Finite Group Databasebeta

← #11233684

Subgroups — #11233684

Order 768.

2582 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5543C211yesmaximal, characteristic, complemented
2C2×#5555C211yesmaximal, characteristic, complemented
3C3×#2397C211yesmaximal, characteristic, complemented
4#82415C211yesmaximal, characteristic, complemented
5#82377C212yesmaximal, complemented
6#82375C211yesmaximal, characteristic, complemented
7#82353C212yesmaximal, complemented
8#82352C212yesmaximal, complemented
9D4×C4⋊Dic3C212yesmaximal, complemented
10#78119C214yesmaximal, complemented
11#78064C214yesmaximal, complemented
12#76675C214yesmaximal, complemented
13#76668C214yesmaximal, complemented
14#73391C211yesmaximal, characteristic, complemented
15#73387C211yesmaximal, characteristic, complemented
16#45356—31nomaximal, Sylow
17C22×#237C2212yes—
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2212yes—
22C2×#939C2214yes—
23C2×D4×Dic3C2212yes—
24C2×D4×Dic3C2214yes—
25C2×#930C2212yes—
26C2×#930C2212yes—
27C22×C4⋊Dic3C2212yes—
28C22×C4⋊Dic3C2212yes—
29C2×#924C2212yes—
30C2×#924C2212yes—
31#5555—28no—
32#5543—28no—
33C2×#897C2212yes—
34C2×#890C2212yes—
35C2×#886C2214yes—
36C2×#878C2214yes—
37C2×#875C2211yescharacteristic
38C2×C4×D12C2211yescharacteristic
39C2×#869C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41C3×#399C2212yes—
42C3×#399C2212yes—
43C3×#392C2212yes—
44C3×#392C2212yes—
45C3×#376C2211yescharacteristic
46#5223C2214yes—
47#5223C2212yes—
48#5222C2212yes—
49#5220C2214yes—
50#5212C2212yes—
51#5212C2212yes—
52#4994C2214yes—
53#4993C2214yes—
54#4992C2212yes—
55#4984C2214yes—
56#4983C2212yes—
57#4982C2214yes—
58#4961C2214yes—
59#4958C2214yes—
60#4957C2214yes—
61#4957C2214yes—
62#4956C2214yes—
63#4950C2214yes—
64#4950C2214yes—
65#4948C2214yes—
66#4944C2214yes—
67#4942C2214yes—
68#4939C2211yescharacteristic
69#4936C2211yescharacteristic
70#4935C2211yescharacteristic
71C4×C4⋊Dic3C2212yes—
72#4929C2211yescharacteristic
73#4928C2212yes—
74#4670C2214yes—
75#4668C2214yes—
76#4654C2214yes—
77#4646C2214yes—
78C2×#554—31no—
79C2×#547—31no—
80#2732—34no—
81#2694—34no—
82#2540—32no—
83#2536—31no—
84#2535—31no—
85#2456—34no—
86#2417—32no—
87#2416—32no—
88#2407—34no—
89#2397D311yescharacteristic, complemented
90D4×C4⋊C4—32no—
91#2331—31no—
92#2326—31no—
93C22×C3×D4C2311yescharacteristic
94C22×C3×D4C2311yescharacteristic
95C22×C3×D4C2312yes—
96C22×C3×D4C2312yes—
97C23×C3×C4C2312yes—
98C23×C3×C4C2312yes—
99C22×#67C2314yes—
100C22×#67C2314yes—
101C23×Dic3C2314yes—
102C23×Dic3C2314yes—
103C2×#237—28no—
104C2×#237—48no—
105C22×D12C2311yescharacteristic
106C22×C4×D3C2312yes—
107C22×Dic6C2311yescharacteristic
108C22×Dic6C2312yes—
109C3×C4×D4—24no—
110C3×C4×D4—24no—
111C3×C4×D4—28no—
112C3×C4×D4—44no—
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C22⋊C4C2312yes—
118C2×C3×C22⋊C4C2312yes—
119C2×C3×C22⋊C4C2312yes—
120C2×C3×C22⋊C4C2312yes—
121C2×C3×C22⋊C4C2312yes—
122C2×C3×C22⋊C4C2312yes—
123C2×C3×C42C2311yescharacteristic
124C2×C3×C42C2311yescharacteristic
125C2×C3×C42C2311yescharacteristic
126C2×#217C2314yes—
127C2×#217C2314yes—
128C2×#217C2314yes—
129C2×#217C2314yes—
130#939—48no—
131D4×Dic3—28no—
132D4×Dic3—48no—
133#930—28no—
134#930—44no—
135C2×(C2×D3)⋊C4C2314yes—
136C2×(C2×D3)⋊C4C2312yes—
137C2×(C2×D3)⋊C4C2314yes—
138C2×(C2×D3)⋊C4C2314yes—
139C2×(C2×D3)⋊C4C2314yes—
140C2×C4⋊Dic3—28no—
141C2×C4⋊Dic3—28no—
142C2×C4⋊Dic3C2312yes—
143C2×C4⋊Dic3C2311yescharacteristic
144C2×C4⋊Dic3C2311yescharacteristic
145C2×C4⋊Dic3C2314yes—
146C2×C4⋊Dic3C2314yes—
147C2×C4⋊Dic3C2314yes—
148C2×C4⋊Dic3C2312yes—
149C2×C4⋊Dic3C2312yes—
150#924—28no—
151#924—44no—
152C2×Dic3⋊C4C2312yes—
153C2×Dic3⋊C4C2314yes—
154C2×Dic3⋊C4C2314yes—
155C2×Dic3⋊C4C2314yes—
156C2×C4×Dic3C2312yes—
157C2×C4×Dic3C2314yes—
158#897—28no—
159#890—28no—
160#886—216no—
161#878—216no—
162#875—24no—
163C4×D12—24no—
164#869D418yescomplemented
165C4×Dic6—24no—
166C3×#95C2312yes—
167C3×#95C2312yes—
168C3⋊#95C2312yes—
169C3⋊#95C2314yes—
170C3⋊#95C2314yes—
171C3⋊#95C2312yes—
172C3⋊#95C2314yes—
173C3⋊#95C2314yes—
174C3⋊#95C2314yes—
175C22×#41—32no—
176#554—68no—
177#547—68no—
178C2×C4⋊Q8—31no—
179C2×#125—32no—
180C2×#124—31no—
181C2×#123—34no—
182C2×C22⋊Q8—32no—
183C2×C22⋊Q8—32no—
184C2×C22⋊Q8—32no—
185C2×C22⋊Q8—34no—
186C2×#121—32no—
187C2×#121—34no—
188C2×#121—32no—
189C2×C4×Q8—31no—
190C2×C4×D4—32no—
191C2×C4×D4—34no—
192C2×C4×D4—31no—
193C2×C4×D4C2×D312yes—
194C2×C4×D4C2×D311yescharacteristic
195C2×C4×D4C2×D311yescharacteristic
196C2×C4×D4C2×D312yes—
197C22×C4⋊C4—32no—
198C22×C4⋊C4—32no—
199#399—34no—
200#399C2×D312yes—
201#399C2×D312yes—
202#398—34no—
203#398—32no—
204#397—32no—
205#396—34no—
206#396—34no—
207#394—32no—
208#394—34no—
209#392—34no—
210#392—32no—
211#392C2×D312yes—
212#392C2×D312yes—
213#391—31no—
214#391—34no—
215#390—34no—
216#390—34no—
217#390—31no—
218#389—34no—
219#389—32no—
220#388—34no—
221#388—34no—
222#387—34no—
223#386—34no—
224#386—34no—
225#386—34no—
226#385—34no—
227#385—34no—
228#385—34no—
229#384—31no—
230#382—31no—
231#380—32no—
232#380—32no—
233#380—34no—
234C4×C4⋊C4—32no—
235#376C2×D311yescharacteristic
236C24×C3C2412yes—
237C24×C3C2412yes—
238C23×D3C2412yes—
239C2×C3×D4—24no—
240C2×C3×D4—24no—
241C2×C3×D4—24no—
242C2×C3×D4—24no—
243C2×C3×D4—24no—
244C2×C3×D4—24no—
245C2×C3×D4—22no—
246C2×C3×D4—22no—
247C2×C3×D4—24no—
248C2×C3×D4—42no—
249C22×C3×C4—24no—
250C22×C3×C4—24no—
251C22×C3×C4—24no—
252C22×C3×C4—24no—
253C22×C3×C4—24no—
254C22×C3×C4—24no—
255C22×C3×C4C2412yes—
256C22×C3×C4C2412yes—
257C22×C3×C4C2412yes—
258C22×C3×C4C2411yescharacteristic
259C22×C3×C4C2411yescharacteristic
260C22×C3×C4C2412yes—
261C22×C3×C4C2411yescharacteristic
262C2×#67—28no—
263C2×#67—28no—
264C2×#67—28no—
265C2×#67—44no—
266C2×#67—44no—
267C2×#67—44no—
268C22×Dic3—28no—
269C22×Dic3—28no—
270C22×Dic3—28no—
271C22×Dic3—44no—
272C22×Dic3—44no—
273C22×Dic3—44no—
274C22×Dic3C2414yes—
275C22×Dic3C2412yes—
276C22×Dic3C2412yes—
277C22×Dic3C2412yes—
278C22×Dic3C2414yes—
279#237—432no—
280C2×D12—22no—
281C2×D12—22no—
282C2×D12—22no—
283C2×C4×D3—24no—
284C2×C4×D3—44no—
285C2×Dic6—22no—
286C2×Dic6—22no—
287C2×Dic6—22no—
288C2×Dic6—44no—
289C2×Dic6C2×D418yes—
290C3×C4⋊C4—22no—
291C3×C4⋊C4—22no—
292C3×C4⋊C4—24no—
293C3×C4⋊C4—42no—
294C3×C22⋊C4—24no—
295C3×C22⋊C4—24no—
296C3×C22⋊C4—24no—
297C3×C22⋊C4—24no—
298C3×C22⋊C4—42no—
299C3×C22⋊C4—42no—
300C3×C42—22no—
301C3×C42C2×D414yes—
302C3×C42#4114yescomplemented
303#217—28no—
304#217—28no—
305#217—44no—
306#217—44no—
307(C2×D3)⋊C4—24no—
308(C2×D3)⋊C4—28no—
309(C2×D3)⋊C4—28no—
310(C2×D3)⋊C4—44no—
311(C2×D3)⋊C4—44no—
312C4⋊Dic3—22no—
313C4⋊Dic3—22no—
314C4⋊Dic3—24no—
315C4⋊Dic3—28no—
316C4⋊Dic3—28no—
317C4⋊Dic3—44no—
318C4⋊Dic3C2×D418yes—
319C4⋊Dic3C2×D418yes—
320Dic3⋊C4—24no—
321Dic3⋊C4—28no—
322Dic3⋊C4—44no—
323Dic3⋊C4—44no—
324C4×Dic3—24no—
325C4×Dic3—44no—
326C2×#41—68no—
327C2×#41—128no—
328C22×Q8—32no—
329C22×Q8—31no—
330C22×D4—34no—
331C22×D4—34no—
332C22×D4—31no—
333C22×D4C22×D311yescharacteristic
334C22×D4C22×D312yes—
335C22×D4C22×D311yescharacteristic
336C22×D4C22×D312yes—
337C23×C4—34no—
338C23×C4—34no—
339C23×C4—32no—
340C23×C4C22×D312yes—
341C23×C4C22×D312yes—
342C4⋊Q8—38no—
343#125—68no—
344#124—64no—
345#123—616no—
346C22⋊Q8—616no—
347C22⋊Q8—68no—
348C22⋊Q8—68no—
349C22⋊Q8—124no—
350#121—68no—
351#121—128no—
352#121—124no—
353C4×Q8—64no—
354C4×D4—24no—
355C4×D4—24no—
356C4×D4—28no—
357C4×D4—44no—
358C4×D4—68no—
359C4×D4—64no—
360C4×D4—128no—
361C2×C4⋊C4—32no—
362C2×C4⋊C4—32no—
363C2×C4⋊C4—31no—
364C2×C4⋊C4—31no—
365C2×C4⋊C4—34no—
366C2×C4⋊C4—34no—
367C2×C4⋊C4—32no—
368C2×C4⋊C4—32no—
369C2×C4⋊C4—34no—
370C2×C4⋊C4—34no—
371C2×C4⋊C4—34no—
372C2×C4⋊C4—34no—
373C2×C4⋊C4—68no—
374C2×C4⋊C4—68no—
375C2×C4⋊C4C22×D311yescharacteristic
376C2×C4⋊C4C22×D311yescharacteristic
377C2×C4⋊C4C22×D312yes—
378C2×C4⋊C4C22×D312yes—
379C2×C22⋊C4—34no—
380C2×C22⋊C4—34no—
381C2×C22⋊C4—32no—
382C2×C22⋊C4—34no—
383C2×C22⋊C4—34no—
384C2×C22⋊C4—34no—
385C2×C22⋊C4—34no—
386C2×C22⋊C4—34no—
387C2×C22⋊C4—34no—
388C2×C22⋊C4C22×D312yes—
389C2×C22⋊C4C22×D312yes—
390C2×C22⋊C4C22×D312yes—
391C2×C22⋊C4C22×D312yes—
392C2×C22⋊C4C22×D312yes—
393C2×C22⋊C4C22×D312yes—
394C2×C42—32no—
395C2×C42—34no—
396C2×C42C22×D311yescharacteristic
397C2×C42C22×D311yescharacteristic
398C2×C42C22×D311yescharacteristic
399#95—34no—
400#95—34no—
401#95—34no—
402#95—34no—
403#95—32no—
404#95—32no—
405#95—34no—
406#95C22×D312yes—
407#95C22×D312yes—
408C23×C3—22no—
409C23×C3—22no—
410C23×C3—22no—
411C23×C3—22no—
412C23×C3—22no—
413C23×C3—22no—
414C23×C3—22no—
415C23×C3—22no—
416C23×C3—22no—
417C23×C3—22no—
418C23×C3—22no—
419C23×C3—22no—
420C23×C3H(5,𝔽2)14yes—
421C23×C32−1+414yes—
422C23×C3C2511yescharacteristic
423C22×D3—22no—
424C22×D3—22no—
425C22×D3—22no—
426C22×D3—24no—
427C22×D3—24no—
428C3×D4—28no—
429C3×D4—28no—
430C3×D4—216no—
431C3×D4—48no—
432C2×C3×C4—22no—
433C2×C3×C4—22no—
434C2×C3×C4—22no—
435C2×C3×C4—22no—
436C2×C3×C4—22no—
437C2×C3×C4—22no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—22no—
442C2×C3×C4—22no—
443C2×C3×C4—22no—
444C2×C3×C4—22no—
445C2×C3×C4—48no—
446C2×C3×C4—48no—
447C2×C3×C4C22×D412yes—
448C2×C3×C4C22×D412yes—
449C2×C3×C4C22×D412yes—
450C2×C3×C4C2×#4114yes—
451C2×C3×C4C2×#4112yes—
452C2×C3×C4H(5,𝔽2)12yes—
453#67—416no—
454#67—88no—
455C2×Dic3—24no—
456C2×Dic3—24no—
457C2×Dic3—22no—
458C2×Dic3—22no—
459C2×Dic3—22no—
460C2×Dic3—22no—
461C2×Dic3—24no—
462C2×Dic3—22no—
463C2×Dic3—24no—
464C2×Dic3—24no—
465C2×Dic3—24no—
466C2×Dic3—24no—
467C2×Dic3—416no—
468C2×Dic3—88no—
469C2×Dic3C22×D418yes—
470D12—44no—
471C4×D3—48no—
472Dic6—48no—
473Dic6—44no—
474C24—32no—
475C24C23×D312yes—
476C24C23×D312yes—
477#41—1232no—
478C2×Q8—38no—
479C2×Q8—62no—
480C2×Q8—62no—
481C2×Q8—62no—
482C2×Q8—124no—
483C2×D4—24no—
484C2×D4—24no—
485C2×D4—24no—
486C2×D4—24no—
487C2×D4—24no—
488C2×D4—24no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—24no—
492C2×D4—42no—
493C2×D4—68no—
494C2×D4—68no—
495C2×D4—68no—
496C2×D4—62no—
497C2×D4—62no—
498C2×D4—62no—
499C2×D4—124no—
500C2×D4—124no—
501C2×D4—124no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—24no—
507C22×C4—24no—
508C22×C4—34no—
509C22×C4—34no—
510C22×C4—32no—
511C22×C4—32no—
512C22×C4—32no—
513C22×C4—68no—
514C22×C4—68no—
515C22×C4—68no—
516C22×C4—64no—
517C22×C4—124no—
518C22×C4—124no—
519C22×C4—124no—
520C22×C4—124no—
521C22×C4C23×D312yes—
522C22×C4C23×D312yes—
523C22×C4C23×D312yes—
524C22×C4C23×D311yescharacteristic
525C22×C4C23×D311yescharacteristic
526C22×C4C23×D312yes—
527C22×C4C23×D311yescharacteristic
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—38no—
532C4⋊C4—38no—
533C4⋊C4—42no—
534C4⋊C4—68no—
535C4⋊C4—68no—
536C4⋊C4—68no—
537C4⋊C4—62no—
538C4⋊C4—62no—
539C4⋊C4—64no—
540C4⋊C4—64no—
541C4⋊C4—124no—
542C4⋊C4—124no—
543C4⋊C4—124no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—68no—
551C22⋊C4—68no—
552C22⋊C4—68no—
553C22⋊C4—64no—
554C22⋊C4—68no—
555C22⋊C4—124no—
556C22⋊C4—124no—
557C22⋊C4—124no—
558C22⋊C4—124no—
559C42—22no—
560C42—64no—
561C42—124no—
562C42D3×D414yes—
563C42#23714yescomplemented
564C22×C3—24no—
565C22×C3—24no—
566C22×C3—24no—
567C22×C3—24no—
568C22×C3—24no—
569C22×C3—24no—
570C22×C3—42no—
571C22×C3—42no—
572C22×C3—42no—
573C22×C3—42no—
574C22×C3—42no—
575C22×C3—42no—
576C22×C3—42no—
577C22×C3—42no—
578C22×C3C23×D411yescharacteristic
579C22×C3C22×#4111yescharacteristic
580C22×C3C2×H(5,𝔽2)11yescharacteristic
581C22×C3C2×H(5,𝔽2)12yes—
582C22×C3C2×2−1+412yes—
583C2×D3—42no—
584C2×D3—42no—
585C2×D3—42no—
586C2×D3—44no—
587C2×D3—44no—
588C3×C4—22no—
589C3×C4—22no—
590C3×C4—24no—
591C3×C4—42no—
592C3×C4—42no—
593C3×C4—42no—
594C3×C4#55014yes—
595Dic3—28no—
596Dic3—42no—
597Dic3—42no—
598Dic3—42no—
599Dic3—44no—
600C23—22no—
601C23—22no—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—22no—
606C23—22no—
607C23—22no—
608C23—22no—
609C23—22no—
610C23—22no—
611C23—22no—
612C23—62no—
613C23—62no—
614C23—62no—
615C23—64no—
616C23—64no—
617C23#100914yes—
618C23#101014yes—
619C23C24×D311yescharacteristic
620Q8—128no—
621Q8—124no—
622D4—28no—
623D4—28no—
624D4—216no—
625D4—48no—
626D4—1216no—
627D4—124no—
628D4—248no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—38no—
643C2×C4—48no—
644C2×C4—48no—
645C2×C4—64no—
646C2×C4—64no—
647C2×C4—62no—
648C2×C4—62no—
649C2×C4—62no—
650C2×C4—62no—
651C2×C4—64no—
652C2×C4—62no—
653C2×C4—64no—
654C2×C4—64no—
655C2×C4—64no—
656C2×C4—64no—
657C2×C4—128no—
658C2×C4—1216no—
659C2×C4—248no—
660C2×C4C2×D3×D412yes—
661C2×C4C2×D3×D412yes—
662C2×C4C2×D3×D412yes—
663C2×C4C2×#23714yes—
664C2×C4C2×#23712yes—
665C2×C4#100412yes—
666C2×C3—44no—
667C2×C3—44no—
668C2×C3C2×#55011yescharacteristic
669C2×C3#348411yescharacteristic
670C2×C3#348611yescharacteristic
671C2×C3#349212yes—
672C2×C3#353612yes—
673D3—82no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—122no—
689C22—124no—
690C22—122no—
691C22—124no—
692C22—122no—
693C22C22×D3×D411yescharacteristic
694C22C22×#23711yescharacteristic
695C22C2×#100411yescharacteristic
696C22C2×#100912yes—
697C22C2×#101012yes—
698C4—22no—
699C4—24no—
700C4—22no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—68no—
705C4—122no—
706C4—122no—
707C4—124no—
708C4—122no—
709C4#571814yes—
710C3#4535611yescharacteristic, complemented, Sylow
711C2—44no—
712C2—44no—
713C2—242no—
714C2#8602211yescharacteristic
715C2#8602412yes—
716C2#8603911yescharacteristic
717C2#8605112yes—
718C2C2×#571811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2397C2
hypercenter#2397D3
hypocenterC3#45356
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. C2×#5543C2semidirect
2. C2×#5555C2semidirect
3. C3×#2397C2semidirect
4. #82415C2semidirect
5. #82377C2semidirect
6. #82375C2semidirect
7. #82353C2semidirect
8. #82352C2semidirect
9. D4×C4⋊Dic3C2semidirect
10. #78119C2semidirect
11. #78064C2semidirect
12. #76675C2semidirect
13. #76668C2semidirect
14. #73391C2semidirect
15. #73387C2semidirect
89. #2397D3semidirect
164. #869D4semidirect
302. C3×C42#41semidirect
563. C42#237semidirect
710. C3#45356semidirect

Maximal subgroups

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

subgroupindex
C2×#55432
C2×#55552
C3×#23972
#824152
#823772
#823752
#823532
#823522
D4×C4⋊Dic32
#781192
#780642
#766752
#766682
#733912
#733872
#453563