Finite Group Databasebeta

← #29832825

Subgroups — #29832825

Order 15616.

1252 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459607C211yesmaximal, characteristic
2#21454280C211yesmaximal, characteristic, complemented
3#21447493C212yesmaximal, complemented
4#21447486C214yesmaximal, complemented
5#21447482C212yesmaximal, complemented
6C2×#16839556C214yesmaximal, complemented
7C61×#2883C211yesmaximal, characteristic, complemented
8C2×#14519824C411yescharacteristic, complemented
9#16840419—21no—
10#16840419—21no—
11#16840406—22no—
12#16840406—21no—
13#16840406—22no—
14#16840406—21no—
15#16840333C412yescomplemented
16#16840333C2211yescharacteristic
17#16840329C411yescharacteristic, complemented
18#16840329C2212yes—
19#16840308C412yescomplemented
20#16840308C2211yescharacteristic
21C2×#14519764C411yescharacteristic, complemented
22C2×#14519764C2212yes—
23#16840237—21no—
24#41×Dic61C411yescharacteristic, complemented
25#16839736—21no—
26#16839736C2212yes—
27C2×#14519692—22no—
28C2×#14519692C2214yes—
29#16839556C2214yescomplemented
30#16839556C2218yescomplemented
31#16839554C2214yescomplemented
32C2×H(5,𝔽2)×C61C2211yescharacteristic
33C61×#518—21no—
34C61×#410—21no—
35C61×#410C2212yes—
36C2×#99×C61—22no—
37C2×#99×C61C2214yes—
38#14519824—44no—
39C2×#11758222C2×C412yescomplemented
40C2×#11758222C2×C411yescharacteristic, complemented
41C2×#11758224—22no—
42C2×#11758224—21no—
43C22×Dic122C2×C411yescharacteristic, complemented
44C22×Dic122C2×C412yescomplemented
45C2×Q8×D61—21no—
46#14519775C2×C412yescomplemented
47#14519775C2×C411yescharacteristic, complemented
48#14519773C2×C412yescomplemented
49#14519773C2311yescharacteristic
50#14519766C2×C411yescharacteristic, complemented
51#14519764—21no—
52#14519764—21no—
53#14519764—22no—
54#14519764—22no—
55#14519764C2×C412yescomplemented
56#14519764C2×C412yescomplemented
57#14519764C2×C412yescomplemented
58#14519764C2311yescharacteristic
59#14519764C2312yes—
60#14519758—22no—
61#14519758—22no—
62#14519758—21no—
63#14519758—21no—
64#14519756—22no—
65#14519756—22no—
66#14519756—21no—
67#14519756—21no—
68#14519749—22no—
69#14519749—22no—
70#14519749—21no—
71#14519749—21no—
72#14519748—21no—
73#14519748—21no—
74#14519748—22no—
75#14519748—22no—
76#14519746—22no—
77#14519746—22no—
78#14519746—22no—
79Q8×Dic61—21no—
80D4×Dic61—22no—
81D4×Dic61—21no—
82D4×Dic61—44no—
83C4×C22⋊D61—22no—
84C4×C22⋊D61—22no—
85C4×C22⋊D61—21no—
86C4×C22⋊D61—21no—
87#14519736—22no—
88#14519736C2×C411yescharacteristic, complemented
89#14519736C2×C412yescomplemented
90C2×C22⋊Dic61—22no—
91C2×C22⋊Dic61—22no—
92C2×C22⋊Dic61—22no—
93C2×C22⋊Dic61—24no—
94C2×C4×Dic61C2×C411yescharacteristic, complemented
95C2×C4×Dic61C2×C412yescomplemented
96#14519692—22no—
97#14519692—24no—
98#14519692C2314yes—
99H(5,𝔽2)×C61—44no—
100C2×#41×C61C2×C412yes—
101C2×#41×C61C2×C411yescharacteristic
102C2×#41×C61D411yescharacteristic
103C2×#41×C61D411yescharacteristic
104C2×#41×C61C2311yescharacteristic
105C22×D4×C61C2×C411yescharacteristic
106C22×D4×C61D412yes—
107C22×D4×C61D411yescharacteristic
108C22×D4×C61D412yes—
109C22×D4×C61D411yescharacteristic
110C22×D4×C61C2312yes—
111C4×D4×C61—44no—
112#117×C61—22no—
113C2×C22⋊C4×C61—24no—
114#99×C61—22no—
115#99×C61—24no—
116#99×C61C2314yes—
117C24×C61C22⋊C412yes—
118C24×C61C2×D412yes—
119C24×C61C2×D412yes—
120#41×C61—24no—
121#41×C61—42no—
122#41×C61—44no—
123#41×C61—42no—
124#41×C61—42no—
125C2×Q8×C61C22⋊C411yescharacteristic
126C2×Q8×C61C22×C411yescharacteristic
127C2×D4×C61—24no—
128C2×D4×C61—24no—
129C2×D4×C61—42no—
130C2×D4×C61—44no—
131C2×D4×C61—44no—
132C2×D4×C61—44no—
133C2×D4×C61C22⋊C412yes—
134C2×D4×C61C22⋊C411yescharacteristic
135C2×D4×C61C22⋊C411yescharacteristic
136C2×D4×C61C22⋊C412yes—
137C2×D4×C61C22⋊C412yes—
138C2×D4×C61C22⋊C411yescharacteristic
139C2×D4×C61C22×C412yes—
140C2×D4×C61C2×D411yescharacteristic
141C2×D4×C61C2×D412yes—
142C2×D4×C61C2×D411yescharacteristic
143C2×D4×C61C2×D412yes—
144C2×D4×C61C2411yescharacteristic
145C22×C4×C61—44no—
146C22×C4×C61C22⋊C412yes—
147C22×C4×C61C22⋊C412yes—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C22×C412yes—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—21no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—21no—
157C2×C22⋊D61—21no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—21no—
161C22×Dic61—22no—
162C22×Dic61—44no—
163Q8×D61—44no—
164#11758224—48no—
165#11758224—44no—
166#11758222—44no—
167#11758222—42no—
168C2×D244C22×C411yescharacteristic
169C2×C4×D61—22no—
170C2×C4×D61—21no—
171C2×Dic122—22no—
172C2×Dic122—21no—
173C2×Dic122—44no—
174C2×Dic122—42no—
175C2×Dic122C22×C412yes—
176C2×Dic122C22×C411yescharacteristic
177C4⋊C4×C61—42no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—44no—
180C42×C61—42no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61—21no—
188C22⋊Dic61—22no—
189C22⋊Dic61—21no—
190C22⋊Dic61—21no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—24no—
194C22⋊Dic61—44no—
195#11758198—21no—
196#11758198—21no—
197C4⋊Dic61—22no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200#11758196—22no—
201#11758196—22no—
202#11758196—21no—
203#11758196—21no—
204C4×Dic61—22no—
205C4×Dic61—21no—
206C4×Dic61—42no—
207C4×Dic61C22×C411yescharacteristic
208C4×Dic61C22×C411yescharacteristic
209C4×Dic61C22×C412yes—
210C23×C61—24no—
211C23×C61—24no—
212C23×C61—42no—
213C23×C61—82no—
214C23×C61—82no—
215C23×C61—82no—
216C23×C61C2×C22⋊C412yes—
217C23×C61C2×C22⋊C412yes—
218C23×C61C2×C22⋊C411yescharacteristic
219C23×C61C2×C22⋊C412yes—
220C23×C61C22×D411yescharacteristic
221C23×C61C22×D411yescharacteristic
222C22×D61—21no—
223Q8×C61—22no—
224Q8×C61—41no—
225D4×C61—24no—
226D4×C61—24no—
227D4×C61—24no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—42no—
231D4×C61—42no—
232D4×C61—42no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61#9914yes—
237C2×C4×C61—24no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—44no—
244C2×C4×C61—164no—
245C2×C4×C61C2×C22⋊C412yes—
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C23×C411yescharacteristic
250C22⋊D61—42no—
251C22⋊D61—44no—
252C22⋊D61—82no—
253C22⋊D61—82no—
254C22⋊D61—81no—
255C22⋊D61—81no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—42no—
262C2×Dic61—44no—
263C2×Dic61—44no—
264C2×Dic61—82no—
265C2×Dic61—81no—
266C2×Dic61—164no—
267D244—41no—
268C4×D61—44no—
269C4×D61—42no—
270Dic122—44no—
271Dic122—42no—
272Dic122—41no—
273Dic122—42no—
274#33361—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—42no—
281C22×C61—41no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—84no—
287C22×C61—84no—
288C22×C61—162no—
289C22×C61C2×#9914yes—
290C22×C61C22×C22⋊C411yescharacteristic
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C4×C61—21no—
295C4×C61—22no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—84no—
299C4×C61C2×#9912yes—
300Dic61—41no—
301Dic61—41no—
302Dic61—42no—
303Dic61—41no—
304Dic61—42no—
305Dic61—84no—
306#3500—611no—
307#2883—611no—
308#2883D6111yescharacteristic, complemented
309#2129—612no—
310#2127—614no—
311#2126—612no—
312C2×#354—614no—
313C2×C61—21no—
314C2×C61—24no—
315C2×C61—41no—
316C2×C61—41no—
317C2×C61—42no—
318C2×C61—42no—
319C2×C61—41no—
320C2×C61C22×#9911yescharacteristic
321D61—81no—
322C2×2−1+4—611no—
323C2×H(5,𝔽2)C2×D6111yescharacteristic
324#548—1221no—
325#548—1221no—
326#546—1222no—
327#546—1221no—
328#546—1222no—
329#546—1221no—
330#536—612no—
331#536—611no—
332#535—611no—
333#535—612no—
334#532—612no—
335#532—611no—
336C2×#124—611no—
337C2×#124—612no—
338#518—21no—
339#518—1221no—
340C4×#41—611no—
341#410—21no—
342#410—612no—
343#410—1221no—
344#410C2×D6112yes—
345C2×#99—22no—
346C2×#99—614no—
347C2×#99—1222no—
348C2×#99C2×D6114yes—
349#354—614no—
350#354—618no—
351#353—614no—
352C61#3336111yescharacteristic, complemented, Sylow
3532−1+4—2444no—
354H(5,𝔽2)—44no—
355C2×#41—611no—
356C2×#41—612no—
357C2×#41—1222no—
358C2×#41—1221no—
359C2×#41C4×D6112yes—
360C2×#41C4×D6111yescharacteristic
361C2×#41D24411yescharacteristic
362C2×#41C22⋊D6111yescharacteristic
363C2×#41C22×D6111yescharacteristic
364C22×Q8—611no—
365C22×Q8—612no—
366C22×Q8—1221no—
367C22×D4C4×D6111yescharacteristic
368C22×D4D24412yes—
369C22×D4D24411yescharacteristic
370C22×D4C22⋊D6112yes—
371C22×D4C22⋊D6111yescharacteristic
372C22×D4C22×D6112yes—
373C4⋊Q8—612no—
374C4⋊Q8—611no—
375#127—612no—
376#127—611no—
377#124—611no—
378#124—612no—
379#124—612no—
380#124—612no—
381#124—611no—
382#124—612no—
383#124—1221no—
384#124—1221no—
385#124—1222no—
386#124—1222no—
387#123—1222no—
388#123—1222no—
389#123—1221no—
390#123—1221no—
391C22⋊Q8—1222no—
392C22⋊Q8—1222no—
393C22⋊Q8—1221no—
394C22⋊Q8—1221no—
395#121—1222no—
396#121—1222no—
397#121—1221no—
398#121—1221no—
399#121—1221no—
400#121—1222no—
401#121—1221no—
402#121—1222no—
403#120—1222no—
404#120—1222no—
405#120—1222no—
406C4×Q8—1221no—
407C4×D4—44no—
408C4×D4—1222no—
409C4×D4—1221no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1221no—
413C4×D4—1221no—
414C4×D4—2444no—
415#117—22no—
416#117—611no—
417#117—612no—
418#117—1222no—
419C2×C22⋊C4—24no—
420C2×C22⋊C4—1222no—
421C2×C22⋊C4—1222no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1224no—
424C2×C42—611no—
425C2×C42—612no—
426#99—22no—
427#99—24no—
428#99—614no—
429#99—1224no—
430#99—1222no—
431#99C22×D6114yes—
432C24#1175819812yes—
433C24C2×D24412yes—
434C24C2×C22⋊D6112yes—
435#41—24no—
436#41—42no—
437#41—44no—
438#41—42no—
439#41—42no—
440#41—2444no—
441#41—2442no—
442#41—2448no—
443#41—2444no—
444C2×Q8—612no—
445C2×Q8—611no—
446C2×Q8—1222no—
447C2×Q8—1221no—
448C2×Q8—2442no—
449C2×Q8—2444no—
450C2×Q8—2444no—
451C2×Q8#1175819811yescharacteristic
452C2×Q8C2×C4×D6111yescharacteristic
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—611no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1221no—
465C2×D4—1222no—
466C2×D4#1175819812yes—
467C2×D4#1175819812yes—
468C2×D4#1175819812yes—
469C2×D4#1175819811yescharacteristic
470C2×D4#1175819811yescharacteristic
471C2×D4#1175819811yescharacteristic
472C2×D4C2×C4×D6112yes—
473C2×D4C2×D24411yescharacteristic
474C2×D4C2×D24412yes—
475C2×D4C2×C22⋊D6111yescharacteristic
476C2×D4C2×C22⋊D6112yes—
477C2×D4C23×D6111yescharacteristic
478C22×C4—44no—
479C22×C4—1221no—
480C22×C4—1222no—
481C22×C4—1221no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1221no—
485C22×C4—2444no—
486C22×C4#1175819812yes—
487C22×C4#1175819812yes—
488C22×C4C2×C4×D6111yescharacteristic
489C22×C4C2×C4×D6112yes—
490C22×C4C2×D24411yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C4⋊C4—42no—
493C4⋊C4—1222no—
494C4⋊C4—1221no—
495C4⋊C4—1222no—
496C4⋊C4—1221no—
497C4⋊C4—1222no—
498C4⋊C4—1221no—
499C4⋊C4—2442no—
500C22⋊C4—24no—
501C22⋊C4—44no—
502C22⋊C4—1221no—
503C22⋊C4—1222no—
504C22⋊C4—1221no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1221no—
513C22⋊C4—1222no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C22⋊C4—1224no—
517C22⋊C4—2444no—
518C42—42no—
519C42—611no—
520C42—611no—
521C42—612no—
522C42—1222no—
523C42—1221no—
524C42—2442no—
525C23—24no—
526C23—24no—
527C23—42no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—1221no—
532C23C2×#1175819812yes—
533C23C2×#1175819812yes—
534C23C2×#1175819811yescharacteristic
535C23C2×#1175819812yes—
536C23C22×C22⋊D6111yescharacteristic
537C23C22×D24411yescharacteristic
538Q8—22no—
539Q8—41no—
540Q8—2442no—
541Q8—2441no—
542Q8—2444no—
543Q8—2442no—
544D4—24no—
545D4—24no—
546D4—24no—
547D4—22no—
548D4—22no—
549D4—42no—
550D4—42no—
551D4—41no—
552D4—42no—
553D4—41no—
554D4—41no—
555D4—4881no—
556D4—4881no—
557D4—4882no—
558D4—4882no—
559D4—2441no—
560D4—2442no—
561D4—2444no—
562D4#1451969114yes—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—164no—
571C2×C4—1221no—
572C2×C4—1222no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1222no—
576C2×C4—9764no—
577C2×C4—4882no—
578C2×C4—4881no—
579C2×C4—2444no—
580C2×C4—2444no—
581C2×C4—2442no—
582C2×C4—2444no—
583C2×C4—2442no—
584C2×C4C2×#1175819812yes—
585C2×C4C2×#1175819811yescharacteristic
586C2×C4C2×#1175819811yescharacteristic
587C2×C4C2×#1175819811yescharacteristic
588C2×C4C22×C4×D6111yescharacteristic
589C22—21no—
590C22—21no—
591C22—21no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—42no—
599C22—42no—
600C22—84no—
601C22—84no—
602C22—162no—
603C22—2441no—
604C22—2441no—
605C22—2441no—
606C22C2×#1451969114yes—
607C22C22×#1175819811yescharacteristic
608C4—21no—
609C4—22no—
610C4—21no—
611C4—21no—
612C4—84no—
613C4—4884no—
614C4—2441no—
615C4—2442no—
616C4—2441no—
617C4—2442no—
618C4—2441no—
619C4C2×#1451969112yes—
620C2—21no—
621C2—24no—
622C2—41no—
623C2—41no—
624C2—42no—
625C2—42no—
626C2—41no—
627C2—4881no—
628C2C22×#1451969111yescharacteristic
subgroupG/H
centerC2C22×#14519691
commutatorC2×C4×C61C23×C4
FrattiniC2×D4C23×D61
FittingC61×#2883C2
socleC2×C61C22×#99
prime radical#21459607C2
cosocleC2×D4×C61C24
hypercenter#2883D61
hypocenterC61#33361
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×D4 ◁ C2×H(5,𝔽2) ◁ #2883

Complements

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

NH ≅ G/Ntype
2. #21454280C2semidirect
3. #21447493C2semidirect
4. #21447486C2semidirect
5. #21447482C2semidirect
6. C2×#16839556C2semidirect
7. C61×#2883C2semidirect
8. C2×#14519824C4semidirect
15. #16840333C4semidirect
17. #16840329C4semidirect
19. #16840308C4semidirect
21. C2×#14519764C4semidirect
24. #41×Dic61C4semidirect
29. #16839556C22semidirect
30. #16839556C22semidirect
31. #16839554C22semidirect
39. C2×#11758222C2×C4semidirect
40. C2×#11758222C2×C4semidirect
43. C22×Dic122C2×C4semidirect
44. C22×Dic122C2×C4semidirect
46. #14519775C2×C4semidirect
47. #14519775C2×C4semidirect
48. #14519773C2×C4semidirect
50. #14519766C2×C4semidirect
55. #14519764C2×C4semidirect
56. #14519764C2×C4semidirect
57. #14519764C2×C4semidirect
88. #14519736C2×C4semidirect
89. #14519736C2×C4semidirect
94. C2×C4×Dic61C2×C4semidirect
95. C2×C4×Dic61C2×C4semidirect
308. #2883D61semidirect
352. C61#33361semidirect

Maximal subgroups

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

subgroupindex
#214596072
#214542802
#214474932
#214474862
#214474822
C2×#168395562
C61×#28832
#3336161