Finite Group Databasebeta

← #11025131

Subgroups — #11025131

Order 768.

2356 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×#5697C211yesmaximal, characteristic, complemented
2C2×D3×#125C211yesmaximal, characteristic, complemented
3C3×#2330C211yesmaximal, characteristic, complemented
4#76211C214yesmaximal, complemented
5#76211C212yesmaximal, complemented
6#76114C214yesmaximal, complemented
7#76016C212yesmaximal, complemented
8#75319C214yesmaximal, complemented
9#75155C214yesmaximal, complemented
10#74375C211yesmaximal, characteristic, complemented
11#74344C211yesmaximal, characteristic, complemented
12#73787C212yesmaximal, complemented
13#73774C212yesmaximal, complemented
14#73429C211yesmaximal, characteristic, complemented
15#73414C211yesmaximal, characteristic, complemented
16#37706—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18#5697—28no—
19D3×#125—28no—
20C2×#895C2212yes—
21C2×#895C2212yes—
22C2×#895C2214yes—
23C2×#895C2214yes—
24C2×#895C2214yes—
25C2×#894C2214yes—
26C2×#893C2212yes—
27C2×D3×C4⋊C4C2212yes—
28C2×D3×C4⋊C4C2214yes—
29C2×#890C2212yes—
30C2×#889C2214yes—
31C2×#874C2211yescharacteristic
32C2×C4×D12C2211yescharacteristic
33C2×C4×D12C2214yes—
34C2×#872C2211yescharacteristic
35C2×C42×D3C2211yescharacteristic
36C2×#870C2211yescharacteristic
37C3×#389C2214yes—
38C3×#389C2212yes—
39C3×#387C2214yes—
40C3×#384C2211yescharacteristic
41C3×C4×C4⋊C4C2212yes—
42C3×#376C2211yescharacteristic
43#4994C2214yes—
44#4994C2214yes—
45#4994C2214yes—
46#4988C2212yes—
47#4988C2214yes—
48#4987C2212yes—
49#4987C2214yes—
50#4984C2212yes—
51#4981C2214yes—
52#4940C2214yes—
53#4940C2214yes—
54#4939C2212yes—
55#4939C2214yes—
56#4939C2212yes—
57C4×(C2×D3)⋊C4C2211yescharacteristic
58C4×(C2×D3)⋊C4C2211yescharacteristic
59C4×C4⋊Dic3C2211yescharacteristic
60#4677C2214yes—
61#4674C2214yes—
62#4672C2214yes—
63#4672C2214yes—
64#4672C2214yes—
65#4672C2214yes—
66#4670C2212yes—
67#4668C2214yes—
68#4668C2212yes—
69#4668C2214yes—
70#4665C2214yes—
71#4665C2214yes—
72#4665C2212yes—
73#4653C2212yes—
74C2×#540—31no—
75C22×#125—31no—
76#2856—31no—
77#2711—34no—
78#2707—34no—
79#2641—31no—
80#2629—31no—
81#2490—32no—
82#2454—32no—
83#2443—34no—
84#2416—32no—
85#2416—34no—
86#2361—32no—
87#2330D311yescharacteristic, complemented
88#2290—31no—
89C22×D12C2311yescharacteristic
90C22×D12C2314yes—
91C22×D12C2312yes—
92C22×D12C2314yes—
93C22×D12C2314yes—
94C22×C4×D3C2311yescharacteristic
95C22×C4×D3C2314yes—
96C22×C4×D3C2312yes—
97C22×C4×D3C2311yescharacteristic
98C22×C4×D3C2314yes—
99C3×#125D418yescomplemented
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2314yes—
108C2×C3×C42C2311yescharacteristic
109C2×(C2×D3)⋊C4C2311yescharacteristic
110C2×(C2×D3)⋊C4C2312yes—
111C2×(C2×D3)⋊C4C2312yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2311yescharacteristic
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×(C2×D3)⋊C4C2314yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×C4⋊Dic3C2311yescharacteristic
125C2×C4⋊Dic3C2314yes—
126C2×C4⋊Dic3C2312yes—
127C2×C4⋊Dic3C2312yes—
128C2×C4⋊Dic3C2314yes—
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2314yes—
133C2×C4×Dic3C2312yes—
134C2×C4×Dic3C2311yescharacteristic
135C2×C4×Dic3C2311yescharacteristic
136#895—28no—
137#895—216no—
138#895—28no—
139#895—48no—
140#895—48no—
141#894—216no—
142#893—28no—
143D3×C4⋊C4—216no—
144D3×C4⋊C4—28no—
145#890—28no—
146#889—216no—
147#874—24no—
148C4×D12—24no—
149C4×D12—48no—
150#872—24no—
151C42×D3—24no—
152#870—24no—
153C3×#95C2312yes—
154C3×#95C2314yes—
155C3×#95C2314yes—
156C3×#95C2312yes—
157C3⋊#95C2312yes—
158C3⋊#95C2312yes—
159C3⋊#95C2312yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162#540—68no—
163C2×#127—31no—
164C2×#125—34no—
165C2×#125—32no—
166C2×#125—31no—
167C2×#125—68no—
168C2×#125C2×D311yescharacteristic
169C2×#123—34no—
170C2×#121—32no—
171C2×#121—34no—
172C2×#121—32no—
173C2×#121—34no—
174C2×#121—34no—
175C2×C4×D4—32no—
176C2×C4×D4—31no—
177C2×C4×D4—34no—
178C2×#117—31no—
179C22×C4⋊C4—32no—
180C22×C4⋊C4—34no—
181C22×C42—31no—
182#399—34no—
183#398—34no—
184#397—34no—
185#396—34no—
186#396—34no—
187#396—34no—
188#394—34no—
189#394—34no—
190#394—34no—
191#394—34no—
192#390—34no—
193#390—32no—
194#390—32no—
195#390—32no—
196#390—34no—
197#389—32no—
198#389C2×D314yes—
199#389C2×D312yes—
200#388—34no—
201#388—34no—
202#388—32no—
203#387—32no—
204#387C2×D314yes—
205#386—34no—
206#386—32no—
207#386—34no—
208#385—34no—
209#385—32no—
210#385—34no—
211#384C2×D311yescharacteristic
212#380—32no—
213#380—34no—
214C4×C4⋊C4—31no—
215C4×C4⋊C4C2×D312yes—
216C4×C22⋊C4—31no—
217C4×C22⋊C4—31no—
218#376C2×D311yescharacteristic
219C23×D3C2411yescharacteristic
220C23×D3C2414yes—
221C23×D3C2411yescharacteristic
222C22×C3×C4C2412yes—
223C22×C3×C4C2411yescharacteristic
224C22×C3×C4C2414yes—
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2412yes—
227C22×C3×C4C2414yes—
228C22×C3×C4C2411yescharacteristic
229C22×Dic3C2411yescharacteristic
230C22×Dic3C2412yes—
231C22×Dic3C2412yes—
232C22×Dic3C2414yes—
233C22×Dic3C2411yescharacteristic
234C2×D12—22no—
235C2×D12—28no—
236C2×D12—28no—
237C2×D12—24no—
238C2×D12—28no—
239C2×D12—24no—
240C2×D12—24no—
241C2×D12—24no—
242C2×D12—28no—
243C2×D12—28no—
244C2×D12—44no—
245C2×D12—44no—
246C2×D12—44no—
247C2×D12—44no—
248C2×C4×D3—22no—
249C2×C4×D3—22no—
250C2×C4×D3—28no—
251C2×C4×D3—24no—
252C2×C4×D3—24no—
253C2×C4×D3—22no—
254C2×C4×D3—28no—
255C2×C4×D3—22no—
256C2×C4×D3—22no—
257C2×C4×D3—28no—
258C2×C4×D3—42no—
259C2×C4×D3—44no—
260C2×C4×D3—44no—
261C2×C4×D3—44no—
262C2×C4×D3#4114yes—
263C3×C4⋊C4—24no—
264C3×C4⋊C4—44no—
265C3×C4⋊C4—44no—
266C3×C4⋊C4C2×D4116yes—
267C3×C4⋊C4C2×D418yes—
268C3×C42—22no—
269C3×C42—22no—
270C3×C42—44no—
271C3×C42C2×D414yes—
272(C2×D3)⋊C4—22no—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—24no—
275(C2×D3)⋊C4—28no—
276(C2×D3)⋊C4—28no—
277(C2×D3)⋊C4—22no—
278(C2×D3)⋊C4—28no—
279(C2×D3)⋊C4—44no—
280(C2×D3)⋊C4—44no—
281(C2×D3)⋊C4—44no—
282(C2×D3)⋊C4—44no—
283(C2×D3)⋊C4—44no—
284(C2×D3)⋊C4—42no—
285(C2×D3)⋊C4—44no—
286(C2×D3)⋊C4—44no—
287C4⋊Dic3—24no—
288C4⋊Dic3—28no—
289C4⋊Dic3—22no—
290C4⋊Dic3—44no—
291C4⋊Dic3—42no—
292Dic3⋊C4—28no—
293Dic3⋊C4—24no—
294Dic3⋊C4—28no—
295Dic3⋊C4—28no—
296C4×Dic3—24no—
297C4×Dic3—22no—
298C4×Dic3—22no—
299C22×D4—31no—
300C22×D4—34no—
301C22×D4—32no—
302C22×D4—34no—
303C22×D4—34no—
304C23×C4—31no—
305C23×C4—31no—
306C23×C4—34no—
307C23×C4—32no—
308C23×C4—34no—
309#127—64no—
310#125—616no—
311#125—68no—
312#125—64no—
313#125D1218yescomplemented
314#123—616no—
315#121—68no—
316#121—616no—
317#121—68no—
318#121—128no—
319#121—128no—
320C4×D4—64no—
321C4×D4—68no—
322C4×D4—128no—
323#117—64no—
324C2×C4⋊C4—32no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—68no—
334C2×C4⋊C4—616no—
335C2×C4⋊C4C22×D312yes—
336C2×C4⋊C4C22×D314yes—
337C2×C4⋊C4C22×D314yes—
338C2×C4⋊C4C22×D312yes—
339C2×C4⋊C4C22×D314yes—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—32no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—31no—
344C2×C22⋊C4—31no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—34no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4—34no—
352C2×C22⋊C4—32no—
353C2×C22⋊C4—34no—
354C2×C22⋊C4—34no—
355C2×C42—31no—
356C2×C42—31no—
357C2×C42—32no—
358C2×C42—64no—
359C2×C42C22×D311yescharacteristic
360C2×C42C22×D311yescharacteristic
361C2×C42C22×D314yes—
362C2×C42C22×D311yescharacteristic
363#95—32no—
364#95—32no—
365#95—32no—
366#95—34no—
367#95—34no—
368#95C22×D312yes—
369#95C22×D312yes—
370#95C22×D314yes—
371#95C22×D314yes—
372C23×C3C2511yescharacteristic
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—22no—
376C22×D3—21no—
377C22×D3—24no—
378C22×D3—21no—
379C22×D3—24no—
380C22×D3—24no—
381C22×D3—22no—
382C22×D3—24no—
383C22×D3—21no—
384C22×D3—24no—
385C22×D3—24no—
386C22×D3—22no—
387C22×D3—24no—
388C22×D3C2×#4112yes—
389C22×D3H(5,𝔽2)14yes—
390C2×C3×C4—24no—
391C2×C3×C4—22no—
392C2×C3×C4—24no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—24no—
397C2×C3×C4—24no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—24no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4C22×D414yes—
405C2×C3×C4C22×D418yes—
406C2×C3×C4C22×D412yes—
407C2×C3×C4C2×#4112yes—
408C2×C3×C42−1+412yes—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—24no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—22no—
417C2×Dic3—24no—
418C2×Dic3—22no—
419C2×Dic3—21no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—24no—
423C2×Dic3C2×#4112yes—
424D12—44no—
425D12—416no—
426D12—48no—
427D12—88no—
428D12—88no—
429C4×D3—28no—
430C4×D3—416no—
431C4×D3—48no—
432C4×D3—44no—
433C4×D3—88no—
434C24—31no—
435C24—31no—
436C24—34no—
437C2×D4—62no—
438C2×D4—68no—
439C2×D4—68no—
440C2×D4—64no—
441C2×D4—68no—
442C2×D4—64no—
443C2×D4—68no—
444C2×D4—68no—
445C2×D4—64no—
446C2×D4—64no—
447C2×D4—124no—
448C2×D4—124no—
449C2×D4—124no—
450C2×D4—124no—
451C22×C4—31no—
452C22×C4—34no—
453C22×C4—31no—
454C22×C4—32no—
455C22×C4—32no—
456C22×C4—34no—
457C22×C4—68no—
458C22×C4—64no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—64no—
462C22×C4—68no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—68no—
467C22×C4—124no—
468C22×C4—122no—
469C22×C4—124no—
470C22×C4—124no—
471C22×C4C23×D312yes—
472C22×C4C23×D311yescharacteristic
473C22×C4C23×D314yes—
474C22×C4C23×D311yescharacteristic
475C22×C4C23×D312yes—
476C22×C4C23×D314yes—
477C22×C4C23×D311yescharacteristic
478C4⋊C4—24no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—62no—
482C4⋊C4—68no—
483C4⋊C4—68no—
484C4⋊C4—68no—
485C4⋊C4—68no—
486C4⋊C4—64no—
487C4⋊C4—64no—
488C4⋊C4—124no—
489C4⋊C4—122no—
490C4⋊C4C2×D1218yes—
491C4⋊C4C2×D12116yes—
492C22⋊C4—62no—
493C22⋊C4—68no—
494C22⋊C4—62no—
495C22⋊C4—64no—
496C22⋊C4—64no—
497C22⋊C4—68no—
498C22⋊C4—68no—
499C22⋊C4—124no—
500C22⋊C4—124no—
501C22⋊C4—124no—
502C22⋊C4—124no—
503C22⋊C4—124no—
504C22⋊C4—124no—
505C22⋊C4—122no—
506C22⋊C4—124no—
507C42—22no—
508C42—22no—
509C42—44no—
510C42—64no—
511C42—62no—
512C42—62no—
513C42C2×D1214yes—
514C22×C3C23×D411yescharacteristic
515C22×C3C22×#4111yescharacteristic
516C22×C3C2×H(5,𝔽2)12yes—
517C22×C3C2×2−1+411yescharacteristic
518C22×C3#58512yes—
519C2×D3—22no—
520C2×D3—24no—
521C2×D3—22no—
522C2×D3—24no—
523C2×D3—41no—
524C2×D3—44no—
525C2×D3—41no—
526C2×D3—44no—
527C2×D3—44no—
528C2×D3—42no—
529C2×D3—44no—
530C2×D3—41no—
531C2×D3—44no—
532C2×D3—44no—
533C2×D3—42no—
534C2×D3—44no—
535C2×D3#54014yescomplemented
536C3×C4—28no—
537C3×C4—24no—
538C3×C4—41no—
539C3×C4—44no—
540C3×C4—41no—
541C3×C4—42no—
542C3×C4#54814yes—
543Dic3—22no—
544Dic3—41no—
545Dic3—42no—
546Dic3—44no—
547Dic3—42no—
548C23—32no—
549C23—34no—
550C23—61no—
551C23—61no—
552C23—62no—
553C23—61no—
554C23—64no—
555C23—61no—
556C23—64no—
557C23—61no—
558C23—64no—
559C23—64no—
560C23—62no—
561C23—64no—
562C23—64no—
563C23—62no—
564C23—64no—
565C23C24×D311yescharacteristic
566D4—124no—
567D4—1216no—
568D4—128no—
569D4—248no—
570D4—248no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—32no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—68no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—61no—
592C2×C4—64no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—64no—
598C2×C4—64no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—1216no—
602C2×C4—128no—
603C2×C4—124no—
604C2×C4—248no—
605C2×C4C22×D1214yes—
606C2×C4C22×D1218yes—
607C2×C4C22×D1212yes—
608C2×C4#100712yes—
609C2×C4D3×#4112yes—
610C2×C3C2×#54011yescharacteristic
611C2×C3C2×#54811yescharacteristic
612C2×C3#348812yes—
613C2×C3#351211yescharacteristic
614C2×C3#355912yes—
615D3—24no—
616D3—81no—
617D3—84no—
618C22—34no—
619C22—62no—
620C22—64no—
621C22—64no—
622C22—62no—
623C22—121no—
624C22—124no—
625C22—122no—
626C22—124no—
627C22—124no—
628C22—124no—
629C22—122no—
630C22—124no—
631C22—121no—
632C22—124no—
633C22—121no—
634C22—124no—
635C22C23×D1211yescharacteristic
636C22C2×#100711yescharacteristic
637C22C2×D3×#4111yescharacteristic
638C22D3×H(5,𝔽2)12yes—
639C22#596812yes—
640C4—28no—
641C4—24no—
642C4—42no—
643C4—41no—
644C4—44no—
645C4—41no—
646C4—62no—
647C4—121no—
648C4—122no—
649C4—124no—
650C4—122no—
651C4#557614yes—
652C3#3770611yescharacteristic, complemented, Sylow
653C2—64no—
654C2—241no—
655C2—244no—
656C2C2×#557611yescharacteristic
657C2#8590012yes—
658C2D3×#54011yescharacteristic
659C2#8649711yescharacteristic
660C2#8660312yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2330C2
hypercenter#2330D3
hypocenterC3#37706
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
1. C2×#5697C2semidirect
2. C2×D3×#125C2semidirect
3. C3×#2330C2semidirect
4. #76211C2semidirect
5. #76211C2semidirect
6. #76114C2semidirect
7. #76016C2semidirect
8. #75319C2semidirect
9. #75155C2semidirect
10. #74375C2semidirect
11. #74344C2semidirect
12. #73787C2semidirect
13. #73774C2semidirect
14. #73429C2semidirect
15. #73414C2semidirect
87. #2330D3semidirect
99. C3×#125D4semidirect
313. #125D12semidirect
535. C2×D3#540semidirect
652. C3#37706semidirect

Maximal subgroups

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

subgroupindex
C2×#56972
C2×D3×#1252
C3×#23302
#762112
#761142
#760162
#753192
#751552
#743752
#743442
#737872
#737742
#734292
#734142
#377063