Finite Group Databasebeta

← #29689350

Subgroups — #29689350

Order 15616.

1474 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#124×D61C211yesmaximal, characteristic
2C2×#16839818C211yesmaximal, characteristic, complemented
3#21448223C211yesmaximal, characteristic, complemented
4#21446752C211yesmaximal, characteristic, complemented
5#21446751C211yesmaximal, characteristic, complemented
6#21446749C211yesmaximal, characteristic, complemented
7#21446634C212yesmaximal, complemented
8#21446632C212yesmaximal, complemented
9#21445752C212yesmaximal, complemented
10#21445748C212yesmaximal
11C61×#1983C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C22×D4×D61C2211yescharacteristic
14C2×#14519766C2211yescharacteristic
15C2×#14519765C2211yescharacteristic
16C2×#14519764C2211yescharacteristic
17C2×#14519763—21no—
18C2×#14519763—21no—
19#124×D61—24no—
20#124×D61C418yescomplemented
21C2×#14519734—21no—
22C2×C22⋊C4×D61—21no—
23C2×C22⋊C4×D61—21no—
24C2×C42×D61C2211yescharacteristic
25#16839818—24no—
26C2×C4⋊#92418C2211yescharacteristic, complemented
27C2×#14519702C2212yescomplemented
28C2×#14519701C2212yescomplemented
29C2×#14519700C2212yescomplemented
30C2×#14519699C2212yes—
31C2×#14519698C2212yes—
32C2×#14519697C2212yes—
33C2×#14519689C2212yes—
34C2×#14519687C2212yescomplemented
35#16839631C2212yescomplemented
36#16839626—21no—
37#16839621—21no—
38#16839619C2212yes—
39C4×#11758196—21no—
40C4×C4⋊Dic61C2211yescharacteristic, complemented
41C2×#124×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic, complemented
43C2×#103×C61C2212yescomplemented
44C2×#102×C61C2212yes—
45C4×C4⋊C4×C61C2211yescharacteristic, complemented
46C24×D61—21no—
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C2×Q8×D61—22no—
50C2×Q8×D61—42no—
51C2×Q8×D61—42no—
52C2×Q8×D61C2×C414yes—
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C2×D4×D61—22no—
56C2×D4×D61—42no—
57C2×D4×D61—42no—
58C2×D4×D61C2×C414yes—
59C22×C4×D61—21no—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2312yes—
64#14519766—22no—
65#14519766C2×C414yes—
66#14519765—22no—
67#14519765C2×C414yes—
68#14519764—22no—
69#14519764C2×C414yes—
70#14519763—28no—
71#14519763—28no—
72#14519734—44no—
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C22⋊C4×D61—28no—
84C22⋊C4×D61—28no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C42×D61C2×C414yes—
90C42×D61D414yes—
91C4⋊#92418—22no—
92#14519702—42no—
93#14519701—42no—
94#14519700—42no—
95#14519699—42no—
96#14519698—42no—
97#14519697—42no—
98#14519689—24no—
99#14519687—24no—
100#14519684—21no—
101#14519684—22no—
102C22×Q8×C61C2311yescharacteristic
103C22×D4×C61C2311yescharacteristic
104C22×C8×C61C2312yes—
105#124×C61—22no—
106#124×C61C2×C414yes—
107C2×C4⋊C4×C61C2312yes—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C42×C61—21no—
111C2×C42×C61C2311yescharacteristic
112C4⋊C8×C61—22no—
113#103×C61—42no—
114#102×C61—42no—
115#95×C61—21no—
116C24×C61—21no—
117C23×D61—21no—
118C23×D61—24no—
119C23×D61—44no—
120C23×D61—44no—
121C23×D61—42no—
122C23×D61—42no—
123C23×D61C2×D411yescharacteristic
124C2×Q8×C61—21no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61C22×C412yes—
128C2×D4×C61—21no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61C22×C412yes—
132C22×C4×C61—21no—
133C22×C4×C61—22no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—24no—
138C2×C22⋊D61—44no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144Q8×D61—48no—
145Q8×D61—44no—
146Q8×D61—44no—
147D4×D61—48no—
148D4×D61—44no—
149D4×D61—44no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—21no—
153C2×D244C22×C412yes—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—24no—
159C2×C4×D61—42no—
160C2×C4×D61—44no—
161C2×C4×D61C22⋊C414yes—
162C2×C4×D61C22⋊C414yes—
163C2×C4×D61C22×C412yes—
164C2×C4×D61C2×D412yes—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—21no—
168C2×Dic122—24no—
169C2×Dic122—44no—
170C2×Dic122C22×C412yes—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C22⋊C4×C61—24no—
177C22⋊C4×C61—24no—
178C42×C61—42no—
179C42×C61C22×C412yes—
180C42×C61C2×D412yes—
181C22⋊Dic61—24no—
182C22⋊Dic61—24no—
183#11758198—24no—
184#11758198—24no—
185#11758198—24no—
186#11758198—24no—
187#11758198—44no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190#11758196—44no—
191#11758196—44no—
192C4×Dic61—42no—
193C4×Dic61C22⋊C414yes—
194C4×Dic61C22⋊C414yes—
195C4×Dic61C22×C412yes—
196C4×Dic61C2×D412yes—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C23×C61—22no—
201C23×C61—42no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—42no—
205C23×C61C22×D411yescharacteristic
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—48no—
210C22×D61—48no—
211C22×D61—42no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—42no—
215C22×D61—48no—
216C22×D61—88no—
217C22×D61—88no—
218C22×D61C2×C22⋊C412yes—
219C22×D61AGL(1,ℤ/8)12yes—
220C22×D61#13712yes—
221Q8×C61—24no—
222Q8×C61—41no—
223Q8×C61—41no—
224D4×C61—24no—
225D4×C61—41no—
226D4×C61—41no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61H(5,𝔽2)12yes—
240C2×C4×C61H(5,𝔽2)12yes—
241C22⋊D61—48no—
242C2×Dic61—22no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C2×C22⋊C412yes—
253C2×Dic61C2×C22⋊C412yes—
254D244—24no—
255D244—41no—
256D244—41no—
257C4×D61—28no—
258C4×D61—24no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—44no—
262C4×D61—48no—
263Dic122—24no—
264Dic122—48no—
265Dic122—41no—
266Dic122—41no—
267C8×C61—42no—
268#92418—42no—
269#21808—611nomaximal, Sylow
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—82no—
274C22×C61—82no—
275C22×C61C22×C22⋊C411yescharacteristic
276C22×C61#53512yes—
277C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C61C2×#13711yescharacteristic
279C22×C61#57512yes—
280C2×D61—22no—
281C2×D61—22no—
282C2×D61—24no—
283C2×D61—24no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—816no—
288C2×D61—82no—
289C2×D61—82no—
290C2×D61#41714yes—
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—42no—
294C4×C61—82no—
295C4×C61#51812yes—
296C4×C61#53511yescharacteristic
297C4×C61#53511yescharacteristic
298Dic61—22no—
299Dic61—22no—
300Dic61—22no—
301Dic61—42no—
302Dic61—82no—
303C22×#124—611no—
304C2×#432—611no—
305#2285—611no—
306#1983—611no—
307#1983—611no—
308#1983—611no—
309#1983D6111yescharacteristic, complemented
310#1959—612no—
311#1959—612no—
312#1787—612no—
313#1786—612no—
314C2×C61—82no—
315C2×C61#230911yescharacteristic
316C2×C61C2×#41711yescharacteristic
317C2×C61#289611yescharacteristic
318C2×C61#321312yes—
319C2×C61#321512yes—
320D61—24no—
321D61—82no—
322C23×Q8—611no—
323C23×D4—611no—
324C2×#124—611no—
325C2×#124—611no—
326C2×#124—618no—
327C2×#124—611no—
328C2×#124—1221no—
329C2×#124—1221no—
330C2×#124—1224no—
331C2×#124C2×D6111yescharacteristic
332C2×#117—1221no—
333C22×C22⋊C4—1221no—
334C22×C22⋊C4—1221no—
335C22×C42—611no—
336#432—1224no—
337C2×C4⋊C8—611no—
338C2×C4⋊C8C2×D6111yescharacteristic, complemented
339C2×#103—612no—
340C2×#103—612no—
341C2×#103—612no—
342C2×#103C2×D6112yescomplemented
343C2×#102—612no—
344C2×#102—612no—
345C2×#102—612no—
346C2×#102C2×D6112yes—
347C2×C22⋊C8—612no—
348C2×#97—612no—
349#384—612no—
350#382—1221no—
351#380—612no—
352#380—1221no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—1221no—
355C4×C4⋊C4C2×D6111yescharacteristic, complemented
356C61#2180811yescharacteristic, complemented, Sylow
357C25—1221no—
358C22×Q8—614no—
359C22×Q8—611no—
360C22×Q8—1221no—
361C22×Q8—1222no—
362C22×Q8—2442no—
363C22×Q8—2442no—
364C22×Q8C22×D6111yescharacteristic
365C22×D4—614no—
366C22×D4—611no—
367C22×D4—1222no—
368C22×D4—1221no—
369C22×D4—2442no—
370C22×D4—2442no—
371C22×D4C22×D6111yescharacteristic
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—611no—
375C23×C4—1221no—
376C22×C8—612no—
377C22×C8C22×D6112yes—
378#124—22no—
379#124—614no—
380#124—614no—
381#124—614no—
382#124—1228no—
383#124—1228no—
384#124—1222no—
385#124—1222no—
386#124—1222no—
387#124C4×D6114yes—
388#117—2444no—
389C2×C4⋊C4—612no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4C22×D6112yes—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—1228no—
396C2×C22⋊C4—1228no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1222no—
404C2×C42—21no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—614no—
409C2×C42—614no—
410C2×C42—1221no—
411C2×C42C22×D6111yescharacteristic
412C4⋊C8—22no—
413C4⋊C8—1222no—
414#103—42no—
415#103—2442no—
416#103—2442no—
417#103—2442no—
418#102—42no—
419#102—2442no—
420#102—2442no—
421#102—2442no—
422C22⋊C8—1224no—
423#97—1224no—
424#95—21no—
425#95—1222no—
426#95—1221no—
427C24—21no—
428C24—611no—
429C24—1224no—
430C24—1221no—
431C24—2442no—
432C24—2442no—
433C24—2444no—
434C24—2444no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—612no—
439C2×Q8—1224no—
440C2×Q8—1221no—
441C2×Q8—1222no—
442C2×Q8—1222no—
443C2×Q8—2444no—
444C2×Q8—2448no—
445C2×Q8—2444no—
446C2×Q8—2444no—
447C2×Q8C2×C4×D6112yes—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—612no—
452C2×D4—1224no—
453C2×D4—1221no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—2448no—
457C2×D4—2444no—
458C2×D4—2444no—
459C2×D4—2444no—
460C2×D4C2×C4×D6112yes—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—614no—
469C22×C4—614no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1224no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1221no—
476C22×C4—1222no—
477C22×C4—2442no—
478C22×C4—2444no—
479C22×C4D4×D6111yescharacteristic
480C22×C4D4×D6111yescharacteristic
481C22×C4C23×D6111yescharacteristic
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—1222no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—2444no—
493C4⋊C4—2444no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—1224no—
497C22⋊C4—1224no—
498C22⋊C4—1224no—
499C22⋊C4—1224no—
500C22⋊C4—1224no—
501C22⋊C4—1224no—
502C22⋊C4—2444no—
503C42—42no—
504C42—612no—
505C42—612no—
506C42—614no—
507C42—614no—
508C42—2442no—
509C42C2×C4×D6112yes—
510C42D4×D6112yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—612no—
517C23—612no—
518C23—612no—
519C23—1222no—
520C23—1222no—
521C23—1222no—
522C23—4888no—
523C23—4888no—
524C23—2448no—
525C23—2448no—
526C23—2448no—
527C23—2441no—
528C23—2441no—
529C23—2442no—
530C23—2442no—
531C23C2×D4×D6111yescharacteristic
532Q8—24no—
533Q8—41no—
534Q8—41no—
535Q8—1224no—
536Q8—2448no—
537Q8—2441no—
538Q8—2441no—
539D4—24no—
540D4—41no—
541D4—41no—
542D4—1224no—
543D4—2448no—
544D4—2441no—
545D4—2441no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—1228no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1224no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—2444no—
567C2×C4—2448no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4C22⋊C4×D6112yes—
573C2×C4C22⋊C4×D6112yes—
574C2×C4C22×C4×D6111yescharacteristic
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4#1451982212yes—
577C2×C4#1451982312yes—
578C8—42no—
579C8—2442no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—82no—
584C22—82no—
585C22—614no—
586C22—1224no—
587C22—1224no—
588C22—1222no—
589C22—1222no—
590C22—48816no—
591C22—4882no—
592C22—4882no—
593C22—2442no—
594C22—2442no—
595C22—2442no—
596C22C2×C22⋊C4×D6111yescharacteristic
597C22#1684032812yes—
598C22AGL(1,ℤ/8)×D6111yescharacteristic
599C22#137×D6111yescharacteristic
600C22#1684055912yes—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—82no—
605C4—1222no—
606C4—1222no—
607C4—1222no—
608C4—4882no—
609C4—2442no—
610C4#1684023412yes—
611C4#1684032511yescharacteristic
612C4#1684033011yescharacteristic
613C2—82no—
614C2—1224no—
615C2—4882no—
616C2#2144839811yescharacteristic
617C2#417×D6111yescharacteristic
618C2#2145433411yescharacteristic
619C2#2145680412yes—
620C2#2145682012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1983C2
socleC23×C61C22×D4
prime radicalC22×D4×D61C22
cosocleC22×C4×C61C24
hypercenter#1983D61
hypocenterC61#21808
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. C2×#16839818C2semidirect
3. #21448223C2semidirect
4. #21446752C2semidirect
5. #21446751C2semidirect
6. #21446749C2semidirect
7. #21446634C2semidirect
8. #21446632C2semidirect
9. #21445752C2semidirect
11. C61×#1983C2semidirect
20. #124×D61C4semidirect
26. C2×C4⋊#92418C22semidirect
27. C2×#14519702C22semidirect
28. C2×#14519701C22semidirect
29. C2×#14519700C22semidirect
34. C2×#14519687C22semidirect
35. #16839631C22semidirect
40. C4×C4⋊Dic61C22semidirect
42. C2×C4⋊C8×C61C22semidirect
43. C2×#103×C61C22semidirect
45. C4×C4⋊C4×C61C22semidirect
309. #1983D61semidirect
338. C2×C4⋊C8C2×D61semidirect
342. C2×#103C2×D61semidirect
355. C4×C4⋊C4C2×D61semidirect
356. C61#21808semidirect

Maximal subgroups

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

subgroupindex
C2×#124×D612
C2×#168398182
#214482232
#214467522
#214467512
#214467492
#214466342
#214466322
#214457522
#214457482
C61×#19832
#2180861