Finite Group Databasebeta

← #30620314

Subgroups — #30620314

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#21462009C212yesmaximal, complemented
3#21461840C214yesmaximal, complemented
4#21461715C212yesmaximal, complemented
5#21461679C211yesmaximal, characteristic, complemented
6C4×#14519835C211yesmaximal, characteristic, complemented
7#21461516C212yesmaximal, complemented
8C2×#16840409C211yesmaximal, characteristic
9C2×D4×#6923—22no—
10C22×C22⋊#6923C2214yes—
11C2×#14519835C2212yes—
12C2×#14519835C2212yes—
13#16840714—22no—
14#16840710—24no—
15#16840704C2212yes—
16#16840703C2214yes—
17C4×C22⋊#6923C2212yes—
18#16840640C2211yescharacteristic
19#16840640C2214yes—
20#16840640C2212yes—
21#16840640C2211yescharacteristic
22#16840640C2214yes—
23C22×#11758222C411yescharacteristic, complemented
24C22×D4×D61C2211yescharacteristic
25#16840409—22no—
26#16840409—24no—
27#16840409—22no—
28C2×#14519764C411yescharacteristic, complemented
29C2×#14519759C412yescomplemented
30C2×#14519757C2211yescharacteristic
31C2×#14519748C412yescomplemented
32C2×#14519745C2212yes—
33C2×#14519739C411yescharacteristic, complemented
34C2×C4×C22⋊D61C2211yescharacteristic
35C2×C22⋊C4×D61C2212yes—
36C2×#121×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#14519835—24no—
50#14519835—42no—
51(C2×#6923)⋊C4—22no—
52(C2×#6923)⋊C4—22no—
53C24×D61C2312yes—
54C2×#11758222—22no—
55C2×#11758222—21no—
56C2×#11758222—21no—
57C2×#11758222—44no—
58C22×Dic122C2×C411yescharacteristic, complemented
59C22×D244C2×C411yescharacteristic
60C22×C22⋊D61C2×C412yes—
61C22×C22⋊D61C2×C412yescomplemented
62C22×C22⋊D61C2311yescharacteristic
63C2×D4×D61—44no—
64C2×D4×D61D412yes—
65C2×D4×D61D414yes—
66C2×D4×D61D412yes—
67C22×C4×D61C2×C412yescomplemented
68C22×C4×D61C2×C412yescomplemented
69C22×C4×D61C2311yescharacteristic
70#14519764—22no—
71#14519764—22no—
72#14519759—24no—
73#14519759—24no—
74#14519757—22no—
75#14519757—22no—
76#14519748—24no—
77#14519748—24no—
78#14519745—24no—
79#14519745—24no—
80#14519739—22no—
81#14519739—21no—
82#14519739—21no—
83C4×C22⋊D61—22no—
84C4×C22⋊D61—22no—
85C2×C4⋊Dic61C2×C412yescomplemented
86C2×#11758196C2×C411yescharacteristic, complemented
87C2×C22⋊Dic61C2×C412yescomplemented
88C2×C22⋊Dic61C2×C411yescharacteristic, complemented
89C2×C22⋊Dic61C2×C412yescomplemented
90C2×#11758198C2×C412yescomplemented
91C2×#11758198C2311yescharacteristic
92C2×#11758198C2312yes—
93C22⋊C4×D61—24no—
94C22⋊C4×D61—24no—
95C2×C4×Dic61C2×C411yescharacteristic, complemented
96C22×D4×C61C2×C411yescharacteristic
97C22×D4×C61C2×C412yescomplemented
98C23×C4×C61C2×C411yescharacteristic, complemented
99#121×C61—22no—
100#121×C61—22no—
101C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
102C2×C22⋊C4×C61C2×C412yescomplemented
103C24×C61C22×C411yescharacteristic
104C24×C61C22×C412yes—
105C23×D61—24no—
106C23×D61—22no—
107C23×D61—22no—
108C23×D61—44no—
109C23×D61—82no—
110C23×D61C22×C412yes—
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×D4×C61—24no—
121C2×D4×C61—42no—
122C2×D4×C61—82no—
123C2×D4×C61C22⋊C412yes—
124C2×D4×C61C22⋊C412yes—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—42no—
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C22×C412yes—
131C2×C22⋊D61—22no—
132C2×C22⋊D61—22no—
133C2×C22⋊D61—22no—
134C2×C22⋊D61—22no—
135C2×C22⋊D61—21no—
136C2×C22⋊D61—21no—
137C2×C22⋊D61—44no—
138C2×C22⋊D61—44no—
139C2×C22⋊D61C22⋊C414yes—
140C2×C22⋊D61C22⋊C414yes—
141C22×Dic61C22×C411yescharacteristic
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C412yes—
144C22×Dic61C22×C411yescharacteristic
145D4×D61—28no—
146D4×D61—44no—
147D4×D61—48no—
148#11758222—48no—
149#11758222—44no—
150#11758222—44no—
151C2×D244—24no—
152C2×D244C22⋊C412yes—
153C2×D244C22⋊C411yescharacteristic
154C2×D244C22⋊C411yescharacteristic
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—42no—
160C2×C4×D61—44no—
161C2×C4×D61—82no—
162C2×C4×D61C2×D412yes—
163C2×C4×D61C2×D412yes—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—42no—
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—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C22⋊C4×C61—24no—
181C22⋊Dic61—24no—
182C22⋊Dic61—24no—
183C22⋊Dic61—22no—
184#11758198—22no—
185#11758198—22no—
186#11758198—22no—
187#11758198—24no—
188C4⋊Dic61—24no—
189#11758196—22no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61C2×C22⋊C414yes—
202C23×C61C23×C411yescharacteristic
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—24no—
206C22×D61—24no—
207C22×D61—28no—
208C22×D61—24no—
209C22×D61—24no—
210C22×D61—24no—
211C22×D61—41no—
212C22×D61—42no—
213C22×D61—44no—
214C22×D61—41no—
215C22×D61—44no—
216C22×D61—44no—
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—
236D4×C61—24no—
237D4×C61—42no—
238D4×C61—84no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C22⋊D61—28no—
249C22⋊D61—44no—
250C22⋊D61—44no—
251C22⋊D61—42no—
252C22⋊D61—42no—
253C22⋊D61—44no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261C2×Dic61—41no—
262C2×Dic61C2×C22⋊C412yes—
263D244—24no—
264D244—22no—
265D244#9914yescomplemented
266C4×D61—24no—
267C4×D61—42no—
268C4×D61—44no—
269C4×D61—44no—
270C4×D61—82no—
271C4×D61—82no—
272Dic122—42no—
273Dic122—42no—
274#22476—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—24no—
278C22×C61—24no—
279C22×C61—24no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—82no—
284C22×C61—82no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61C22×C22⋊C411yescharacteristic
288C22×C61C4×#4111yescharacteristic
289C22×C61#52011yescharacteristic
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—42no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—48no—
301C2×D61—48no—
302C2×D61—44no—
303C2×D61—44no—
304C2×D61—84no—
305C2×D61—84no—
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×C61C2×#9912yes—
316Dic61—22no—
317Dic61—41no—
318Dic61—42no—
319Dic61—41no—
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—81no—
332C2×C61#233911yescharacteristic
333C2×C61C22×#9911yescharacteristic
334C2×C61#288311yescharacteristic
335D61—22no—
336D61—22no—
337D61—24no—
338D61—42no—
339C22×#41—611no—
340C23×D4—611no—
341#546—1222no—
342#546—1224no—
343#546—1222no—
344C2×#124—611no—
345C2×#123—612no—
346C2×C22⋊Q8—611no—
347C2×#121—612no—
348C2×#121#692311yescharacteristic, complemented
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×Q8—611no—
374C22×D4—612no—
375C22×D4—612no—
376C22×D4—614no—
377C22×D4—612no—
378C22×D4—611no—
379C22×D4—612no—
380C22×D4—611no—
381C22×D4—2444no—
382C22×D4C2×#692311yescharacteristic
383C22×D4C2×#692312yescomplemented
384C23×C4—611no—
385C23×C4—612no—
386C23×C4—612no—
387C23×C4—1224no—
388C23×C4C2×#692311yescharacteristic, complemented
389#124—1222no—
390#124—1222no—
391#123—1224no—
392#123—1224no—
393C22⋊Q8—1222no—
394C22⋊Q8—1222no—
395#121—22no—
396#121—22no—
397#121—1224no—
398#121—1224no—
399#120—1224no—
400#120—1224no—
401C4×D4—1221no—
402C4×D4—1222no—
403C4×D4—1221no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—2448no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—612no—
409C2×C4⋊C4—1222no—
410C2×C4⋊C4C2×#692311yescharacteristic, complemented
411C2×C22⋊C4—612no—
412C2×C22⋊C4—612no—
413C2×C22⋊C4—611no—
414C2×C22⋊C4—612no—
415C2×C22⋊C4—611no—
416C2×C22⋊C4—612no—
417C2×C22⋊C4—612no—
418C2×C22⋊C4—614no—
419C2×C22⋊C4—612no—
420C2×C22⋊C4—1224no—
421C2×C22⋊C4—1224no—
422C2×C22⋊C4—1228no—
423C2×C22⋊C4—1224no—
424C2×C22⋊C4—1224no—
425C2×C22⋊C4—1224no—
426C2×C22⋊C4—2444no—
427C2×C22⋊C4C2×#692312yescomplemented
428C2×C42—611no—
429C2×C42—1222no—
430#99—1224no—
431#99—2442no—
432#95—1222no—
433#95—1222no—
434C24—611no—
435C24—614no—
436C24—612no—
437C24—614no—
438C24—1224no—
439C24—1222no—
440C24—1222no—
441C24—4882no—
442C24—2444no—
443C24C22×#692311yescharacteristic
444C24C22×#692312yes—
445#41—2444no—
446#41—2448no—
447#41—2444no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—2442no—
451C2×D4—24no—
452C2×D4—42no—
453C2×D4—82no—
454C2×D4—614no—
455C2×D4—614no—
456C2×D4—611no—
457C2×D4—612no—
458C2×D4—611no—
459C2×D4—1224no—
460C2×D4—1228no—
461C2×D4—1222no—
462C2×D4—1221no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—1222no—
466C2×D4—1221no—
467C2×D4—2444no—
468C2×D4—2448no—
469C2×D4—2444no—
470C2×D4—2444no—
471C2×D4C22⋊#692312yes—
472C2×D4C22⋊#692312yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—42no—
476C22×C4—611no—
477C22×C4—612no—
478C22×C4—612no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—612no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1224no—
489C22×C4—4882no—
490C22×C4—4884no—
491C22×C4—2442no—
492C22×C4—2444no—
493C22×C4—2448no—
494C22×C4—2448no—
495C22×C4C22×#692311yescharacteristic
496C22×C4C22×#692312yes—
497C22×C4C22×#692311yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—1222no—
501C4⋊C4—1224no—
502C4⋊C4—2444no—
503C22⋊C4—24no—
504C22⋊C4—614no—
505C22⋊C4—1222no—
506C22⋊C4—1224no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1224no—
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—1221no—
521C42—1221no—
522C42—2444no—
523C23—22no—
524C23—22no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—612no—
533C23—612no—
534C23—612no—
535C23—611no—
536C23—611no—
537C23—612no—
538C23—1224no—
539C23—1224no—
540C23—1228no—
541C23—1224no—
542C23—1224no—
543C23—1224no—
544C23—1222no—
545C23—1222no—
546C23—4884no—
547C23—4888no—
548C23—4884no—
549C23—2444no—
550C23—2444no—
551C23—2442no—
552C23—2444no—
553C23—2441no—
554C23—2441no—
555C23C2×C22⋊#692314yes—
556C23C23×#692311yescharacteristic
557Q8—2442no—
558Q8—2442no—
559D4—24no—
560D4—42no—
561D4—84no—
562D4—614no—
563D4—1224no—
564D4—1228no—
565D4—1222no—
566D4—2444no—
567D4—2442no—
568D4—2442no—
569D4—2444no—
570D4—2444no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—612no—
579C2×C4—1224no—
580C2×C4—1224no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1222no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—4882no—
587C2×C4—4882no—
588C2×C4—48816no—
589C2×C4—2442no—
590C2×C4—2444no—
591C2×C4—2444no—
592C2×C4—2441no—
593C2×C4—2444no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2444no—
598C2×C4—2444no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2441no—
602C2×C4—2442no—
603C2×C4C2×C22⋊#692311yescharacteristic
604C2×C4C2×C22⋊#692311yescharacteristic
605C22—21no—
606C22—21no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—82no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22—612no—
618C22—612no—
619C22—614no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—1222no—
624C22—1222no—
625C22—1222no—
626C22—1222no—
627C22—4884no—
628C22—4884no—
629C22—2442no—
630C22—2441no—
631C22—2441no—
632C22—2444no—
633C22—2444no—
634C22—2448no—
635C22—2448no—
636C22C22×C22⋊#692311yescharacteristic
637C22#41×#692311yescharacteristic
638C22#1684083111yescharacteristic
639C4—21no—
640C4—41no—
641C4—42no—
642C4—1222no—
643C4—4884no—
644C4—4882no—
645C4—2444no—
646C4—2441no—
647C4—2442no—
648C4—2441no—
649C4D61⋊#9912yes—
650C2—21no—
651C2—21no—
652C2—44no—
653C2—81no—
654C2—1222no—
655C2—1222no—
656C2—1224no—
657C2—2442no—
658C2#2146232211yescharacteristic
659C2C2×D61⋊#9911yescharacteristic
660C2#2146241811yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#121×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840409C2
cosocleC23×D61C24
hypercenterC2×#121#6923
hypocenterC61#22476
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. D4×C22⋊#6923C2semidirect
2. #21462009C2semidirect
3. #21461840C2semidirect
4. #21461715C2semidirect
5. #21461679C2semidirect
6. C4×#14519835C2semidirect
7. #21461516C2semidirect
23. C22×#11758222C4semidirect
28. C2×#14519764C4semidirect
29. C2×#14519759C4semidirect
31. C2×#14519748C4semidirect
33. C2×#14519739C4semidirect
36. C2×#121×C61C4semidirect
58. C22×Dic122C2×C4semidirect
61. C22×C22⋊D61C2×C4semidirect
67. C22×C4×D61C2×C4semidirect
68. C22×C4×D61C2×C4semidirect
85. C2×C4⋊Dic61C2×C4semidirect
86. C2×#11758196C2×C4semidirect
87. C2×C22⋊Dic61C2×C4semidirect
88. C2×C22⋊Dic61C2×C4semidirect
89. C2×C22⋊Dic61C2×C4semidirect
90. C2×#11758198C2×C4semidirect
95. C2×C4×Dic61C2×C4semidirect
97. C22×D4×C61C2×C4semidirect
98. C23×C4×C61C2×C4semidirect
101. C2×C4⋊C4×C61C2×C4semidirect
102. C2×C22⋊C4×C61C2×C4semidirect
175. C22⋊#6923#41semidirect
265. D244#99semidirect
348. C2×#121#6923semidirect
367. C61#22476semidirect
383. C22×D4C2×#6923semidirect
388. C23×C4C2×#6923semidirect
410. C2×C4⋊C4C2×#6923semidirect
427. C2×C22⋊C4C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D4×C22⋊#69232
#214620092
#214618402
#214617152
#214616792
C4×#145198352
#214615162
C2×#168404092
#2247661