Finite Group Databasebeta

← #30620316

Subgroups — #30620316

Order 15616.

1696 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
1D4×C22⋊#6923C212yesmaximal, complemented
2#21462008C212yesmaximal, complemented
3#21461841C214yesmaximal, complemented
4#21461715C212yesmaximal, complemented
5#21461680C211yesmaximal, characteristic, complemented
6C4×#14519836C211yesmaximal, characteristic, complemented
7#21461516C212yesmaximal, complemented
8C2×#16840405C211yesmaximal, characteristic
9C2×D4×#6923—22no—
10C22×C22⋊#6923C2214yes—
11C2×#14519836C2212yes—
12C2×#14519836C2212yes—
13#16840714—22no—
14#16840710—24no—
15#16840704C2212yes—
16#16840703C2214yes—
17C4×C22⋊#6923C2212yes—
18#16840640C2211yescharacteristic
19#16840640C2214yes—
20#16840640C2212yes—
21#16840640C2211yescharacteristic
22#16840640C2214yes—
23C22×#11758226C411yescharacteristic, complemented
24C22×D4×D61C2211yescharacteristic
25#16840405—22no—
26#16840405—24no—
27#16840405—22no—
28C2×#14519759C412yescomplemented
29C2×#14519753C2211yescharacteristic
30C2×#14519750C412yescomplemented
31C2×#14519748C411yescharacteristic, complemented
32C2×#14519747C2212yes—
33C2×C4×D244C411yescharacteristic, complemented
34C2×C4×D244C2211yescharacteristic
35C2×C22⋊C4×D61C2212yes—
36C2×#124×C61C411yescharacteristic, complemented
37C23×#6923—24no—
38D4×#6923—48no—
39C2×C4⋊#6923—22no—
40C2×C22⋊#6923—28no—
41C2×C22⋊#6923—24no—
42C2×C22⋊#6923—24no—
43C2×C22⋊#6923—24no—
44C2×C22⋊#6923—44no—
45C2×C22⋊#6923C2314yes—
46C2×C22⋊#6923C2312yes—
47C2×C22⋊#6923C2312yes—
48C2×C4×#6923—22no—
49#14519836—24no—
50#14519836—42no—
51(C2×#6923)⋊C4—22no—
52(C2×#6923)⋊C4—22no—
53C24×D61C2312yes—
54C2×#11758226—21no—
55C2×#11758226—22no—
56C2×#11758226—21no—
57C2×#11758226—44no—
58C22×D244C2×C411yescharacteristic
59C22×D244C2×C412yescomplemented
60C22×D244C2311yescharacteristic
61C22×C22⋊D61C2×C412yes—
62C22×C22⋊D61C2×C412yescomplemented
63C2×D4×D61—44no—
64C2×D4×D61D412yes—
65C2×D4×D61D414yes—
66C2×D4×D61D412yes—
67C22×C4×D61C2×C412yescomplemented
68C22×C4×D61C2×C411yescharacteristic, complemented
69C22×C4×D61C2×C412yescomplemented
70C22×C4×D61C2311yescharacteristic
71#14519759—24no—
72#14519759—24no—
73#14519753—22no—
74#14519753—22no—
75#14519750—24no—
76#14519750—24no—
77#14519748—22no—
78#14519748—22no—
79#14519747—24no—
80#14519747—24no—
81C4×D244—22no—
82C4×D244—21no—
83C4×D244—21no—
84C4×D244—22no—
85C4×D244—22no—
86C2×C4⋊Dic61C2×C411yescharacteristic, complemented
87C2×C4⋊Dic61C2×C411yescharacteristic, complemented
88C2×#11758196C2×C412yescomplemented
89C2×C22⋊Dic61C2×C412yescomplemented
90C2×#11758198C2×C412yescomplemented
91C2×#11758198C2×C411yescharacteristic, complemented
92C2×#11758198C2311yescharacteristic
93C2×#11758198C2312yes—
94C22⋊C4×D61—24no—
95C22⋊C4×D61—24no—
96C22×Q8×C61C2×C411yescharacteristic, complemented
97C22×D4×C61C2×C411yescharacteristic
98#124×C61—22no—
99#124×C61—22no—
100C2×C22⋊C4×C61C2×C412yescomplemented
101C2×C22⋊C4×C61C2×C412yescomplemented
102C2×C42×C61C2×C411yescharacteristic, complemented
103C24×C61C22×C412yes—
104C23×D61—24no—
105C23×D61—22no—
106C23×D61—22no—
107C23×D61—44no—
108C23×D61—82no—
109C23×D61C22×C412yes—
110C23×D61C22×C411yescharacteristic
111C23×D61C2×D414yes—
112C23×D61C2×D414yes—
113C23×D61C2411yescharacteristic
114C22×#6923—22no—
115C22×#6923—22no—
116C22×#6923—24no—
117C22×#6923—48no—
118C22×#6923—48no—
119C22×#6923—84no—
120C2×Q8×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—42no—
123C2×D4×C61—42no—
124C2×D4×C61C22⋊C412yes—
125C2×D4×C61C22⋊C412yes—
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C412yes—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—82no—
133C2×C22⋊D61C22⋊C414yes—
134C2×C22⋊D61C22⋊C414yes—
135C22×Dic61C22×C411yescharacteristic
136C22×Dic61C22×C412yes—
137C22×Dic61C22×C411yescharacteristic
138#11758226—48no—
139#11758226—44no—
140#11758226—44no—
141D4×D61—28no—
142D4×D61—44no—
143D4×D61—48no—
144C2×D244—24no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—44no—
152C2×D244C22⋊C412yes—
153C2×D244C22⋊C411yescharacteristic
154C2×D244C22⋊C411yescharacteristic
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—44no—
164C2×C4×D61—82no—
165C2×C4×D61C2×D412yes—
166C2×C4×D61C2×D412yes—
167C22⋊#6923—22no—
168C22⋊#6923—22no—
169C22⋊#6923—24no—
170C22⋊#6923—24no—
171C22⋊#6923—22no—
172C22⋊#6923—48no—
173C22⋊#6923—48no—
174C22⋊#6923—84no—
175C22⋊#6923#4114yescomplemented
176C4⋊#6923—44no—
177C4×#6923—44no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—24no—
180C42×C61—21no—
181C42×C61—21no—
182C22⋊Dic61—24no—
183#11758198—22no—
184#11758198—22no—
185#11758198—22no—
186#11758198—24no—
187#11758198—22no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—22no—
191#11758196—24no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61C2×C22⋊C414yes—
197C23×C61C23×C411yescharacteristic
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—24no—
201C22×D61—24no—
202C22×D61—28no—
203C22×D61—24no—
204C22×D61—24no—
205C22×D61—24no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—41no—
210C22×D61—42no—
211C22×D61—44no—
212C22×D61—41no—
213C22×D61—44no—
214C22×D61—44no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—88no—
218C22×D61—84no—
219C22×D61—84no—
220C22×D61C2×C22⋊C412yes—
221C22×D61C2×C22⋊C412yes—
222C22×D61C22×D412yes—
223C22×D61C2×#4112yes—
224C22×D61H(5,𝔽2)11yescharacteristic
225C22×D612−1+411yescharacteristic
226C2×#6923—24no—
227C2×#6923—42no—
228C2×#6923—42no—
229C2×#6923—42no—
230C2×#6923—44no—
231C2×#6923—42no—
232C2×#6923—44no—
233C2×#6923—44no—
234C2×#6923—42no—
235C2×#6923—816no—
236Q8×C61—42no—
237Q8×C61—42no—
238D4×C61—24no—
239D4×C61—42no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—42no—
247C2×C4×C61—41no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C22⋊D61—28no—
251C22⋊D61—44no—
252C22⋊D61—84no—
253C2×Dic61—22no—
254C2×Dic61—21no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—42no—
258C2×Dic61C2×C22⋊C412yes—
259D244—24no—
260D244—22no—
261D244—44no—
262D244—42no—
263D244—42no—
264D244—44no—
265D244#9914yescomplemented
266C4×D61—24no—
267C4×D61—42no—
268C4×D61—42no—
269C4×D61—44no—
270C4×D61—44no—
271C4×D61—42no—
272C4×D61—82no—
273C4×D61—82no—
274#22476—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—24no—
278C22×C61—24no—
279C22×C61—24no—
280C22×C61—82no—
281C22×C61—82no—
282C22×C61C22×C22⋊C411yescharacteristic
283C22×C61C4×#4111yescharacteristic
284C22×C61#52011yescharacteristic
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—42no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—48no—
296C2×D61—48no—
297C2×D61—44no—
298C2×D61—44no—
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—84no—
303C2×D61—84no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61C2×#9914yes—
307C2×D61#54612yes—
308C2×D61#54712yes—
309#6923—44no—
310#6923—82no—
311#6923—84no—
312C4×C61—21no—
313C4×C61—41no—
314C4×C61—42no—
315C4×C61—41no—
316C4×C61C2×#9912yes—
317Dic61—22no—
318Dic61—41no—
319Dic61—42no—
320C2×#546—611no—
321D4×C22⋊C4—612no—
322#1996—612no—
323#1898—614no—
324#1819—612no—
325#1799—611no—
326C4×#99—611no—
327C2×#342—612no—
328C2×C61—21no—
329C2×C61—21no—
330C2×C61—44no—
331C2×C61#233911yescharacteristic
332C2×C61C22×#9911yescharacteristic
333C2×C61#288311yescharacteristic
334D61—22no—
335D61—22no—
336D61—24no—
337D61—42no—
338D61—81no—
339C22×#41—611no—
340C23×D4—611no—
341#546—1222no—
342#546—1224no—
343#546—1222no—
344C2×#124#692311yescharacteristic, complemented
345C2×#123—612no—
346C2×C22⋊Q8—611no—
347C2×#121—611no—
348C2×#121—612no—
349C2×#120—612no—
350C2×C4×D4—611no—
351C2×C4×D4—611no—
352C2×C4×D4—1222no—
353C22×C22⋊C4—612no—
354C22×C22⋊C4—614no—
355C2×#99—612no—
356C2×#99—612no—
357#390—1222no—
358#386—1224no—
359#380—612no—
360#379—614no—
361C4×C22⋊C4—612no—
362#342—612no—
363#342—611no—
364#342—614no—
365#342—614no—
366#342—611no—
367C61#2247611yescharacteristic, complemented, Sylow
368C25—612no—
369C2×#41—1221no—
370C2×#41—1222no—
371C2×#41—1221no—
372C2×#41—2444no—
373C22×Q8C2×#692311yescharacteristic, complemented
374C22×D4—612no—
375C22×D4—612no—
376C22×D4—614no—
377C22×D4—612no—
378C22×D4—612no—
379C22×D4—611no—
380C22×D4—611no—
381C22×D4—612no—
382C22×D4—2444no—
383C22×D4C2×#692311yescharacteristic
384C23×C4—611no—
385C23×C4—612no—
386C23×C4—612no—
387C23×C4—611no—
388C23×C4—1224no—
389#124—22no—
390#124—22no—
391#123—1224no—
392#123—1224no—
393C22⋊Q8—1222no—
394C22⋊Q8—1222no—
395#121—1224no—
396#121—1224no—
397#121—1222no—
398#121—1222no—
399#120—1224no—
400#120—1224no—
401C4×D4—1222no—
402C4×D4—1222no—
403C4×D4—1221no—
404C4×D4—1222no—
405C4×D4—1221no—
406C4×D4—2448no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4—1222no—
411C2×C22⋊C4—612no—
412C2×C22⋊C4—612no—
413C2×C22⋊C4—611no—
414C2×C22⋊C4—612no—
415C2×C22⋊C4—612no—
416C2×C22⋊C4—611no—
417C2×C22⋊C4—614no—
418C2×C22⋊C4—612no—
419C2×C22⋊C4—1224no—
420C2×C22⋊C4—1224no—
421C2×C22⋊C4—1228no—
422C2×C22⋊C4—1224no—
423C2×C22⋊C4—1224no—
424C2×C22⋊C4—1224no—
425C2×C22⋊C4—2444no—
426C2×C22⋊C4C2×#692312yescomplemented
427C2×C22⋊C4C2×#692312yescomplemented
428C2×C42—1222no—
429C2×C42C2×#692311yescharacteristic, complemented
430#99—1224no—
431#99—2442no—
432#95—1222no—
433#95—1222no—
434C24—611no—
435C24—614no—
436C24—612no—
437C24—614no—
438C24—611no—
439C24—1224no—
440C24—1222no—
441C24—1222no—
442C24—4882no—
443C24—2444no—
444C24C22×#692312yes—
445#41—2444no—
446#41—2448no—
447#41—2444no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—42no—
451C2×D4—42no—
452C2×D4—614no—
453C2×D4—614no—
454C2×D4—611no—
455C2×D4—612no—
456C2×D4—611no—
457C2×D4—1224no—
458C2×D4—1222no—
459C2×D4—1221no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1221no—
464C2×D4—1228no—
465C2×D4—1224no—
466C2×D4—4882no—
467C2×D4—2444no—
468C2×D4—2444no—
469C2×D4—2448no—
470C2×D4—2444no—
471C2×D4C22⋊#692312yes—
472C2×D4C22⋊#692312yes—
473C22×C4—611no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—611no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1222no—
486C22×C4—1224no—
487C22×C4—4882no—
488C22×C4—4884no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4—2444no—
492C22×C4—2448no—
493C22×C4—2448no—
494C22×C4C22×#692311yescharacteristic
495C22×C4C22×#692311yescharacteristic
496C22×C4C22×#692311yescharacteristic
497C22×C4C22×#692312yes—
498C4⋊C4—1224no—
499C4⋊C4—1222no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—2444no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—614no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1224no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1224no—
513C22⋊C4—1222no—
514C22⋊C4—1224no—
515C22⋊C4—1224no—
516C22⋊C4—1222no—
517C22⋊C4—4884no—
518C22⋊C4—2448no—
519C22⋊C4—2448no—
520C42—21no—
521C42—21no—
522C42—2444no—
523C23—22no—
524C23—22no—
525C23—42no—
526C23—42no—
527C23—612no—
528C23—612no—
529C23—612no—
530C23—611no—
531C23—611no—
532C23—612no—
533C23—1224no—
534C23—1224no—
535C23—1228no—
536C23—1224no—
537C23—1224no—
538C23—1224no—
539C23—1222no—
540C23—1222no—
541C23—1221no—
542C23—1221no—
543C23—1221no—
544C23—4884no—
545C23—4888no—
546C23—4884no—
547C23—2444no—
548C23—2444no—
549C23—2442no—
550C23—2444no—
551C23—2441no—
552C23—2441no—
553C23—2441no—
554C23—2441no—
555C23C2×C22⋊#692314yes—
556C23C23×#692311yescharacteristic
557Q8—42no—
558Q8—42no—
559D4—24no—
560D4—42no—
561D4—614no—
562D4—1224no—
563D4—1228no—
564D4—1222no—
565D4—4884no—
566D4—2442no—
567D4—2442no—
568D4—2444no—
569D4—2444no—
570D4—2444no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—612no—
580C2×C4—1224no—
581C2×C4—1224no—
582C2×C4—1222no—
583C2×C4—1221no—
584C2×C4—4882no—
585C2×C4—4882no—
586C2×C4—48816no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4—2441no—
592C2×C4—2444no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2441no—
603C2×C4C2×C22⋊#692311yescharacteristic
604C2×C4C2×C22⋊#692311yescharacteristic
605C22—21no—
606C22—21no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—82no—
611C22—82no—
612C22—612no—
613C22—612no—
614C22—614no—
615C22—1222no—
616C22—1222no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—1222no—
621C22—1222no—
622C22—4881no—
623C22—4881no—
624C22—4884no—
625C22—4884no—
626C22—2442no—
627C22—2441no—
628C22—2441no—
629C22—2444no—
630C22—2444no—
631C22—2448no—
632C22—2448no—
633C22—2441no—
634C22—2441no—
635C22—2441no—
636C22C22×C22⋊#692311yescharacteristic
637C22#41×#692311yescharacteristic
638C22#1684083111yescharacteristic
639C4—21no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—1222no—
644C4—4884no—
645C4—4882no—
646C4—2444no—
647C4—2441no—
648C4—2442no—
649C4D61⋊#9912yes—
650C2—21no—
651C2—21no—
652C2—44no—
653C2—1222no—
654C2—1222no—
655C2—1224no—
656C2—4881no—
657C2—2442no—
658C2#2146232211yescharacteristic
659C2C2×D61⋊#9911yescharacteristic
660C2#2146241911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#124×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840405C2
cosocleC23×D61C24
hypercenterC2×#124#6923
hypocenterC61#22476
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. D4×C22⋊#6923C2semidirect
2. #21462008C2semidirect
3. #21461841C2semidirect
4. #21461715C2semidirect
5. #21461680C2semidirect
6. C4×#14519836C2semidirect
7. #21461516C2semidirect
23. C22×#11758226C4semidirect
28. C2×#14519759C4semidirect
30. C2×#14519750C4semidirect
31. C2×#14519748C4semidirect
33. C2×C4×D244C4semidirect
36. C2×#124×C61C4semidirect
59. C22×D244C2×C4semidirect
62. C22×C22⋊D61C2×C4semidirect
67. C22×C4×D61C2×C4semidirect
68. C22×C4×D61C2×C4semidirect
69. C22×C4×D61C2×C4semidirect
86. C2×C4⋊Dic61C2×C4semidirect
87. C2×C4⋊Dic61C2×C4semidirect
88. C2×#11758196C2×C4semidirect
89. C2×C22⋊Dic61C2×C4semidirect
90. C2×#11758198C2×C4semidirect
91. C2×#11758198C2×C4semidirect
96. C22×Q8×C61C2×C4semidirect
100. C2×C22⋊C4×C61C2×C4semidirect
101. C2×C22⋊C4×C61C2×C4semidirect
102. C2×C42×C61C2×C4semidirect
175. C22⋊#6923#41semidirect
265. D244#99semidirect
344. C2×#124#6923semidirect
367. C61#22476semidirect
373. C22×Q8C2×#6923semidirect
426. C2×C22⋊C4C2×#6923semidirect
427. C2×C22⋊C4C2×#6923semidirect
429. C2×C42C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D4×C22⋊#69232
#214620082
#214618412
#214617152
#214616802
C4×#145198362
#214615162
C2×#168404052
#2247661