Finite Group Databasebeta

← #11233328

Subgroups — #11233328

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×#5579C211yesmaximal, characteristic, complemented
2C2×D4×Dic6C211yesmaximal, characteristic, complemented
3C3×#2326C211yesmaximal, characteristic, complemented
4#82996C212yesmaximal, complemented
5#82481C211yesmaximal, characteristic, complemented
6#82416C211yesmaximal, characteristic
7#76632C212yesmaximal, complemented
8#76433C212yesmaximal, complemented
9#75856C214yesmaximal, complemented
10#75672C214yesmaximal, complemented
11#74641C214yesmaximal, complemented
12#74405C214yesmaximal, complemented
13#74326C212yesmaximal, complemented
14#73419C211yesmaximal, characteristic, complemented
15#73391C211yesmaximal, characteristic, complemented
16#45337—31nomaximal, Sylow
17C23×Dic6C2212yes—
18C2×C3×#124C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×#946C2212yes—
22C2×#938C2212yes—
23C2×D4×Dic3C2212yes—
24C2×#924C2212yes—
25#5579—28no—
26D4×Dic6—28no—
27C2×#896C2214yes—
28C2×#896C2214yes—
29C2×D3×C4⋊C4C2212yes—
30C2×#888C2212yes—
31C2×#67⋊C4C2214yes—
32C2×#878C2214yes—
33C2×C4×D12C2212yes—
34C2×C4×D12C2211yescharacteristic
35C2×#870C2212yes—
36C2×#869C2211yescharacteristic
37C2×C4×Dic6C2211yescharacteristic
38C3×#391C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#388C2214yes—
41C3×#385C2214yes—
42C3×#384C2211yescharacteristic
43C3×#376C2211yescharacteristic
44#5229C2212yes—
45#5213C2214yes—
46#4989C2214yes—
47#4987C2212yes—
48#4986C2212yes—
49#4983C2214yes—
50#4979C2212yes—
51#4979C2214yes—
52#4972C2212yes—
53#4960C2212yes—
54#4959C2214yes—
55#4958C2214yes—
56#4956C2214yes—
57#4953C2212yes—
58#4952C2214yes—
59#4946C2214yes—
60#4944C2214yes—
61#4942C2214yes—
62C4×(C2×D3)⋊C4C2212yes—
63C4×(C2×D3)⋊C4C2212yes—
64#4936C2211yescharacteristic
65C4×C4⋊Dic3C2212yes—
66#4933C2214yes—
67#4928C2212yes—
68#4674C2212yes—
69#4673C2214yes—
70#4673C2212yes—
71#4672C2212yes—
72#4669C2214yes—
73#4659C2212yes—
74#4658C2214yes—
75#4654C2214yes—
76#4646C2214yes—
77#4646C2214yes—
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#2331—31no—
92#2326D311yescharacteristic, complemented
93C22×C3×Q8C2311yescharacteristic
94C22×C3×D4C2311yescharacteristic
95C23×C3×C4C2312yes—
96C22×#67C2314yes—
97C23×Dic3C2314yes—
98C22×D12C2311yescharacteristic
99C22×C4×D3C2312yes—
100C22×C4×D3C2314yes—
101C22×Dic6—28no—
102C22×Dic6—48no—
103C22×Dic6C2311yescharacteristic
104C22×Dic6C2314yes—
105C22×Dic6C2312yes—
106C3×#124—24no—
107C3×C4×Q8—24no—
108C3×C4×D4—24no—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C4⋊C4C2314yes—
113C2×C3×C22⋊C4C2314yes—
114C2×C3×C22⋊C4C2312yes—
115C2×C3×C22⋊C4C2314yes—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2312yes—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×#217C2314yes—
121C2×#217C2314yes—
122#946—28no—
123#938—28no—
124D4×Dic3—28no—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×C4⋊Dic3C2311yescharacteristic
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2312yes—
137#924—28no—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2314yes—
144C2×C4×Dic3C2312yes—
145C2×C4×Dic3C2314yes—
146#896—216no—
147#896—48no—
148D3×C4⋊C4—28no—
149#888—28no—
150#67⋊C4—48no—
151#878—216no—
152C4×D12—24no—
153C4×D12—28no—
154#870—28no—
155#869D418yescomplemented
156C4×Dic6—24no—
157C3×#95C2312yes—
158C3×#95C2314yes—
159C3×#95C2312yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162C3⋊#95C2314yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2312yes—
167C3⋊#95C2312yes—
168C3⋊#95C2314yes—
169C3⋊#95C2312yes—
170C3⋊#95C2312yes—
171C3⋊#95C2314yes—
172C23×Q8—32no—
173#554—68no—
174D4×Q8—68no—
175C2×C4⋊Q8—32no—
176C2×C4⋊Q8—31no—
177C2×#125—32no—
178C2×#124C2×D311yescharacteristic
179C2×C22⋊Q8—32no—
180C2×C22⋊Q8—32no—
181C2×C22⋊Q8—34no—
182C2×C22⋊Q8—34no—
183C2×C22⋊Q8—34no—
184C2×#121—32no—
185C2×C4×Q8—31no—
186C2×C4×Q8C2×D311yescharacteristic
187C2×C4×D4—32no—
188C2×C4×D4—34no—
189C2×C4×D4—31no—
190C2×C4×D4—32no—
191C2×C4×D4C2×D311yescharacteristic
192C22×C4⋊C4—32no—
193#400—32no—
194#400—34no—
195#400—32no—
196#398—34no—
197#397—34no—
198#397—32no—
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—34no—
209#391—34no—
210#391C2×D311yescharacteristic
211#390C2×D311yescharacteristic
212#389—32no—
213#389—32no—
214#388—34no—
215#388—34no—
216#388C2×D314yes—
217#387—34no—
218#387—34no—
219#386—34no—
220#386—34no—
221#386—34no—
222#385—34no—
223#385—34no—
224#385C2×D314yes—
225#384C2×D311yescharacteristic
226#382—31no—
227#380—32no—
228#380—34no—
229C4×C4⋊C4—32no—
230C4×C22⋊C4—32no—
231C4×C22⋊C4—32no—
232#376C2×D311yescharacteristic
233C24×C3C2412yes—
234C23×D3C2412yes—
235C2×C3×Q8—22no—
236C2×C3×Q8—22no—
237C2×C3×Q8—22no—
238C2×C3×D4—22no—
239C2×C3×D4—22no—
240C2×C3×D4#4114yes—
241C22×C3×C4—24no—
242C22×C3×C4—44no—
243C22×C3×C4C2412yes—
244C22×C3×C4C2412yes—
245C22×C3×C4C2412yes—
246C22×C3×C4C2412yes—
247C22×C3×C4C2412yes—
248C22×C3×C4C2411yescharacteristic
249C22×C3×C4C2411yescharacteristic
250C22×C3×C4C2411yescharacteristic
251C2×#67—28no—
252C2×#67—28no—
253C2×#67—44no—
254C22×Dic3—28no—
255C22×Dic3—28no—
256C22×Dic3—44no—
257C22×Dic3C2414yes—
258C22×Dic3C2412yes—
259C22×Dic3C2412yes—
260C22×Dic3C2414yes—
261C22×Dic3C2412yes—
262C2×D12—24no—
263C2×D12—22no—
264C2×C4×D3—24no—
265C2×C4×D3—24no—
266C2×C4×D3—24no—
267C2×C4×D3—28no—
268C2×C4×D3—44no—
269C2×C4×D3—44no—
270C2×Dic6—22no—
271C2×Dic6—24no—
272C2×Dic6—28no—
273C2×Dic6—28no—
274C2×Dic6—432no—
275C2×Dic6—44no—
276C2×Dic6—44no—
277C2×Dic6C2×D418yes—
278C3×C4⋊C4—22no—
279C3×C4⋊C4—22no—
280C3×C4⋊C4—24no—
281C3×C4⋊C4—44no—
282C3×C22⋊C4—28no—
283C3×C22⋊C4—24no—
284C3×C22⋊C4—44no—
285C3×C42—24no—
286C3×C42—22no—
287C3×C42—22no—
288C3×C42C2×D414yes—
289#217—28no—
290#217—28no—
291(C2×D3)⋊C4—24no—
292(C2×D3)⋊C4—24no—
293(C2×D3)⋊C4—24no—
294(C2×D3)⋊C4—28no—
295(C2×D3)⋊C4—24no—
296(C2×D3)⋊C4—24no—
297(C2×D3)⋊C4—44no—
298C4⋊Dic3—22no—
299C4⋊Dic3—22no—
300C4⋊Dic3—24no—
301C4⋊Dic3C2×D418yes—
302C4⋊Dic3C2×D418yes—
303Dic3⋊C4—24no—
304Dic3⋊C4—28no—
305Dic3⋊C4—28no—
306Dic3⋊C4—28no—
307Dic3⋊C4—28no—
308Dic3⋊C4—44no—
309C4×Dic3—24no—
310C4×Dic3—44no—
311C22×Q8—32no—
312C22×Q8—31no—
313C22×Q8—34no—
314C22×Q8—68no—
315C22×Q8—128no—
316C22×Q8C22×D311yescharacteristic
317C22×D4—34no—
318C22×D4—31no—
319C22×D4C22×D311yescharacteristic
320C23×C4—34no—
321C23×C4—32no—
322C23×C4—34no—
323C23×C4C22×D312yes—
324C4⋊Q8—38no—
325C4⋊Q8—68no—
326#125—68no—
327#124—24no—
328C22⋊Q8—616no—
329C22⋊Q8—616no—
330C22⋊Q8—68no—
331C22⋊Q8—68no—
332C22⋊Q8—128no—
333#121—68no—
334C4×Q8—24no—
335C4×Q8—64no—
336C4×D4—24no—
337C4×D4—68no—
338C4×D4—68no—
339C4×D4—64no—
340C4×D4—128no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—34no—
344C2×C4⋊C4—34no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—32no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—32no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—34no—
352C2×C4⋊C4—68no—
353C2×C4⋊C4C22×D311yescharacteristic
354C2×C4⋊C4C22×D311yescharacteristic
355C2×C4⋊C4C22×D312yes—
356C2×C4⋊C4C22×D314yes—
357C2×C22⋊C4—34no—
358C2×C22⋊C4—34no—
359C2×C22⋊C4—32no—
360C2×C22⋊C4—32no—
361C2×C22⋊C4—34no—
362C2×C22⋊C4—32no—
363C2×C22⋊C4—32no—
364C2×C22⋊C4—32no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4C22×D312yes—
367C2×C22⋊C4C22×D314yes—
368C2×C22⋊C4C22×D314yes—
369C2×C42—32no—
370C2×C42—34no—
371C2×C42C22×D312yes—
372C2×C42C22×D311yescharacteristic
373C2×C42C22×D311yescharacteristic
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—32no—
383#95—32no—
384#95—34no—
385#95—34no—
386#95—34no—
387#95C22×D312yes—
388#95C22×D312yes—
389#95C22×D314yes—
390C23×C3—22no—
391C23×C3—24no—
392C23×C3—24no—
393C23×C3—22no—
394C23×C3C2×#4114yes—
395C23×C3C2511yescharacteristic
396C22×D3—22no—
397C22×D3—22no—
398C22×D3—22no—
399C22×D3—22no—
400C22×D3—22no—
401C22×D3—22no—
402C22×D32−1+414yes—
403C3×Q8—44no—
404C3×D4—28no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—24no—
414C2×C3×C4—22no—
415C2×C3×C4—24no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—48no—
421C2×C3×C4C22×D412yes—
422C2×C3×C4C22×D412yes—
423C2×C3×C4C22×D412yes—
424C2×C3×C4C2×#4112yes—
425C2×C3×C42−1+412yes—
426#67—416no—
427C2×Dic3—22no—
428C2×Dic3—24no—
429C2×Dic3—24no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—24no—
434C2×Dic3—22no—
435C2×Dic3—24no—
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3—24no—
440C2×Dic3—416no—
441C2×Dic3C22×D418yes—
442C2×Dic3H(5,𝔽2)14yes—
443D12—28no—
444C4×D3—48no—
445C4×D3—88no—
446Dic6—28no—
447Dic6—48no—
448Dic6—416no—
449C24—32no—
450C24C23×D312yes—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—38no—
455C2×Q8—64no—
456C2×Q8—68no—
457C2×Q8—68no—
458C2×Q8—62no—
459C2×Q8—124no—
460C2×Q8—124no—
461C2×Q8—1232no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—64no—
465C2×D4—68no—
466C2×D4—68no—
467C2×D4—62no—
468C2×D4—124no—
469C2×D4#23514yes—
470C22×C4—24no—
471C22×C4—32no—
472C22×C4—34no—
473C22×C4—34no—
474C22×C4—32no—
475C22×C4—32no—
476C22×C4—44no—
477C22×C4—68no—
478C22×C4—68no—
479C22×C4—68no—
480C22×C4—64no—
481C22×C4—64no—
482C22×C4—64no—
483C22×C4—124no—
484C22×C4—124no—
485C22×C4—124no—
486C22×C4C23×D312yes—
487C22×C4C23×D312yes—
488C22×C4C23×D312yes—
489C22×C4C23×D312yes—
490C22×C4C23×D312yes—
491C22×C4C23×D311yescharacteristic
492C22×C4C23×D311yescharacteristic
493C22×C4C23×D311yescharacteristic
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—38no—
498C4⋊C4—38no—
499C4⋊C4—44no—
500C4⋊C4—68no—
501C4⋊C4—68no—
502C4⋊C4—68no—
503C4⋊C4—68no—
504C4⋊C4—64no—
505C4⋊C4—62no—
506C4⋊C4—64no—
507C4⋊C4—62no—
508C4⋊C4—124no—
509C22⋊C4—28no—
510C22⋊C4—24no—
511C22⋊C4—44no—
512C22⋊C4—68no—
513C22⋊C4—64no—
514C22⋊C4—64no—
515C22⋊C4—64no—
516C22⋊C4—64no—
517C22⋊C4—68no—
518C22⋊C4—68no—
519C22⋊C4—64no—
520C22⋊C4—124no—
521C42—22no—
522C42—24no—
523C42—22no—
524C42—64no—
525C42—124no—
526C42D3×D414yes—
527C22×C3—24no—
528C22×C3—28no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—44no—
532C22×C3C23×D411yescharacteristic
533C22×C3C22×#4111yescharacteristic
534C22×C3C2×H(5,𝔽2)12yes—
535C22×C3C2×2−1+411yescharacteristic
536C22×C3C2×2−1+412yes—
537C2×D3—24no—
538C2×D3—24no—
539C2×D3—24no—
540C2×D3—42no—
541C2×D3—42no—
542C2×D3—42no—
543C2×D3—42no—
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—42no—
547C3×C4—42no—
548C3×C4—42no—
549C3×C4—42no—
550C3×C4—42no—
551C3×C4#54814yes—
552Dic3—24no—
553Dic3—28no—
554Dic3—42no—
555Dic3—42no—
556Dic3—44no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—24no—
561C23—34no—
562C23—62no—
563C23—62no—
564C23—62no—
565C23—62no—
566C23—62no—
567C23—62no—
568C23C2×#23514yes—
569C23C24×D311yescharacteristic
570Q8—44no—
571Q8—68no—
572Q8—128no—
573Q8—1216no—
574D4—28no—
575D4—68no—
576D4—1216no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—38no—
593C2×C4—34no—
594C2×C4—48no—
595C2×C4—62no—
596C2×C4—64no—
597C2×C4—64no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—64no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—64no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—64no—
608C2×C4—1216no—
609C2×C4—128no—
610C2×C4—248no—
611C2×C4C2×#23512yes—
612C2×C4C2×D3×D412yes—
613C2×C4C2×D3×D412yes—
614C2×C4C2×D3×D412yes—
615C2×C4#101012yes—
616C2×C3—44no—
617C2×C3C2×#54811yescharacteristic
618C2×C3#348411yescharacteristic
619C2×C3#348611yescharacteristic
620C2×C3#352012yes—
621C2×C3#355212yes—
622D3—44no—
623C22—28no—
624C22—24no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—64no—
629C22—64no—
630C22—64no—
631C22—122no—
632C22—122no—
633C22—122no—
634C22—122no—
635C22C22×#23511yescharacteristic
636C22C22×D3×D411yescharacteristic
637C22C2×#101011yescharacteristic
638C22D3×H(5,𝔽2)12yes—
639C22D3×2−1+412yes—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—64no—
648C4—68no—
649C4—122no—
650C4—124no—
651C4—122no—
652C4#555214yes—
653C3#4533711yescharacteristic, complemented, Sylow
654C2—44no—
655C2—124no—
656C2C2×#555211yescharacteristic
657C2#8590712yes—
658C2#8633011yescharacteristic
659C2#8635511yescharacteristic
660C2#8656612yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2326C2
prime radical#82416C2
hypercenter#2326D3
hypocenterC3#45337
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. C2×#5579C2semidirect
2. C2×D4×Dic6C2semidirect
3. C3×#2326C2semidirect
4. #82996C2semidirect
5. #82481C2semidirect
7. #76632C2semidirect
8. #76433C2semidirect
9. #75856C2semidirect
10. #75672C2semidirect
11. #74641C2semidirect
12. #74405C2semidirect
13. #74326C2semidirect
14. #73419C2semidirect
15. #73391C2semidirect
92. #2326D3semidirect
155. #869D4semidirect
653. C3#45337semidirect

Maximal subgroups

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

subgroupindex
C2×#55792
C2×D4×Dic62
C3×#23262
#829962
#824812
#824162
#766322
#764332
#758562
#756722
#746412
#744052
#743262
#734192
#733912
#453373