Finite Group Databasebeta

← #29729921

Subgroups — #29729921

Order 15616.

806 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840111C211yesmaximal, characteristic
2C2×#16839971C211yesmaximal, characteristic
3#21451025C211yesmaximal, characteristic
4#21448873C211yesmaximal, characteristic
5#21447230C211yesmaximal, characteristic
6#21447228C211yesmaximal, characteristic
7#21447142C211yesmaximal, characteristic
8#21447140C211yesmaximal, characteristic
9#21446765C211yesmaximal, characteristic
10#21446534C211yesmaximal, characteristic
11#21446161C211yesmaximal, characteristic
12#21446160C211yesmaximal, characteristic
13#21445823C211yesmaximal, characteristic
14#21445638C211yesmaximal, characteristic
15C61×#1803C211yesmaximal, characteristic
16C22×Dic244C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×#14519776C2211yescharacteristic
19C2×#14519775C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22#16840111—24no—
23#16839971—24no—
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839682—21no—
34#16839682—21no—
35#16839669C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839667C2211yescharacteristic
39#16839657—21no—
40#16839657—21no—
41#16839657—21no—
42#16839646—21no—
43#16839643—21no—
44#16839631C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4×#11758196—21no—
47C4×#11758196C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#384C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C22×Dic122C2311yescharacteristic
64C2×Dic244—22no—
65C2×Dic244—22no—
66C2×Dic244—41no—
67C2×#11758202—22no—
68C2×#11758202—22no—
69C2×#11758202—41no—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519776—22no—
73#14519776—22no—
74#14519775—22no—
75#14519775D414yes—
76#14519742—22no—
77#14519742—22no—
78#14519742—22no—
79#14519742—41no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196D411yescharacteristic
90C2×#11758196D411yescharacteristic
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519706—22no—
97#14519706—41no—
98#14519702—22no—
99#14519702—22no—
100#14519701—22no—
101#14519700—22no—
102#14519700—22no—
103#14519700—22no—
104C8×Dic61—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116C22×Q8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C4×Q8×C61—22no—
120C4×Q8×C61—41no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—41no—
130#103×C61—22no—
131#103×C61—22no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61C2×D412yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—42no—
164C2×Dic122—81no—
165C2×Dic122—81no—
166C2×Dic122C2×D412yes—
167C2×Dic122#4112yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—42no—
180C4⋊C4×C61—81no—
181C4⋊C4×C61—81no—
182C4⋊C4×C61#4112yes—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—42no—
186#11758202—44no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191#11758196—24no—
192#11758196—24no—
193#11758196—22no—
194#11758196—42no—
195#11758196—42no—
196#11758196—42no—
197#11758196—81no—
198#11758196—81no—
199#11758196Q1614yescomplemented
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—42no—
203C4×Dic61—42no—
204C4×Dic61—42no—
205C4×Dic61C2×D412yes—
206C4×Dic61#4112yes—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213Dic244—44no—
214C23×C61C22×D411yescharacteristic
215Q8×C61—22no—
216Q8×C61—22no—
217Q8×C61—41no—
218Q8×C61—41no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61C2×Q1612yes—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61H(5,𝔽2)11yescharacteristic
242C2×C4×C612−1+411yescharacteristic
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—22no—
248C2×Dic61—21no—
249C2×Dic61—22no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61#12112yes—
263C2×Dic61#12112yes—
264C2×Dic61C2×Q1612yes—
265C2×Dic61C2×Q1612yes—
266C2×Dic61#13712yes—
267Dic122—22no—
268Dic122—22no—
269Dic122—22no—
270Dic122—22no—
271Dic122—41no—
272Dic122—41no—
273Dic122—41no—
274Dic122—41no—
275Dic122—84no—
276C8×C61—41no—
277C8×C61—41no—
278#92418—41no—
279#92418—41no—
280#25324—611nomaximal, Sylow
281C22×C61C2×#12111yescharacteristic
282C22×C61#54611yescharacteristic
283C22×C61#54811yescharacteristic
284C22×C61C22×Q1611yescharacteristic
285C22×C61C2×#13711yescharacteristic
286C22×C61#57711yescharacteristic
287C22×C61#57711yescharacteristic
288C4×C61—22no—
289C4×C61—41no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61—81no—
293C4×C61—81no—
294C4×C61#52911yescharacteristic
295C4×C61#54211yescharacteristic
296C4×C61#54611yescharacteristic
297C4×C61#54711yescharacteristic
298Dic61—22no—
299Dic61—22no—
300Dic61—41no—
301Dic61—42no—
302Dic61—42no—
303Dic61—81no—
304Dic61—81no—
305Dic61#46214yes—
306C2×#487—611no—
307C2×#462—611no—
308#2567—611no—
309#2365—611no—
310#2072—611no—
311#2072—611no—
312#2056—611no—
313#2056—611no—
314#1987—611no—
315#1939—611no—
316#1878—611no—
317#1878—611no—
318#1803—611no—
319#1803D6111yescharacteristic
320C4×#103—611no—
321C2×C61#242711yescharacteristic
322C2×C61C2×#46211yescharacteristic
323C2×C61#304311yescharacteristic
324C2×C61#329911yescharacteristic
325C2×C61#333111yescharacteristic
326C2×C61#336311yescharacteristic
327C2×C61#336911yescharacteristic
328C22×Q16—611no—
329C22×Q16—611no—
330C2×C4⋊Q8—611no—
331C2×C4⋊Q8—611no—
332C2×C4×Q8—611no—
333C2×C4×Q8—611no—
334C2×C4×Q8C2×D6111yescharacteristic
335#487—1224no—
336#462—1224no—
337C2×C4⋊C8—611no—
338C2×C4⋊C8—611no—
339C2×C4⋊C8C2×D6111yescharacteristic
340C2×#103—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103C2×D6111yescharacteristic
347C2×#103C2×D6111yescharacteristic
348C2×C4×C8—611no—
349#395—1221no—
350#395—1221no—
351#391—611no—
352#391—611no—
353#391—611no—
354#391—611no—
355#389—1221no—
356#389—1221no—
357#389—1221no—
358#387—1221no—
359#387—1221no—
360#384—611no—
361#384C2×D6111yescharacteristic
362C4×C4⋊C4—611no—
363C4×C4⋊C4—611no—
364C4×C4⋊C4—1221no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340C2×D6111yescharacteristic
370#340C2×D6111yescharacteristic
371#336—611no—
372#336—611no—
373C61#2532411yescharacteristic, complemented, Sylow
374C22×Q8—611no—
375C22×Q8—611no—
376C22×Q8—1221no—
377C22×Q8C22×D6111yescharacteristic
378C2×Q16—1222no—
379C2×Q16—1222no—
380C2×Q16—1222no—
381C2×Q16—1222no—
382C2×Q16—2441no—
383C2×Q16—2441no—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388C4⋊Q8—614no—
389C4⋊Q8—1222no—
390C4⋊Q8—1222no—
391C4⋊Q8—1222no—
392C4×Q8—22no—
393C4×Q8—41no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×Q8—1222no—
397C4×Q8—2441no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C42—21no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—1221no—
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42C22×D6111yescharacteristic
422C4⋊C8—41no—
423C4⋊C8—1222no—
424C4⋊C8—2441no—
425#103—22no—
426#103—22no—
427#103—1222no—
428#103—1222no—
429#103—1222no—
430#103—1222no—
431#103—1222no—
432#103—1222no—
433C4×C8—1222no—
434#95—611no—
435#95—611no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—612no—
451C2×Q8—612no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—4881no—
463C2×Q8—4881no—
464C2×Q8—2442no—
465C2×Q8D4×D6112yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4D4×D6111yescharacteristic
479C22×C4D4×D6111yescharacteristic
480C22×C4C23×D6111yescharacteristic
481Q16—2444no—
482Q16—2444no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—614no—
504C4⋊C4—1222no—
505C4⋊C4—1224no—
506C4⋊C4—1224no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—4881no—
512C4⋊C4—4881no—
513C4⋊C4—2442no—
514C4⋊C4—2442no—
515C4⋊C4—2442no—
516C4⋊C4#1175822212yes—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—612no—
521C42—612no—
522C42—1221no—
523C42—1221no—
524C42—2442no—
525C42—2442no—
526C42—2442no—
527C23C2×D4×D6111yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532Q8—1222no—
533Q8—1222no—
534Q8—1222no—
535Q8—1222no—
536Q8—4884no—
537Q8—2441no—
538Q8—2441no—
539Q8—2441no—
540Q8—2441no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—612no—
559C2×C4—612no—
560C2×C4—612no—
561C2×C4—612no—
562C2×C4—612no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1222no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4Q16×D6112yes—
583C2×C4C2×D4×D6111yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4#41×D6111yescharacteristic
586C2×C4C2×#1175822211yescharacteristic
587C2×C4#1451982211yescharacteristic
588C2×C4#1451982511yescharacteristic
589C8—41no—
590C8—41no—
591C8—2441no—
592C8—2441no—
593C22#121×D6111yescharacteristic
594C22#1684040211yescharacteristic
595C22#1684042211yescharacteristic
596C22C2×Q16×D6111yescharacteristic
597C22#137×D6111yescharacteristic
598C22#1684056811yescharacteristic
599C22#1684057211yescharacteristic
600C4—22no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—81no—
605C4—81no—
606C4—614no—
607C4—1222no—
608C4—1222no—
609C4—4881no—
610C4—4881no—
611C4—2442no—
612C4—2442no—
613C4—2441no—
614C4#1684029011yescharacteristic
615C4#1684038211yescharacteristic
616C4#1684040111yescharacteristic
617C4#1684041311yescharacteristic
618C2#2144939511yescharacteristic
619C2#462×D6111yescharacteristic
620C2#2145533511yescharacteristic
621C2#2145769111yescharacteristic
622C2#2145813111yescharacteristic
623C2#2145849611yescharacteristic
624C2#2145856211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socle, prime radicalC23×C61C22×D4
hypercenter#1803D61
hypocenterC61#25324
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
199. #11758196Q16semidirect
373. C61#25324semidirect

Maximal subgroups

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

subgroupindex
C2×#168401112
C2×#168399712
#214510252
#214488732
#214472302
#214472282
#214471422
#214471402
#214467652
#214465342
#214461612
#214461602
#214458232
#214456382
C61×#18032
#2532461