Finite Group Databasebeta

← #29685383

Subgroups — #29685383

Order 15616.

1380 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840313C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic, complemented
3C4×M16×D61C211yesmaximal, characteristic, complemented
4#21446809C211yesmaximal, characteristic, complemented
5#21446659C211yesmaximal, characteristic, complemented
6#21446658C211yesmaximal, characteristic, complemented
7#21446281C214yesmaximal, complemented
8#21445610C214yesmaximal, complemented
9C61×#1997C211yesmaximal, characteristic, complemented
10C22×#11758226C412yes—
11C2×M16×D61C2212yes—
12#16840313—21no—
13#16840313—21no—
14#16840313—21no—
15#16840313—21no—
16#16840313—21no—
17#16840313—21no—
18#16840313—21no—
19#16840313—21no—
20C2×#14519774C2211yescharacteristic
21C2×#14519766C412yes—
22C2×#14519752C412yes—
23C2×#14519752C2212yes—
24C2×#14519739C412yes—
25C2×#14519739C2212yes—
26C2×C42×D61C2211yescharacteristic
27#16839818—21no—
28#16839818—21no—
29#16839814—21no—
30#16839814—21no—
31#16839786C2211yescharacteristic
32C2×#14519693C2214yes—
33C2×#14519693C2214yes—
34#16839725C2212yes—
35C4×#11758194C2211yescharacteristic
36#16839714—21no—
37M16×Dic61C2212yes—
38M16×Dic61C2212yes—
39C4×#11758189—21no—
40#97×D61—21no—
41C4×C8×D61—21no—
42#16839531C2214yes—
43C2×C4⋊Q8×C61C2211yescharacteristic
44C61×#423C2211yescharacteristic
45C4×M16×C61C2211yescharacteristic
46C61×#343C2214yes—
47C2×#11758226—22no—
48C2×#11758226—22no—
49C2×#11758226—22no—
50C2×#11758226—22no—
51C2×#11758226—48no—
52C22×D244C2×C412yes—
53C22×D244C2×C414yes—
54C22×D244C2×C412yes—
55C22×D244C2312yes—
56C22×C4×D61C2×C411yescharacteristic
57C22×C4×D61C2×C414yes—
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61C2311yescharacteristic
60C2×#11758194C2312yes—
61C2×#11758194C2312yes—
62C2×#11758189—22no—
63M16×D61—24no—
64M16×D61—24no—
65C2×C8×D61—22no—
66#14519774—21no—
67#14519774—21no—
68#14519774—21no—
69#14519774—21no—
70#14519766—22no—
71#14519766—22no—
72#14519766—22no—
73#14519766—22no—
74#14519752—22no—
75#14519752—22no—
76#14519752—22no—
77#14519752—22no—
78#14519752—22no—
79#14519752—22no—
80#14519752—22no—
81#14519752—22no—
82#14519739—22no—
83#14519739—22no—
84#14519739—22no—
85#14519739—22no—
86#14519739—22no—
87#14519739—22no—
88#14519739—22no—
89#14519739—22no—
90C2×#11758198C2×C414yes—
91C2×#11758198C2×C414yes—
92C2×C4×Dic61C2×C411yescharacteristic
93C2×C4×Dic61C2×C411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61D411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61D411yescharacteristic
99C42×D61D411yescharacteristic
100C42×D61D411yescharacteristic
101C42×D61D411yescharacteristic
102C42×D61D411yescharacteristic
103C4⋊#92418—21no—
104C4⋊#92418—21no—
105#14519693—28no—
106#14519693—28no—
107#14519689—22no—
108#14519689—22no—
109#14519688—21no—
110#14519687—22no—
111#14519687—22no—
112C4×#92418—21no—
113C8×Dic61—22no—
114C8×Dic61—22no—
115C22×Q8×C61C2×C412yes—
116C2×M16×C61C2312yes—
117C2×M16×C61C2312yes—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—21no—
122C2×C4⋊C4×C61C2×C412yes—
123C2×C4⋊C4×C61C2312yes—
124C2×C42×C61C2311yescharacteristic
125C4⋊C8×C61—21no—
126C4⋊C8×C61—21no—
127#97×C61—21no—
128C4×C8×C61—21no—
129C23×D61C22×C411yescharacteristic
130C23×D61C22×C414yes—
131C2×Q8×C61—22no—
132C2×Q8×C61—22no—
133C2×Q8×C61—44no—
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C414yes—
137C22×C4×C61C2411yescharacteristic
138C22×Dic61C22×C411yescharacteristic
139C22×Dic61C22×C411yescharacteristic
140C22×Dic61C22×C411yescharacteristic
141#11758226—48no—
142#11758226—48no—
143#11758226—48no—
144#11758226—48no—
145C2×D244—24no—
146C2×D244—24no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—22no—
153C2×D244—24no—
154C2×D244—24no—
155C2×D244—44no—
156C2×D244—44no—
157C2×D244—48no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—24no—
163C2×C4×D61—42no—
164C2×C4×D61—48no—
165C2×C4×D61C22⋊C411yescharacteristic
166C2×C4×D61C22⋊C411yescharacteristic
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×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61C2×D411yescharacteristic
177M16×C61—22no—
178M16×C61—24no—
179M16×C61#4114yescomplemented
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190#11758198—24no—
191#11758198—24no—
192#11758198—24no—
193#11758198—24no—
194C4×Dic61C22⋊C411yescharacteristic
195C4×Dic61C22⋊C411yescharacteristic
196C4×Dic61C22⋊C411yescharacteristic
197C4×Dic61C22⋊C411yescharacteristic
198C4×Dic61C22⋊C411yescharacteristic
199C4×Dic61C22⋊C411yescharacteristic
200C4×Dic61C22⋊C411yescharacteristic
201C4×Dic61C22⋊C411yescharacteristic
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61C2×D411yescharacteristic
205C4×Dic61C2×D411yescharacteristic
206#11758194—22no—
207#11758194—24no—
208#11758194#4114yescomplemented
209C2×#92418—22no—
210C2×#92418—22no—
211#11758189—28no—
212C8×D61—28no—
213C23×C61C23×C411yescharacteristic
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—24no—
217C22×D61—28no—
218C22×D61—41no—
219C22×D61—44no—
220C22×D61—44no—
221C22×D61C2×C22⋊C411yescharacteristic
222C22×D61C2×C22⋊C411yescharacteristic
223Q8×C61—44no—
224Q8×C61—44no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—44no—
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4112yes—
236C2×C4×C61C2×#4112yes—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—41no—
240C2×Dic61C2×C22⋊C411yescharacteristic
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61C2×C22⋊C411yescharacteristic
243C2×Dic61C2×C22⋊C411yescharacteristic
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C2×C22⋊C411yescharacteristic
246D244—44no—
247D244—44no—
248D244—48no—
249D244—48no—
250D244—44no—
251D244—42no—
252D244—42no—
253D244—44no—
254D244—44no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—21no—
259C4×D61—21no—
260C4×D61—21no—
261C4×D61—21no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—41no—
267C4×D61—41no—
268C4×D61—48no—
269C4×D61—48no—
270C4×D61#10014yescomplemented
271C8×C61—24no—
272C8×C61—42no—
273#92418—24no—
274#92418—42no—
275#21424—611nomaximal, Sylow
276C22×C61—21no—
277C22×C61—21no—
278C22×C61C22×C22⋊C411yescharacteristic
279C22×C61C4×#4112yes—
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—44no—
286C2×D61—44no—
287C2×D61—44no—
288C2×D61—44no—
289C2×D61—88no—
290C2×D61C2×#10012yes—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—44no—
294C4×C61C2×#10012yes—
295C4×C61#41312yes—
296C4×C61#52512yes—
297C4×C61#53211yescharacteristic
298C4×C61#53211yescharacteristic
299Dic61—21no—
300Dic61—21no—
301Dic61—21no—
302Dic61—21no—
303Dic61—21no—
304Dic61C2×#10012yes—
305C2×#532—611no—
306#2920—611no—
307C2×C4×M16—611no—
308#1997—611no—
309#1997D6111yescharacteristic, complemented
310#1965—611no—
311#1965—611no—
312#1900—614no—
313C4×#100—614no—
314C2×C61—22no—
315C2×C61#229711yescharacteristic
316C2×C61C22×#10011yescharacteristic
317C2×C61C2×#41311yescharacteristic
318D61—22no—
319D61—41no—
320D61—84no—
321C22×#41—612no—
322C22×M16—612no—
323#532—1221no—
324#532—1221no—
325#532—1221no—
326#532—1221no—
327#532—1221no—
328#532—1221no—
329#532—1221no—
330#532—1221no—
331C2×C4⋊Q8C2×D6111yescharacteristic
332C2×#127—611no—
333C2×#124—612no—
334C2×#121—612no—
335C2×#121—612no—
336C2×C4×D4—612no—
337C2×C4×D4—612no—
338C22×C42—611no—
339#432—1221no—
340#432—1221no—
341#431—1221no—
342#431—1221no—
343#423—611no—
344#423C2×D6111yescharacteristic
345C2×#100—614no—
346C2×#100—614no—
347#407—612no—
348C4×M16—612no—
349C4×M16—612no—
350C4×M16—611no—
351C4×M16—1221no—
352C4×M16—1221no—
353C4×M16C2×D6111yescharacteristic
354C2×#97—1221no—
355C2×C4×C8—1221no—
356#343—614no—
357#343C2×D6114yes—
358C61#2142411yescharacteristic, complemented, Sylow
359C2×#41—1222no—
360C2×#41—1222no—
361C2×#41—1222no—
362C2×#41—1222no—
363C2×#41—2448no—
364C22×Q8C4×D6112yes—
365C22×D4—612no—
366C22×D4—614no—
367C22×D4—612no—
368C22×D4—612no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—611no—
372C23×C4—614no—
373C2×M16—612no—
374C2×M16—612no—
375C2×M16—1222no—
376C2×M16—1224no—
377C2×M16—1224no—
378C2×M16C22×D6112yes—
379C2×M16C22×D6112yes—
380C22×C8—1222no—
381C4⋊Q8—21no—
382C4⋊Q8—21no—
383C4⋊Q8—21no—
384C4⋊Q8—21no—
385#127—1221no—
386#127—1221no—
387#127—1221no—
388#127—1221no—
389#124—1222no—
390#124—1222no—
391#124—1222no—
392#124—1222no—
393#121—1222no—
394#121—1222no—
395#121—1222no—
396#121—1222no—
397#121—1222no—
398#121—1222no—
399#121—1222no—
400#121—1222no—
401C4×D4—1222no—
402C4×D4—1222no—
403C4×D4—1222no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C4×D4—1222no—
408C4×D4—1222no—
409C2×C4⋊C4C4×D6112yes—
410C2×C4⋊C4C22×D6112yes—
411C2×C22⋊C4—614no—
412C2×C22⋊C4—614no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42C22×D6111yescharacteristic
425C4⋊C8—21no—
426C4⋊C8—21no—
427C4⋊C8—1221no—
428C4⋊C8—1221no—
429#100—1228no—
430#100—1228no—
431C22⋊C8—1222no—
432C22⋊C8—1222no—
433#97—21no—
434#97—1222no—
435#97—1222no—
436#97—1221no—
437C4×C8—21no—
438C4×C8—1222no—
439C4×C8—1222no—
440C4×C8—1221no—
441C24—611no—
442C24—614no—
443#41—2448no—
444#41—2448no—
445#41—2448no—
446#41—2448no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—44no—
450C2×D4—1224no—
451C2×D4—1224no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—1224no—
455C2×D4—1224no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—2444no—
461C2×D4—2444no—
462C2×D4—2448no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
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—1224no—
479C22×C4—1224no—
480C22×C4—1224no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—2442no—
484C22×C4—2448no—
485C22×C4C2×C4×D6111yescharacteristic
486C22×C4C2×C4×D6111yescharacteristic
487C22×C4C2×C4×D6114yes—
488C22×C4C23×D6111yescharacteristic
489M16—22no—
490M16—24no—
491M16—614no—
492M16—1228no—
493M16—1222no—
494M16—1224no—
495M16#1175822614yescomplemented
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—1228no—
499C2×C8—1222no—
500C2×C8—1222no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C22⋊C4—1224no—
506C22⋊C4—1224no—
507C22⋊C4—1224no—
508C22⋊C4—1224no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42D4×D6111yescharacteristic
522C42D4×D6111yescharacteristic
523C42D4×D6111yescharacteristic
524C42D4×D6111yescharacteristic
525C23—611no—
526C23—611no—
527C23—1222no—
528C23—1222no—
529C23—1228no—
530C23—1224no—
531C23—2441no—
532C23—2444no—
533C23—2444no—
534C23C22×C4×D6111yescharacteristic
535Q8—44no—
536Q8—44no—
537D4—2444no—
538D4—2444no—
539D4—2448no—
540D4—2448no—
541D4—2444no—
542D4—2444no—
543D4—2444no—
544D4—2442no—
545D4—2442no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—44no—
550C2×C4—611no—
551C2×C4—614no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—611no—
555C2×C4—611no—
556C2×C4—611no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2448no—
573C2×C4—2448no—
574C2×C4—2441no—
575C2×C4C22⋊C4×D6111yescharacteristic
576C2×C4C22⋊C4×D6111yescharacteristic
577C2×C4C22⋊C4×D6111yescharacteristic
578C2×C4C22⋊C4×D6111yescharacteristic
579C2×C4C2×D4×D6111yescharacteristic
580C2×C4C2×D4×D6111yescharacteristic
581C2×C4#41×D6112yes—
582C2×C4C2×#1175822612yes—
583C8—24no—
584C8—42no—
585C8—1224no—
586C8—2442no—
587C22—21no—
588C22—21no—
589C22—612no—
590C22—1221no—
591C22—1221no—
592C22—1221no—
593C22—1221no—
594C22—1221no—
595C22—4888no—
596C22—2444no—
597C22—2444no—
598C22—2444no—
599C22—2444no—
600C22C2×C22⋊C4×D6111yescharacteristic
601C22C4×#1175822612yes—
602C4—21no—
603C4—21no—
604C4—44no—
605C4—612no—
606C4—1221no—
607C4—1221no—
608C4—1221no—
609C4—1221no—
610C4—1221no—
611C4#100×D6112yes—
612C4#1683974512yes—
613C4#1684027512yes—
614C4#1684031111yescharacteristic
615C4#1684031311yescharacteristic
616C2—22no—
617C2—1222no—
618C2—4884no—
619C2—2441no—
620C2#2144830811yescharacteristic
621C2C2×#100×D6111yescharacteristic
622C2C2×#1683974511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1997C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840313C2
cosocleC22×C4×C61C24
hypercenter#1997D61
hypocenterC61#21424
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
2. #21454451C2semidirect
3. C4×M16×D61C2semidirect
4. #21446809C2semidirect
5. #21446659C2semidirect
6. #21446658C2semidirect
7. #21446281C2semidirect
8. #21445610C2semidirect
9. C61×#1997C2semidirect
179. M16×C61#41semidirect
208. #11758194#41semidirect
270. C4×D61#100semidirect
309. #1997D61semidirect
358. C61#21424semidirect
495. M16#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168403132
#214544512
C4×M16×D612
#214468092
#214466592
#214466582
#214462812
#214456102
C61×#19972
#2142461