Finite Group Databasebeta

← #29653572

Subgroups — #29653572

Order 15616.

1132 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455052C211yesmaximal, characteristic
2C2×#16839917C211yesmaximal, characteristic, complemented
3#21449464C211yesmaximal, characteristic, complemented
4#21444793C214yesmaximal, complemented
5#21444782C214yesmaximal, complemented
6#21444339C211yesmaximal, characteristic, complemented
7#21444310C211yesmaximal, characteristic
8#21444183C211yesmaximal, characteristic, complemented
9C61×#3538C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C2×#14519807—21no—
12C22×#11758202—21no—
13C2×#14519775—21no—
14#16839917—22no—
15#16839917—21no—
16#16839917C2214yes—
17#16839917C2212yes—
18#16839900—21no—
19#16839900C2212yes—
20#16839770—21no—
21C2×#14519700—21no—
22#16839726C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24#16839682C2214yes—
25#16839667—21no—
26#16839634—22no—
27C4×C22⋊Dic61C2211yescharacteristic
28#16839499C2214yes—
29#16839499C2212yes—
30#16839496C2212yes—
31#16839488C2214yes—
32C61×#563—22no—
33C61×#552—21no—
34C61×#552—21no—
35C61×#551—22no—
36D4×Q8×C61—21no—
37D4×Q8×C61—21no—
38C61×#544—21no—
39C61×#544C2212yes—
40C61×#541—21no—
41C61×#539—21no—
42C61×#539C2212yes—
43C61×#536—21no—
44C61×#525C2211yescharacteristic
45C2×C22⋊Q8×C61C2211yescharacteristic
46C2×C22⋊Q8×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×Dic122—21no—
49C22×Dic122—21no—
50C22×Dic122—21no—
51C22×Dic122—21no—
52C22×Dic122—44no—
53C22×Dic122C2312yes—
54C23×Dic61—21no—
55#14519807—44no—
56C2×#11758202—24no—
57C2×#11758202—22no—
58C2×#11758202—44no—
59C2×#11758200—22no—
60C2×#11758194—21no—
61C22×#92418—21no—
62#14519775—44no—
63C2×#11758196—24no—
64C2×C22⋊Dic61—21no—
65C2×C4×Dic61—21no—
66#14519700—24no—
67#14519700—22no—
68#14519700—22no—
69C22⋊#92418—21no—
70C22⋊#92418—21no—
71C22⋊#92418C2312yes—
72C22⋊#92418C2312yes—
73#14519684—24no—
74#14519684C2312yes—
75C2×#41×C61D411yescharacteristic
76C22×Q8×C61—21no—
77C22×Q8×C61D411yescharacteristic
78C22×Q8×C61D411yescharacteristic
79C23×C4×C61C2311yescharacteristic
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—22no—
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61—22no—
84#126×C61—22no—
85#126×C61—22no—
86#126×C61—22no—
87#126×C61—22no—
88#125×C61—21no—
89#125×C61—22no—
90#125×C61—22no—
91#125×C61—22no—
92#125×C61—22no—
93#124×C61—22no—
94#123×C61—22no—
95#123×C61—22no—
96#123×C61—22no—
97#123×C61—22no—
98#123×C61—21no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—21no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61—21no—
109C22⋊Q8×C61—21no—
110C22⋊Q8×C61—21no—
111C22⋊Q8×C61D411yescharacteristic
112C22⋊Q8×C61D412yes—
113C22⋊Q8×C61C2312yes—
114C22⋊Q8×C61C2311yescharacteristic
115#121×C61D411yescharacteristic
116#121×C61C2311yescharacteristic
117#120×C61—21no—
118C4×Q8×C61—21no—
119C4×Q8×C61—21no—
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C4×D4×C61—21no—
123C4×D4×C61—21no—
124#117×C61—21no—
125#117×C61—22no—
126#117×C61—22no—
127#117×C61D411yescharacteristic
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61D411yescharacteristic
130C2×C4⋊C4×C61D411yescharacteristic
131C2×C4⋊C4×C61C2312yes—
132C2×C22⋊C4×C61—21no—
133C2×C22⋊C4×C61D412yes—
134C24×C61C2×D411yescharacteristic
135#41×C61—42no—
136C2×Q8×C61—22no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—42no—
141C2×Q8×C61—42no—
142C2×Q8×C61—82no—
143C2×Q8×C61C2×D412yes—
144C2×Q8×C61C2×D411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—22no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—42no—
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—44no—
165C22×Dic61—41no—
166C22×Dic61—41no—
167C2×Dic122—24no—
168C2×Dic122—24no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—24no—
172C2×Dic122—44no—
173C2×Dic122—44no—
174C2×Dic122—44no—
175C2×Dic122—42no—
176C2×Dic122—42no—
177C2×Dic122—42no—
178C2×Dic122—41no—
179C2×Dic122—41no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61C2×D412yes—
198C4⋊C4×C61C2×D411yescharacteristic
199C4⋊C4×C61C2×D411yescharacteristic
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—21no—
207C22⋊C4×C61—21no—
208C22⋊C4×C61—22no—
209C22⋊C4×C61—21no—
210C22⋊C4×C61—21no—
211C22⋊C4×C61—21no—
212C42×C61—22no—
213C42×C61—22no—
214C42×C61—22no—
215C42×C61—21no—
216C22⋊Dic61—24no—
217#11758202—48no—
218#11758202—44no—
219#11758200—44no—
220#11758196—48no—
221C4×Dic61—42no—
222#11758194—42no—
223C2×#92418—22no—
224C2×#92418—22no—
225C2×#92418—42no—
226C23×C61—22no—
227C23×C61—22no—
228C23×C61—21no—
229C23×C61—41no—
230C23×C61#12011yescharacteristic
231C23×C61#12011yescharacteristic
232C23×C61C22×D411yescharacteristic
233Q8×C61—42no—
234Q8×C61—41no—
235Q8×C61—42no—
236Q8×C61—42no—
237Q8×C61—81no—
238Q8×C61—82no—
239D4×C61—41no—
240D4×C61—42no—
241D4×C61—42no—
242D4×C61—81no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—24no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61—81no—
261C2×C4×C61—82no—
262C2×C4×C61#12011yescharacteristic
263C2×C4×C61#12011yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×Dic61—24no—
267C2×Dic61—24no—
268C2×Dic61—42no—
269C2×Dic61—44no—
270C2×Dic61—44no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274Dic122—24no—
275Dic122—24no—
276Dic122—42no—
277Dic122—42no—
278Dic122—44no—
279Dic122—44no—
280#92418—42no—
281#92418—42no—
282#18259—611nomaximal, Sylow
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—44no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61#45612yes—
295C22×C61C2×#12011yescharacteristic
296C22×C61#57812yes—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—42no—
301C4×C61—41no—
302C4×C61—42no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—42no—
306C4×C61—42no—
307C4×C61B(4,𝔽2)12yes—
308Dic61—44no—
309Dic61—44no—
310#3538D6111yescharacteristic
311#3004—611no—
312C2×#451—611no—
313#2434—611no—
314#1605—614no—
315#1603—614no—
316#1508—611no—
317#1501—611no—
318#1472—611no—
319C2×C61—22no—
320C2×C61—22no—
321C2×C61—41no—
322C2×C61—81no—
323C2×C61#301111yescharacteristic
324C2×C61C2×#45611yescharacteristic
325C2×C61C2×B(4,𝔽2)11yescharacteristic
326C23×Q8—611no—
327C2×#137—1221no—
328C22×Q16—1221no—
329#563—22no—
330#552—21no—
331#552—21no—
332#551—22no—
333D4×Q8—21no—
334D4×Q8—21no—
335#544—21no—
336#544C2×D6112yes—
337#541—21no—
338#539—21no—
339#539C2×D6112yes—
340#536—21no—
341C2×C4⋊Q8—1221no—
342#525C2×D6111yescharacteristic
343C2×C22⋊Q8C2×D6111yescharacteristic
344C2×C22⋊Q8C2×D6111yescharacteristic
345#451—614no—
346#451—612no—
347#451—1222no—
348#451—1221no—
349#448—612no—
350#448—1221no—
351#419—1221no—
352C2×#103—1221no—
353#407—611no—
354C2×C22⋊C8—611no—
355#395—614no—
356#391—1221no—
357#385—1222no—
358C4×C22⋊C4—611no—
359#328—614no—
360#328—612no—
361#327—612no—
362#324—614no—
363C61#1825911yescharacteristic, complemented, Sylow
364C2×#41C22⋊D6111yescharacteristic
365C22×Q8—21no—
366C22×Q8—612no—
367C22×Q8—1221no—
368C22×Q8—1221no—
369C22×Q8—1221no—
370C22×Q8—1221no—
371C22×Q8—1221no—
372C22×Q8—2444no—
373C22×Q8C22⋊D6111yescharacteristic
374C22×Q8C22⋊D6111yescharacteristic
375C23×C4—1221no—
376C23×C4C22×D6111yescharacteristic
377#137—2444no—
378C2×Q16—1224no—
379C2×Q16—1222no—
380C2×Q16—2444no—
381C2×SD16—1222no—
382C2×M16—1221no—
383C22×C8—1221no—
384C4⋊Q8—21no—
385C4⋊Q8—22no—
386C4⋊Q8—22no—
387C4⋊Q8—22no—
388C4⋊Q8—2444no—
389#126—22no—
390#126—22no—
391#126—22no—
392#126—22no—
393#125—21no—
394#125—22no—
395#125—22no—
396#125—22no—
397#125—22no—
398#124—22no—
399#123—22no—
400#123—22no—
401#123—22no—
402#123—21no—
403#123—22no—
404C22⋊Q8—21no—
405C22⋊Q8—21no—
406C22⋊Q8—22no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409C22⋊Q8—21no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—21no—
414C22⋊Q8—21no—
415C22⋊Q8—22no—
416C22⋊Q8C22⋊D6112yes—
417C22⋊Q8C22⋊D6111yescharacteristic
418C22⋊Q8C22×D6112yes—
419C22⋊Q8C22×D6111yescharacteristic
420#121C22⋊D6111yescharacteristic
421#121C22×D6111yescharacteristic
422#120—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×D4—22no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—22no—
429#117—21no—
430#117—22no—
431#117—22no—
432#117C22⋊D6111yescharacteristic
433C2×C4⋊C4—21no—
434C2×C4⋊C4—1224no—
435C2×C4⋊C4C22⋊D6111yescharacteristic
436C2×C4⋊C4C22⋊D6111yescharacteristic
437C2×C4⋊C4C22×D6112yes—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4C22⋊D6112yes—
441C2×C42—1221no—
442#103—1224no—
443#103—1222no—
444#103—1222no—
445C22⋊C8—612no—
446C22⋊C8—612no—
447C22⋊C8—1221no—
448C22⋊C8—1221no—
449#95—612no—
450#95—1224no—
451C24C2×C22⋊D6111yescharacteristic
452#41—42no—
453C2×Q8—22no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—82no—
460C2×Q8—1224no—
461C2×Q8—1224no—
462C2×Q8—1224no—
463C2×Q8—1222no—
464C2×Q8—1222no—
465C2×Q8—2442no—
466C2×Q8—2442no—
467C2×Q8—2442no—
468C2×Q8—2444no—
469C2×Q8—2444no—
470C2×Q8—2444no—
471C2×Q8—2441no—
472C2×Q8—2441no—
473C2×Q8C2×C22⋊D6112yes—
474C2×Q8C2×C22⋊D6111yescharacteristic
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4C2×C22⋊D6111yescharacteristic
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—42no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—2444no—
487C22×C4—2441no—
488C22×C4—2441no—
489C22×C4C2×C22⋊D6112yes—
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C2×C22⋊D6112yes—
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497Q16—2448no—
498Q16—2444no—
499SD16—2444no—
500M16—2442no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—2442no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—2448no—
522C4⋊C4C2×C22⋊D6112yes—
523C4⋊C4C2×C22⋊D6111yescharacteristic
524C4⋊C4C2×C22⋊D6111yescharacteristic
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—21no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—1224no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—21no—
542C42—2442no—
543C23—22no—
544C23—22no—
545C23—21no—
546C23—41no—
547C23#1451974611yescharacteristic
548C23#1451974611yescharacteristic
549C23C22×C22⋊D6111yescharacteristic
550Q8—42no—
551Q8—41no—
552Q8—42no—
553Q8—42no—
554Q8—81no—
555Q8—82no—
556Q8—1224no—
557Q8—1224no—
558Q8—2442no—
559Q8—2442no—
560Q8—2444no—
561Q8—2444no—
562D4—41no—
563D4—42no—
564D4—42no—
565D4—81no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4—1224no—
586C2×C4—1224no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4#1451974611yescharacteristic
594C2×C4#1451974611yescharacteristic
595C2×C4C22×C22⋊D6111yescharacteristic
596C2×C4C22×C22⋊D6111yescharacteristic
597C8—2442no—
598C8—2442no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—44no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22#1683994512yes—
611C22C2×#1451974611yescharacteristic
612C22#1684057712yes—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—42no—
623C4—2444no—
624C4—2444no—
625C4#1683994912yes—
626C2—22no—
627C2—22no—
628C2—41no—
629C2—81no—
630C2#2145511411yescharacteristic
631C2C2×#1683994511yescharacteristic
632C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3538C2
socleC22×C61C2×#120
prime radicalC61×#525C22
hypercenter#3538D61
hypocenterC61#18259
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3538

Complements

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

NH ≅ G/Ntype
2. C2×#16839917C2semidirect
3. #21449464C2semidirect
4. #21444793C2semidirect
5. #21444782C2semidirect
6. #21444339C2semidirect
8. #21444183C2semidirect
363. C61#18259semidirect

Maximal subgroups

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

subgroupindex
#214550522
C2×#168399172
#214494642
#214447932
#214447822
#214443392
#214443102
#214441832
C61×#35382
#1825961