Finite Group Databasebeta

← #11002002

Subgroups — #11002002

Order 768.

2326 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5623C211yesmaximal, characteristic, complemented
2C22×#896C211yesmaximal, characteristic
3C3×#2627C211yesmaximal, characteristic, complemented
4#82834C211yesmaximal, characteristic
5#82183C211yesmaximal, characteristic, complemented
6#78368C214yesmaximal, complemented
7#77974C214yesmaximal, complemented
8#77968C214yesmaximal, complemented
9#77261C214yesmaximal, complemented
10#74614C212yesmaximal
11#74412C212yesmaximal, complemented
12#74132C212yesmaximal
13#74073C212yesmaximal, complemented
14#74049C211yesmaximal, characteristic
15#73867C211yesmaximal, characteristic, complemented
16#36708—31nomaximal, Sylow
17C23×C4×D3C2211yescharacteristic
18C23×Dic6C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C2×#946C2212yes—
22C2×#944C2212yescomplemented
23C22×(C2×D3)⋊C4C2211yescharacteristic
24C22×(C2×D3)⋊C4C2211yescharacteristic
25C2×#924C2211yescharacteristic
26C22×Dic3⋊C4C2211yescharacteristic
27C22×Dic3⋊C4C2211yescharacteristic
28#5623—28no—
29C2×#897C2212yes—
30C2×#896—28no—
31C2×#896C2214yes—
32C2×#896C2214yes—
33C2×#889C2212yescomplemented
34C2×#889C2212yescomplemented
35C2×#888C2212yescomplemented
36C2×#888C2214yescomplemented
37C2×#878C2212yes—
38C2×#877C2212yes—
39C3×#400C2212yescomplemented
40C3×#398C2214yescomplemented
41C3×#395C2212yescomplemented
42C3×#385C2212yes—
43C3×#381C2212yes—
44C3×#380C2211yescharacteristic
45#5212C2211yescharacteristic
46#5211C2212yes—
47#5210C2212yes—
48#4993C2214yes—
49#4990C2214yes—
50#4989C2214yes—
51#4987C2212yes—
52#4987C2212yes—
53#4986C2212yescomplemented
54#4982C2214yescomplemented
55#4979C2212yescomplemented
56#4979C2214yescomplemented
57#4978C2214yescomplemented
58#4955C2212yes—
59#4955C2212yes—
60#4950C2212yes—
61#4946C2212yes—
62#4943C2212yes—
63Dic3×C22⋊C4C2212yes—
64#4675C2214yes—
65#4675C2214yes—
66#4673C2214yes—
67#4670C2214yes—
68#4669C2214yes—
69#4669C2214yes—
70#4668C2214yes—
71#4658C2214yescomplemented
72#4657C2214yescomplemented
73#4656C2214yescomplemented
74#4655C2214yescomplemented
75#4654C2214yescomplemented
76#4652C2214yescomplemented
77#4646C2214yescomplemented
78C2×#544—31no—
79C22×C22⋊Q8—31no—
80#2848—31no—
81#2848—31no—
82#2690—34no—
83#2690—34no—
84#2677—34no—
85#2677—34no—
86#2627—31no—
87#2627D311yescharacteristic, complemented
88#2561—32no—
89#2484—32no—
90#2482—32no—
91#2434—32no—
92#2313—31no—
93C24×D3C2311yescharacteristic
94C22×C3×Q8C2311yescharacteristic
95C23×C3×C4C2311yescharacteristic
96C23×C3×C4C2311yescharacteristic
97C23×C3×C4C2311yescharacteristic
98C23×Dic3C2311yescharacteristic
99C23×Dic3C2311yescharacteristic
100C23×Dic3C2311yescharacteristic
101C22×C4×D3—24no—
102C22×C4×D3—44no—
103C22×C4×D3C2312yes—
104C22×C4×D3C2312yes—
105C22×Dic6—24no—
106C22×Dic6—44no—
107C22×Dic6C2314yes—
108C22×Dic6C2312yes—
109C22×Dic6C2311yescharacteristic
110C3×C22⋊Q8—24no—
111C2×C3×C4⋊C4—24no—
112C2×C3×C4⋊C4C2312yes—
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C4⋊C4C2311yescharacteristic
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C4⋊C4C2314yes—
118C2×C3×C22⋊C4C2312yes—
119C2×C3×C22⋊C4C2312yes—
120C2×#217C2312yes—
121C2×#217C2312yes—
122#946—44no—
123#944—28no—
124C2×(C2×D3)⋊C4—24no—
125C2×(C2×D3)⋊C4—24no—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2312yes—
136#924—24no—
137C2×Dic3⋊C4—24no—
138C2×Dic3⋊C4D418yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2312yes—
143C2×Dic3⋊C4C2312yes—
144C2×Dic3⋊C4C2312yes—
145C2×Dic3⋊C4C2312yes—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2312yes—
149C2×C4×Dic3C2312yes—
150C2×C4×Dic3C2312yes—
151C2×C4×Dic3C2314yes—
152#897—44no—
153#896—216no—
154#896—216no—
155#889—28no—
156#889—28no—
157#888—28no—
158#888—48no—
159#878—28no—
160#877—28no—
161C3×#95C2314yes—
162C3×#95C2314yes—
163C3×#95C2314yes—
164C3×#95C2312yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2314yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C3⋊#95C2314yes—
173C3⋊#95C2312yes—
174C23×Q8—31no—
175C24×C4—31no—
176#544—68no—
177C2×C4⋊Q8—32no—
178C2×C4⋊Q8—32no—
179C2×C4⋊Q8—34no—
180C2×#125—32no—
181C2×#125—32no—
182C2×C22⋊Q8—31no—
183C2×C22⋊Q8—32no—
184C2×C22⋊Q8—34no—
185C2×C22⋊Q8—34no—
186C2×C22⋊Q8—32no—
187C2×C22⋊Q8—32no—
188C2×C22⋊Q8—68no—
189C2×C22⋊Q8C2×D311yescharacteristic
190C2×#117—32no—
191C22×C4⋊C4—31no—
192C22×C4⋊C4—31no—
193C22×C4⋊C4C2×D311yescharacteristic
194C22×C22⋊C4—31no—
195C22×C22⋊C4—31no—
196#400—32no—
197#400—34no—
198#400C2×D312yescomplemented
199#399—34no—
200#398—34no—
201#398—34no—
202#398C2×D314yescomplemented
203#397—34no—
204#397—34no—
205#395—32no—
206#395—34no—
207#395—34no—
208#395C2×D312yescomplemented
209#393—34no—
210#391—34no—
211#388—34no—
212#388—34no—
213#388—34no—
214#388—34no—
215#387—34no—
216#387—34no—
217#387—34no—
218#387—34no—
219#385—32no—
220#385—34no—
221#385—32no—
222#385—32no—
223#385—34no—
224#385C2×D312yes—
225#381—32no—
226#381—32no—
227#381C2×D312yes—
228#380—31no—
229#380—32no—
230#380—32no—
231#380C2×D311yescharacteristic
232#379—32no—
233#379—32no—
234C4×C22⋊C4—32no—
235C24×C3C2411yescharacteristic
236C23×D3—22no—
237C23×D3—24no—
238C23×D3—22no—
239C23×D3—22no—
240C23×D3—42no—
241C23×D3C2412yes—
242C2×C3×Q8—22no—
243C2×C3×Q8—42no—
244C22×C3×C4—22no—
245C22×C3×C4—22no—
246C22×C3×C4—22no—
247C22×C3×C4—22no—
248C22×C3×C4—42no—
249C22×C3×C4—42no—
250C22×C3×C4C2×D414yes—
251C22×C3×C4C2414yes—
252C22×C3×C4C2412yes—
253C22×C3×C4C2412yes—
254C22×C3×C4C2412yes—
255C22×C3×C4C2412yes—
256C22×C3×C4C2411yescharacteristic
257C22×C3×C4C2411yescharacteristic
258C22×Dic3—22no—
259C22×Dic3—22no—
260C22×Dic3—22no—
261C22×Dic3—24no—
262C22×Dic3—42no—
263C22×Dic3C2×D414yes—
264C22×Dic3C2×D414yes—
265C22×Dic3C2414yes—
266C22×Dic3C2412yes—
267C22×Dic3C2412yes—
268C22×Dic3C2412yes—
269C22×Dic3C2412yes—
270C22×Dic3C2412yes—
271C2×C4×D3—24no—
272C2×C4×D3—24no—
273C2×C4×D3—24no—
274C2×C4×D3—24no—
275C2×C4×D3—42no—
276C2×C4×D3—42no—
277C2×C4×D3—416no—
278C2×Dic6—28no—
279C2×Dic6—28no—
280C2×Dic6—22no—
281C2×Dic6—24no—
282C2×Dic6—44no—
283C2×Dic6—416no—
284C2×Dic6—44no—
285C2×Dic6—42no—
286C3×C4⋊C4—24no—
287C3×C4⋊C4—22no—
288C3×C4⋊C4—24no—
289C3×C4⋊C4—24no—
290C3×C4⋊C4—44no—
291C3×C4⋊C4—42no—
292C3×C22⋊C4—24no—
293C3×C22⋊C4—24no—
294#217—24no—
295#217—24no—
296(C2×D3)⋊C4—28no—
297(C2×D3)⋊C4—28no—
298(C2×D3)⋊C4—28no—
299(C2×D3)⋊C4—28no—
300(C2×D3)⋊C4—44no—
301(C2×D3)⋊C4—44no—
302C4⋊Dic3—22no—
303C4⋊Dic3—24no—
304C4⋊Dic3—28no—
305C4⋊Dic3—42no—
306Dic3⋊C4—24no—
307Dic3⋊C4—24no—
308Dic3⋊C4—24no—
309Dic3⋊C4—24no—
310Dic3⋊C4—24no—
311Dic3⋊C4—28no—
312Dic3⋊C4—28no—
313Dic3⋊C4—44no—
314Dic3⋊C4C2×D418yes—
315Dic3⋊C4C2×D418yes—
316C4×Dic3—24no—
317C4×Dic3—24no—
318C4×Dic3—44no—
319C25—31no—
320C22×Q8—34no—
321C22×Q8—31no—
322C22×Q8—32no—
323C22×Q8—64no—
324C22×Q8—124no—
325C22×Q8C22×D311yescharacteristic
326C23×C4—31no—
327C23×C4—31no—
328C23×C4—31no—
329C23×C4—32no—
330C23×C4—32no—
331C23×C4—64no—
332C23×C4—124no—
333C23×C4C22×D311yescharacteristic
334C23×C4C22×D311yescharacteristic
335C23×C4C22×D311yescharacteristic
336C4⋊Q8—68no—
337C4⋊Q8—68no—
338C4⋊Q8—128no—
339#125—68no—
340#125—68no—
341C22⋊Q8—24no—
342C22⋊Q8—68no—
343C22⋊Q8—616no—
344C22⋊Q8—616no—
345C22⋊Q8—64no—
346C22⋊Q8—124no—
347C22⋊Q8—124no—
348#117—68no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—38no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—32no—
353C2×C4⋊C4—34no—
354C2×C4⋊C4—32no—
355C2×C4⋊C4—32no—
356C2×C4⋊C4—31no—
357C2×C4⋊C4—32no—
358C2×C4⋊C4—32no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—32no—
361C2×C4⋊C4—34no—
362C2×C4⋊C4—34no—
363C2×C4⋊C4—32no—
364C2×C4⋊C4—34no—
365C2×C4⋊C4—64no—
366C2×C4⋊C4C22×D312yes—
367C2×C4⋊C4C22×D312yes—
368C2×C4⋊C4C22×D311yescharacteristic
369C2×C4⋊C4C22×D312yes—
370C2×C4⋊C4C22×D312yes—
371C2×C4⋊C4C22×D314yes—
372C2×C22⋊C4—32no—
373C2×C22⋊C4—32no—
374C2×C22⋊C4—34no—
375C2×C22⋊C4—34no—
376C2×C22⋊C4—34no—
377C2×C22⋊C4—34no—
378C2×C22⋊C4—34no—
379C2×C22⋊C4—34no—
380C2×C22⋊C4—64no—
381C2×C22⋊C4—64no—
382C2×C22⋊C4C22×D312yes—
383C2×C22⋊C4C22×D312yes—
384C2×C42—32no—
385C2×C42—32no—
386C2×C42—34no—
387#95—34no—
388#95—32no—
389#95—34no—
390#95—34no—
391#95—34no—
392#95—34no—
393#95—34no—
394#95—34no—
395#95—34no—
396#95C22×D312yes—
397#95C22×D314yes—
398#95C22×D314yes—
399#95C22×D314yes—
400C23×C3—22no—
401C23×C3—22no—
402C23×C3C22×D412yes—
403C23×C32−1+412yes—
404C23×C32−1+412yes—
405C23×C3C2511yescharacteristic
406C22×D3—28no—
407C22×D3—22no—
408C22×D3—22no—
409C22×D3—22no—
410C22×D3—22no—
411C22×D3—22no—
412C22×D3—22no—
413C22×D3—48no—
414C22×D3—48no—
415C22×D3—44no—
416C22×D3—44no—
417C22×D32−1+414yes—
418C3×Q8—82no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—22no—
425C2×C3×C4—24no—
426C2×C3×C4—24no—
427C2×C3×C4—24no—
428C2×C3×C4—24no—
429C2×C3×C4—24no—
430C2×C3×C4—24no—
431C2×C3×C4—22no—
432C2×C3×C4—22no—
433C2×C3×C4—44no—
434C2×C3×C4—44no—
435C2×C3×C4—44no—
436C2×C3×C4C22×D412yes—
437C2×C3×C4C22×D412yes—
438C2×Dic3—28no—
439C2×Dic3—28no—
440C2×Dic3—22no—
441C2×Dic3—24no—
442C2×Dic3—22no—
443C2×Dic3—22no—
444C2×Dic3—22no—
445C2×Dic3—22no—
446C2×Dic3—22no—
447C2×Dic3—24no—
448C2×Dic3—24no—
449C2×Dic3—24no—
450C2×Dic3—24no—
451C2×Dic3—24no—
452C2×Dic3—44no—
453C2×Dic3C22×D414yes—
454C2×Dic3C22×D414yes—
455C2×Dic3H(5,𝔽2)14yes—
456C4×D3—48no—
457C4×D3—48no—
458Dic6—416no—
459Dic6—84no—
460Dic6—82no—
461C24—32no—
462C24—64no—
463C24—62no—
464C24—62no—
465C24—62no—
466C24—122no—
467C24C23×D311yescharacteristic
468C2×Q8—22no—
469C2×Q8—42no—
470C2×Q8—68no—
471C2×Q8—68no—
472C2×Q8—64no—
473C2×Q8—62no—
474C2×Q8—124no—
475C2×Q8—122no—
476C2×Q8—124no—
477C2×Q8—1216no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—34no—
483C22×C4—32no—
484C22×C4—32no—
485C22×C4—32no—
486C22×C4—34no—
487C22×C4—34no—
488C22×C4—32no—
489C22×C4—32no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—64no—
493C22×C4—64no—
494C22×C4—64no—
495C22×C4—62no—
496C22×C4—62no—
497C22×C4—62no—
498C22×C4—64no—
499C22×C4—64no—
500C22×C4—122no—
501C22×C4—122no—
502C22×C4—1216no—
503C22×C4—122no—
504C22×C4D3×D414yes—
505C22×C4C23×D314yes—
506C22×C4C23×D312yes—
507C22×C4C23×D312yes—
508C22×C4C23×D312yes—
509C22×C4C23×D312yes—
510C22×C4C23×D311yescharacteristic
511C22×C4C23×D311yescharacteristic
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—38no—
517C4⋊C4—38no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—68no—
521C4⋊C4—64no—
522C4⋊C4—64no—
523C4⋊C4—64no—
524C4⋊C4—64no—
525C4⋊C4—64no—
526C4⋊C4—68no—
527C4⋊C4—68no—
528C4⋊C4—62no—
529C4⋊C4—64no—
530C4⋊C4—122no—
531C4⋊C4—124no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—68no—
535C22⋊C4—68no—
536C22⋊C4—68no—
537C22⋊C4—68no—
538C22⋊C4—64no—
539C22⋊C4—64no—
540C22⋊C4—124no—
541C22⋊C4—124no—
542C42—64no—
543C42—64no—
544C42—124no—
545C22×C3—22no—
546C22×C3—22no—
547C22×C3—24no—
548C22×C3—24no—
549C22×C3#53614yes—
550C22×C3C23×D411yescharacteristic
551C22×C3C2×H(5,𝔽2)12yes—
552C22×C3C2×2−1+411yescharacteristic
553C22×C3C2×2−1+411yescharacteristic
554C22×C3C2×2−1+412yes—
555C2×D3—24no—
556C2×D3—24no—
557C2×D3—24no—
558C2×D3—416no—
559C2×D3—42no—
560C2×D3—42no—
561C2×D3—42no—
562C2×D3—42no—
563C3×C4—22no—
564C3×C4—22no—
565C3×C4—42no—
566C3×C4—42no—
567C3×C4—42no—
568C3×C4—42no—
569C3×C4—44no—
570Dic3—24no—
571Dic3—24no—
572Dic3—24no—
573Dic3—42no—
574Dic3—42no—
575Dic3—44no—
576C23—22no—
577C23—22no—
578C23—34no—
579C23—68no—
580C23—62no—
581C23—62no—
582C23—62no—
583C23—62no—
584C23—62no—
585C23—62no—
586C23—128no—
587C23—128no—
588C23—124no—
589C23—124no—
590C23C2×D3×D412yes—
591C23#100712yes—
592C23#101012yes—
593C23C24×D311yescharacteristic
594Q8—82no—
595Q8—1216no—
596Q8—242no—
597Q8—244no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—34no—
613C2×C4—34no—
614C2×C4—34no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—68no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—64no—
623C2×C4—62no—
624C2×C4—68no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—64no—
628C2×C4—64no—
629C2×C4—64no—
630C2×C4—64no—
631C2×C4—64no—
632C2×C4—128no—
633C2×C4—128no—
634C2×C4—124no—
635C2×C4C2×D3×D412yes—
636C2×C4C2×D3×D412yes—
637C2×C3—24no—
638C2×C3C2×#53611yescharacteristic
639C2×C3#348411yescharacteristic
640C2×C3#348611yescharacteristic
641C2×C3#353812yes—
642C2×C3#353812yes—
643D3—44no—
644C22—22no—
645C22—24no—
646C22—22no—
647C22—24no—
648C22—64no—
649C22—64no—
650C22—64no—
651C22—1216no—
652C22—122no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22#562814yes—
657C22C22×D3×D411yescharacteristic
658C22C2×#100711yescharacteristic
659C22C2×#101011yescharacteristic
660C22D3×H(5,𝔽2)12yes—
661C22D3×2−1+412yes—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—64no—
670C4—64no—
671C4—64no—
672C4—122no—
673C4—124no—
674C4—122no—
675C3#3670811yescharacteristic, complemented, Sylow
676C2—24no—
677C2—124no—
678C2C2×#562811yescharacteristic
679C2#8630811yescharacteristic
680C2#8635511yescharacteristic
681C2#8643512yes—
682C2#8658812yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2627C2
prime radicalC24×D3C23
hypercenter#2627D3
hypocenterC3#36708
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2627

Complements

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

NH ≅ G/Ntype
1. C2×#5623C2semidirect
3. C3×#2627C2semidirect
5. #82183C2semidirect
6. #78368C2semidirect
7. #77974C2semidirect
8. #77968C2semidirect
9. #77261C2semidirect
11. #74412C2semidirect
13. #74073C2semidirect
15. #73867C2semidirect
22. C2×#944C22semidirect
33. C2×#889C22semidirect
34. C2×#889C22semidirect
35. C2×#888C22semidirect
36. C2×#888C22semidirect
39. C3×#400C22semidirect
40. C3×#398C22semidirect
41. C3×#395C22semidirect
53. #4986C22semidirect
54. #4982C22semidirect
55. #4979C22semidirect
56. #4979C22semidirect
57. #4978C22semidirect
71. #4658C22semidirect
72. #4657C22semidirect
73. #4656C22semidirect
74. #4655C22semidirect
75. #4654C22semidirect
76. #4652C22semidirect
77. #4646C22semidirect
87. #2627D3semidirect
198. #400C2×D3semidirect
202. #398C2×D3semidirect
208. #395C2×D3semidirect
675. C3#36708semidirect

Maximal subgroups

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

subgroupindex
C2×#56232
C22×#8962
C3×#26272
#828342
#821832
#783682
#779742
#779682
#772612
#746142
#744122
#741322
#740732
#740492
#738672
#367083