Finite Group Databasebeta

← #11233695

Subgroups — #11233695

Order 768.

2406 conjugacy classes of subgroups.

The subgroup lattice has 654 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#76656C214yesmaximal, complemented
5#76619C214yesmaximal, complemented
6#76016C212yesmaximal, complemented
7#76011C212yesmaximal, complemented
8#75394C214yesmaximal, complemented
9#75321C214yesmaximal, complemented
10#74190C211yesmaximal, characteristic, complemented
11#74190C211yesmaximal, characteristic, complemented
12#73714C212yesmaximal, complemented
13#73653C212yesmaximal
14#73397C211yesmaximal, characteristic, complemented
15#73389C211yesmaximal, characteristic
16#45357—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×#893C2212yes—
27C2×#893C2212yes—
28C2×D3×C4⋊C4C2212yes—
29C2×#890C2212yes—
30C2×#888C2212yes—
31C2×#874C2211yescharacteristic
32C2×C4×D12C2211yescharacteristic
33C2×C4×D12C2211yescharacteristic
34C2×C4×D12C2214yes—
35C2×C4×Dic6C2211yescharacteristic
36C2×C4×Dic6C2211yescharacteristic
37C3×#391C2214yes—
38C3×#389C2212yes—
39C3×#387C2214yes—
40C3×#382C2211yescharacteristic
41C3×C4×C4⋊C4C2212yes—
42C3×#376C2211yescharacteristic
43#4994C2214yes—
44#4994C2214yes—
45#4988C2212yes—
46#4988C2212yes—
47#4987C2214yes—
48#4984C2212yes—
49#4984C2212yes—
50#4939C2214yes—
51#4939C2211yescharacteristic
52#4939C2211yescharacteristic
53#4936C2211yescharacteristic
54C4×C4⋊Dic3C2212yes—
55C4×C4⋊Dic3C2212yes—
56#4929C2214yes—
57#4928C2214yes—
58#4677C2212yes—
59#4677C2212yes—
60#4677C2214yes—
61#4673C2212yes—
62#4673C2214yes—
63#4673C2212yes—
64#4668C2214yes—
65#4668C2214yes—
66#4668C2214yes—
67#4668C2214yes—
68#4665C2214yes—
69#4665C2214yes—
70#4665C2214yes—
71#4665C2214yes—
72#4661C2212yes—
73#4661C2212yes—
74#4661C2214yes—
75#4656C2214yes—
76#4656C2212yes—
77#4656C2212yes—
78C2×#554—31no—
79C2×D4×Q8—31no—
80#2732—34no—
81#2692—34no—
82#2580—31no—
83#2580—31no—
84#2550—32no—
85#2454—32no—
86#2452—32no—
87#2449—34no—
88#2409—34no—
89#2399—31no—
90#2366—32no—
91#2331D311yescharacteristic, complemented
92#2328—31no—
93C22×C3×Q8C2312yes—
94C2×D3×Q8—28no—
95C2×D3×Q8—48no—
96C22×D12C2311yescharacteristic
97C22×D12C2311yescharacteristic
98C22×D12C2314yes—
99C22×C4×D3C2314yes—
100C22×C4×D3C2312yes—
101C22×C4×D3C2312yes—
102C22×C4×D3C2314yes—
103C22×Dic6C2311yescharacteristic
104C22×Dic6C2314yes—
105C22×Dic6C2311yescharacteristic
106C3×C4⋊Q8D418yescomplemented
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2314yes—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2314yes—
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115#946—28no—
116#946—216no—
117C2×(C2×D3)⋊C4C2312yes—
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×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×C4⋊Dic3C2312yes—
128C2×C4⋊Dic3C2312yes—
129C2×C4⋊Dic3C2311yescharacteristic
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4⋊Dic3C2314yes—
132C2×C4⋊Dic3C2311yescharacteristic
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2314yes—
137C2×C4⋊Dic3C2314yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×C4×Dic3C2312yes—
143C2×C4×Dic3C2312yes—
144#897—28no—
145#897—48no—
146#896—216no—
147#893—28no—
148#893—28no—
149D3×C4⋊C4—28no—
150#890—28no—
151#888—28no—
152#874Q818yescomplemented
153C4×D12—24no—
154C4×D12—24no—
155C4×D12—48no—
156C4×Dic6—24no—
157C4×Dic6—24no—
158C3×#95C2312yes—
159C3×#95C2312yes—
160C3×#95C2314yes—
161C3×#95C2312yes—
162C3×#95C2312yes—
163C3×#95C2314yes—
164C3⋊#95C2312yes—
165C3⋊#95C2312yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2314yes—
171C23×Q8—32no—
172#554—68no—
173D4×Q8—68no—
174C2×C4⋊Q8—32no—
175C2×C4⋊Q8C2×D311yescharacteristic
176C2×#127—31no—
177C2×#125—32no—
178C2×C22⋊Q8—34no—
179C2×C22⋊Q8—32no—
180C2×C22⋊Q8—32no—
181C2×C22⋊Q8—34no—
182C2×C22⋊Q8—34no—
183C2×C4×Q8—31no—
184C2×C4×Q8—31no—
185C2×C4×D4—32no—
186C2×C4×D4—32no—
187C2×C4×D4—31no—
188C2×C4×D4—31no—
189C2×C4×D4—34no—
190C22×C4⋊C4—32no—
191#399—32no—
192#399—32no—
193#399—34no—
194#398—32no—
195#398—32no—
196#398—34no—
197#396—34no—
198#396—34no—
199#395—32no—
200#395—32no—
201#395—34no—
202#393—32no—
203#393—32no—
204#393—34no—
205#391—34no—
206#391C2×D314yes—
207#390—32no—
208#390—32no—
209#390—31no—
210#390—31no—
211#390—34no—
212#389—32no—
213#389—32no—
214#389—34no—
215#389C2×D312yes—
216#387C2×D314yes—
217#386—34no—
218#386—34no—
219#386—34no—
220#386—34no—
221#385—34no—
222#385—34no—
223#385—34no—
224#385—34no—
225#382—31no—
226#382C2×D311yescharacteristic
227#380—34no—
228C4×C4⋊C4—32no—
229C4×C4⋊C4—32no—
230C4×C4⋊C4C2×D312yes—
231#376C2×D311yescharacteristic
232C23×D3C2412yes—
233C23×D3C2412yes—
234C2×C3×Q8—44no—
235C2×C3×Q8C2×D418yes—
236C22×C3×C4C2412yes—
237C22×C3×C4C2414yes—
238C22×C3×C4C2411yescharacteristic
239C22×C3×C4C2411yescharacteristic
240C22×C3×C4C2414yes—
241C22×C3×C4C2412yes—
242C22×C3×C4C2411yescharacteristic
243C22×Dic3C2412yes—
244C22×Dic3C2412yes—
245C22×Dic3C2412yes—
246C22×Dic3C2412yes—
247C22×Dic3C2412yes—
248C22×Dic3C2412yes—
249D3×Q8—432no—
250C2×D12—22no—
251C2×D12—22no—
252C2×D12—22no—
253C2×D12—22no—
254C2×D12—44no—
255C2×D12—44no—
256C2×D12C2×Q814yes—
257C2×D12C2×Q814yescomplemented
258C2×D12C2×Q8116yescomplemented
259C2×C4×D3—24no—
260C2×C4×D3—24no—
261C2×C4×D3—28no—
262C2×C4×D3—28no—
263C2×C4×D3—28no—
264C2×C4×D3—44no—
265C2×C4×D3—44no—
266C2×C4×D3—44no—
267C2×C4×D3—44no—
268C2×C4×D3—44no—
269C2×Dic6—22no—
270C2×Dic6—24no—
271C2×Dic6—22no—
272C2×Dic6—28no—
273C2×Dic6—28no—
274C2×Dic6—42no—
275C2×Dic6—44no—
276C3×C4⋊C4—24no—
277C3×C4⋊C4—44no—
278C3×C4⋊C4C2×D418yes—
279C3×C4⋊C4C2×D418yes—
280C3×C42—22no—
281C3×C42—44no—
282C3×C42C2×D414yes—
283C3×C42C2×Q814yescomplemented
284(C2×D3)⋊C4—24no—
285(C2×D3)⋊C4—24no—
286(C2×D3)⋊C4—28no—
287(C2×D3)⋊C4—28no—
288(C2×D3)⋊C4—28no—
289(C2×D3)⋊C4—44no—
290(C2×D3)⋊C4—44no—
291(C2×D3)⋊C4—44no—
292(C2×D3)⋊C4—44no—
293(C2×D3)⋊C4—44no—
294C4⋊Dic3—22no—
295C4⋊Dic3—22no—
296C4⋊Dic3—22no—
297C4⋊Dic3—22no—
298C4⋊Dic3—28no—
299C4⋊Dic3—24no—
300C4⋊Dic3—24no—
301C4⋊Dic3—44no—
302C4⋊Dic3—42no—
303C4⋊Dic3—44no—
304C4⋊Dic3—42no—
305Dic3⋊C4—24no—
306Dic3⋊C4—24no—
307Dic3⋊C4—28no—
308Dic3⋊C4—28no—
309C4×Dic3—24no—
310C4×Dic3—24no—
311C22×Q8—31no—
312C22×Q8—34no—
313C22×Q8—31no—
314C22×Q8—68no—
315C22×Q8—128no—
316C22×Q8C22×D312yes—
317C22×D4—31no—
318C22×D4—31no—
319C22×D4—34no—
320C23×C4—34no—
321C23×C4—34no—
322C23×C4—32no—
323C23×C4—32no—
324C4⋊Q8—68no—
325C4⋊Q8D1218yescomplemented
326#127—38no—
327#125—68no—
328C22⋊Q8—68no—
329C22⋊Q8—616no—
330C22⋊Q8—616no—
331C22⋊Q8—68no—
332C22⋊Q8—128no—
333C4×Q8—64no—
334C4×Q8—64no—
335C4×D4—68no—
336C4×D4—68no—
337C4×D4—64no—
338C4×D4—64no—
339C4×D4—128no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—34no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—31no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—32no—
346C2×C4⋊C4—32no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—32no—
351C2×C4⋊C4—34no—
352C2×C4⋊C4—32no—
353C2×C4⋊C4—32no—
354C2×C4⋊C4—34no—
355C2×C4⋊C4—68no—
356C2×C4⋊C4C22×D312yes—
357C2×C4⋊C4C22×D314yes—
358C2×C4⋊C4C22×D312yes—
359C2×C4⋊C4C22×D312yes—
360C2×C22⋊C4—32no—
361C2×C22⋊C4—32no—
362C2×C22⋊C4—34no—
363C2×C22⋊C4—34no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4—34no—
367C2×C22⋊C4—34no—
368C2×C22⋊C4—34no—
369C2×C22⋊C4—34no—
370C2×C42—32no—
371C2×C42—32no—
372C2×C42C22×D311yescharacteristic
373C2×C42C22×D314yes—
374C2×C42C22×D311yescharacteristic
375C2×C42C22×D311yescharacteristic
376#95—32no—
377#95—32no—
378#95—32no—
379#95—32no—
380#95—34no—
381#95—34no—
382#95—34no—
383#95C22×D312yes—
384#95C22×D312yes—
385#95C22×D314yes—
386#95C22×D312yes—
387#95C22×D312yes—
388#95C22×D314yes—
389C23×C3C2511yescharacteristic
390C22×D3—22no—
391C22×D3—22no—
392C22×D3—24no—
393C22×D3—24no—
394C22×D3—22no—
395C22×D3—22no—
396C22×D3—24no—
397C22×D3—24no—
398C22×D3C22×Q814yes—
399C22×D3C22×Q814yes—
400C3×Q8—48no—
401C2×C3×C4—22no—
402C2×C3×C4—24no—
403C2×C3×C4—24no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—24no—
409C2×C3×C4—22no—
410C2×C3×C4—24no—
411C2×C3×C4—22no—
412C2×C3×C4—24no—
413C2×C3×C4C22×D412yes—
414C2×C3×C4C22×D412yes—
415C2×C3×C4C22×D418yes—
416C2×C3×C4C22×D412yes—
417C2×C3×C4C22×Q814yes—
418C2×C3×C4C22×Q812yes—
419C2×C3×C42−1+412yes—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—24no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—24no—
434C2×Dic3—22no—
435C2×Dic3H(5,𝔽2)14yes—
436D12—28no—
437D12—28no—
438D12—416no—
439C4×D3—416no—
440C4×D3—48no—
441C4×D3—48no—
442C4×D3—88no—
443Dic6—28no—
444Dic6—416no—
445Dic6—44no—
446C24—32no—
447C24—32no—
448C2×Q8—44no—
449C2×Q8—64no—
450C2×Q8—68no—
451C2×Q8—62no—
452C2×Q8—62no—
453C2×Q8—68no—
454C2×Q8—124no—
455C2×Q8—122no—
456C2×Q8—1232no—
457C2×Q8C2×D1218yes—
458C2×D4—34no—
459C2×D4—34no—
460C2×D4—316no—
461C2×D4—62no—
462C2×D4—62no—
463C2×D4—62no—
464C2×D4—62no—
465C2×D4—124no—
466C2×D4—124no—
467C22×C4—32no—
468C22×C4—32no—
469C22×C4—32no—
470C22×C4—32no—
471C22×C4—32no—
472C22×C4—32no—
473C22×C4—68no—
474C22×C4—64no—
475C22×C4—68no—
476C22×C4—64no—
477C22×C4—68no—
478C22×C4—124no—
479C22×C4—124no—
480C22×C4—124no—
481C22×C4—124no—
482C22×C4—124no—
483C22×C4C23×D312yes—
484C22×C4C23×D314yes—
485C22×C4C23×D311yescharacteristic
486C22×C4C23×D311yescharacteristic
487C22×C4C23×D314yes—
488C22×C4C23×D312yes—
489C22×C4C23×D311yescharacteristic
490C4⋊C4—24no—
491C4⋊C4—44no—
492C4⋊C4—64no—
493C4⋊C4—64no—
494C4⋊C4—62no—
495C4⋊C4—68no—
496C4⋊C4—62no—
497C4⋊C4—62no—
498C4⋊C4—64no—
499C4⋊C4—64no—
500C4⋊C4—62no—
501C4⋊C4—68no—
502C4⋊C4—68no—
503C4⋊C4—124no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—124no—
507C4⋊C4C2×D1218yes—
508C4⋊C4C2×D1218yes—
509C22⋊C4—68no—
510C22⋊C4—68no—
511C22⋊C4—68no—
512C22⋊C4—64no—
513C22⋊C4—64no—
514C22⋊C4—124no—
515C22⋊C4—124no—
516C22⋊C4—124no—
517C22⋊C4—124no—
518C22⋊C4—124no—
519C42—22no—
520C42—44no—
521C42—64no—
522C42—64no—
523C42C2×D1214yes—
524C42D3×Q814yescomplemented
525C22×C3C23×D411yescharacteristic
526C22×C3C23×Q811yescharacteristic
527C22×C3C2×H(5,𝔽2)12yes—
528C22×C3C2×2−1+411yescharacteristic
529C22×C3#58512yes—
530C2×D3—24no—
531C2×D3—24no—
532C2×D3—28no—
533C2×D3—28no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—44no—
539C2×D3—44no—
540C3×C4—22no—
541C3×C4—28no—
542C3×C4—22no—
543C3×C4—24no—
544C3×C4—42no—
545C3×C4—44no—
546C3×C4D4×Q814yes—
547Dic3—24no—
548Dic3—42no—
549Dic3—42no—
550Dic3—42no—
551Dic3—42no—
552Dic3—42no—
553C23—34no—
554C23—34no—
555C23—62no—
556C23—62no—
557C23—62no—
558C23—62no—
559C23—64no—
560C23—64no—
561C23—64no—
562C23—64no—
563C23C24×D311yescharacteristic
564Q8—48no—
565Q8—68no—
566Q8—1216no—
567Q8—124no—
568D4—68no—
569D4—68no—
570D4—1216no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—34no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—64no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—64no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—1216no—
600C2×C4—128no—
601C2×C4—128no—
602C2×C4—248no—
603C2×C4C22×D1212yes—
604C2×C4C22×D1212yes—
605C2×C4C22×D1218yes—
606C2×C4C22×D1212yes—
607C2×C4C2×D3×Q814yes—
608C2×C4C2×D3×Q812yes—
609C2×C4#101012yes—
610C2×C3C2×D4×Q811yescharacteristic
611C2×C3#348411yescharacteristic
612C2×C3#348711yescharacteristic
613C2×C3#350112yes—
614C2×C3#355412yes—
615D3—44no—
616D3—44no—
617C22—64no—
618C22—64no—
619C22—68no—
620C22—68no—
621C22—124no—
622C22—124no—
623C22—122no—
624C22—122no—
625C22—122no—
626C22—122no—
627C22C23×D1211yescharacteristic
628C22C22×D3×Q811yescharacteristic
629C22C2×#101011yescharacteristic
630C22D3×H(5,𝔽2)12yes—
631C22#596812yes—
632C4—24no—
633C4—22no—
634C4—28no—
635C4—22no—
636C4—42no—
637C4—44no—
638C4—64no—
639C4—122no—
640C4—122no—
641C4—122no—
642C4—122no—
643C4—122no—
644C4Q8×D1214yes—
645C3#4535711yescharacteristic, complemented, Sylow
646C2—124no—
647C2—124no—
648C2C2×Q8×D1211yescharacteristic
649C2#8586711yescharacteristic
650C2#8587411yescharacteristic
651C2#8653512yes—
652C2#8659912yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2331C2
prime radicalC2×#874C22
hypercenter#2331D3
hypocenterC3#45357
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. #76656C2semidirect
5. #76619C2semidirect
6. #76016C2semidirect
7. #76011C2semidirect
8. #75394C2semidirect
9. #75321C2semidirect
10. #74190C2semidirect
11. #74190C2semidirect
12. #73714C2semidirect
14. #73397C2semidirect
91. #2331D3semidirect
106. C3×C4⋊Q8D4semidirect
152. #874Q8semidirect
257. C2×D12C2×Q8semidirect
258. C2×D12C2×Q8semidirect
283. C3×C42C2×Q8semidirect
325. C4⋊Q8D12semidirect
524. C42D3×Q8semidirect
645. C3#45357semidirect

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
#766562
#766192
#760162
#760112
#753942
#753212
#741902
#737142
#736532
#733972
#733892
#453573