Finite Group Databasebeta

← #11904789

Subgroups — #11904789

Order 1152.

3191 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1010C211yesmaximal, characteristic, complemented
2C2×C3×D3×#41C211yesmaximal, characteristic, complemented
3#10596298C211yesmaximal, characteristic, complemented
4#10595409C212yesmaximal, complemented
5Q8×#625C212yesmaximal, complemented
6#10595252C216yesmaximal, complemented
7#10595244C216yesmaximal, complemented
8#10595044C213yesmaximal, complemented
9#10595022C213yesmaximal, complemented
10#10594855C213yesmaximal, complemented
11C2×#63856C213yesmaximal, complemented
12#87295—31nomaximal
13#87264—31nomaximal
14C2×C32×#41C2211yescharacteristic
15C3×#1010—28no—
16C3×D3×#41—28no—
17C2×C3×#239C2211yescharacteristic
18C2×C3×D3×Q8C2211yescharacteristic
19C2×C3×D3×Q8C2212yescomplemented
20C2×C3×#237C2213yes—
21C2×C3×#237C2216yescomplemented
22C2×C3×D3×D4C2213yes—
23C2×C3×#235C2213yes—
24C2×C3×#235C2213yescomplemented
25C22×C3×C4×D3C2213yes—
26C22×C3×Dic6C2213yescomplemented
27Q8×C3⋊Dic3C2212yescomplemented
28D4×C3⋊Dic3C2216yescomplemented
29#64135C2213yescomplemented
30C2×C4⋊(C3⋊Dic3)C2213yescomplemented
31#63980C22112yescomplemented
32#63976C2216yes—
33C2×#3724C2216yescomplemented
34#63961C22112yescomplemented
35#63961C2216yescomplemented
36C2×(C2×D3)⋊Dic3C2216yes—
37#63908C2211yescharacteristic
38#63908C2216yescomplemented
39#63908C2213yescomplemented
40C4×#625C2216yes—
41C4×#625C2216yescomplemented
42#63873C2216yescomplemented
43#63863C2216yescomplemented
44#63856C2213yes—
45#63856C2216yes—
46#63856C2216yescomplemented
47#63856C2216yescomplemented
48#63856C2213yescomplemented
49#63856C2216yescomplemented
50C2×C3×2−1+4—31no—
51C22×C3×#41—31no—
52C2×#1010D311yescharacteristic, complemented
53C2×D3×#41D311yescharacteristic, complemented
54#5833—33no—
55#5833—33no—
56#5831—32no—
57#5829—36no—
58#5828—33no—
59#5827—31no—
60#5827—31no—
61#5827—61no—
62#5825—33no—
63#5824—36no—
64#5824—36no—
65#5817—32no—
66#5816—33no—
67Q8×#67—32no—
68Q8×#67—32no—
69C2×#946—33no—
70#5806—36no—
71#5791—33no—
72C2×#924—33no—
73C23×C3×D3C2313yes—
74C32×#41D418yescomplemented
75C2×C32×Q8C2311yescharacteristic
76C2×C32×D4C2313yes—
77C22×C32×C4C2313yes—
78C22×C3⋊Dic3C2316yes—
79C2×C3×#67C2316yes—
80C2×C3×#67C2316yes—
81C22×C3×Dic3C2316yes—
82C22×C3×Dic3C2313yes—
83C3×#239—24no—
84C3×D3×Q8—24no—
85C3×D3×Q8—28no—
86C3×#237—212no—
87C3×#237—224no—
88C3×D3×D4—212no—
89C3×#235—212no—
90C3×#235—212no—
91C2×C3×D12C2313yes—
92C2×C3×D12C2311yescharacteristic
93C2×C3×C4×D3—212no—
94C2×C3×C4×D3C2311yescharacteristic
95C2×C3×C4×D3C2313yes—
96C2×C3×C4×D3C2313yes—
97C2×C3×C4×D3C2313yes—
98C2×C3×C4×D3C2316yes—
99C2×C3×Dic6—212no—
100C2×C3×Dic6C2313yes—
101C2×C3×Dic6C2316yes—
102C2×C3×Dic6C2313yes—
103C2×#625C2312yes—
104C2×#625C2316yes—
105#3757C2316yes—
106C4⋊(C3⋊Dic3)C2313yes—
107C4⋊(C3⋊Dic3)C2316yes—
108C4⋊(C3⋊Dic3)C2311yescharacteristic
109C4×C3⋊Dic3C2316yes—
110#3724C23112yes—
111#3724C2316yes—
112#3724C2316yes—
113(C2×D3)⋊Dic3C2316yes—
114(C2×D3)⋊Dic3C23112yes—
115(C2×D3)⋊Dic3C2316yes—
116(C2×D3)⋊Dic3C2316yes—
117(C2×D3)⋊Dic3C2312yes—
118#3489—91noSylow
119C3×2−1+4—68no—
120C2×C3×#41—21no—
121C2×C3×#41—31no—
122C2×C3×#41—33no—
123C2×C3×#41—36no—
124C2×C3×#41—33no—
125C2×C3×#41—33no—
126C2×C3×#41—68no—
127C2×C3×#41C2×D311yescharacteristic
128C2×C3×#41C2×D311yescharacteristic
129C22×C3×Q8—31no—
130C22×C3×Q8—32no—
131C22×C3×Q8—33no—
132C22×C3×D4—33no—
133C23×C3×C4—33no—
134#1010—28no—
135D3×#41—28no—
136C2×#239C2×D311yescharacteristic
137C2×D3×Q8C2×D311yescharacteristic
138C2×D3×Q8C2×D312yescomplemented
139C2×#237C2×D313yes—
140C2×#237C2×D316yescomplemented
141C2×D3×D4C2×D313yes—
142C2×#235C2×D313yes—
143C2×#235C2×D313yescomplemented
144C22×C4×D3C2×D313yes—
145C22×Dic6C2×D313yescomplemented
146#953—36no—
147C2×#217—36no—
148#947—36no—
149#946—33no—
150#946—36no—
151#946—36no—
152#946—36no—
153#946—36no—
154#946—33no—
155Q8×Dic3—32no—
156Q8×Dic3—32no—
157Q8×Dic3—62no—
158#944—36no—
159#944—36no—
160#939—312no—
161#939—312no—
162#938—31no—
163#938—36no—
164#938—33no—
165#937—36no—
166#936—36no—
167#935—312no—
168#935—36no—
169D4×Dic3—36no—
170D4×Dic3—36no—
171D4×Dic3—66no—
172#930—31no—
173#930—33no—
174#930—36no—
175#929—36no—
176#929—312no—
177C4×#67—36no—
178C4×#67—36no—
179C4×#67—36no—
180C4×#67—36no—
181C2×(C2×D3)⋊C4—36no—
182#926—33no—
183#926—33no—
184#926—63no—
185C2×C4⋊Dic3—33no—
186C2×C4⋊Dic3—33no—
187C2×C4⋊Dic3—63no—
188#924—33no—
189#924—36no—
190#924—36no—
191#924—36no—
192#924—33no—
193#924—36no—
194C2×Dic3⋊C4—36no—
195C2×Dic3⋊C4—36no—
196C23×C32C2413yes—
197C22×C3×D3—26no—
198C22×C3×D3—26no—
199C22×C3×D3—43no—
200C22×C3×D3C2413yes—
201C22×C3×D3C2412yes—
202C22×C3×D3C2411yescharacteristic
203C32×Q8C2×D414yes—
204C32×D4C2×D4112yes—
205C2×C32×C4C2×D4112yes—
206C2×C32×C4C2413yes—
207C2×C32×C4C2411yescharacteristic
208C2×C3⋊Dic3—46no—
209C2×C3⋊Dic3C2412yes—
210C2×C3⋊Dic3C2416yes—
211C3×#67—212no—
212C3×#67—212no—
213C2×C3×Dic3—212no—
214C2×C3×Dic3—26no—
215C2×C3×Dic3C2416yes—
216C2×C3×Dic3C2413yes—
217C2×C3×Dic3C2411yescharacteristic
218C3×D12—22no—
219C3×D12—26no—
220C3×C4×D3—22no—
221C3×C4×D3—26no—
222C3×C4×D3—26no—
223C3×C4×D3—26no—
224C3×C4×D3—212no—
225C3×Dic6—26no—
226C3×Dic6—212no—
227C3×Dic6—26no—
228#625—24no—
229#625—46no—
230C2×2−1+4—31no—
231C22×#41—31no—
232D4×Q8—92no—
233#548—92no—
234#547—96no—
235#546—96no—
236#537—93no—
237#536—93no—
238#525—93no—
239C2×C22⋊Q8—93no—
240#520—91no—
241C24×C3—33no—
242C23×D3C22×D313yes—
243C3×#41—28no—
244C3×#41—64no—
245C3×#41—612no—
246C3×#41—624no—
247C3×#41—612no—
248C3×#41—612no—
249C3×#41#6718yescomplemented
250C3×#41#6718yescomplemented
251C2×C3×Q8—21no—
252C2×C3×Q8—33no—
253C2×C3×Q8—36no—
254C2×C3×Q8—33no—
255C2×C3×Q8—64no—
256C2×C3×Q8—68no—
257C2×C3×Q8—612no—
258C2×C3×Q8C22×D311yescharacteristic
259C2×C3×Q8C22×D311yescharacteristic
260C2×C3×D4—23no—
261C2×C3×D4—36no—
262C2×C3×D4—36no—
263C2×C3×D4—33no—
264C2×C3×D4—31no—
265C2×C3×D4—612no—
266C2×C3×D4C22×D313yes—
267C2×C3×D4C22×D313yes—
268C22×C3×C4—23no—
269C22×C3×C4—33no—
270C22×C3×C4—36no—
271C22×C3×C4—31no—
272C22×C3×C4—33no—
273C22×C3×C4—33no—
274C22×C3×C4—33no—
275C22×C3×C4—36no—
276C22×C3×C4—612no—
277C22×C3×C4C22×D313yes—
278C22×C3×C4C22×D313yes—
279C2×#67—32no—
280C2×#67—32no—
281C2×#67—36no—
282C2×#67—36no—
283C2×#67C22×D316yes—
284C2×#67C22×D316yes—
285C22×Dic3—36no—
286C22×Dic3—36no—
287C22×Dic3—66no—
288C22×Dic3C22×D313yes—
289C22×Dic3C22×D316yes—
290#239—24no—
291D3×Q8—24no—
292D3×Q8—28no—
293#237—212no—
294#237—224no—
295D3×D4—212no—
296#235—212no—
297#235—212no—
298C2×D12C22×D311yescharacteristic
299C2×D12C22×D313yes—
300C2×C4×D3—212no—
301C2×C4×D3C22×D316yes—
302C2×C4×D3C22×D313yes—
303C2×C4×D3C22×D313yes—
304C2×C4×D3C22×D311yescharacteristic
305C2×C4×D3C22×D313yes—
306C2×Dic6—212no—
307C2×Dic6C22×D313yes—
308C2×Dic6C22×D316yes—
309C2×Dic6C22×D313yes—
310#217—36no—
311#217—36no—
312#217—36no—
313#217—36no—
314#217—312no—
315#217—36no—
316#217—32no—
317#217—66no—
318(C2×D3)⋊C4—36no—
319(C2×D3)⋊C4—312no—
320(C2×D3)⋊C4—32no—
321(C2×D3)⋊C4—36no—
322(C2×D3)⋊C4—36no—
323C4⋊Dic3—33no—
324C4⋊Dic3—31no—
325C4⋊Dic3—36no—
326C4⋊Dic3—36no—
327C4⋊Dic3—33no—
328C4⋊Dic3—31no—
329C4⋊Dic3—63no—
330C4⋊Dic3—66no—
331C4⋊Dic3—61no—
332Dic3⋊C4—36no—
333Dic3⋊C4—36no—
334Dic3⋊C4—312no—
335Dic3⋊C4—312no—
336Dic3⋊C4—36no—
337Dic3⋊C4—36no—
338C4×Dic3—36no—
339C4×Dic3—36no—
340C4×Dic3—66no—
341C22×C32—23no—
342C22×C32—23no—
343C22×C32C22×D416yes—
344C22×C32C2511yescharacteristic
345C2×C3×D3—21no—
346C2×C3×D3—21no—
347C2×C3×D3—22no—
348C2×C3×D3—23no—
349C2×C3×D3—22no—
350C2×C3×D3—23no—
351C2×C3×D3—22no—
352C2×C3×D3—23no—
353C2×C3×D3—412no—
354C2×C3×D32−1+412yescomplemented
355C32×C4C22×D412yes—
356C32×C4C22×D416yes—
357C3⋊Dic3—22no—
358C3⋊Dic3—26no—
359C3×Dic3—26no—
360C3×Dic3—23no—
361C3×Dic3—21no—
3622−1+4—68no—
363C2×#41—31no—
364C2×#41—36no—
365C2×#41—33no—
366C2×#41—33no—
367C2×#41—33no—
368C2×#41—68no—
369C2×#41D3211yescharacteristic
370C22×Q8—32no—
371C22×Q8—31no—
372C22×Q8—33no—
373C22×D4—33no—
374C23×C4—33no—
375C4⋊Q8—96no—
376#124—96no—
377#123—96no—
378#123—912no—
379C22⋊Q8—93no—
380C22⋊Q8—96no—
381C22⋊Q8—96no—
382C22⋊Q8—96no—
383C22⋊Q8—93no—
384C22⋊Q8—96no—
385#121—91no—
386#121—93no—
387#121—96no—
388#121—912no—
389#120—96no—
390C4×Q8—92no—
391C4×D4—96no—
392C4×D4—96no—
393C4×D4—96no—
394#117—93no—
395C2×C4⋊C4—96no—
396C2×C4⋊C4—93no—
397C2×C22⋊C4—96no—
398C23×C3—23no—
399C23×C3—31no—
400C23×C3—33no—
401C23×C3—32no—
402C23×C3—66no—
403C23×C3—66no—
404C23×C3—123no—
405C23×C3C23×D313yes—
406C23×C3C23×D313yes—
407C22×D3—26no—
408C22×D3—26no—
409C22×D3—43no—
410C22×D3C23×D313yes—
411C22×D3C23×D312yes—
412C22×D3C23×D311yescharacteristic
413C3×Q8—24no—
414C3×Q8—612no—
415C3×Q8—66no—
416C3×Q8—66no—
417C3×Q8C2×#6714yes—
418C3×Q8C2×#6714yes—
419C3×D4—212no—
420C3×D4—612no—
421C3×D4—612no—
422C3×D4—66no—
423C3×D4—62no—
424C3×D4C2×#67112yes—
425C3×D4C2×#67112yes—
426C2×C3×C4—23no—
427C2×C3×C4—21no—
428C2×C3×C4—212no—
429C2×C3×C4—31no—
430C2×C3×C4—33no—
431C2×C3×C4—36no—
432C2×C3×C4—66no—
433C2×C3×C4—612no—
434C2×C3×C4—62no—
435C2×C3×C4—66no—
436C2×C3×C4—66no—
437C2×C3×C4—66no—
438C2×C3×C4—612no—
439C2×C3×C4C2×#67112yes—
440C2×C3×C4C2×#67112yes—
441C2×C3×C4C23×D313yes—
442C2×C3×C4C23×D313yes—
443C2×C3×C4C23×D311yescharacteristic
444C2×C3×C4C23×D311yescharacteristic
445#67—212no—
446#67—212no—
447#67—64no—
448#67—64no—
449#67—126no—
450#67—126no—
451C2×Dic3—212no—
452C2×Dic3—26no—
453C2×Dic3—32no—
454C2×Dic3—32no—
455C2×Dic3—36no—
456C2×Dic3—36no—
457C2×Dic3—62no—
458C2×Dic3—66no—
459C2×Dic3—126no—
460C2×Dic3—126no—
461C2×Dic3—246no—
462C2×Dic3C23×D313yes—
463C2×Dic3C23×D311yescharacteristic
464C2×Dic3C23×D316yes—
465D12—22no—
466D12—26no—
467C4×D3—22no—
468C4×D3—26no—
469C4×D3—26no—
470C4×D3—26no—
471C4×D3—212no—
472Dic6—26no—
473Dic6—212no—
474Dic6—26no—
475C2×C32—26no—
476C2×C32C23×D411yescharacteristic
477C2×C32C2×2−1+411yescharacteristic
478C2×C32#58511yescharacteristic
479C3×D3—22no—
480C3×D3—42no—
481C3×D3—43no—
482C24—33no—
483#41—64no—
484#41—612no—
485#41—624no—
486#41—612no—
487#41—612no—
488#41#62518yescomplemented
489C2×Q8—33no—
490C2×Q8—36no—
491C2×Q8—33no—
492C2×Q8—64no—
493C2×Q8—68no—
494C2×Q8—612no—
495C2×Q8C2×D3211yescharacteristic
496C2×D4—36no—
497C2×D4—36no—
498C2×D4—31no—
499C2×D4—33no—
500C2×D4—612no—
501C2×D4—92no—
502C2×D4—96no—
503C2×D4C2×D3213yes—
504C22×C4—36no—
505C22×C4—33no—
506C22×C4—36no—
507C22×C4—33no—
508C22×C4—33no—
509C22×C4—31no—
510C22×C4—33no—
511C22×C4—612no—
512C22×C4—96no—
513C22×C4C2×D3213yes—
514C4⋊C4—96no—
515C4⋊C4—912no—
516C4⋊C4—96no—
517C4⋊C4—93no—
518C4⋊C4—96no—
519C4⋊C4—91no—
520C22⋊C4—96no—
521C22⋊C4—912no—
522C22⋊C4—92no—
523C22⋊C4—96no—
524C22⋊C4—96no—
525C22⋊C4—96no—
526C42—96no—
527C22×C3—23no—
528C22×C3—23no—
529C22×C3—23no—
530C22×C3—23no—
531C22×C3—26no—
532C22×C3—21no—
533C22×C3—32no—
534C22×C3—43no—
535C22×C3—43no—
536C22×C3—61no—
537C22×C3—61no—
538C22×C3—62no—
539C22×C3—62no—
540C22×C3—63no—
541C22×C3—63no—
542C22×C3—62no—
543C22×C3—63no—
544C22×C3—1212no—
545C22×C3C22×#6716yes—
546C22×C3C22×#6716yes—
547C22×C3C24×D311yescharacteristic
548C22×C3C24×D311yescharacteristic
549C2×D3—21no—
550C2×D3—21no—
551C2×D3—23no—
552C2×D3—23no—
553C2×D3—22no—
554C2×D3—22no—
555C2×D3—22no—
556C2×D3—23no—
557C2×D3—412no—
558C2×D3#101012yescomplemented
559C3×C4—22no—
560C3×C4—26no—
561C3×C4—61no—
562C3×C4—63no—
563C3×C4—66no—
564C3×C4C22×#6712yes—
565C3×C4C22×#6716yes—
566C3×C4C22×#6716yes—
567C3×C4C22×#6712yes—
568Dic3—26no—
569Dic3—23no—
570Dic3—21no—
571Dic3—62no—
572Dic3—66no—
573Dic3—62no—
574Dic3—66no—
575Dic3—126no—
576Dic3—122no—
577C32#348911yescharacteristic, complemented, Sylow
578C23—31no—
579C23—33no—
580C23—32no—
581C23—66no—
582C23—66no—
583C23—123no—
584C23C22×D3213yes—
585Q8—66no—
586Q8—612no—
587Q8—66no—
588Q8C2×#62514yes—
589D4—612no—
590D4—612no—
591D4—66no—
592D4—62no—
593D4—184no—
594D4—366no—
595D4C2×#625112yes—
596C2×C4—36no—
597C2×C4—31no—
598C2×C4—33no—
599C2×C4—66no—
600C2×C4—612no—
601C2×C4—62no—
602C2×C4—66no—
603C2×C4—66no—
604C2×C4—66no—
605C2×C4—612no—
606C2×C4—92no—
607C2×C4—96no—
608C2×C4—366no—
609C2×C4C2×#625112yes—
610C2×C4C22×D3213yes—
611C2×C4C22×D3211yescharacteristic
612C2×C3—26no—
613C2×C3—26no—
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—46no—
618C2×C3—62no—
619C2×C3—122no—
620C2×C3—123no—
621C2×C3C2×#101011yescharacteristic
622C2×C3#596411yescharacteristic
623C2×C3D3×2−1+411yescharacteristic
624C2×C3#596811yescharacteristic
625C2×C3C23×#6711yescharacteristic
626C2×C3C23×#6711yescharacteristic
627D3—22no—
628D3—43no—
629D3—42no—
630C22—23no—
631C22—23no—
632C22—32no—
633C22—61no—
634C22—61no—
635C22—62no—
636C22—63no—
637C22—62no—
638C22—63no—
639C22—62no—
640C22—63no—
641C22—1212no—
642C22C22×#62516yes—
643C22C23×D3211yescharacteristic
644C4—61no—
645C4—63no—
646C4—66no—
647C4—182no—
648C4—186no—
649C4C22×#62512yes—
650C4C22×#62516yes—
651C3—21no—
652C3#8726411yescharacteristic, complemented
653C3#8729511yescharacteristic, complemented
654C2—26no—
655C2—62no—
656C2—123no—
657C2—122no—
658C2D3×#101011yescharacteristic
659C2#1059669711yescharacteristic
660C2C23×#62511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D32
commutator, socle, cosocleC22×C32C25
FittingC2×C32×#41C22
hypercenterC2×#41D32
hypocenterC32#3489
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1010C2semidirect
2. C2×C3×D3×#41C2semidirect
3. #10596298C2semidirect
4. #10595409C2semidirect
5. Q8×#625C2semidirect
6. #10595252C2semidirect
7. #10595244C2semidirect
8. #10595044C2semidirect
9. #10595022C2semidirect
10. #10594855C2semidirect
11. C2×#63856C2semidirect
19. C2×C3×D3×Q8C22semidirect
21. C2×C3×#237C22semidirect
24. C2×C3×#235C22semidirect
26. C22×C3×Dic6C22semidirect
27. Q8×C3⋊Dic3C22semidirect
28. D4×C3⋊Dic3C22semidirect
29. #64135C22semidirect
30. C2×C4⋊(C3⋊Dic3)C22semidirect
31. #63980C22semidirect
33. C2×#3724C22semidirect
34. #63961C22semidirect
35. #63961C22semidirect
38. #63908C22semidirect
39. #63908C22semidirect
41. C4×#625C22semidirect
42. #63873C22semidirect
43. #63863C22semidirect
46. #63856C22semidirect
47. #63856C22semidirect
48. #63856C22semidirect
49. #63856C22semidirect
52. C2×#1010D3semidirect
53. C2×D3×#41D3semidirect
74. C32×#41D4semidirect
138. C2×D3×Q8C2×D3semidirect
140. C2×#237C2×D3semidirect
143. C2×#235C2×D3semidirect
145. C22×Dic6C2×D3semidirect
249. C3×#41#67semidirect
250. C3×#41#67semidirect
354. C2×C3×D32−1+4semidirect
488. #41#625semidirect
558. C2×D3#1010semidirect
577. C32#3489semidirect
652. C3#87264semidirect
653. C3#87295semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10102
C2×C3×D3×#412
#105962982
#105954092
Q8×#6252
#105952522
#105952442
#105950442
#105950222
#105948552
C2×#638562
#872953
#872643