Finite Group Databasebeta

← #29711995

Subgroups — #29711995

Order 15616.

1382 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839912C211yesmaximal, characteristic, complemented
2C2×#16839895C211yesmaximal, characteristic
3#21448399C211yesmaximal, characteristic, complemented
4#21446952C212yesmaximal, complemented
5#21446896C212yesmaximal, complemented
6#21446174C212yesmaximal, complemented
7#21446167C212yesmaximal, complemented
8#21446142C211yesmaximal, characteristic, complemented
9#21446132C211yesmaximal, characteristic, complemented
10#21445890C211yesmaximal, characteristic, complemented
11C2×C61×#542C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758201C2212yes—
14C22×#11758199C2212yes—
15C2×#14519736C2211yescharacteristic
16#16839912—24no—
17#16839895—24no—
18C2×#14519701C2212yescomplemented
19C2×#14519698C2212yescomplemented
20C2×#14519697C2212yes—
21C2×#14519697C2212yes—
22C2×C22⋊#92418C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24#16839664C2212yes—
25#16839664C2212yes—
26#16839663—21no—
27#16839663—21no—
28#16839617—21no—
29#16839617—21no—
30#16839524C2212yescomplemented
31#16839524C2212yescomplemented
32#16839517C2212yescomplemented
33#16839517C2212yescomplemented
34C61×#542—24no—
35C61×#542—24no—
36C2×C4⋊Q8×C61C2211yescharacteristic, complemented
37C2×#126×C61—21no—
38C2×#124×C61—21no—
39C2×#124×C61C2211yescharacteristic, complemented
40C2×#123×C61—21no—
41C2×C22⋊Q8×C61—21no—
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×#121×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic, complemented
45C2×C4×D4×C61C2211yescharacteristic, complemented
46C2×#117×C61C2211yescharacteristic
47C24×D61—21no—
48C22×D244—21no—
49C22×D244—24no—
50C22×D244—22no—
51C22×D244—22no—
52C22×D244—42no—
53C22×D244C2312yes—
54C2×#11758201—24no—
55C2×#11758201—24no—
56C2×#11758201—42no—
57C2×#11758199—24no—
58C2×#11758199—24no—
59C2×#11758199—42no—
60C22×#92418C2312yes—
61C22×#92418C2312yes—
62#14519736—42no—
63C2×C4⋊Dic61C2312yes—
64C2×C22⋊Dic61—21no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×C4×Dic61—21no—
70#14519701—42no—
71#14519698—42no—
72#14519697—24no—
73#14519697—24no—
74C22⋊#92418—22no—
75C22⋊#92418—22no—
76C22×Q8×C61—21no—
77C22×Q8×C61C2311yescharacteristic
78C22×D4×C61—21no—
79C22×D4×C61C2311yescharacteristic
80C23×C4×C61—21no—
81C23×C4×C61C2311yescharacteristic
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61—22no—
84#126×C61—28no—
85#124×C61—28no—
86#124×C61—22no—
87#124×C61—22no—
88#123×C61—28no—
89C22⋊Q8×C61—28no—
90C22⋊Q8×C61—22no—
91C22⋊Q8×C61D414yes—
92#121×C61—22no—
93#121×C61D414yes—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98#117×C61—22no—
99#117×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C24×C61—21no—
118C24×C61C2×D411yescharacteristic
119C23×D61—22no—
120C23×D61—24no—
121C23×D61—42no—
122C23×D61—42no—
123C23×D61—82no—
124C23×D61—82no—
125C2×Q8×C61—24no—
126C2×Q8×C61—21no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—82no—
130C2×Q8×C61C2×D412yes—
131C2×D4×C61—24no—
132C2×D4×C61—21no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—82no—
136C2×D4×C61C2×D412yes—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—24no—
143C22×C4×C61—82no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—22no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—24no—
158C2×D244—22no—
159C2×D244—48no—
160C2×D244—48no—
161C2×D244—42no—
162C2×D244—42no—
163C2×D244#4114yes—
164C4⋊C4×C61—24no—
165C4⋊C4×C61—24no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C2×D412yes—
172C4⋊C4×C61C2×D412yes—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—24no—
176C22⋊C4×C61—24no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—24no—
180C42×C61—24no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C22⋊Dic61—42no—
186#11758201—48no—
187#11758199—48no—
188#11758198—82no—
189#11758198—82no—
190#11758198—82no—
191#11758198—82no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4×Dic61—42no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61#12012yes—
211C23×C61C22×D411yescharacteristic
212C22×D61—24no—
213C22×D61—48no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—88no—
221C22×D61—84no—
222C22×D61—84no—
223C22×D61—88no—
224Q8×C61—24no—
225Q8×C61—42no—
226Q8×C61—84no—
227D4×C61—24no—
228D4×C61—42no—
229D4×C61—84no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—84no—
250C2×C4×C61#12012yes—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4112yes—
254C2×C4×C61H(5,𝔽2)12yes—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258D244—24no—
259D244—24no—
260D244—48no—
261D244—42no—
262D244—42no—
263#92418—42no—
264#92418—42no—
265#23945—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—82no—
275C22×C61—82no—
276C22×C61C2×#12011yescharacteristic
277C22×C61#52511yescharacteristic
278C22×C61#53511yescharacteristic
279C22×C61#57612yes—
280C22×C61#57712yes—
281C2×D61—44no—
282C2×D61—44no—
283C2×D61—44no—
284C2×D61—816no—
285C2×D61—82no—
286C2×D61—82no—
287C2×D61—82no—
288C2×D61—82no—
289C4×C61—22no—
290C4×C61—42no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—42no—
294C4×C61—42no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61#54612yes—
298C4×C61#54612yes—
299Dic61—82no—
300C2×#542D6111yescharacteristic, complemented
301C2×#450—611no—
302C2×#447—611no—
303#2309—611no—
304#2023—612no—
305#2012—612no—
306#1880—612no—
307#1879—612no—
308#1875—611no—
309#1873—611no—
310#1818—611no—
311C2×C61—42no—
312C2×C61—82no—
313C2×C61#240411yescharacteristic
314C2×C61#300811yescharacteristic
315C2×C61#301011yescharacteristic
316C2×C61#311812yes—
317C2×C61#322412yes—
318D61—84no—
319C23×D4—611no—
320C22×SD16—612no—
321C22×D8—612no—
322#542—24no—
323#542—24no—
324C2×C4⋊Q8C2×D6111yescharacteristic, complemented
325C2×#126—21no—
326C2×#124—21no—
327C2×#124C2×D6111yescharacteristic, complemented
328C2×#123—21no—
329C2×C22⋊Q8—21no—
330C2×C22⋊Q8C2×D6111yescharacteristic
331C2×#121C2×D6111yescharacteristic
332C2×C4×Q8C2×D6111yescharacteristic, complemented
333C2×C4×D4C2×D6111yescharacteristic, complemented
334C2×#117—611no—
335C2×#117C2×D6111yescharacteristic
336#450—1224no—
337#447—1224no—
338C2×#103—612no—
339C2×#102—612no—
340C2×#102—612no—
341C2×#102—612no—
342C2×C22⋊C8—611no—
343C2×C22⋊C8—611no—
344#390—612no—
345#390—612no—
346#390—1221no—
347#390—1221no—
348#379—1221no—
349#379—1221no—
350#340—612no—
351#340—612no—
352#336—612no—
353#336—612no—
354C61#2394511yescharacteristic, complemented, Sylow
355C25—1221no—
356C22×Q8—21no—
357C22×Q8C22×D6111yescharacteristic
358C22×D4—21no—
359C22×D4—612no—
360C22×D4—1224no—
361C22×D4—1221no—
362C22×D4—1222no—
363C22×D4—1222no—
364C22×D4—2442no—
365C22×D4C22×D6111yescharacteristic
366C23×C4—21no—
367C23×C4C22×D6111yescharacteristic
368C2×SD16—1224no—
369C2×SD16—1224no—
370C2×SD16—2442no—
371C2×D8—1224no—
372C2×D8—1224no—
373C2×D8—2442no—
374C22×C8—612no—
375C22×C8—612no—
376C4⋊Q8—22no—
377C4⋊Q8—22no—
378#126—28no—
379#124—28no—
380#124—22no—
381#124—22no—
382#123—28no—
383C22⋊Q8—28no—
384C22⋊Q8—22no—
385C22⋊Q8C22⋊D6114yes—
386#121—22no—
387#121C22⋊D6114yes—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×D4—22no—
391C4×D4—22no—
392#117—22no—
393#117—22no—
394#117—2442no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4C22⋊D6111yescharacteristic
400C2×C4⋊C4C22⋊D6111yescharacteristic
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1222no—
409C2×C22⋊C4—1222no—
410C2×C22⋊C4—1222no—
411C2×C22⋊C4—1222no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4C22⋊D6111yescharacteristic
414C2×C22⋊C4C22⋊D6111yescharacteristic
415C2×C42—21no—
416C2×C42—1221no—
417C2×C42C22×D6111yescharacteristic
418C2×C42C22×D6111yescharacteristic
419#103—2442no—
420#102—1224no—
421#102—1224no—
422#102—2442no—
423C22⋊C8—1222no—
424C22⋊C8—1222no—
425C24—21no—
426C24—1224no—
427C24—1222no—
428C24—4882no—
429C24—4882no—
430C24—2442no—
431C24—2442no—
432C24C2×C22⋊D6111yescharacteristic
433C2×Q8—24no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—82no—
438C2×Q8C2×C22⋊D6112yes—
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—82no—
444C2×D4—614no—
445C2×D4—1224no—
446C2×D4—1222no—
447C2×D4—1222no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—1222no—
451C2×D4—2448no—
452C2×D4—2448no—
453C2×D4—2442no—
454C2×D4—2442no—
455C2×D4C2×C22⋊D6112yes—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—82no—
463C22×C4—1222no—
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6112yes—
469C22×C4C2×C22⋊D6112yes—
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C23×D6111yescharacteristic
472SD16—2448no—
473D8—2448no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—1222no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4C2×C22⋊D6112yes—
490C4⋊C4C2×C22⋊D6112yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—24no—
498C22⋊C4—4882no—
499C22⋊C4—4882no—
500C22⋊C4—4882no—
501C22⋊C4—4882no—
502C22⋊C4—2442no—
503C42—24no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—2442no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—22no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—1224no—
519C23—4884no—
520C23—4884no—
521C23—4888no—
522C23—4888no—
523C23—2448no—
524C23—2442no—
525C23—2442no—
526C23—2442no—
527C23—2442no—
528C23—2442no—
529C23—2442no—
530C23#1451974612yes—
531C23C22×C22⋊D6111yescharacteristic
532Q8—24no—
533Q8—42no—
534Q8—84no—
535D4—24no—
536D4—42no—
537D4—84no—
538D4—1224no—
539D4—1224no—
540D4—2448no—
541D4—2442no—
542D4—2442no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—84no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4#1451974612yes—
567C2×C4C22×C22⋊D6111yescharacteristic
568C2×C4C22×C22⋊D6111yescharacteristic
569C2×C4#41×D6112yes—
570C2×C4#1451982312yes—
571C8—2442no—
572C8—2442no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—82no—
582C22—82no—
583C22—48816no—
584C22—4882no—
585C22—4882no—
586C22—4882no—
587C22—4882no—
588C22—2444no—
589C22—2444no—
590C22—2444no—
591C22C2×#1451974611yescharacteristic
592C22#1684026911yescharacteristic
593C22#1684032611yescharacteristic
594C22#1684056512yes—
595C22#1684057012yes—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—42no—
604C4—4882no—
605C4#1684040012yes—
606C4#1684040012yes—
607C2—42no—
608C2—82no—
609C2—4884no—
610C2#2144913411yescharacteristic
611C2#2145508711yescharacteristic
612C2#2145510511yescharacteristic
613C2#2145599712yes—
614C2#2145690712yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#542C2
socleC23×C61C22×D4
prime radicalC2×#16839895C2
hypercenterC2×#542D61
hypocenterC61#23945
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. C2×#16839912C2semidirect
3. #21448399C2semidirect
4. #21446952C2semidirect
5. #21446896C2semidirect
6. #21446174C2semidirect
7. #21446167C2semidirect
8. #21446142C2semidirect
9. #21446132C2semidirect
10. #21445890C2semidirect
11. C2×C61×#542C2semidirect
18. C2×#14519701C22semidirect
19. C2×#14519698C22semidirect
30. #16839524C22semidirect
31. #16839524C22semidirect
32. #16839517C22semidirect
33. #16839517C22semidirect
36. C2×C4⋊Q8×C61C22semidirect
39. C2×#124×C61C22semidirect
44. C2×C4×Q8×C61C22semidirect
45. C2×C4×D4×C61C22semidirect
300. C2×#542D61semidirect
324. C2×C4⋊Q8C2×D61semidirect
327. C2×#124C2×D61semidirect
332. C2×C4×Q8C2×D61semidirect
333. C2×C4×D4C2×D61semidirect
354. C61#23945semidirect

Maximal subgroups

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

subgroupindex
C2×#168399122
C2×#168398952
#214483992
#214469522
#214468962
#214461742
#214461672
#214461422
#214461322
#214458902
C2×C61×#5422
#2394561