Finite Group Databasebeta

← #10975279

Subgroups — #10975279

Order 768.

2570 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5634C211yesmaximal, characteristic
2C22×#896C211yesmaximal, characteristic
3C3×#2502C211yesmaximal, characteristic, complemented
4#82785C211yesmaximal, characteristic, complemented
5#82178C211yesmaximal, characteristic, complemented
6#77958C214yesmaximal, complemented
7#77745C214yesmaximal, complemented
8#76637C214yesmaximal, complemented
9#74591C212yesmaximal
10D3×#385C212yesmaximal
11#74489C212yesmaximal, complemented
12#74412C212yesmaximal, complemented
13#74326C214yesmaximal, complemented
14#73560C211yesmaximal, characteristic
15#73547C211yesmaximal, characteristic, complemented
16#35391—31nomaximal, Sylow
17C23×C4×D3C2211yescharacteristic
18C23×Dic6C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C2×#946C2212yes—
22C22×(C2×D3)⋊C4C2211yescharacteristic
23C22×(C2×D3)⋊C4C2211yescharacteristic
24C2×#924C2211yescharacteristic
25C22×Dic3⋊C4C2211yescharacteristic
26C22×Dic3⋊C4C2211yescharacteristic
27#5634—28no—
28C2×#897C2212yes—
29C2×#896—28no—
30C2×#896C2212yes—
31C2×#896C2214yes—
32C2×#896C2212yes—
33C2×#896C2214yes—
34C2×D3×C4⋊C4C2214yes—
35C2×#888C2214yescomplemented
36C2×D3×C22⋊C4C2212yes—
37C2×D3×C22⋊C4C2212yes—
38C2×#878C2212yes—
39C3×#398C2214yescomplemented
40C3×#391C2212yescomplemented
41C3×#385C2212yes—
42C3×#385C2212yes—
43C3×#384C2212yescomplemented
44C3×C4×C22⋊C4C2211yescharacteristic
45#5230C2212yescomplemented
46#5210C2212yes—
47#5210C2212yes—
48C4×#217C2211yescharacteristic
49#4989C2214yes—
50#4989C2214yes—
51#4987C2214yes—
52#4983C2214yescomplemented
53#4983C2212yescomplemented
54#4979C2214yescomplemented
55#4979C2214yescomplemented
56#4976C2214yescomplemented
57#4975C2212yescomplemented
58#4973C2212yescomplemented
59#4972C2214yescomplemented
60#4950C2212yes—
61#4950C2212yes—
62#4946C2212yes—
63#4946C2212yes—
64Dic3×C22⋊C4C2212yes—
65C4×(C2×D3)⋊C4C2214yes—
66#4928C2214yescomplemented
67#4668C2214yes—
68#4668C2214yes—
69#4667C2214yes—
70#4666C2214yes—
71D3×#95C2214yes—
72#4661C2212yescomplemented
73#4657C2214yescomplemented
74#4656C2212yescomplemented
75#4656C2214yescomplemented
76#4647C2214yescomplemented
77#4646C2214yescomplemented
78C2×#543—31no—
79C22×C22⋊Q8—31no—
80#2583—32no—
81#2583—31no—
82#2583—31no—
83#2549—34no—
84#2549—34no—
85#2502—31no—
86#2502D311yescharacteristic, complemented
87#2440—34no—
88#2440—34no—
89#2434—32no—
90#2315—32no—
91C2×#385—32no—
92#2273—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×D3C2314yes—
104C22×C4×D3C2312yes—
105C22×C4×D3C2312yes—
106C22×C4×D3C2314yes—
107C22×Dic6—24no—
108C22×Dic6—44no—
109C22×Dic6C2314yes—
110C22×Dic6C2311yescharacteristic
111C22×Dic6C2312yes—
112C3×C22⋊Q8—24no—
113C2×C3×C4⋊C4—24no—
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C4⋊C4C2311yescharacteristic
117C2×C3×C4⋊C4C2312yes—
118C2×C3×C4⋊C4C2314yes—
119C2×C3×C4⋊C4C2314yes—
120C2×C3×C22⋊C4C2312yes—
121C2×C3×C22⋊C4C2312yes—
122C2×C3×C42C2312yes—
123C2×#217C2312yes—
124C2×#217C2312yes—
125#946—28no—
126C2×(C2×D3)⋊C4—24no—
127C2×(C2×D3)⋊C4D418yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×(C2×D3)⋊C4C2314yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2314yes—
136C2×C4⋊Dic3C2311yescharacteristic
137C2×C4⋊Dic3C2314yes—
138#924—24no—
139C2×Dic3⋊C4—24no—
140C2×Dic3⋊C4—24no—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2312yes—
144C2×Dic3⋊C4C2312yes—
145C2×Dic3⋊C4C2312yes—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2314yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2314yes—
151C2×C4×Dic3C2314yes—
152C2×C4×Dic3C2312yes—
153#897—28no—
154#896—28no—
155#896—28no—
156#896—216no—
157#896—216no—
158D3×C4⋊C4—48no—
159#888—48no—
160D3×C22⋊C4—28no—
161D3×C22⋊C4Q8116yescomplemented
162#878—28no—
163C3×#95C2314yes—
164C3×#95C2314yes—
165C3×#95C2312yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2314yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C3⋊#95C2312yes—
173C3⋊#95C2314yes—
174C23×Q8—31no—
175C24×C4—31no—
176#543—68no—
177C2×C4⋊Q8—34no—
178C2×C22⋊Q8—32no—
179C2×C22⋊Q8—31no—
180C2×C22⋊Q8—34no—
181C2×C22⋊Q8—34no—
182C2×C22⋊Q8—32no—
183C2×C22⋊Q8—32no—
184C2×C22⋊Q8—32no—
185C2×C22⋊Q8—32no—
186C2×C22⋊Q8—68no—
187C2×C22⋊Q8C2×D311yescharacteristic
188C22×C4⋊C4—31no—
189C22×C4⋊C4—31no—
190C22×C4⋊C4—34no—
191C22×C4⋊C4C2×D311yescharacteristic
192C22×C22⋊C4—31no—
193C22×C22⋊C4—32no—
194C22×C22⋊C4—31no—
195C22×C22⋊C4—32no—
196#398—32no—
197#398—34no—
198#398—34no—
199#398—32no—
200#398C2×D314yescomplemented
201#395—32no—
202#395—34no—
203#395—34no—
204#395—32no—
205#395—34no—
206#391—32no—
207#391—34no—
208#391C2×D312yescomplemented
209#389—34no—
210#389—34no—
211#389—34no—
212#389—34no—
213#385—32no—
214#385—32no—
215#385—34no—
216#385—34no—
217#385—32no—
218#385—32no—
219#385—32no—
220#385—32no—
221#385—34no—
222#385—34no—
223#385C2×D312yes—
224#385C2×D312yes—
225#384—32no—
226#384C2×D312yescomplemented
227#380—34no—
228#380—34no—
229C4×C22⋊C4—31no—
230C4×C22⋊C4—32no—
231C4×C22⋊C4—34no—
232C4×C22⋊C4—34no—
233C4×C22⋊C4C2×D311yescharacteristic
234C2×#95—34no—
235C24×C3C2411yescharacteristic
236C23×D3—42no—
237C23×D3—42no—
238C23×D3C2×D414yes—
239C23×D3C2×Q818yes—
240C23×D3C2412yes—
241C2×C3×Q8—22no—
242C2×C3×Q8—42no—
243C22×C3×C4—22no—
244C22×C3×C4—22no—
245C22×C3×C4—42no—
246C22×C3×C4—42no—
247C22×C3×C4—42no—
248C22×C3×C4C2×D414yes—
249C22×C3×C4C2412yes—
250C22×C3×C4C2412yes—
251C22×C3×C4C2412yes—
252C22×C3×C4C2412yes—
253C22×C3×C4C2411yescharacteristic
254C22×C3×C4C2411yescharacteristic
255C22×C3×C4C2414yes—
256C22×Dic3—22no—
257C22×Dic3—24no—
258C22×Dic3—22no—
259C22×Dic3—42no—
260C22×Dic3—42no—
261C22×Dic3C2×D414yes—
262C22×Dic3C2412yes—
263C22×Dic3C2412yes—
264C22×Dic3C2412yes—
265C22×Dic3C2412yes—
266C22×Dic3C2414yes—
267C22×Dic3C2412yes—
268C2×C4×D3—28no—
269C2×C4×D3—28no—
270C2×C4×D3—24no—
271C2×C4×D3—24no—
272C2×C4×D3—44no—
273C2×C4×D3—44no—
274C2×C4×D3—44no—
275C2×C4×D3—416no—
276C2×C4×D3—44no—
277C2×C4×D3—44no—
278C2×C4×D3C2×Q8116yescomplemented
279C2×Dic6—28no—
280C2×Dic6—28no—
281C2×Dic6—22no—
282C2×Dic6—24no—
283C2×Dic6—44no—
284C2×Dic6—416no—
285C2×Dic6—44no—
286C2×Dic6—42no—
287C3×C4⋊C4—24no—
288C3×C4⋊C4—24no—
289C3×C4⋊C4—22no—
290C3×C4⋊C4—24no—
291C3×C4⋊C4—44no—
292C3×C4⋊C4—44no—
293C3×C22⋊C4—24no—
294C3×C22⋊C4C2×Q818yes—
295C3×C42—42no—
296#217—24no—
297#217C2×Q818yes—
298(C2×D3)⋊C4—24no—
299(C2×D3)⋊C4—24no—
300(C2×D3)⋊C4—28no—
301(C2×D3)⋊C4C2×D418yes—
302(C2×D3)⋊C4C2×D418yes—
303(C2×D3)⋊C4C2×Q8116yescomplemented
304C4⋊Dic3—22no—
305C4⋊Dic3—24no—
306C4⋊Dic3—28no—
307C4⋊Dic3—44no—
308Dic3⋊C4—24no—
309Dic3⋊C4—24no—
310Dic3⋊C4—24no—
311Dic3⋊C4—24no—
312Dic3⋊C4—24no—
313Dic3⋊C4—28no—
314Dic3⋊C4—28no—
315Dic3⋊C4—44no—
316Dic3⋊C4—44no—
317Dic3⋊C4—44no—
318C4×Dic3—44no—
319C4×Dic3—42no—
320C25—31no—
321C22×Q8—34no—
322C22×Q8—32no—
323C22×Q8—31no—
324C22×Q8—64no—
325C22×Q8—124no—
326C22×Q8C22×D311yescharacteristic
327C23×C4—31no—
328C23×C4—34no—
329C23×C4—31no—
330C23×C4—31no—
331C23×C4—34no—
332C23×C4—32no—
333C23×C4—32no—
334C23×C4—64no—
335C23×C4—124no—
336C23×C4C22×D311yescharacteristic
337C23×C4C22×D311yescharacteristic
338C23×C4C22×D311yescharacteristic
339C4⋊Q8—128no—
340C22⋊Q8—24no—
341C22⋊Q8—68no—
342C22⋊Q8—616no—
343C22⋊Q8—616no—
344C22⋊Q8—68no—
345C22⋊Q8—68no—
346C22⋊Q8—68no—
347C22⋊Q8—68no—
348C22⋊Q8—64no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—32no—
353C2×C4⋊C4—34no—
354C2×C4⋊C4—34no—
355C2×C4⋊C4—31no—
356C2×C4⋊C4—32no—
357C2×C4⋊C4—32no—
358C2×C4⋊C4—32no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—34no—
361C2×C4⋊C4—34no—
362C2×C4⋊C4—34no—
363C2×C4⋊C4—34no—
364C2×C4⋊C4—64no—
365C2×C4⋊C4—64no—
366C2×C4⋊C4—128no—
367C2×C4⋊C4C22×D312yes—
368C2×C4⋊C4C22×D312yes—
369C2×C4⋊C4C22×D311yescharacteristic
370C2×C4⋊C4C22×D312yes—
371C2×C4⋊C4C22×D314yes—
372C2×C4⋊C4C22×D314yes—
373C2×C22⋊C4—32no—
374C2×C22⋊C4—38no—
375C2×C22⋊C4—32no—
376C2×C22⋊C4—32no—
377C2×C22⋊C4—34no—
378C2×C22⋊C4—316no—
379C2×C22⋊C4—32no—
380C2×C22⋊C4—32no—
381C2×C22⋊C4—34no—
382C2×C22⋊C4—32no—
383C2×C22⋊C4—68no—
384C2×C22⋊C4—64no—
385C2×C22⋊C4C22×D312yes—
386C2×C22⋊C4C22×D312yes—
387C2×C42—34no—
388C2×C42—32no—
389C2×C42C22×D312yes—
390#95—34no—
391#95—34no—
392#95—34no—
393#95—34no—
394#95—32no—
395#95—34no—
396#95—34no—
397#95—34no—
398#95C22×D314yes—
399#95C22×D312yes—
400#95C22×D314yes—
401C23×C3—22no—
402C23×C3—22no—
403C23×C3C22×D412yes—
404C23×C3C22×Q814yes—
405C23×C3C2511yescharacteristic
406C22×D3—24no—
407C22×D3—24no—
408C22×D3—44no—
409C22×D3—44no—
410C22×D3—416no—
411C22×D3C4⋊Q8116yescomplemented
412C22×D3C22×D414yes—
413C22×D3C22×Q814yes—
414C22×D3C22×Q814yes—
415C3×Q8—82no—
416C2×C3×C4—24no—
417C2×C3×C4—22no—
418C2×C3×C4—24no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—24no—
425C2×C3×C4—24no—
426C2×C3×C4—24no—
427C2×C3×C4—24no—
428C2×C3×C4—44no—
429C2×C3×C4—44no—
430C2×C3×C4—44no—
431C2×C3×C4C22×D412yes—
432C2×C3×C4C22×D412yes—
433C2×C3×C4C22×Q818yes—
434C2×Dic3—28no—
435C2×Dic3—22no—
436C2×Dic3—24no—
437C2×Dic3—24no—
438C2×Dic3—22no—
439C2×Dic3—24no—
440C2×Dic3—22no—
441C2×Dic3—22no—
442C2×Dic3—22no—
443C2×Dic3—24no—
444C2×Dic3—24no—
445C2×Dic3—24no—
446C2×Dic3—44no—
447C2×Dic3—44no—
448C2×Dic3C22×D414yes—
449C2×Dic3C22×Q818yes—
450C2×Dic3H(5,𝔽2)14yes—
451C4×D3—416no—
452C4×D3—416no—
453C4×D3—84no—
454C4×D3—84no—
455Dic6—416no—
456Dic6—84no—
457Dic6—82no—
458C24—32no—
459C24—34no—
460C24—38no—
461C24—122no—
462C24—122no—
463C24C23×D311yescharacteristic
464C2×Q8—22no—
465C2×Q8—42no—
466C2×Q8—68no—
467C2×Q8—68no—
468C2×Q8—64no—
469C2×Q8—62no—
470C2×Q8—124no—
471C2×Q8—124no—
472C2×Q8—122no—
473C2×Q8—1216no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—34no—
477C22×C4—32no—
478C22×C4—32no—
479C22×C4—34no—
480C22×C4—316no—
481C22×C4—32no—
482C22×C4—32no—
483C22×C4—32no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—68no—
488C22×C4—68no—
489C22×C4—64no—
490C22×C4—64no—
491C22×C4—64no—
492C22×C4—62no—
493C22×C4—62no—
494C22×C4—124no—
495C22×C4—124no—
496C22×C4—124no—
497C22×C4—124no—
498C22×C4—124no—
499C22×C4—1216no—
500C22×C4—122no—
501C22×C4—122no—
502C22×C4D3×D414yes—
503C22×C4C23×D312yes—
504C22×C4C23×D312yes—
505C22×C4C23×D312yes—
506C22×C4C23×D312yes—
507C22×C4C23×D311yescharacteristic
508C22×C4C23×D311yescharacteristic
509C22×C4C23×D314yes—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—68no—
517C4⋊C4—68no—
518C4⋊C4—68no—
519C4⋊C4—64no—
520C4⋊C4—64no—
521C4⋊C4—64no—
522C4⋊C4—64no—
523C4⋊C4—64no—
524C4⋊C4—64no—
525C4⋊C4—62no—
526C4⋊C4—124no—
527C4⋊C4—124no—
528C4⋊C4—124no—
529C4⋊C4—124no—
530C22⋊C4—24no—
531C22⋊C4—38no—
532C22⋊C4—38no—
533C22⋊C4—38no—
534C22⋊C4—316no—
535C22⋊C4—64no—
536C22⋊C4—64no—
537C22⋊C4—68no—
538C22⋊C4—64no—
539C22⋊C4D3×Q818yes—
540C42—42no—
541C42—124no—
542C42—122no—
543C22×C3—22no—
544C22×C3—22no—
545C22×C3—28no—
546C22×C3C2×C4⋊Q814yes—
547C22×C3C23×D411yescharacteristic
548C22×C3C23×Q812yes—
549C22×C3C2×H(5,𝔽2)12yes—
550C22×C3#58512yes—
551C2×D3—24no—
552C2×D3—24no—
553C2×D3—28no—
554C2×D3—28no—
555C2×D3—416no—
556C2×D3C2×C4⋊Q818yes—
557C3×C4—22no—
558C3×C4—22no—
559C3×C4—28no—
560C3×C4—42no—
561C3×C4—42no—
562C3×C4—42no—
563C3×C4—42no—
564Dic3—24no—
565Dic3—24no—
566Dic3—28no—
567Dic3—42no—
568Dic3—42no—
569Dic3—42no—
570C23—22no—
571C23—22no—
572C23—34no—
573C23—34no—
574C23—34no—
575C23—316no—
576C23—64no—
577C23—64no—
578C23—124no—
579C23—1216no—
580C23—124no—
581C23C2×D3×D412yes—
582C23C2×D3×Q814yes—
583C23C24×D311yescharacteristic
584Q8—82no—
585Q8—1216no—
586Q8—242no—
587Q8—244no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—34no—
601C2×C4—38no—
602C2×C4—34no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—68no—
607C2×C4—62no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—64no—
616C2×C4—64no—
617C2×C4—64no—
618C2×C4—1216no—
619C2×C4—1216no—
620C2×C4—124no—
621C2×C4—124no—
622C2×C4—244no—
623C2×C4—244no—
624C2×C4C2×D3×D412yes—
625C2×C4C2×D3×D412yes—
626C2×C4C2×D3×Q818yes—
627C2×C3—24no—
628C2×C3C22×C4⋊Q811yescharacteristic
629C2×C3#348411yescharacteristic
630C2×C3#348511yescharacteristic
631C2×C3#350114yes—
632D3—28no—
633C22—22no—
634C22—28no—
635C22—22no—
636C22—38no—
637C22—64no—
638C22—68no—
639C22—68no—
640C22—64no—
641C22—1216no—
642C22D3×C4⋊Q814yes—
643C22C22×D3×D411yescharacteristic
644C22C22×D3×Q812yes—
645C22#596412yes—
646C22D3×H(5,𝔽2)12yes—
647C4—28no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—64no—
655C4—64no—
656C4—68no—
657C4—122no—
658C4—122no—
659C4—122no—
660C3#3539111yescharacteristic, complemented, Sylow
661C2—24no—
662C2—68no—
663C2#8601211yescharacteristic
664C2C2×D3×C4⋊Q811yescharacteristic
665C2#8633011yescharacteristic
666C2#8694914yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2502C2
prime radicalC24×D3C23
hypercenter#2502D3
hypocenterC3#35391
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2502

Complements

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

NH ≅ G/Ntype
3. C3×#2502C2semidirect
4. #82785C2semidirect
5. #82178C2semidirect
6. #77958C2semidirect
7. #77745C2semidirect
8. #76637C2semidirect
11. #74489C2semidirect
12. #74412C2semidirect
13. #74326C2semidirect
15. #73547C2semidirect
35. C2×#888C22semidirect
39. C3×#398C22semidirect
40. C3×#391C22semidirect
43. C3×#384C22semidirect
45. #5230C22semidirect
52. #4983C22semidirect
53. #4983C22semidirect
54. #4979C22semidirect
55. #4979C22semidirect
56. #4976C22semidirect
57. #4975C22semidirect
58. #4973C22semidirect
59. #4972C22semidirect
66. #4928C22semidirect
72. #4661C22semidirect
73. #4657C22semidirect
74. #4656C22semidirect
75. #4656C22semidirect
76. #4647C22semidirect
77. #4646C22semidirect
86. #2502D3semidirect
161. D3×C22⋊C4Q8semidirect
200. #398C2×D3semidirect
208. #391C2×D3semidirect
226. #384C2×D3semidirect
278. C2×C4×D3C2×Q8semidirect
303. (C2×D3)⋊C4C2×Q8semidirect
411. C22×D3C4⋊Q8semidirect
660. C3#35391semidirect

Maximal subgroups

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

subgroupindex
C2×#56342
C22×#8962
C3×#25022
#827852
#821782
#779582
#777452
#766372
#745912
D3×#3852
#744892
#744122
#743262
#735602
#735472
#353913