Finite Group Databasebeta

← #30571046

Subgroups — #30571046

Order 15616.

1958 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459585C211yesmaximal, characteristic
2C2×#16840487C211yesmaximal, characteristic, complemented
3#21458797C214yesmaximal, complemented
4#21458795C214yesmaximal, complemented
5#21458458C214yesmaximal, complemented
6#21458456C214yesmaximal, complemented
7#21456744C211yesmaximal, characteristic, complemented
8#21456742C214yesmaximal, complemented
9#21455851C211yesmaximal, characteristic, complemented
10#21455846C211yesmaximal, characteristic, complemented
11#21455845C214yesmaximal, complemented
12#21454899C211yesmaximal, characteristic, complemented
13C61×#3206C211yesmaximal, characteristic, complemented
14C2×#14519824—21no—
15#16840487—22no—
16#16840487—22no—
17#16840487C2214yes—
18#16840487C2214yes—
19#16840433C2212yes—
20#16840432C2212yes—
21D4×Dic122C2212yes—
22#16840427C2212yes—
23#16840421—22no—
24#16840420—22no—
25#16840340—21no—
26#16840339—22no—
27#16840312C2211yescharacteristic
28#16840307C2214yes—
29C2×#14519775C2212yes—
30C2×#14519768—21no—
31C2×#14519767—22no—
32C2×Q8×Dic61C2212yes—
33C2×C4×Dic122C2211yescharacteristic
34C2×C4×Dic122C2211yescharacteristic
35C2×#14519742—21no—
36#16840136C2212yes—
37#16840133C2218yes—
38#16840112C2212yes—
39#16840111C2218yes—
40#16840081C2212yes—
41#16840078C2212yes—
42#16840048C2212yes—
43#16840045C2218yes—
44#16840044C2212yes—
45#16839966C2218yes—
46#16839965C2218yes—
47#16839964C2214yes—
48#16839963C2212yes—
49#16839960C2218yes—
50#16839959C2212yes—
51#16839903C2212yes—
52#16839901C2214yes—
53#16839900C2212yes—
54C8×Dic122C2214yes—
55#16839886C2214yes—
56C4×#11758190C2214yes—
57SD16×Dic61C2218yes—
58C4×#11758200C2214yes—
59#16839847C2218yes—
60#16839815C2211yescharacteristic
61#16839812C2214yes—
62C61×#532C2211yescharacteristic
63C2×C4⋊Q8×C61C2211yescharacteristic
64C61×#492C2212yes—
65C61×#487C2212yes—
66C61×#461C2214yes—
67C61×#448C2214yes—
68C61×#431C2211yescharacteristic
69#14519824—48no—
70C2×#11758222—21no—
71C2×#11758222—22no—
72C2×#11758224—24no—
73C2×#11758224—22no—
74C22×Dic122—22no—
75C22×Dic122D411yescharacteristic
76C22×Dic122C2311yescharacteristic
77C2×Q8×D61—22no—
78C2×#11758190C2314yes—
79C2×#11758200C2314yes—
80C2×#11758201C2318yes—
81#14519777C2311yescharacteristic
82#14519776C2314yes—
83#14519775—24no—
84#14519775C2318yes—
85#14519774C2311yescharacteristic
86#14519773C2314yes—
87#14519768—24no—
88#14519768—24no—
89#14519767—28no—
90#14519767—28no—
91#14519765—21no—
92#14519763—24no—
93#14519756—22no—
94#14519755—24no—
95#14519754—24no—
96#14519753—22no—
97#14519751—21no—
98#14519751—22no—
99#14519751C2312yes—
100#14519750—24no—
101#14519749—24no—
102#14519749—24no—
103#14519748—22no—
104Q8×Dic61—24no—
105Q8×Dic61C2318yes—
106C4×Dic122—21no—
107C4×Dic122—21no—
108C4×Dic122D412yes—
109C4×Dic122D412yes—
110C4×Dic122C2312yes—
111C4×Dic122C2312yes—
112C4×Dic122C2312yes—
113C4×Dic122C2312yes—
114#14519742—24no—
115#14519742—24no—
116C4×D244—21no—
117C4×D244C2312yes—
118D4×Dic61C2314yes—
119#14519739C2314yes—
120#14519737—24no—
121C2×C4⋊Dic61—22no—
122C2×C4⋊Dic61D411yescharacteristic
123C2×C4⋊Dic61C2311yescharacteristic
124C2×#11758196—21no—
125C2×#11758196—22no—
126C2×#11758196—22no—
127C2×#11758196D412yes—
128C2×C4×Dic61—21no—
129C2×C4×Dic61—22no—
130C2×C4×Dic61C2312yes—
131#14519709C2314yes—
132#14519708C2314yes—
133C4⋊#92418C2312yes—
134#14519706C2318yes—
135#14519702C2314yes—
136#14519701C2318yes—
137#14519700C2314yes—
138#14519699C2314yes—
139#14519698C2314yes—
140C22⋊#92418C2312yes—
141C4×#92418C2312yes—
142C8×Dic61C2318yes—
143C2×#41×C61—21no—
144C22×Q8×C61C2312yes—
145C2×SD16×C61C2318yes—
146C4⋊Q8×C61—21no—
147C4⋊Q8×C61—21no—
148C4⋊Q8×C61C2312yes—
149C4⋊Q8×C61C2312yes—
150C4⋊Q8×C61C2311yescharacteristic
151#127×C61C2311yescharacteristic
152#124×C61—21no—
153#121×C61—21no—
154#121×C61C2312yes—
155C4×D4×C61—21no—
156C4×D4×C61C2312yes—
157C2×C4⋊C4×C61—21no—
158C2×C4⋊C4×C61—21no—
159C2×C42×C61C2311yescharacteristic
160C4⋊C8×C61C2312yes—
161#103×C61C2318yes—
162C22⋊C8×C61C2312yes—
163C4×C8×C61C2312yes—
164#41×C61—44no—
165C2×Q8×C61—21no—
166C2×Q8×C61—22no—
167C2×Q8×C61—44no—
168C2×Q8×C61C2414yes—
169C2×D4×C61—22no—
170C2×D4×C61—21no—
171C2×D4×C61—21no—
172C2×D4×C61C2412yes—
173C22×C4×C61—22no—
174C22×C4×C61—22no—
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61C2411yescharacteristic
178C2×C22⋊D61—22no—
179C2×C22⋊D61—24no—
180C2×C22⋊D61—24no—
181C22×Dic61—21no—
182C22×Dic61—21no—
183C22×Dic61—24no—
184C22×Dic61C2×D412yes—
185C22×Dic61C2×D412yes—
186Q8×D61—48no—
187#11758224—416no—
188#11758224—48no—
189#11758222—44no—
190#11758222—48no—
191C2×D244—21no—
192C2×D244—21no—
193C2×D244C2412yes—
194C2×C4×D61—22no—
195C2×C4×D61—24no—
196C2×Dic122—21no—
197C2×Dic122—21no—
198C2×Dic122—24no—
199C2×Dic122—42no—
200C2×Dic122—48no—
201C2×Dic122SD1618yescomplemented
202C2×Dic122C2×D412yes—
203C2×Dic122C2×D412yes—
204C2×Dic122C2412yes—
205SD16×C61—48no—
206C2×C8×C61C2414yes—
207C4⋊C4×C61—22no—
208C4⋊C4×C61—22no—
209C4⋊C4×C61—22no—
210C4⋊C4×C61—22no—
211C4⋊C4×C61—21no—
212C4⋊C4×C61C2412yes—
213C22⋊C4×C61—22no—
214C22⋊C4×C61—22no—
215C42×C61C2×D411yescharacteristic
216C42×C61C2×D411yescharacteristic
217C42×C61C2411yescharacteristic
218C42×C61C2411yescharacteristic
219C22⋊Dic61—24no—
220#11758201—48no—
221#11758200—44no—
222#11758198—22no—
223#11758198—22no—
224#11758198—24no—
225C4⋊Dic61—21no—
226C4⋊Dic61—21no—
227C4⋊Dic61—28no—
228C4⋊Dic61C2×D412yes—
229C4⋊Dic61C2×D412yes—
230C4⋊Dic61C2412yes—
231C4⋊Dic61C2412yes—
232#11758196—22no—
233#11758196—22no—
234#11758196—28no—
235#11758196—28no—
236#11758196—24no—
237#11758196C2×D414yes—
238#11758196C2×D414yes—
239C4×Dic61—22no—
240C4×Dic61—22no—
241C4×Dic61—28no—
242C4×Dic61C2×D414yes—
243C4×Dic61C2414yes—
244C2×#92418C2414yes—
245#11758190—44no—
246C23×C61—22no—
247C23×C61C22×D411yescharacteristic
248C22×D61—22no—
249Q8×C61—28no—
250Q8×C61—44no—
251Q8×C61—42no—
252D4×C61—24no—
253D4×C61—44no—
254D4×C61—42no—
255D4×C61—81no—
256C2×C4×C61—22no—
257C2×C4×C61—24no—
258C2×C4×C61—22no—
259C2×C4×C61—41no—
260C2×C4×C61—44no—
261C2×C4×C61—44no—
262C2×C4×C61C2×SD1614yes—
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C2511yescharacteristic
269C22⋊D61—44no—
270C22⋊D61—48no—
271C22⋊D61—84no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—48no—
279C2×Dic61C2×SD1618yes—
280C2×Dic61C22×D414yes—
281C2×Dic61C22×D414yes—
282D244—24no—
283D244—42no—
284D244—81no—
285C4×D61—44no—
286C4×D61—48no—
287Dic122—22no—
288Dic122—22no—
289Dic122—42no—
290Dic122—42no—
291Dic122—48no—
292Dic122C2×SD1618yes—
293Dic122C2×SD1618yes—
294C8×C61—24no—
295#92418—24no—
296#61883—611nomaximal, Sylow
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61C22×SD1612yes—
303C22×C61C23×D411yescharacteristic
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C4×C61—21no—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—44no—
311C4×C61—42no—
312C4×C61C22×SD1612yes—
313C4×C61C22×SD1612yes—
314C4×C61#58512yes—
315Dic61—24no—
316Dic61—42no—
317Dic61—42no—
318Dic61—42no—
319Dic61C22×SD1618yes—
320#3497—611no—
321C2×#557—611no—
322#3392—614no—
323#3392—614no—
324#3359—614no—
325#3359—614no—
326#3206—614no—
327#3206—611no—
328#3206D6111yescharacteristic, complemented
329#3103—614no—
330#3103—611no—
331#3103—611no—
332#2982—611no—
333C2×C61—22no—
334C2×C61—82no—
335C2×C61#348711yescharacteristic
336C2×C61C23×SD1611yescharacteristic
337C2×C61#358611yescharacteristic
338D61—82no—
339C2×2−1+4—1221no—
340#557—614no—
341#557—614no—
342#557—1222no—
343#557—1222no—
344#550—612no—
345#550—612no—
346D4×Q8—612no—
347D4×Q8—612no—
348#548—1222no—
349#548—1222no—
350#537—1222no—
351#537—1221no—
352#532—614no—
353#532—611no—
354#532C2×D6111yescharacteristic
355C2×C4⋊Q8—612no—
356C2×C4⋊Q8C2×D6111yescharacteristic
357C2×#125—1222no—
358C2×#125—1221no—
359C2×C4×Q8—611no—
360C2×C4×Q8—612no—
361C2×C4×Q8—611no—
362C2×C4×Q8—1221no—
363#492—618no—
364#492—612no—
365#492C2×D6112yes—
366#487—612no—
367#487—618no—
368#487C2×D6112yes—
369#481—612no—
370#481—612no—
371#475—612no—
372#475—612no—
373#475—618no—
374#461—618no—
375#461—614no—
376#461—618no—
377#461C2×D6114yes—
378#460—618no—
379#460—612no—
380#460—612no—
381#448—614no—
382#448—612no—
383#448—612no—
384#448C2×D6114yes—
385C8×Q8—614no—
386C8×Q8—614no—
387C4×SD16—618no—
388C4×SD16—614no—
389C4×SD16—618no—
390C4×SD16—614no—
391#431—611no—
392#431—614no—
393#431C2×D6111yescharacteristic
394C61#6188311yescharacteristic, complemented, Sylow
3952−1+4—2448no—
396C2×#41—21no—
397C2×#41—1222no—
398C2×#41—1224no—
399C2×#41—1222no—
400C2×#41—1221no—
401C22×Q8—611no—
402C22×Q8—611no—
403C22×Q8—1222no—
404C22×Q8—1222no—
405C22×Q8C22×D6112yes—
406C2×SD16—618no—
407C2×SD16—614no—
408C2×SD16—614no—
409C2×SD16C22×D6118yes—
410C4⋊Q8—21no—
411C4⋊Q8—21no—
412C4⋊Q8—614no—
413C4⋊Q8—618no—
414C4⋊Q8—611no—
415C4⋊Q8—1224no—
416C4⋊Q8C22×D6112yes—
417C4⋊Q8C22×D6112yes—
418C4⋊Q8C22×D6111yescharacteristic
419#127—614no—
420#127—611no—
421#127C22×D6111yescharacteristic
422#125—1228no—
423#125—1228no—
424#125—1224no—
425#125—1224no—
426#124—21no—
427#124—1224no—
428#124—1221no—
429C22⋊Q8—1224no—
430C22⋊Q8—1224no—
431C22⋊Q8—1222no—
432C22⋊Q8—1222no—
433#121—21no—
434#121—612no—
435#121—1224no—
436#121—1222no—
437#121—1222no—
438#121—1224no—
439#121—1224no—
440#121—1221no—
441#121C22×D6112yes—
442C4×Q8—612no—
443C4×Q8—612no—
444C4×Q8—612no—
445C4×Q8—612no—
446C4×Q8—618no—
447C4×Q8—612no—
448C4×Q8—612no—
449C4×Q8—1224no—
450C4×Q8—1224no—
451C4×Q8—1224no—
452C4×Q8—1221no—
453C4×Q8—1221no—
454C4×D4—21no—
455C4×D4—614no—
456C4×D4—614no—
457C4×D4—612no—
458C4×D4—1224no—
459C4×D4—1221no—
460C4×D4C22×D6112yes—
461C2×C4⋊C4—21no—
462C2×C4⋊C4—21no—
463C2×C4⋊C4—612no—
464C2×C4⋊C4—611no—
465C2×C4⋊C4—611no—
466C2×C4⋊C4—1222no—
467C2×C4⋊C4—1222no—
468C2×C4⋊C4—1222no—
469C2×C4⋊C4—1221no—
470C2×C42—612no—
471C2×C42—1222no—
472C2×C42—1221no—
473C2×C42C22×D6111yescharacteristic
474#106—614no—
475#106—614no—
476C4⋊C8—618no—
477C4⋊C8—612no—
478C4⋊C8C22×D6112yes—
479#103—618no—
480#103—614no—
481#103—614no—
482#103C22×D6118yes—
483#102—614no—
484#102—614no—
485C22⋊C8—612no—
486C22⋊C8C22×D6112yes—
487C4×C8—618no—
488C4×C8—612no—
489C4×C8C22×D6112yes—
490#41—44no—
491#41—2448no—
492#41—24416no—
493#41—2448no—
494#41—2444no—
495C2×Q8—22no—
496C2×Q8—21no—
497C2×Q8—44no—
498C2×Q8—612no—
499C2×Q8—612no—
500C2×Q8—618no—
501C2×Q8—612no—
502C2×Q8—1224no—
503C2×Q8—1221no—
504C2×Q8—1221no—
505C2×Q8—2442no—
506C2×Q8—2448no—
507C2×Q8—2448no—
508C2×Q8C23×D6114yes—
509C2×D4—22no—
510C2×D4—21no—
511C2×D4—21no—
512C2×D4—612no—
513C2×D4—1224no—
514C2×D4—1224no—
515C2×D4—1222no—
516C2×D4—1221no—
517C2×D4—1221no—
518C2×D4C23×D6112yes—
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—612no—
522C22×C4—612no—
523C22×C4—1224no—
524C22×C4—1224no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1222no—
528C22×C4D4×D6111yescharacteristic
529C22×C4D4×D6111yescharacteristic
530C22×C4C23×D6111yescharacteristic
531SD16—48no—
532SD16—2444no—
533SD16—2444no—
534SD16—2448no—
535C2×C8—614no—
536C2×C8C23×D6114yes—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—614no—
543C4⋊C4—614no—
544C4⋊C4—612no—
545C4⋊C4—612no—
546C4⋊C4—612no—
547C4⋊C4—612no—
548C4⋊C4—1228no—
549C4⋊C4—1228no—
550C4⋊C4—1228no—
551C4⋊C4—1224no—
552C4⋊C4—1221no—
553C4⋊C4—1222no—
554C4⋊C4—1222no—
555C4⋊C4—1221no—
556C4⋊C4C23×D6112yes—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—1224no—
560C22⋊C4—1224no—
561C22⋊C4—1222no—
562C22⋊C4—1222no—
563C42—614no—
564C42—614no—
565C42—1228no—
566C42—1222no—
567C42—1222no—
568C42D4×D6111yescharacteristic
569C42D4×D6111yescharacteristic
570C42C23×D6111yescharacteristic
571C42C23×D6111yescharacteristic
572C23—22no—
573C23—1222no—
574C23C2×D4×D6111yescharacteristic
575Q8—28no—
576Q8—44no—
577Q8—42no—
578Q8—618no—
579Q8—618no—
580Q8—1222no—
581Q8—1222no—
582Q8—2442no—
583Q8—2448no—
584Q8—2442no—
585D4—24no—
586D4—44no—
587D4—42no—
588D4—81no—
589D4—1224no—
590D4—4881no—
591D4—4884no—
592D4—2448no—
593D4—2444no—
594D4—2442no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—618no—
602C2×C4—614no—
603C2×C4—614no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—2448no—
608C2×C4—2448no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2444no—
613C2×C4SD16×D6114yes—
614C2×C4C2×D4×D6111yescharacteristic
615C2×C4C2×D4×D6111yescharacteristic
616C2×C4C2×D4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4C2×D4×D6111yescharacteristic
619C2×C4C24×D6111yescharacteristic
620C8—24no—
621C8—1224no—
622C22—21no—
623C22—21no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—2442no—
628C22—2442no—
629C22—2442no—
630C22C2×SD16×D6112yes—
631C22C22×D4×D6111yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—44no—
636C4—42no—
637C4—618no—
638C4—1224no—
639C4—2442no—
640C4—2442no—
641C4—2442no—
642C4C2×SD16×D6112yes—
643C4C2×SD16×D6112yes—
644C4#1684059712yes—
645C2—22no—
646C2—82no—
647C2—4882no—
648C2#2145947511yescharacteristic
649C2C22×SD16×D6111yescharacteristic
650C2#2146060211yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3206C2
socleC22×C61C23×D4
prime radical#21459585C2
hypercenter#3206D61
hypocenterC61#61883
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
2. C2×#16840487C2semidirect
3. #21458797C2semidirect
4. #21458795C2semidirect
5. #21458458C2semidirect
6. #21458456C2semidirect
7. #21456744C2semidirect
8. #21456742C2semidirect
9. #21455851C2semidirect
10. #21455846C2semidirect
11. #21455845C2semidirect
12. #21454899C2semidirect
13. C61×#3206C2semidirect
201. C2×Dic122SD16semidirect
328. #3206D61semidirect
394. C61#61883semidirect

Maximal subgroups

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

subgroupindex
#214595852
C2×#168404872
#214587972
#214587952
#214584582
#214584562
#214567442
#214567422
#214558512
#214558462
#214558452
#214548992
C61×#32062
#6188361