Finite Group Databasebeta

← #30571060

Subgroups — #30571060

Order 15616.

1814 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840497C211yesmaximal, characteristic, complemented
2#21458827C214yesmaximal, complemented
3#21458815C214yesmaximal, complemented
4#21458515C214yesmaximal, complemented
5#21458479C214yesmaximal, complemented
6#21456780C214yesmaximal, complemented
7#21456757C211yesmaximal, characteristic, complemented
8#21456734C211yesmaximal, characteristic
9#21455862C214yesmaximal, complemented
10#21455854C211yesmaximal, characteristic, complemented
11#21455846C211yesmaximal, characteristic, complemented
12#21454900C211yesmaximal, characteristic, complemented
13C61×#3507C211yesmaximal, characteristic, complemented
14#16840497—22no—
15#16840497—22no—
16#16840497C2214yes—
17#16840497C2214yes—
18C2×#14519774C2211yescharacteristic
19C2×#14519766—21no—
20C2×#14519765C2212yes—
21C2×#14519762—22no—
22C2×C4×Dic122C2211yescharacteristic
23C2×C4×D244C2212yes—
24C2×#14519739—21no—
25#16840144C2218yescomplemented
26#16840138C2212yes—
27#16840136C2212yes—
28#16840119C2218yescomplemented
29#16840110C2212yes—
30#16840110C2212yes—
31#16840095C2212yes—
32#16840078C2212yes—
33#16840059C2218yescomplemented
34#16840054C2212yescomplemented
35#16840044C2212yescomplemented
36#16839984C2218yescomplemented
37#16839977C2218yescomplemented
38#16839974C2218yescomplemented
39#16839969C2212yescomplemented
40#16839962C2214yes—
41#16839959C2212yescomplemented
42#16839957C2214yes—
43#16839917C2212yes—
44#16839912C2214yes—
45#16839900C2212yes—
46#16839895C2214yes—
47Q8×#92418C2214yescomplemented
48Q8×#92418C2214yescomplemented
49C4×#11758202C2214yescomplemented
50C4×#11758201C2218yescomplemented
51C4×#11758200C2214yescomplemented
52C4×#11758199C2218yescomplemented
53#16839815C2211yescharacteristic
54#16839815C2211yescharacteristic
55#16839815C2214yes—
56C61×#559C2212yescomplemented
57C61×#557C2212yescomplemented
58C61×#556—22no—
59C61×#556C2212yescomplemented
60C61×#554C2212yescomplemented
61C61×#552—22no—
62C61×#539—22no—
63C61×#536—21no—
64C61×#533C2211yescharacteristic
65C61×#532C2211yescharacteristic
66C61×#529C2214yes—
67C61×#520—21no—
68C2×C4×Q8×C61C2211yescharacteristic
69C22×Dic122C2311yescharacteristic
70C22×D244—21no—
71C22×D244—21no—
72C22×D244C2312yes—
73C22×C4×D61—22no—
74C2×#11758202C2314yes—
75C2×#11758200C2314yes—
76C2×#11758201C2318yes—
77C2×#11758199C2318yes—
78#14519774—21no—
79#14519774—21no—
80#14519774C2312yes—
81#14519774C2312yes—
82#14519766—24no—
83#14519766—24no—
84#14519765—24no—
85#14519765C2318yes—
86#14519762—28no—
87#14519762—28no—
88C4×Dic122—21no—
89C4×Dic122—21no—
90C4×Dic122C2312yes—
91C4×Dic122C2312yes—
92C4×D244—24no—
93C4×D244C2318yes—
94#14519739—24no—
95#14519739—24no—
96C2×C4⋊Dic61C2311yescharacteristic
97C2×#11758196—21no—
98C2×#11758198—22no—
99C2×#11758198—22no—
100C2×#11758198—22no—
101C2×C4×Dic61—21no—
102#14519710C2314yes—
103#14519708C2314yes—
104C4⋊#92418C2312yes—
105C4⋊#92418C2312yes—
106C4⋊#92418C2318yes—
107#14519701C2314yes—
108#14519700C2314yes—
109#14519700C2314yes—
110#14519698C2314yes—
111#14519697C2318yes—
112#14519697C2318yes—
113C22⋊#92418C2312yes—
114C22⋊#92418C2312yes—
115C4×#92418C2318yes—
116C4×#92418C2312yes—
117C4×#92418C2312yes—
118C2×#41×C61—21no—
119C22×Q8×C61D411yescharacteristic
120C4⋊Q8×C61—22no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124#127×C61C2311yescharacteristic
125#126×C61—24no—
126#126×C61—24no—
127#125×C61—24no—
128#125×C61—21no—
129#125×C61C2314yes—
130#124×C61—22no—
131#124×C61—21no—
132#124×C61C2314yes—
133#123×C61—24no—
134#123×C61—24no—
135C22⋊Q8×C61—22no—
136C22⋊Q8×C61—22no—
137C22⋊Q8×C61—21no—
138C22⋊Q8×C61—24no—
139C22⋊Q8×C61C2312yes—
140#121×C61—21no—
141#121×C61C2312yes—
142C4×Q8×C61—22no—
143C4×Q8×C61—21no—
144C4×Q8×C61D412yes—
145C4×Q8×C61D412yes—
146C4×Q8×C61C2312yes—
147C4×Q8×C61C2314yes—
148C4×Q8×C61C2312yes—
149C4×Q8×C61C2312yes—
150C4×D4×C61—24no—
151C4×D4×C61—22no—
152C4×D4×C61—21no—
153C4×D4×C61C2312yes—
154C4×D4×C61C2314yes—
155#117×C61—21no—
156#117×C61—22no—
157#117×C61—24no—
158C2×C4⋊C4×C61—22no—
159C2×C4⋊C4×C61D412yes—
160C2×C4⋊C4×C61D411yescharacteristic
161C2×C42×C61C2311yescharacteristic
162C2×C42×C61C2312yes—
163C23×D61—22no—
164#41×C61—44no—
165C2×Q8×C61—24no—
166C2×Q8×C61—21no—
167C2×Q8×C61—21no—
168C2×Q8×C61C2×D412yes—
169C2×Q8×C61C2×D412yes—
170C2×Q8×C61C2412yes—
171C2×D4×C61—22no—
172C2×D4×C61—21no—
173C2×D4×C61—21no—
174C2×D4×C61C2412yes—
175C22×C4×C61—22no—
176C22×C4×C61—24no—
177C22×C4×C61C2×D412yes—
178C22×C4×C61C2×D412yes—
179C22×C4×C61C2×D411yescharacteristic
180C22×C4×C61C2×D411yescharacteristic
181C22×C4×C61C2411yescharacteristic
182C22×Dic61—21no—
183C22×Dic61—21no—
184C2×D244—22no—
185C2×D244—22no—
186C2×D244—22no—
187C2×D244—22no—
188C2×D244—22no—
189C2×D244—44no—
190C2×D244—44no—
191C2×D244—82no—
192C2×D244C2414yes—
193C2×C4×D61—28no—
194C2×C4×D61—84no—
195C2×Dic122—21no—
196C2×Dic122—42no—
197C2×Dic122C2412yes—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—24no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—22no—
202C4⋊C4×C61—24no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C4⋊C4×C61—22no—
206C4⋊C4×C61—24no—
207C4⋊C4×C61C2×D412yes—
208C4⋊C4×C61C2×D412yes—
209C4⋊C4×C61C2×D414yes—
210C4⋊C4×C61C2×D414yes—
211C4⋊C4×C61C2412yes—
212C4⋊C4×C61C2412yes—
213C22⋊C4×C61—22no—
214C22⋊C4×C61—22no—
215C22⋊C4×C61—24no—
216C22⋊C4×C61—24no—
217C22⋊C4×C61—22no—
218C22⋊C4×C61—24no—
219C42×C61—24no—
220C42×C61—22no—
221C42×C61C2×D414yes—
222C42×C61C2×D411yescharacteristic
223C42×C61C2×D411yescharacteristic
224C42×C61C2411yescharacteristic
225C42×C61C2414yes—
226C42×C61C2411yescharacteristic
227#11758202—44no—
228#11758201—48no—
229#11758200—44no—
230#11758199—48no—
231#11758198—28no—
232#11758198—28no—
233#11758198—28no—
234C4⋊Dic61—21no—
235C4⋊Dic61C2412yes—
236#11758196—22no—
237#11758196—22no—
238C4×Dic61—22no—
239C4×Dic61—22no—
240C2×#92418C2414yes—
241C2×#92418C2414yes—
242C23×C61—22no—
243C23×C61C22×D411yescharacteristic
244C22×D61—24no—
245C22×D61—44no—
246C22×D61—44no—
247C22×D61—44no—
248Q8×C61—24no—
249Q8×C61—24no—
250Q8×C61—24no—
251Q8×C61—42no—
252Q8×C61—81no—
253D4×C61—24no—
254D4×C61—44no—
255D4×C61—42no—
256D4×C61—81no—
257C2×C4×C61—24no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61—42no—
263C2×C4×C61—41no—
264C2×C4×C61—44no—
265C2×C4×C61—84no—
266C2×C4×C61#13514yes—
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D414yes—
271C2×C4×C61C22×D414yes—
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C2511yescharacteristic
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279D244—28no—
280D244—44no—
281D244—44no—
282D244—44no—
283D244—82no—
284D244—82no—
285C4×D61—88no—
286Dic122—22no—
287Dic122—22no—
288Dic122—42no—
289#92418—24no—
290#92418—24no—
291#61885—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61C2×#13512yes—
298C22×C61C23×D411yescharacteristic
299C2×D61—44no—
300C2×D61—44no—
301C2×D61—44no—
302C2×D61—88no—
303C4×C61—24no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—24no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61C2×#13512yes—
312C4×C61C2×#13512yes—
313C4×C61#58512yes—
314Dic61—42no—
315Dic61—42no—
316#3507D6111yescharacteristic, complemented
317C2×#559—611no—
318#3396—614no—
319#3394—614no—
320#3365—614no—
321#3361—614no—
322#3211—614no—
323#3208—611no—
324#3205—611no—
325#3105—614no—
326#3104—611no—
327#3103—611no—
328#2982—611no—
329C2×C61—22no—
330C2×C61—82no—
331C2×C61#348711yescharacteristic
332C2×C61C22×#13511yescharacteristic
333C2×C61#358511yescharacteristic
334D61—84no—
335#559—614no—
336#559—614no—
337#559—1222no—
338#559—1222no—
339#559C2×D6112yescomplemented
340#557C2×D6112yescomplemented
341#556—22no—
342#556C2×D6112yescomplemented
343#554C2×D6112yescomplemented
344#552—22no—
345#539—22no—
346#536—21no—
347#533C2×D6111yescharacteristic
348#532C2×D6111yescharacteristic
349C2×#127—611no—
350#529C2×D6114yes—
351C2×#124—612no—
352C2×#124—1221no—
353C2×#123—1222no—
354#520—21no—
355C2×C4×Q8—611no—
356C2×C4×Q8C2×D6111yescharacteristic
357C2×C4×D4—612no—
358C2×C4×D4—1221no—
359#495—618no—
360#493—612no—
361#492—612no—
362#488—618no—
363#486—612no—
364#486—612no—
365#484—612no—
366#481—612no—
367#478—618no—
368#477—612no—
369#475—612no—
370#464—618no—
371#463—618no—
372#463—618no—
373#462—612no—
374#460—612no—
375#460—614no—
376#459—614no—
377#451—612no—
378#450—614no—
379#448—612no—
380#447—614no—
381C8×Q8—614no—
382C8×Q8—614no—
383C4×Q16—614no—
384C4×SD16—618no—
385C4×SD16—614no—
386C4×D8—618no—
387#431—611no—
388#431—611no—
389#431—614no—
390C61#6188511yescharacteristic, complemented, Sylow
391C2×#41—21no—
392C22×Q8—611no—
393C22×Q8C22⋊D6111yescharacteristic
394C22×D4—612no—
395C22×D4—1221no—
396C22×D4—1221no—
397C23×C4—1222no—
398C2×Q16—614no—
399C2×SD16—618no—
400C2×SD16—614no—
401C2×D8—618no—
402C4⋊Q8—22no—
403C4⋊Q8C22×D6111yescharacteristic
404C4⋊Q8C22×D6111yescharacteristic
405C4⋊Q8C22×D6111yescharacteristic
406#127—612no—
407#127—612no—
408#127—1221no—
409#127—1221no—
410#127C22×D6111yescharacteristic
411#126—24no—
412#126—24no—
413#125—24no—
414#125—21no—
415#125C22×D6114yes—
416#124—22no—
417#124—21no—
418#124—618no—
419#124—1224no—
420#124—1224no—
421#124—1224no—
422#124C22×D6114yes—
423#123—24no—
424#123—24no—
425#123—1228no—
426#123—1228no—
427C22⋊Q8—22no—
428C22⋊Q8—22no—
429C22⋊Q8—21no—
430C22⋊Q8—24no—
431C22⋊Q8C22×D6112yes—
432#121—21no—
433#121C22×D6112yes—
434C4×Q8—22no—
435C4×Q8—21no—
436C4×Q8—612no—
437C4×Q8—612no—
438C4×Q8—1221no—
439C4×Q8—1221no—
440C4×Q8C22⋊D6112yes—
441C4×Q8C22⋊D6112yes—
442C4×Q8C22×D6114yes—
443C4×Q8C22×D6112yes—
444C4×Q8C22×D6112yes—
445C4×Q8C22×D6112yes—
446C4×D4—24no—
447C4×D4—22no—
448C4×D4—21no—
449C4×D4—618no—
450C4×D4—1224no—
451C4×D4—1224no—
452C4×D4—1224no—
453C4×D4C22×D6114yes—
454C4×D4C22×D6112yes—
455#117—21no—
456#117—22no—
457#117—24no—
458C2×C4⋊C4—22no—
459C2×C4⋊C4—611no—
460C2×C4⋊C4—1221no—
461C2×C4⋊C4C22⋊D6111yescharacteristic
462C2×C4⋊C4C22⋊D6112yes—
463C2×C22⋊C4—1222no—
464C2×C22⋊C4—1222no—
465C2×C22⋊C4—1222no—
466C2×C42—1221no—
467C2×C42C22×D6112yes—
468C2×C42C22×D6111yescharacteristic
469#107—614no—
470#106—614no—
471C4⋊C8—612no—
472C4⋊C8—612no—
473C4⋊C8—618no—
474#103—614no—
475#103—614no—
476#103—614no—
477#102—614no—
478#102—618no—
479#102—618no—
480C22⋊C8—612no—
481C22⋊C8—612no—
482C4×C8—618no—
483C4×C8—612no—
484C4×C8—612no—
485C24—1222no—
486#41—44no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—24no—
490C2×Q8—612no—
491C2×Q8—1221no—
492C2×Q8—2442no—
493C2×Q8C2×C22⋊D6112yes—
494C2×Q8C2×C22⋊D6112yes—
495C2×Q8C23×D6112yes—
496C2×D4—22no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—614no—
500C2×D4—1222no—
501C2×D4—1222no—
502C2×D4—1222no—
503C2×D4—1222no—
504C2×D4—1222no—
505C2×D4—4882no—
506C2×D4—2444no—
507C2×D4—2444no—
508C2×D4C23×D6112yes—
509C22×C4—22no—
510C22×C4—24no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1228no—
514C22×C4—4884no—
515C22×C4C2×C22⋊D6112yes—
516C22×C4C2×C22⋊D6112yes—
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C2×C22⋊D6111yescharacteristic
519C22×C4C23×D6111yescharacteristic
520Q16—2444no—
521SD16—2444no—
522SD16—2448no—
523D8—2448no—
524C2×C8—614no—
525C2×C8—614no—
526C4⋊C4—21no—
527C4⋊C4—24no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—612no—
536C4⋊C4—1221no—
537C4⋊C4—1222no—
538C4⋊C4—1222no—
539C4⋊C4C2×C22⋊D6112yes—
540C4⋊C4C2×C22⋊D6112yes—
541C4⋊C4C2×C22⋊D6114yes—
542C4⋊C4C2×C22⋊D6114yes—
543C4⋊C4C23×D6112yes—
544C4⋊C4C23×D6112yes—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C22⋊C4—24no—
551C22⋊C4—1228no—
552C22⋊C4—1228no—
553C22⋊C4—1228no—
554C42—24no—
555C42—22no—
556C42—1222no—
557C42—1222no—
558C42C2×C22⋊D6114yes—
559C42C2×C22⋊D6111yescharacteristic
560C42C2×C22⋊D6111yescharacteristic
561C42C23×D6114yes—
562C42C23×D6111yescharacteristic
563C42C23×D6111yescharacteristic
564C23—22no—
565C23—1224no—
566C23—2444no—
567C23—2444no—
568C23—2444no—
569C23C22×C22⋊D6111yescharacteristic
570Q8—24no—
571Q8—24no—
572Q8—24no—
573Q8—42no—
574Q8—81no—
575Q8—1222no—
576Q8—1222no—
577Q8—2442no—
578D4—24no—
579D4—44no—
580D4—42no—
581D4—81no—
582D4—1228no—
583D4—4882no—
584D4—4882no—
585D4—2444no—
586D4—2444no—
587D4—2444no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—84no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—4888no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4#1451979714yes—
603C2×C4C22×C22⋊D6114yes—
604C2×C4C22×C22⋊D6111yescharacteristic
605C2×C4C22×C22⋊D6114yes—
606C2×C4C22×C22⋊D6111yescharacteristic
607C2×C4C22×C22⋊D6111yescharacteristic
608C2×C4C22×C22⋊D6111yescharacteristic
609C2×C4C22×C22⋊D6111yescharacteristic
610C2×C4C24×D6111yescharacteristic
611C8—1224no—
612C8—1224no—
613C22—21no—
614C22—21no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—4888no—
619C22—2444no—
620C22—2444no—
621C22—2444no—
622C22C2×#1451979712yes—
623C22C23×C22⋊D6111yescharacteristic
624C4—21no—
625C4—24no—
626C4—24no—
627C4—21no—
628C4—21no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—2442no—
633C4—2442no—
634C4C2×#1451979712yes—
635C4C2×#1451979712yes—
636C4#1684059912yes—
637C2—22no—
638C2—82no—
639C2—4884no—
640C2#2145947411yescharacteristic
641C2C22×#1451979711yescharacteristic
642C2#2146059811yescharacteristic
subgroupG/H
centerC22C23×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3507C2
socleC22×C61C23×D4
prime radical#21456734C2
hypercenter#3507D61
hypocenterC61#61885
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3507

Complements

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

NH ≅ G/Ntype
1. C2×#16840497C2semidirect
2. #21458827C2semidirect
3. #21458815C2semidirect
4. #21458515C2semidirect
5. #21458479C2semidirect
6. #21456780C2semidirect
7. #21456757C2semidirect
9. #21455862C2semidirect
10. #21455854C2semidirect
11. #21455846C2semidirect
12. #21454900C2semidirect
13. C61×#3507C2semidirect
25. #16840144C22semidirect
28. #16840119C22semidirect
33. #16840059C22semidirect
34. #16840054C22semidirect
35. #16840044C22semidirect
36. #16839984C22semidirect
37. #16839977C22semidirect
38. #16839974C22semidirect
39. #16839969C22semidirect
41. #16839959C22semidirect
47. Q8×#92418C22semidirect
48. Q8×#92418C22semidirect
49. C4×#11758202C22semidirect
50. C4×#11758201C22semidirect
51. C4×#11758200C22semidirect
52. C4×#11758199C22semidirect
56. C61×#559C22semidirect
57. C61×#557C22semidirect
59. C61×#556C22semidirect
60. C61×#554C22semidirect
316. #3507D61semidirect
339. #559C2×D61semidirect
340. #557C2×D61semidirect
342. #556C2×D61semidirect
343. #554C2×D61semidirect
390. C61#61885semidirect

Maximal subgroups

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

subgroupindex
C2×#168404972
#214588272
#214588152
#214585152
#214584792
#214567802
#214567572
#214567342
#214558622
#214558542
#214558462
#214549002
C61×#35072
#6188561