Finite Group Databasebeta

← #30610476

Subgroups — #30610476

Order 15616.

1248 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462497C211yesmaximal, characteristic, complemented
2D61⋊#432C211yesmaximal, characteristic, complemented
3#21461161C212yesmaximal, complemented
4#21461160C212yesmaximal, complemented
5#21461153C214yesmaximal, complemented
6#21461117C214yesmaximal, complemented
7C2×C4×#11758226C211yesmaximal, characteristic
8#16840785—21no—
9#16840785—21no—
10#16840784—21no—
11#16840784C2212yescomplemented
12#16840783C2212yes—
13#16840782—21no—
14#16840781C2212yescomplemented
15#16840780C2212yescomplemented
16#16840765—21no—
17D61⋊(C4⋊C8)C2212yescomplemented
18#16840734—21no—
19D61⋊(C22⋊C8)—21no—
20C4×#11758217C2212yes—
21#16840721—21no—
22Q8⋊#92420C2218yescomplemented
23#16840610C2218yescomplemented
24C22×#11758226C411yescharacteristic
25C4×#11758226—21no—
26C4×#11758226—21no—
27C4×#11758226—21no—
28C4×#11758226—21no—
29C4×#11758226C412yes—
30C4×#11758226C412yes—
31C4×#11758226C2212yes—
32C4×#11758226C2212yes—
33C2×Q8×Dic61C2211yescharacteristic
34C2×C4×D244—21no—
35C2×C4×D244C411yescharacteristic
36C2×#14519739—21no—
37C2×#14519739C2211yescharacteristic
38C2×#14519735—21no—
39C2×#14519735C411yescharacteristic
40C2×C42×D61—21no—
41C2×C42×D61C2211yescharacteristic
42C2×C4×Q8×C61C411yescharacteristic
43C2×#11758217—22no—
44C2×D61⋊C8—22no—
45#14519844—22no—
46#14519844—22no—
47C4⋊#92420—22no—
48C4⋊#92420C2314yes—
49#14519834—22no—
50#14519834—22no—
51#14519834—22no—
52C22⋊#92420—22no—
53#14519832—22no—
54#14519832—22no—
55#14519832—22no—
56C4×#92420C2314yes—
57C2×#11758226—28no—
58C2×#11758226—21no—
59C2×#11758226—21no—
60C2×#11758226C2×C412yes—
61C2×#11758226C2×C412yes—
62C22×D244—21no—
63C22×D244C2×C411yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61C2×C411yescharacteristic
68C22×C4×D61D411yescharacteristic
69C22×C4×D61D411yescharacteristic
70Q8×Dic61—21no—
71Q8×Dic61—21no—
72Q8×Dic61C2×C412yes—
73Q8×Dic61C2312yes—
74C4×D244—22no—
75C4×D244—22no—
76C4×D244—22no—
77C4×D244—22no—
78C4×D244—21no—
79C4×D244—21no—
80C4×D244C2×C412yes—
81C4×D244C2×C412yes—
82#14519739—22no—
83#14519739—22no—
84#14519739—22no—
85#14519739—22no—
86#14519739—21no—
87#14519739—21no—
88#14519739C2×C412yes—
89#14519739C2312yes—
90#14519735—21no—
91#14519735—21no—
92#14519735—22no—
93#14519735—22no—
94#14519735—22no—
95#14519735—22no—
96#14519735C2×C412yes—
97#14519735C2×C412yes—
98C2×C4⋊Dic61—21no—
99C2×C4⋊Dic61C2×C411yescharacteristic
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C42×D61—22no—
109C42×D61—22no—
110C42×D61—22no—
111C42×D61—22no—
112C42×D61C2×C411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61C2311yescharacteristic
119C42×D61C2311yescharacteristic
120C22×Q8×C61C2×C411yescharacteristic
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123C4×Q8×C61C2×C412yes—
124C4×Q8×C61C2×C412yes—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2×C411yescharacteristic
127C2×C42×C61—21no—
128C2×C42×C61C2×C411yescharacteristic
129C23×D61—21no—
130C23×D61C22⋊C411yescharacteristic
131C2×Q8×C61—24no—
132C2×Q8×C61—21no—
133C2×Q8×C61C22×C412yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61C22⋊C411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143#11758226—28no—
144#11758226—28no—
145#11758226—24no—
146#11758226—24no—
147#11758226—24no—
148#11758226M1618yescomplemented
149C2×D244—28no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—21no—
153C2×D244—24no—
154C2×D244C22×C412yes—
155C2×C4×D61—28no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—24no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—44no—
166C2×C4×D61—44no—
167C2×C4×D61C22⋊C411yescharacteristic
168C2×C4×D61C22⋊C411yescharacteristic
169C2×C4×D61C22⋊C411yescharacteristic
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22⋊C411yescharacteristic
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22×C411yescharacteristic
174C2×C4×D61C22×C411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61C2×D411yescharacteristic
177C2×C4×D61C2×D411yescharacteristic
178C2×C4×D61C2×D411yescharacteristic
179#11758217—44no—
180C2×#92420—24no—
181C2×#92420—24no—
182D61⋊C8—44no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61C22×C412yes—
187C42×C61—22no—
188C42×C61—22no—
189C42×C61C22⋊C411yescharacteristic
190C42×C61C22⋊C411yescharacteristic
191C42×C61C22×C411yescharacteristic
192C42×C61C22×C411yescharacteristic
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199C4⋊Dic61—21no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61C22×C412yes—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22×C411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2411yescharacteristic
219C23×C61C2×C22⋊C411yescharacteristic
220C22×D61—24no—
221C22×D61—22no—
222C22×D61—22no—
223C22×D61—22no—
224C22×D61—41no—
225C22×D61—42no—
226C22×D61—42no—
227C22×D61C2×C22⋊C411yescharacteristic
228C22×D61C2×C22⋊C411yescharacteristic
229Q8×C61—24no—
230Q8×C61—22no—
231Q8×C61C2×M1614yes—
232C2×C4×C61—24no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C23×C411yescharacteristic
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—42no—
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C22×D411yescharacteristic
255C2×Dic61C22×D411yescharacteristic
256D244—28no—
257D244—24no—
258D244—28no—
259D244—22no—
260D244C2×M1614yes—
261C4×D61—28no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—28no—
267C4×D61—22no—
268C4×D61—41no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—44no—
275C4×D61—44no—
276C4×D61—44no—
277C4×D61—44no—
278C4×D61C2×M1614yes—
279#92420—44no—
280#92420—44no—
281#17452—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61#42112yes—
285C22×C61C22×C22⋊C411yescharacteristic
286C2×D61—21no—
287C2×D61—21no—
288C2×D61—21no—
289C2×D61—24no—
290C2×D61—21no—
291C2×D61—21no—
292C2×D61—44no—
293C2×D61—48no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61#40712yes—
297C4×C61—24no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—42no—
304C4×C61C22×M1612yes—
305Dic61—21no—
306Dic61—21no—
307Dic61—21no—
308Dic61—21no—
309Dic61—42no—
310Dic61#40712yes—
311Dic61#57512yes—
312C2×C4×#41—611no—
313#2920—611no—
314C2×#432—611no—
315#1491—612no—
316#1490—612no—
317#1488—614no—
318#1477—614no—
319C2×C61—22no—
320C2×C61C2×#40711yescharacteristic
321C2×C61#289611yescharacteristic
322C2×C61C2×#42111yescharacteristic
323D61—22no—
324D61—41no—
325D61—44no—
326C22×#41—611no—
327C4×#41—612no—
328C4×#41—612no—
329C4×#41—612no—
330C4×#41—612no—
331C4×#41—1221no—
332C4×#41—1221no—
333C4×#41—1221no—
334C4×#41—1221no—
335C2×C4×Q8—611no—
336C2×C4×Q8#692311yescharacteristic
337C2×C4×D4—611no—
338C2×C4×D4—611no—
339C2×C4×D4—1221no—
340C2×C4×D4—1221no—
341C2×#117—611no—
342C2×#117—1221no—
343C22×C42—611no—
344C22×C42—1221no—
345#432—612no—
346#432—612no—
347#432—1221no—
348#432—1221no—
349#432—1221no—
350#432—1221no—
351#431—612no—
352#431—612no—
353#423—1221no—
354C2×C4⋊C8—612no—
355#407—1221no—
356C2×C22⋊C8—1221no—
357C4×M16—612no—
358C2×#97—1221no—
359Q8⋊C8—618no—
360D4⋊C8—618no—
361C61#1745211yescharacteristic, complemented, Sylow
362C2×#41—612no—
363C2×#41—612no—
364C2×#41—1221no—
365C2×#41—1221no—
366C2×#41—1228no—
367C22×Q8C2×#692311yescharacteristic
368C22×D4—611no—
369C22×D4—1221no—
370C23×C4—611no—
371C23×C4—611no—
372C23×C4—611no—
373C23×C4—1221no—
374C23×C4—1221no—
375C23×C4—1221no—
376C2×M16—1222no—
377C22×C8—1222no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—612no—
381C4×Q8—612no—
382C4×Q8—1221no—
383C4×Q8—1221no—
384C4×Q8C2×#692312yes—
385C4×Q8C2×#692312yes—
386C4×D4—612no—
387C4×D4—612no—
388C4×D4—612no—
389C4×D4—612no—
390C4×D4—1222no—
391C4×D4—1222no—
392C4×D4—1222no—
393C4×D4—1222no—
394C4×D4—1221no—
395C4×D4—1221no—
396C4×D4—1222no—
397C4×D4—1222no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1221no—
401C4×D4—1221no—
402#117—612no—
403#117—612no—
404#117—1222no—
405#117—1222no—
406#117—1222no—
407#117—1222no—
408#117—1221no—
409#117—1221no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4C2×#692311yescharacteristic
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C42—21no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—1222no—
430C2×C42—1222no—
431C2×C42—1222no—
432C2×C42—1222no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42C2×#692311yescharacteristic
436C4⋊C8—614no—
437C4⋊C8—1222no—
438C4⋊C8—1222no—
439C4⋊C8—1222no—
440C22⋊C8—1222no—
441C22⋊C8—1222no—
442C22⋊C8—1222no—
443C22⋊C8—1222no—
444#97—1222no—
445#97—1222no—
446#97—1222no—
447C4×C8—614no—
448C24—611no—
449C24—1221no—
450#41—618no—
451#41—1224no—
452#41—1224no—
453#41—1228no—
454#41—1228no—
455#41—1224no—
456C2×Q8—24no—
457C2×Q8—21no—
458C2×Q8C22×#692312yes—
459C2×D4—612no—
460C2×D4—1224no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1228no—
464C2×D4—1221no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1224no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1228no—
488C22×C4—1221no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—2442no—
492C22×C4—2444no—
493C22×C4—2442no—
494C22×C4—2444no—
495C22×C4C22⋊#692311yescharacteristic
496C22×C4C22⋊#692311yescharacteristic
497C22×C4C22×#692311yescharacteristic
498M16—2444no—
499C2×C8—1224no—
500C2×C8—1224no—
501C2×C8—2444no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—612no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1221no—
509C4⋊C4C22×#692312yes—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C42—22no—
517C42—22no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—611no—
524C42—611no—
525C42—611no—
526C42—611no—
527C42—611no—
528C42—611no—
529C42—611no—
530C42—1222no—
531C42—1222no—
532C42—1222no—
533C42—1222no—
534C42C22⋊#692311yescharacteristic
535C42C22⋊#692311yescharacteristic
536C42C22×#692311yescharacteristic
537C42C22×#692311yescharacteristic
538C23—611no—
539C23—611no—
540C23—1222no—
541C23—1224no—
542C23—1222no—
543C23—1222no—
544C23—2441no—
545C23—2442no—
546C23—2442no—
547C23C2×C22⋊#692311yescharacteristic
548Q8—24no—
549Q8—22no—
550Q8D61⋊M1614yes—
551D4—614no—
552D4—1228no—
553D4—1228no—
554D4—1224no—
555D4—1222no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—614no—
566C2×C4—611no—
567C2×C4—611no—
568C2×C4—611no—
569C2×C4—611no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1228no—
574C2×C4—1222no—
575C2×C4—1228no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2444no—
588C2×C4—2444no—
589C2×C4—2444no—
590C2×C4—2444no—
591C2×C4—2442no—
592C2×C4C2×C22⋊#692311yescharacteristic
593C2×C4C2×C22⋊#692311yescharacteristic
594C2×C4C2×C22⋊#692311yescharacteristic
595C2×C4C2×C22⋊#692311yescharacteristic
596C2×C4C2×C22⋊#692311yescharacteristic
597C2×C4C23×#692311yescharacteristic
598C8—2444no—
599C8—2444no—
600C22—21no—
601C22—21no—
602C22—612no—
603C22—1221no—
604C22—1221no—
605C22—1224no—
606C22—1221no—
607C22—1221no—
608C22—1221no—
609C22—2444no—
610C22—2448no—
611C22—2442no—
612C22—2442no—
613C22#1684076112yes—
614C22C22×C22⋊#692311yescharacteristic
615C4—24no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—42no—
622C4—612no—
623C4—612no—
624C4—1221no—
625C4—1221no—
626C4—1221no—
627C4—1221no—
628C4—2442no—
629C4C2×D61⋊M1612yes—
630C2—22no—
631C2—1222no—
632C2—2441no—
633C2—2444no—
634C2D61⋊#40711yescharacteristic
635C2#2146243911yescharacteristic
636C2C2×#1684076111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758226C4
cosocleC4×Dic61C24
hypercenterC2×C4×Q8#6923
hypocenterC61#17452
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #21462497C2semidirect
2. D61⋊#432C2semidirect
3. #21461161C2semidirect
4. #21461160C2semidirect
5. #21461153C2semidirect
6. #21461117C2semidirect
11. #16840784C22semidirect
14. #16840781C22semidirect
15. #16840780C22semidirect
17. D61⋊(C4⋊C8)C22semidirect
22. Q8⋊#92420C22semidirect
23. #16840610C22semidirect
148. #11758226M16semidirect
361. C61#17452semidirect

Maximal subgroups

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

subgroupindex
#214624972
D61⋊#4322
#214611612
#214611602
#214611532
#214611172
C2×C4×#117582262
#1745261