Finite Group Databasebeta

← #11233331

Subgroups — #11233331

Order 768.

2410 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×#5730C211yesmaximal, characteristic, complemented
2C2×#5727C211yesmaximal, characteristic, complemented
3C3×#2331C211yesmaximal, characteristic, complemented
4#76654C214yesmaximal, complemented
5#76619C214yesmaximal, complemented
6#76219C212yesmaximal, complemented
7#75999C212yesmaximal, complemented
8#75393C214yesmaximal, complemented
9#75327C214yesmaximal, complemented
10#74239C211yesmaximal, characteristic
11#74191C211yesmaximal, characteristic, complemented
12#73789C212yesmaximal, complemented
13#73774C212yesmaximal, complemented
14#73419C211yesmaximal, characteristic, complemented
15#73387C211yesmaximal, characteristic, complemented
16#45337—31nomaximal, Sylow
17C22×D3×Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×#946C2214yes—
20C2×#946C2212yes—
21#5730—28no—
22#5727—28no—
23C2×#897C2212yes—
24C2×#897C2214yes—
25C2×#896C2214yes—
26C2×#895C2212yes—
27C2×#893C2212yes—
28C2×#893C2212yes—
29C2×D3×C4⋊C4C2212yes—
30C2×#890C2212yes—
31C2×#888C2212yes—
32C2×#875C2211yescharacteristic
33C2×C4×D12C2211yescharacteristic
34C2×C4×D12C2211yescharacteristic
35C2×C4×D12C2214yes—
36C2×C4×Dic6C2211yescharacteristic
37C2×C4×Dic6C2211yescharacteristic
38C3×#391C2214yes—
39C3×#389C2212yes—
40C3×#387C2214yes—
41C3×#382C2211yescharacteristic
42C3×C4×C4⋊C4C2212yes—
43C3×#376C2211yescharacteristic
44#4991C2214yes—
45#4987C2212yes—
46#4987C2214yes—
47#4984C2212yes—
48#4981C2214yes—
49#4973C2212yes—
50#4940C2214yes—
51#4940C2214yes—
52#4939C2211yescharacteristic
53C4×(C2×D3)⋊C4C2212yes—
54C4×(C2×D3)⋊C4C2212yes—
55#4935C2211yescharacteristic
56#4929C2211yescharacteristic
57#4929C2214yes—
58#4674C2212yes—
59#4674C2214yes—
60#4674C2212yes—
61#4673C2214yes—
62#4673C2212yes—
63#4672C2212yes—
64#4671C2212yes—
65#4670C2214yes—
66#4668C2214yes—
67#4668C2214yes—
68#4668C2214yes—
69#4665C2214yes—
70#4665C2214yes—
71#4665C2214yes—
72#4659C2214yes—
73#4659C2212yes—
74#4659C2212yes—
75#4656C2212yes—
76#4656C2214yes—
77#4653C2214yes—
78C2×#554—31no—
79C2×D4×Q8—31no—
80#2728—34no—
81#2690—34no—
82#2581—31no—
83#2535—31no—
84#2492—32no—
85#2463—34no—
86#2440—32no—
87#2427—32no—
88#2409—34no—
89#2398—31no—
90#2361—32no—
91#2331D311yescharacteristic, complemented
92#2326—31no—
93C22×C3×Q8C2312yes—
94C2×D3×Q8—28no—
95C2×D3×Q8—48no—
96C22×D12C2311yescharacteristic
97C22×D12C2314yes—
98C22×D12C2311yescharacteristic
99C22×C4×D3C2312yes—
100C22×C4×D3C2314yes—
101C22×C4×D3C2312yes—
102C22×C4×D3C2314yes—
103C22×Dic6C2311yescharacteristic
104C22×Dic6C2311yescharacteristic
105C22×Dic6C2314yes—
106C3×C4⋊Q8D418yescomplemented
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2314yes—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2314yes—
114C2×C3×C42C2311yescharacteristic
115#946—28no—
116#946—216no—
117C2×(C2×D3)⋊C4C2312yes—
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2312yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×(C2×D3)⋊C4C2314yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2312yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×C4⋊Dic3C2314yes—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4⋊Dic3C2311yescharacteristic
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2314yes—
136C2×Dic3⋊C4C2312yes—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2314yes—
140C2×C4×Dic3C2312yes—
141C2×C4×Dic3C2312yes—
142#897—28no—
143#897—48no—
144#896—216no—
145#895—28no—
146#893—28no—
147#893—28no—
148D3×C4⋊C4—28no—
149#890—28no—
150#888—28no—
151#875—24no—
152C4×D12—24no—
153C4×D12—24no—
154C4×D12—48no—
155C4×Dic6—24no—
156C4×Dic6—24no—
157C3×#95C2312yes—
158C3×#95C2312yes—
159C3×#95C2314yes—
160C3×#95C2314yes—
161C3×#95C2312yes—
162C3×#95C2312yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2312yes—
167C3⋊#95C2314yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2314yes—
171C3⋊#95C2312yes—
172C23×Q8—32no—
173#554—68no—
174D4×Q8—68no—
175C2×C4⋊Q8—32no—
176C2×C4⋊Q8C2×D311yescharacteristic
177C2×#125—32no—
178C2×#124—31no—
179C2×C22⋊Q8—34no—
180C2×C22⋊Q8—32no—
181C2×C22⋊Q8—32no—
182C2×C22⋊Q8—34no—
183C2×C22⋊Q8—34no—
184C2×#121—32no—
185C2×C4×Q8—31no—
186C2×C4×Q8—31no—
187C2×C4×D4—32no—
188C2×C4×D4—32no—
189C2×C4×D4—31no—
190C2×C4×D4—31no—
191C2×C4×D4—34no—
192C22×C4⋊C4—32no—
193#400—32no—
194#400—32no—
195#400—34no—
196#398—34no—
197#397—32no—
198#397—34no—
199#397—32no—
200#395—32no—
201#395—34no—
202#394—34no—
203#394—32no—
204#393—32no—
205#393—34no—
206#392—32no—
207#391—32no—
208#391—31no—
209#391—34no—
210#391C2×D314yes—
211#390—31no—
212#389—32no—
213#389C2×D312yes—
214#388—34no—
215#388—34no—
216#388—34no—
217#387—34no—
218#387C2×D314yes—
219#386—34no—
220#386—34no—
221#386—34no—
222#385—34no—
223#385—34no—
224#385—34no—
225#384—31no—
226#382C2×D311yescharacteristic
227#380—34no—
228#380—32no—
229C4×C4⋊C4C2×D312yes—
230C4×C22⋊C4—32no—
231C4×C22⋊C4—32no—
232#376C2×D311yescharacteristic
233C23×D3C2412yes—
234C23×D3C2412yes—
235C2×C3×Q8—44no—
236C2×C3×Q8C2×D418yes—
237C22×C3×C4C2414yes—
238C22×C3×C4C2412yes—
239C22×C3×C4C2412yes—
240C22×C3×C4C2411yescharacteristic
241C22×C3×C4C2411yescharacteristic
242C22×C3×C4C2414yes—
243C22×C3×C4C2411yescharacteristic
244C22×Dic3C2412yes—
245C22×Dic3C2412yes—
246C22×Dic3C2412yes—
247C22×Dic3C2412yes—
248C22×Dic3C2412yes—
249C22×Dic3C2412yes—
250D3×Q8—432no—
251C2×D12—24no—
252C2×D12—22no—
253C2×D12—22no—
254C2×D12—22no—
255C2×D12—28no—
256C2×D12—28no—
257C2×D12—44no—
258C2×D12#4114yes—
259C2×C4×D3—24no—
260C2×C4×D3—24no—
261C2×C4×D3—28no—
262C2×C4×D3—24no—
263C2×C4×D3—24no—
264C2×C4×D3—28no—
265C2×C4×D3—28no—
266C2×C4×D3—44no—
267C2×C4×D3—44no—
268C2×C4×D3—44no—
269C2×C4×D3—44no—
270C2×Dic6—22no—
271C2×Dic6—24no—
272C2×Dic6—22no—
273C2×Dic6—28no—
274C2×Dic6—22no—
275C2×Dic6—22no—
276C2×Dic6—28no—
277C2×Dic6—44no—
278C3×C4⋊C4—24no—
279C3×C4⋊C4—44no—
280C3×C4⋊C4C2×D418yes—
281C3×C4⋊C4C2×D418yes—
282C3×C42—22no—
283C3×C42—22no—
284C3×C42—44no—
285C3×C42C2×D414yes—
286(C2×D3)⋊C4—28no—
287(C2×D3)⋊C4—28no—
288(C2×D3)⋊C4—24no—
289(C2×D3)⋊C4—24no—
290(C2×D3)⋊C4—24no—
291(C2×D3)⋊C4—24no—
292(C2×D3)⋊C4—28no—
293(C2×D3)⋊C4—28no—
294(C2×D3)⋊C4—24no—
295(C2×D3)⋊C4—24no—
296(C2×D3)⋊C4—44no—
297(C2×D3)⋊C4—44no—
298C4⋊Dic3—22no—
299C4⋊Dic3—22no—
300C4⋊Dic3—22no—
301C4⋊Dic3—22no—
302C4⋊Dic3—28no—
303C4⋊Dic3—28no—
304C4⋊Dic3—44no—
305Dic3⋊C4—24no—
306Dic3⋊C4—24no—
307Dic3⋊C4—28no—
308Dic3⋊C4—28no—
309C4×Dic3—24no—
310C4×Dic3—24no—
311C22×Q8—31no—
312C22×Q8—31no—
313C22×Q8—34no—
314C22×Q8—68no—
315C22×Q8—128no—
316C22×Q8C22×D312yes—
317C22×D4—31no—
318C22×D4—31no—
319C22×D4—34no—
320C23×C4—32no—
321C23×C4—34no—
322C23×C4—34no—
323C23×C4—32no—
324C4⋊Q8—68no—
325C4⋊Q8D1218yescomplemented
326#125—68no—
327#124—64no—
328C22⋊Q8—68no—
329C22⋊Q8—616no—
330C22⋊Q8—616no—
331C22⋊Q8—68no—
332C22⋊Q8—128no—
333#121—68no—
334C4×Q8—64no—
335C4×Q8—64no—
336C4×D4—68no—
337C4×D4—68no—
338C4×D4—64no—
339C4×D4—64no—
340C4×D4—128no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4—34no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—32no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—31no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—34no—
352C2×C4⋊C4—68no—
353C2×C4⋊C4C22×D312yes—
354C2×C4⋊C4C22×D314yes—
355C2×C4⋊C4C22×D312yes—
356C2×C4⋊C4C22×D312yes—
357C2×C22⋊C4—32no—
358C2×C22⋊C4—32no—
359C2×C22⋊C4—34no—
360C2×C22⋊C4—34no—
361C2×C22⋊C4—32no—
362C2×C22⋊C4—34no—
363C2×C22⋊C4—32no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4—32no—
366C2×C22⋊C4—32no—
367C2×C22⋊C4—34no—
368C2×C22⋊C4—34no—
369C2×C42—32no—
370C2×C42—32no—
371C2×C42C22×D311yescharacteristic
372C2×C42C22×D311yescharacteristic
373C2×C42C22×D314yes—
374C2×C42C22×D311yescharacteristic
375#95—32no—
376#95—32no—
377#95—32no—
378#95—32no—
379#95—34no—
380#95—34no—
381#95—34no—
382#95—34no—
383#95—34no—
384#95C22×D312yes—
385#95C22×D312yes—
386#95C22×D314yes—
387#95C22×D312yes—
388#95C22×D312yes—
389#95C22×D314yes—
390C23×C3C2511yescharacteristic
391C22×D3—22no—
392C22×D3—22no—
393C22×D3—22no—
394C22×D3—22no—
395C22×D3—22no—
396C22×D3—24no—
397C22×D3—22no—
398C22×D3—22no—
399C22×D3—22no—
400C22×D3—24no—
401C22×D3C2×#4114yes—
402C22×D32−1+414yes—
403C3×Q8—48no—
404C2×C3×C4—24no—
405C2×C3×C4—24no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—24no—
410C2×C3×C4—24no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—24no—
417C2×C3×C4C22×D412yes—
418C2×C3×C4C22×D412yes—
419C2×C3×C4C22×D418yes—
420C2×C3×C4C22×D412yes—
421C2×C3×C4C2×#4112yes—
422C2×C3×C42−1+412yes—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—24no—
433C2×Dic3—24no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3H(5,𝔽2)14yes—
439D12—28no—
440D12—28no—
441D12—416no—
442C4×D3—416no—
443C4×D3—48no—
444C4×D3—48no—
445C4×D3—88no—
446Dic6—28no—
447Dic6—416no—
448Dic6—44no—
449C24—32no—
450C24—32no—
451C2×Q8—44no—
452C2×Q8—64no—
453C2×Q8—68no—
454C2×Q8—62no—
455C2×Q8—62no—
456C2×Q8—68no—
457C2×Q8—62no—
458C2×Q8—62no—
459C2×Q8—124no—
460C2×Q8—1232no—
461C2×Q8C2×D1218yes—
462C2×D4—34no—
463C2×D4—64no—
464C2×D4—68no—
465C2×D4—68no—
466C2×D4—62no—
467C2×D4—62no—
468C2×D4—62no—
469C2×D4—124no—
470C22×C4—32no—
471C22×C4—32no—
472C22×C4—32no—
473C22×C4—32no—
474C22×C4—32no—
475C22×C4—32no—
476C22×C4—68no—
477C22×C4—64no—
478C22×C4—68no—
479C22×C4—64no—
480C22×C4—64no—
481C22×C4—64no—
482C22×C4—68no—
483C22×C4—124no—
484C22×C4—124no—
485C22×C4—124no—
486C22×C4—124no—
487C22×C4C23×D312yes—
488C22×C4C23×D314yes—
489C22×C4C23×D312yes—
490C22×C4C23×D311yescharacteristic
491C22×C4C23×D311yescharacteristic
492C22×C4C23×D314yes—
493C22×C4C23×D311yescharacteristic
494C4⋊C4—24no—
495C4⋊C4—44no—
496C4⋊C4—68no—
497C4⋊C4—62no—
498C4⋊C4—68no—
499C4⋊C4—62no—
500C4⋊C4—62no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—62no—
504C4⋊C4—68no—
505C4⋊C4—68no—
506C4⋊C4—124no—
507C4⋊C4C2×D1218yes—
508C4⋊C4C2×D1218yes—
509C22⋊C4—64no—
510C22⋊C4—64no—
511C22⋊C4—68no—
512C22⋊C4—68no—
513C22⋊C4—64no—
514C22⋊C4—64no—
515C22⋊C4—68no—
516C22⋊C4—68no—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—124no—
520C22⋊C4—124no—
521C42—22no—
522C42—22no—
523C42—44no—
524C42—64no—
525C42—64no—
526C42C2×D1214yes—
527C22×C3C23×D411yescharacteristic
528C22×C3C22×#4111yescharacteristic
529C22×C3C2×H(5,𝔽2)12yes—
530C22×C3C2×2−1+411yescharacteristic
531C22×C3C2×2−1+412yes—
532C2×D3—24no—
533C2×D3—24no—
534C2×D3—24no—
535C2×D3—28no—
536C2×D3—24no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—42no—
542C2×D3—42no—
543C2×D3—44no—
544C3×C4—22no—
545C3×C4—28no—
546C3×C4—22no—
547C3×C4—44no—
548C3×C4—42no—
549C3×C4—42no—
550C3×C4#54814yes—
551Dic3—24no—
552Dic3—42no—
553Dic3—42no—
554Dic3—42no—
555Dic3—42no—
556Dic3—42no—
557C23—34no—
558C23—34no—
559C23—62no—
560C23—62no—
561C23—62no—
562C23—62no—
563C23—62no—
564C23—62no—
565C23—62no—
566C23—64no—
567C23—64no—
568C23—62no—
569C23C24×D311yescharacteristic
570Q8—48no—
571Q8—68no—
572Q8—1216no—
573Q8—124no—
574D4—68no—
575D4—68no—
576D4—1216no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—34no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—64no—
603C2×C4—64no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—1216no—
607C2×C4—128no—
608C2×C4—128no—
609C2×C4—248no—
610C2×C4C22×D1212yes—
611C2×C4C22×D1212yes—
612C2×C4C22×D1218yes—
613C2×C4C22×D1212yes—
614C2×C4#100712yes—
615C2×C4D3×#4112yes—
616C2×C3C2×#54811yescharacteristic
617C2×C3#348411yescharacteristic
618C2×C3#348611yescharacteristic
619C2×C3#352012yes—
620C2×C3#355212yes—
621D3—44no—
622D3—44no—
623C22—64no—
624C22—68no—
625C22—64no—
626C22—64no—
627C22—64no—
628C22—122no—
629C22—122no—
630C22—122no—
631C22—124no—
632C22—122no—
633C22—122no—
634C22—122no—
635C22C23×D1211yescharacteristic
636C22C2×#100711yescharacteristic
637C22C2×D3×#4111yescharacteristic
638C22D3×H(5,𝔽2)12yes—
639C22D3×2−1+412yes—
640C4—22no—
641C4—28no—
642C4—22no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—64no—
647C4—122no—
648C4—122no—
649C4—122no—
650C4—122no—
651C4—122no—
652C4#557614yes—
653C3#4533711yescharacteristic, complemented, Sylow
654C2—124no—
655C2—124no—
656C2C2×#557611yescharacteristic
657C2#8586711yescharacteristic
658C2#8588111yescharacteristic
659C2#8655612yes—
660C2#8660212yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2331C2
prime radical#74239C2
hypercenter#2331D3
hypocenterC3#45337
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2331

Complements

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

NH ≅ G/Ntype
1. C2×#5730C2semidirect
2. C2×#5727C2semidirect
3. C3×#2331C2semidirect
4. #76654C2semidirect
5. #76619C2semidirect
6. #76219C2semidirect
7. #75999C2semidirect
8. #75393C2semidirect
9. #75327C2semidirect
11. #74191C2semidirect
12. #73789C2semidirect
13. #73774C2semidirect
14. #73419C2semidirect
15. #73387C2semidirect
91. #2331D3semidirect
106. C3×C4⋊Q8D4semidirect
325. C4⋊Q8D12semidirect
653. C3#45337semidirect

Maximal subgroups

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

subgroupindex
C2×#57302
C2×#57272
C3×#23312
#766542
#766192
#762192
#759992
#753932
#753272
#742392
#741912
#737892
#737742
#734192
#733872
#453373