36no—
150C4×#67—36no—
151C4×#67—312no—
152C2×(C2×D3)⋊C4—36no—
153C2×(C2×D3)⋊C4—36no—
154#926C2×D313yescomplemented
155#926C2×D313yescomplemented
156C2×C4⋊Dic3C2×D313yescomplemented
157C2×C4×Dic3C2×D313yescomplemented
158#893—33no—
159#893—36no—
160#893—36no—
161#893—36no—
162#893—33no—
163#893—36no—
164#893—36no—
165#892—36no—
166#892—36no—
167D3×C4⋊C4—36no—
168D3×C4⋊C4—36no—
169#887—33no—
170#887—31no—
171#67⋊C4—36no—
172#67⋊C4—36no—
173#67⋊C4—312no—
174D3×C22⋊C4—312no—
175D3×C22⋊C4—312no—
176C4×D12—33no—
177C4×D12—36no—
178C4×D12—36no—
179C4×D12—36no—
180C4×D12—33no—
181C4×D12—36no—
182C4×D12—36no—
183#872—36no—
184#872—36no—
185C42×D3—36no—
186C42×D3—36no—
187C4×Dic6—31no—
188C4×Dic6—33no—
189C23×C32C2413yes—
190C22×C3⋊D3—46no—
191C22×C3⋊D3—46no—
192C22×C3⋊D3C22×C4124yescomplemented
193C22×C3⋊D3C2416yes—
194C32×Q8C22×C414yescomplemented
195C32×D4C22×C4112yescomplemented
196C2×C32×C4C22×C4112yescomplemented
197C2×C32×C4C2411yescharacteristic
198C2×C32×C4C2413yes—
199#638C22×C4124yescomplemented
200C2×C3⋊Dic3C2412yes—
201C4⋊(C3⋊D3)C22×C4112yescomplemented
202C4⋊(C3⋊D3)C22×C4124yescomplemented
203C4×C3⋊D3C22×C4124yescomplemented
204#634C22×C414yescomplemented
205C2×C3×Dic3—26no—
206C2×C3×Dic3—46no—
207C2×C3×Dic3—43no—
208C2×C3×Dic3C2413yes—
209C2×C3×Dic3C2412yes—
210C2×C3×Dic3C2416yes—
211C2×C3×Dic3C2411yescharacteristic
212C2×C3×Dic3C2413yes—
213C2×C3×Dic3C2411yescharacteristic
214#624—212no—
215#624—412no—
216#624—46no—
217C2×H(5,𝔽2)—91no—
218#520—31no—
219#520—93no—
220#520—92no—
221#518—93no—
222#518—96no—
223C4×#41—31no—
224C4×#41—93no—
225C4×#41—92no—
226C2×C4×D4—93no—
227C2×C4×D4—96no—
228C23×D3—36no—
229C23×D3—36no—
230C23×D3—66no—
231C3×#41—28no—
232C3×#41C4×D318yescomplemented
233C3×#41C4×D318yescomplemented
234C2×C3×Q8—21no—
235C2×C3×Q8C22×D311yescharacteristic
236C2×C3×Q8C22×D311yescharacteristic
237C2×C3×D4—23no—
238C2×C3×D4C22×D313yes—
239C2×C3×D4C22×D313yes—
240C22×C3×C4—23no—
241C22×C3×C4—33no—
242C22×C3×C4—33no—
243C22×C3×C4—36no—
244C22×C3×C4C22×D313yes—
245C22×C3×C4C22×D313yes—
246C2×#67—36no—
247C2×#67—36no—
248C2×#67—66no—
249C22×Dic3C22×D313yes—
250C22×Dic3C22×D313yes—
251C22×Dic3C22×D316yes—
252#239—316no—
253#239—316no—
254#239—616no—
255D3×D4—348no—
256D3×D4—348no—
257D3×D4—648no—
258#235—324no—
259#235—324no—
260#235—624no—
261C2×D12—324no—
262C2×D12—324no—
263C2×D12—33no—
264C2×D12—36no—
265C2×D12—36no—
266C2×D12—33no—
267C2×D12—63no—
268C2×D12—66no—
269C2×D12—624no—
270C2×C4×D3—36no—
271C2×C4×D3—36no—
272C2×C4×D3—36no—
273C2×C4×D3—36no—
274C2×C4×D3—36no—
275C2×C4×D3—36no—
276C2×C4×D3—312no—
277C2×C4×D3—312no—
278C2×C4×D3—66no—
279C2×C4×D3—1212no—
280C2×C4×D3—1212no—
281C2×Dic6—31no—
282C2×Dic6—31no—
283C2×Dic6—61no—
284C3×C4⋊C4—33no—
285C3×C4⋊C4—31no—
286C3×C4⋊C4—33no—
287C3×C4⋊C4—36no—
288C3×C4⋊C4—33no—
289C3×C22⋊C4—36no—
290C3×C22⋊C4—36no—
291C3×C42—31no—
292C3×C42—33no—
293C3×C42—33no—
294C3×C42—36no—
295C3×C42—33no—
296#217C22×D316yes—
297#217C22×D316yes—
298(C2×D3)⋊C4—36no—
299(C2×D3)⋊C4—36no—
300(C2×D3)⋊C4—36no—
301(C2×D3)⋊C4—36no—
302(C2×D3)⋊C4—312no—
303(C2×D3)⋊C4—312no—
304C4⋊Dic3C22×D313yes—
305C4⋊Dic3C22×D311yescharacteristic
306C4⋊Dic3C22×D313yes—
307C4⋊Dic3C22×D316yes—
308C4⋊Dic3C22×D313yes—
309Dic3⋊C4—32no—
310Dic3⋊C4—36no—
311Dic3⋊C4—32no—
312Dic3⋊C4—36no—
313C4×Dic3—32no—
314C4×Dic3—36no—
315C4×Dic3—32no—
316C4×Dic3—36no—
317C4×Dic3C22×D313yes—
318C4×Dic3C22×D316yes—
319C4×Dic3C22×D313yes—
320C4×Dic3C22×D311yescharacteristic
321C4×Dic3C22×D313yes—
322C22×C32—23no—
323C22×C32—23no—
324C22×C32C23×C416yes—
325C22×C32C2511yescharacteristic
326C2×C3⋊D3—26no—
327C2×C3⋊D3—26no—
328C2×C3⋊D3—424no—
329C2×C3⋊D3C23×C4112yes—
330C32×C4C23×C412yes—
331C32×C4C23×C416yes—
332C3⋊Dic3C23×C414yes—
333C3×Dic3—26no—
334C3×Dic3—23no—
335C3×Dic3—22no—
336C3×Dic3—23no—
337C3×Dic3—21no—
338C3×Dic3H(5,𝔽2)12yescomplemented
339H(5,𝔽2)—916no—
340C2×#41—92no—
341C2×#41—93no—
342C2×#41D3211yescharacteristic
343C22×D4—96no—
344C22×D4—93no—
345C23×C4—96no—
346C4×Q8—31no—
347C4×Q8—31no—
348C4×Q8—32no—
349C4×Q8—91no—
350C4×Q8—93no—
351C4×D4—33no—
352C4×D4—36no—
353C4×D4—33no—
354C4×D4—93no—
355C4×D4—96no—
356C4×D4—96no—
357C4×D4—96no—
358C4×D4—96no—
359C4×D4—93no—
360C4×D4—96no—
361C4×D4—96no—
362C4×D4—912no—
363C4×D4—96no—
364#117—33no—
365#117—33no—
366#117—96no—
367#117—96no—
368C2×C4⋊C4—33no—
369C2×C4⋊C4—96no—
370C2×C22⋊C4—912no—
371C2×C22⋊C4—96no—
372C2×C42—33no—
373C2×C42—96no—
374C23×C3—23no—
375C23×C3C23×D313yes—
376C23×C3C23×D313yes—
377C22×D3—36no—
378C22×D3—36no—
379C22×D3—324no—
380C22×D3—324no—
381C22×D3—66no—
382C22×D3—624no—
383C22×D3—126no—
384C22×D3—126no—
385C22×D3—126no—
386C22×D3—126no—
387C22×D3—246no—
388C22×D3—246no—
389C3×Q8—24no—
390C3×Q8C2×C4×D314yescomplemented
391C3×Q8C2×C4×D314yescomplemented
392C3×D4—212no—
393C3×D4C2×C4×D3112yescomplemented
394C3×D4C2×C4×D3112yescomplemented
395C2×C3×C4—21no—
396C2×C3×C4—23no—
397C2×C3×C4—212no—
398C2×C3×C4—31no—
399C2×C3×C4—33no—
400C2×C3×C4—32no—
401C2×C3×C4—36no—
402C2×C3×C4—31no—
403C2×C3×C4—33no—
404C2×C3×C4—66no—
405C2×C3×C4—126no—
406C2×C3×C4—123no—
407C2×C3×C4C2×C4×D3112yescomplemented
408C2×C3×C4C2×C4×D3112yescomplemented
409C2×C3×C4C23×D311yescharacteristic
410C2×C3×C4C23×D313yes—
411C2×C3×C4C23×D313yes—
412C2×C3×C4C23×D311yescharacteristic
413#67—324no—
414#67—324no—
415#67—624no—
416C2×Dic3—26no—
417C2×Dic3—32no—
418C2×Dic3—32no—
419C2×Dic3—46no—
420C2×Dic3—43no—
421C2×Dic3—62no—
422C2×Dic3C23×D313yes—
423C2×Dic3C23×D312yes—
424C2×Dic3C23×D316yes—
425C2×Dic3C23×D311yescharacteristic
426C2×Dic3C23×D313yes—
427C2×Dic3C23×D311yescharacteristic
428D12—312no—
429D12—312no—
430D12—324no—
431D12—324no—
432D12—612no—
433D12—624no—
434C4×D3—324no—
435C4×D3—324no—
436C4×D3—624no—
437C4×D3—612no—
438C4×D3—612no—
439C4×D3—126no—
440C4×D3—126no—
441C4×D3—1212no—
442C4×D3—1212no—
443Dic6—34no—
444Dic6—34no—
445Dic6—64no—
446C2×C32—26no—
447C2×C32C24×C411yescharacteristic
448C2×C32C2×H(5,𝔽2)11yescharacteristic
449C2×C32#58511yescharacteristic
450C3⋊D3—212no—
451C24—96no—
452#41—916no—
453#41—924no—
454#41#62418yescomplemented
455C2×Q8—91no—
456C2×Q8C2×D3211yescharacteristic
457C2×D4—96no—
458C2×D4—948no—
459C2×D4—96no—
460C2×D4—93no—
461C2×D4—924no—
462C2×D4C2×D3213yes—
463C22×C4—33no—
464C22×C4—33no—
465C22×C4—36no—
466C22×C4—96no—
467C22×C4—96no—
468C22×C4—912no—
469C22×C4—96no—
470C22×C4—3612no—
471C22×C4C2×D3213yes—
472C4⋊C4—33no—
473C4⋊C4—36no—
474C4⋊C4—33no—
475C4⋊C4—31no—
476C4⋊C4—33no—
477C4⋊C4—96no—
478C4⋊C4—92no—
479C22⋊C4—36no—
480C22⋊C4—36no—
481C22⋊C4—96no—
482C22⋊C4—96no—
483C22⋊C4—912no—
484C42—31no—
485C42—33no—
486C42—33no—
487C42—33no—
488C42—36no—
489C42—92no—
490C42—96no—
491C22×C3—23no—
492C22×C3—23no—
493C22×C3—23no—
494C22×C3—23no—
495C22×C3—26no—
496C22×C3—21no—
497C22×C3—43no—
498C22×C3—43no—
499C22×C3C22×C4×D316yes—
500C22×C3C22×C4×D316yes—
501C22×C3C24×D311yescharacteristic
502C22×C3C24×D311yescharacteristic
503C2×D3—312no—
504C2×D3—312no—
505C2×D3—612no—
506C2×D3—66no—
507C2×D3—66no—
508C2×D3—66no—
509C2×D3—66no—
510C2×D3—126no—
511C2×D3—126no—
512C2×D3—1224no—
513C2×D3—1224no—
514C2×D3—2424no—
515C3×C4—22no—
516C3×C4—26no—
517C3×C4—32no—
518C3×C4—66no—
519C3×C4—62no—
520C3×C4—63no—
521C3×C4—63no—
522C3×C4—61no—
523C3×C4C22×C4×D312yes—
524C3×C4C22×C4×D312yes—
525C3×C4C22×C4×D316yes—
526C3×C4C22×C4×D316yes—
527Dic3—23no—
528Dic3—23no—
529Dic3—21no—
530Dic3—26no—
531Dic3—22no—
532Dic3—34no—
533Dic3—34no—
534Dic3—64no—
535Dic3#100912yescomplemented
536C32C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
537C23—96no—
538C23—924no—
539C23—366no—
540C23—366no—
541C23C22×D3213yes—
542Q8—94no—
543Q8C2×#62414yescomplemented
544D4—924no—
545D4—924no—
546D4—912no—
547D4C2×#624112yescomplemented
548C2×C4—31no—
549C2×C4—31no—
550C2×C4—32no—
551C2×C4—33no—
552C2×C4—33no—
553C2×C4—36no—
554C2×C4—66no—
555C2×C4—92no—
556C2×C4—924no—
557C2×C4—126no—
558C2×C4—123no—
559C2×C4—1812no—
560C2×C4—3612no—
561C2×C4—366no—
562C2×C4C2×#624112yescomplemented
563C2×C4C22×D3213yes—
564C2×C4C22×D3211yescharacteristic
565C2×C3—26no—
566C2×C3—26no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—46no—
571C2×C3C23×C4×D311yescharacteristic
572C2×C3C23×C4×D311yescharacteristic
573C2×C3C2×#100911yescharacteristic
574C2×C3#596411yescharacteristic
575C2×C3D3×H(5,𝔽2)11yescharacteristic
576C2×C3#596811yescharacteristic
577D3—612no—
578D3—612no—
579D3—1212no—
580C22—23no—
581C22—23no—
582C22—912no—
583C22—186no—
584C22—186no—
585C22—3624no—
586C22C22×#62416yes—
587C22C23×D3211yescharacteristic
588C4—32no—
589C4—63no—
590C4—66no—
591C4—61no—
592C4—63no—
593C4—62no—
594C4—94no—
595C4C22×#62416yes—
596C4C22×#62412yes—
597C3—21no—
598C3C4×#100911yescharacteristic, complemented
599C3#8588611yescharacteristic, complemented
600C2—26no—
601C2—1812no—
602C2D3×#100911yescharacteristic
603C2#1059669711yescharacteristic
604C2C23×#62411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D32
commutatorC2×C32C24×C4
FittingC2×C32×#41C22
socle, cosocleC22×C32C25
prime radicalC2×#64366C2
hypercenterC2×#41D32
hypocenterC32C4×H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C3×#5827C2semidirect
3. C3×Dic3×#41C2semidirect
4. #10595386C2semidirect
5. #10595353C2semidirect
6. #10595223C2semidirect
7. D4×#624C2semidirect
8. #10594996C2semidirect
9. #10594990C2semidirect
10. #10594954C2semidirect
11. C2×#63894C2semidirect
15. #64366C4semidirect
20. C22×#624C22semidirect
21. C3×Q8×Dic3C22semidirect
22. C3×Q8×Dic3C22semidirect
23. C3×Q8×Dic3C22semidirect
24. C3×D4×Dic3C22semidirect
25. C3×D4×Dic3C22semidirect
26. C3×D4×Dic3C22semidirect
27. C3×#926C22semidirect
28. C3×#926C22semidirect
29. C2×C3×C4⋊Dic3C22semidirect
30. C2×C3×C4×Dic3C22semidirect
31. #63974C22semidirect
32. C2×#3722C22semidirect
33. #63952C22semidirect
34. #63952C22semidirect
35. #63952C22semidirect
36. #63900C22semidirect
37. #63894C22semidirect
38. #63894C22semidirect
39. #63894C22semidirect
40. #63894C22semidirect
41. #63894C22semidirect
42. #63894C22semidirect
43. #63894C22semidirect
44. C4×#624C22semidirect
45. #63874C22semidirect
46. #63849C22semidirect
47. #63844C22semidirect
48. #63844C22semidirect
54. #5827D3semidirect
55. Dic3×#41D3semidirect
73. C32×#41C2×C4semidirect
78. #3832C2×C4semidirect
79. D4×C3⋊D3C2×C4semidirect
80. #3828C2×C4semidirect
81. C2×C4⋊(C3⋊D3)C2×C4semidirect
143. Q8×Dic3C2×D3semidirect
144. Q8×Dic3C2×D3semidirect
145. Q8×Dic3C2×D3semidirect
146. D4×Dic3C2×D3semidirect
147. D4×Dic3C2×D3semidirect
148. D4×Dic3C2×D3semidirect
154. #926C2×D3semidirect
155. #926C2×D3semidirect
156. C2×C4⋊Dic3C2×D3semidirect
157. C2×C4×Dic3C2×D3semidirect
192. C22×C3⋊D3C22×C4semidirect
194. C32×Q8C22×C4semidirect
195. C32×D4C22×C4semidirect
196. C2×C32×C4C22×C4semidirect
199. #638C22×C4semidirect
201. C4⋊(C3⋊D3)C22×C4semidirect
202. C4⋊(C3⋊D3)C22×C4semidirect
203. C4×C3⋊D3C22×C4semidirect
204. #634C22×C4semidirect
232. C3×#41C4×D3semidirect
233. C3×#41C4×D3semidirect
338. C3×Dic3H(5,𝔽2)semidirect
390. C3×Q8C2×C4×D3semidirect
391. C3×Q8C2×C4×D3semidirect
393. C3×D4C2×C4×D3semidirect
394. C3×D4C2×C4×D3semidirect
407. C2×C3×C4C2×C4×D3semidirect
408. C2×C3×C4C2×C4×D3semidirect
454. #41#624semidirect
535. Dic3#1009semidirect
536. C32C4×H(5,𝔽2)semidirect
543. Q8C2×#624semidirect
547. D4C2×#624semidirect
562. C2×C4C2×#624semidirect
598. C3C4×#1009semidirect
599. C3#85886semidirect

Maximal subgroups

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

subgroupindex
C2×#643662
C3×#58272
C3×Dic3×#412
#105953862
#105953532
#105952232
D4×#6242
#105949962
#105949902
#105949542
C2×#638942
#858863
C4×#10093