Finite Group Databasebeta

← #10968221

Subgroups — #10968221

Order 768.

2650 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#946C211yesmaximal, characteristic
2C2×D3×C22⋊Q8C211yesmaximal, characteristic
3C3×#2434C211yesmaximal, characteristic, complemented
4#84065C211yesmaximal, characteristic, complemented
5#83269C212yesmaximal, complemented
6#83232C212yesmaximal, complemented
7#82775C211yesmaximal, characteristic, complemented
8#77258C214yesmaximal, complemented
9#77255C214yesmaximal, complemented
10#76664C212yesmaximal, complemented
11#76519C212yesmaximal, complemented
12#74591C214yesmaximal
13#74489C214yesmaximal, complemented
14C4×D3×C22⋊C4C211yesmaximal, characteristic
15#73517C211yesmaximal, characteristic, complemented
16#35011—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C22×D3×Q8C2211yescharacteristic
19C23×C4×D3C2211yescharacteristic
20C2×C3×C4⋊Q8C2212yescomplemented
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×#946—28no—
23C2×#946C2212yes—
24C2×#946C2211yescharacteristic
25C2×#946C2212yes—
26C2×#946C2212yes—
27C2×#946C2212yes—
28C2×#944C2212yescomplemented
29C22×(C2×D3)⋊C4C2212yes—
30C22×C4⋊Dic3C2211yescharacteristic
31C2×#924C2211yescharacteristic
32C22×Dic3⋊C4C2212yes—
33D3×C22⋊Q8—28no—
34C2×#897C2211yescharacteristic
35C2×#896C2212yes—
36C2×D3×C4⋊C4C2212yes—
37C2×D3×C4⋊C4C2211yescharacteristic
38C2×D3×C22⋊C4C2212yes—
39C2×D3×C22⋊C4C2212yes—
40C2×#878C2212yes—
41C2×C42×D3C2212yes—
42C3×#395C2214yescomplemented
43C3×#391C2212yescomplemented
44C3×#385C2214yes—
45C3×C4×C22⋊C4C2211yescharacteristic
46#5230C2214yescomplemented
47#5229C2212yescomplemented
48#5229C2212yescomplemented
49#5210C2214yes—
50C4×#217C2211yescharacteristic
51#4989C2218yes—
52#4987C2218yes—
53#4981C2214yescomplemented
54#4979C2218yescomplemented
55#4975C2212yescomplemented
56#4972C2218yescomplemented
57#4950C2214yes—
58#4946C2214yes—
59Dic3×C22⋊C4C2212yes—
60C4×(C2×D3)⋊C4C2212yes—
61#4935C2212yescomplemented
62#4929C2212yescomplemented
63#4668C2218yes—
64#4667C2218yes—
65#4666C2212yes—
66D3×#95C2212yes—
67#4661C2214yescomplemented
68#4657C2218yescomplemented
69#4647C2218yescomplemented
70C22×C22⋊Q8—31no—
71C22×C22⋊Q8—31no—
72#2857—32no—
73#2857—32no—
74#2659—34no—
75#2583—34no—
76#2583—34no—
77#2502—32no—
78#2502—31no—
79#2502—31no—
80#2434—31no—
81#2434—32no—
82#2434D311yescharacteristic, complemented
83#2315—34no—
84C2×C4×C22⋊C4—31no—
85C24×D3C2311yescharacteristic
86C22×C3×Q8—24no—
87C22×C3×Q8—44no—
88C22×C3×Q8C2311yescharacteristic
89C22×C3×Q8C2312yes—
90C22×C3×Q8C2312yes—
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2312yes—
93C23×Dic3C2312yes—
94C23×Dic3C2311yescharacteristic
95C2×D3×Q8—24no—
96C2×D3×Q8—44no—
97C22×C4×D3—44no—
98C22×C4×D3D418yes—
99C22×C4×D3C2311yescharacteristic
100C22×C4×D3C2311yescharacteristic
101C22×C4×D3C2312yes—
102C22×C4×D3C2312yes—
103C22×C4×D3C2311yescharacteristic
104C22×C4×D3C2311yescharacteristic
105C22×C4×D3C2314yes—
106C22×Dic6C2311yescharacteristic
107C22×Dic6C2312yes—
108C3×C4⋊Q8—44no—
109C3×C22⋊Q8—24no—
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C4⋊C4C2318yes—
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C22⋊C4C2312yes—
115C2×C3×C42C2312yes—
116C2×#217C2312yes—
117C2×#217C2312yes—
118#946—28no—
119#946—28no—
120#946—28no—
121#946—28no—
122#946—24no—
123#944—44no—
124C2×(C2×D3)⋊C4—28no—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×C4⋊Dic3—24no—
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2318yes—
136#924—24no—
137C2×Dic3⋊C4—28no—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2312yes—
143C2×Dic3⋊C4C2318yes—
144C2×Dic3⋊C4C2318yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2314yes—
147#897—24no—
148#896—28no—
149D3×C4⋊C4—28no—
150D3×C4⋊C4—24no—
151D3×C22⋊C4—28no—
152D3×C22⋊C4Q8116yescomplemented
153#878—28no—
154C42×D3—44no—
155C3×#95C2318yes—
156C3×#95C2312yes—
157C3⋊#95C2318yes—
158C3⋊#95C2314yes—
159C3⋊#95C2318yes—
160C3⋊#95C2312yes—
161C3⋊#95C2318yes—
162C23×Q8—31no—
163C23×Q8C2×D311yescharacteristic
164C24×C4—31no—
165C2×C4⋊Q8—32no—
166C2×C4⋊Q8C2×D312yescomplemented
167C2×C22⋊Q8—32no—
168C2×C22⋊Q8—32no—
169C2×C22⋊Q8—31no—
170C2×C22⋊Q8—32no—
171C2×C22⋊Q8—32no—
172C2×C22⋊Q8—31no—
173C2×C22⋊Q8—31no—
174C2×C22⋊Q8—32no—
175C2×C22⋊Q8—32no—
176C2×C22⋊Q8—68no—
177C2×C22⋊Q8—68no—
178C2×C22⋊Q8C2×D311yescharacteristic
179C22×C4⋊C4—32no—
180C22×C4⋊C4—31no—
181C22×C4⋊C4—32no—
182C22×C4⋊C4—31no—
183C22×C22⋊C4—32no—
184C22×C22⋊C4—32no—
185C22×C22⋊C4—32no—
186C22×C42—32no—
187#398—34no—
188#398—38no—
189#398—34no—
190#395—34no—
191#395—38no—
192#395C2×D314yescomplemented
193#391—32no—
194#391—32no—
195#391—32no—
196#391C2×D312yescomplemented
197#389—38no—
198#389—38no—
199#385—34no—
200#385—38no—
201#385—34no—
202#385—34no—
203#385—38no—
204#385C2×D314yes—
205#384—32no—
206#384—32no—
207#380—38no—
208#380—38no—
209C4×C22⋊C4—31no—
210C4×C22⋊C4—32no—
211C4×C22⋊C4—32no—
212C4×C22⋊C4—32no—
213C4×C22⋊C4C2×D311yescharacteristic
214C2×#95—32no—
215C24×C3C2411yescharacteristic
216C23×D3—42no—
217C23×D3—42no—
218C23×D3C2×D414yes—
219C23×D3C2×Q818yes—
220C23×D3C2411yescharacteristic
221C23×D3C2411yescharacteristic
222C2×C3×Q8—24no—
223C2×C3×Q8—24no—
224C2×C3×Q8—22no—
225C2×C3×Q8—24no—
226C2×C3×Q8—24no—
227C2×C3×Q8—24no—
228C2×C3×Q8—44no—
229C2×C3×Q8—416no—
230C22×C3×C4—24no—
231C22×C3×C4—24no—
232C22×C3×C4—44no—
233C22×C3×C4C2×D414yes—
234C22×C3×C4C2412yes—
235C22×C3×C4C2414yes—
236C22×C3×C4C2412yes—
237C22×C3×C4C2411yescharacteristic
238C22×C3×C4C2414yes—
239C22×C3×C4C2411yescharacteristic
240C22×Dic3—24no—
241C22×Dic3—42no—
242C22×Dic3—44no—
243C22×Dic3C2×D414yes—
244C22×Dic3C2412yes—
245C22×Dic3C2414yes—
246C22×Dic3C2412yes—
247C22×Dic3C2411yescharacteristic
248C22×Dic3C2411yescharacteristic
249C22×Dic3C2414yes—
250D3×Q8—416no—
251C2×C4×D3—24no—
252C2×C4×D3—24no—
253C2×C4×D3—24no—
254C2×C4×D3—24no—
255C2×C4×D3—416no—
256C2×C4×D3—44no—
257C2×C4×D3—42no—
258C2×C4×D3—42no—
259C2×C4×D3—44no—
260C2×C4×D3—44no—
261C2×C4×D3—44no—
262C2×C4×D3C2×D414yes—
263C2×C4×D3C2×D414yes—
264C2×C4×D3C2×D414yes—
265C2×C4×D3C2×D414yes—
266C2×C4×D3C2×Q8116yescomplemented
267C2×Dic6—22no—
268C2×Dic6—24no—
269C2×Dic6—24no—
270C2×Dic6—44no—
271C3×C4⋊C4—24no—
272C3×C4⋊C4—22no—
273C3×C4⋊C4—48no—
274C3×C22⋊C4—24no—
275C3×C22⋊C4C2×Q818yescomplemented
276C3×C42—24no—
277#217—24no—
278#217C2×Q818yescomplemented
279(C2×D3)⋊C4—24no—
280(C2×D3)⋊C4—24no—
281(C2×D3)⋊C4—28no—
282(C2×D3)⋊C4—28no—
283(C2×D3)⋊C4C2×Q8116yescomplemented
284C4⋊Dic3—22no—
285C4⋊Dic3—24no—
286C4⋊Dic3—24no—
287C4⋊Dic3—24no—
288C4⋊Dic3—48no—
289Dic3⋊C4—24no—
290Dic3⋊C4—24no—
291Dic3⋊C4—24no—
292Dic3⋊C4—28no—
293Dic3⋊C4—28no—
294Dic3⋊C4—48no—
295Dic3⋊C4—48no—
296C4×Dic3—24no—
297C4×Dic3—44no—
298C25—31no—
299C22×Q8—24no—
300C22×Q8—31no—
301C22×Q8—32no—
302C22×Q8—44no—
303C22×Q8—64no—
304C22×Q8—124no—
305C22×Q8C22×D312yes—
306C22×Q8C22×D311yescharacteristic
307C22×Q8C22×D312yes—
308C23×C4—31no—
309C23×C4—31no—
310C23×C4—31no—
311C23×C4—38no—
312C23×C4—34no—
313C23×C4—31no—
314C23×C4—31no—
315C23×C4—32no—
316C23×C4—32no—
317C23×C4—32no—
318C23×C4—124no—
319C23×C4C22×D311yescharacteristic
320C23×C4C22×D312yes—
321C4⋊Q8—44no—
322C4⋊Q8—124no—
323C22⋊Q8—24no—
324C22⋊Q8—68no—
325C22⋊Q8—68no—
326C22⋊Q8—64no—
327C22⋊Q8—68no—
328C22⋊Q8—64no—
329C22⋊Q8—68no—
330C22⋊Q8—68no—
331C22⋊Q8—68no—
332C22⋊Q8—64no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—34no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—34no—
339C2×C4⋊C4—32no—
340C2×C4⋊C4—32no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—38no—
343C2×C4⋊C4—38no—
344C2×C4⋊C4—38no—
345C2×C4⋊C4—64no—
346C2×C4⋊C4—68no—
347C2×C4⋊C4—68no—
348C2×C4⋊C4—64no—
349C2×C4⋊C4C22×D311yescharacteristic
350C2×C4⋊C4C22×D312yes—
351C2×C4⋊C4C22×D318yes—
352C2×C22⋊C4—32no—
353C2×C22⋊C4—34no—
354C2×C22⋊C4—316no—
355C2×C22⋊C4—32no—
356C2×C22⋊C4—32no—
357C2×C22⋊C4—34no—
358C2×C22⋊C4—32no—
359C2×C22⋊C4—34no—
360C2×C22⋊C4—68no—
361C2×C22⋊C4—68no—
362C2×C22⋊C4C22×D312yes—
363C2×C22⋊C4C22×D312yes—
364C2×C42—32no—
365C2×C42—34no—
366C2×C42—124no—
367C2×C42C22×D312yes—
368#95—32no—
369#95—34no—
370#95—38no—
371#95—38no—
372#95—38no—
373#95C22×D312yes—
374#95C22×D318yes—
375C23×C3—22no—
376C23×C3—22no—
377C23×C3C22×D412yes—
378C23×C3C22×Q814yes—
379C23×C3C2511yescharacteristic
380C22×D3—22no—
381C22×D3—22no—
382C22×D3—22no—
383C22×D3—22no—
384C22×D3—44no—
385C22×D3—44no—
386C22×D3—416no—
387C22×D3C4⋊Q8116yescomplemented
388C22×D3C22×D412yes—
389C22×D3C22×D412yes—
390C22×D3C22×Q814yes—
391C22×D3C22×Q814yes—
392C3×Q8—48no—
393C3×Q8—44no—
394C3×Q8—48no—
395C2×C3×C4—24no—
396C2×C3×C4—22no—
397C2×C3×C4—24no—
398C2×C3×C4—28no—
399C2×C3×C4—24no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—24no—
403C2×C3×C4—24no—
404C2×C3×C4—24no—
405C2×C3×C4—24no—
406C2×C3×C4—48no—
407C2×C3×C4C22×D412yes—
408C2×C3×C4C22×D412yes—
409C2×C3×C4C22×Q818yes—
410C2×C3×C4H(5,𝔽2)14yes—
411C2×Dic3—24no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—24no—
415C2×Dic3—22no—
416C2×Dic3—24no—
417C2×Dic3—22no—
418C2×Dic3—24no—
419C2×Dic3—28no—
420C2×Dic3—24no—
421C2×Dic3—44no—
422C2×Dic3—48no—
423C2×Dic3C22×D412yes—
424C2×Dic3C22×D412yes—
425C2×Dic3C22×Q818yes—
426C4×D3—28no—
427C4×D3—28no—
428C4×D3—44no—
429C4×D3—44no—
430C4×D3—416no—
431C4×D3—48no—
432C4×D3—48no—
433Dic6—48no—
434Dic6—44no—
435C24—31no—
436C24—31no—
437C24—34no—
438C24—38no—
439C24—122no—
440C24—122no—
441C24C23×D311yescharacteristic
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8—44no—
449C2×Q8—416no—
450C2×Q8—64no—
451C2×Q8—62no—
452C2×Q8—64no—
453C2×Q8—124no—
454C2×Q8—1216no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—34no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—34no—
461C22×C4—34no—
462C22×C4—34no—
463C22×C4—34no—
464C22×C4—34no—
465C22×C4—316no—
466C22×C4—32no—
467C22×C4—34no—
468C22×C4—32no—
469C22×C4—44no—
470C22×C4—64no—
471C22×C4—64no—
472C22×C4—64no—
473C22×C4—64no—
474C22×C4—64no—
475C22×C4—1216no—
476C22×C4—124no—
477C22×C4—122no—
478C22×C4—122no—
479C22×C4—124no—
480C22×C4—124no—
481C22×C4—124no—
482C22×C4—122no—
483C22×C4—124no—
484C22×C4D3×D414yes—
485C22×C4C23×D312yes—
486C22×C4C23×D314yes—
487C22×C4C23×D312yes—
488C22×C4C23×D311yescharacteristic
489C22×C4C23×D314yes—
490C22×C4C23×D311yescharacteristic
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—48no—
494C4⋊C4—64no—
495C4⋊C4—64no—
496C4⋊C4—68no—
497C4⋊C4—68no—
498C4⋊C4—64no—
499C4⋊C4—64no—
500C4⋊C4—64no—
501C4⋊C4—62no—
502C4⋊C4—64no—
503C4⋊C4—128no—
504C4⋊C4—128no—
505C4⋊C4—128no—
506C22⋊C4—24no—
507C22⋊C4—38no—
508C22⋊C4—316no—
509C22⋊C4—64no—
510C22⋊C4—64no—
511C22⋊C4—68no—
512C22⋊C4—68no—
513C22⋊C4—64no—
514C22⋊C4D3×Q818yescomplemented
515C42—24no—
516C42—64no—
517C42—124no—
518C22×C3—22no—
519C22×C3—22no—
520C22×C3—28no—
521C22×C3C2×C4⋊Q814yes—
522C22×C3C23×D411yescharacteristic
523C22×C3C23×Q812yes—
524C22×C3C2×H(5,𝔽2)12yes—
525C22×C3#58512yes—
526C2×D3—22no—
527C2×D3—22no—
528C2×D3—22no—
529C2×D3—22no—
530C2×D3—28no—
531C2×D3—28no—
532C2×D3—416no—
533C2×D3C2×C4⋊Q814yes—
534C2×D3C2×C4⋊Q814yes—
535C3×C4—22no—
536C3×C4—28no—
537C3×C4—42no—
538C3×C4—44no—
539C3×C4—42no—
540C3×C4#53414yes—
541Dic3—22no—
542Dic3—22no—
543Dic3—28no—
544Dic3—42no—
545Dic3—42no—
546Dic3—44no—
547C23—22no—
548C23—22no—
549C23—32no—
550C23—32no—
551C23—34no—
552C23—34no—
553C23—316no—
554C23—62no—
555C23—62no—
556C23—62no—
557C23—62no—
558C23—124no—
559C23—124no—
560C23—1216no—
561C23C2×D3×D412yes—
562C23C2×D3×Q814yes—
563C23C24×D311yescharacteristic
564Q8—48no—
565Q8—48no—
566Q8—44no—
567Q8—128no—
568Q8—124no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—32no—
581C2×C4—32no—
582C2×C4—38no—
583C2×C4—48no—
584C2×C4—64no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—64no—
588C2×C4—68no—
589C2×C4—68no—
590C2×C4—64no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—64no—
594C2×C4—68no—
595C2×C4—64no—
596C2×C4—124no—
597C2×C4—124no—
598C2×C4—1216no—
599C2×C4—128no—
600C2×C4—128no—
601C2×C4—124no—
602C2×C4—128no—
603C2×C4C2×D3×D412yes—
604C2×C4C2×D3×D412yes—
605C2×C4#100414yes—
606C2×C4C2×D3×Q818yes—
607C2×C3—24no—
608C2×C3C22×C4⋊Q811yescharacteristic
609C2×C3C2×#53411yescharacteristic
610C2×C3#344911yescharacteristic
611C2×C3#350114yes—
612D3—24no—
613D3—24no—
614C22—22no—
615C22—22no—
616C22—28no—
617C22—34no—
618C22—34no—
619C22—62no—
620C22—62no—
621C22—62no—
622C22—62no—
623C22—68no—
624C22—68no—
625C22—1216no—
626C22D3×C4⋊Q814yes—
627C22C22×D3×D411yescharacteristic
628C22C2×#100412yes—
629C22C22×D3×Q812yes—
630C22#596812yes—
631C4—22no—
632C4—28no—
633C4—42no—
634C4—44no—
635C4—42no—
636C4—62no—
637C4—62no—
638C4—68no—
639C4—122no—
640C4—124no—
641C4—122no—
642C4#558914yes—
643C3#3501111yescharacteristic, complemented, Sylow
644C2—24no—
645C2—64no—
646C2—64no—
647C2C2×#558911yescharacteristic
648C2C2×D3×C4⋊Q811yescharacteristic
649C2#8633811yescharacteristic
650C2#8676114yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2434C2
prime radicalC24×D3C23
hypercenter#2434D3
hypocenterC3#35011
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2434

Complements

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

NH ≅ G/Ntype
3. C3×#2434C2semidirect
4. #84065C2semidirect
5. #83269C2semidirect
6. #83232C2semidirect
7. #82775C2semidirect
8. #77258C2semidirect
9. #77255C2semidirect
10. #76664C2semidirect
11. #76519C2semidirect
13. #74489C2semidirect
15. #73517C2semidirect
20. C2×C3×C4⋊Q8C22semidirect
28. C2×#944C22semidirect
42. C3×#395C22semidirect
43. C3×#391C22semidirect
46. #5230C22semidirect
47. #5229C22semidirect
48. #5229C22semidirect
53. #4981C22semidirect
54. #4979C22semidirect
55. #4975C22semidirect
56. #4972C22semidirect
61. #4935C22semidirect
62. #4929C22semidirect
67. #4661C22semidirect
68. #4657C22semidirect
69. #4647C22semidirect
82. #2434D3semidirect
152. D3×C22⋊C4Q8semidirect
166. C2×C4⋊Q8C2×D3semidirect
192. #395C2×D3semidirect
196. #391C2×D3semidirect
266. C2×C4×D3C2×Q8semidirect
275. C3×C22⋊C4C2×Q8semidirect
278. #217C2×Q8semidirect
283. (C2×D3)⋊C4C2×Q8semidirect
387. C22×D3C4⋊Q8semidirect
514. C22⋊C4D3×Q8semidirect
643. C3#35011semidirect

Maximal subgroups

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

subgroupindex
C22×#9462
C2×D3×C22⋊Q82
C3×#24342
#840652
#832692
#832322
#827752
#772582
#772552
#766642
#765192
#745912
#744892
C4×D3×C22⋊C42
#735172
#350113