Finite Group Databasebeta

← #29689330

Subgroups — #29689330

Order 15616.

1418 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×#16840311C211yesmaximal, characteristic
2C2×#16839814C211yesmaximal, characteristic, complemented
3#21448223C211yesmaximal, characteristic, complemented
4#21446752C211yesmaximal, characteristic, complemented
5#21446751C211yesmaximal, characteristic, complemented
6#21446749C211yesmaximal, characteristic, complemented
7#21446623C212yesmaximal, complemented
8#21446619C212yesmaximal, complemented
9#21445758C212yesmaximal, complemented
10#21445756C212yesmaximal, complemented
11C61×#1983C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C22×#11758224C2211yescharacteristic
14#16840311—24no—
15#16840311C418yescomplemented
16C2×#14519775C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519750—21no—
20C2×#14519750—21no—
21C2×#14519737—21no—
22C2×#14519737—21no—
23C2×#14519734—21no—
24C2×C42×D61C2211yescharacteristic
25#16839814—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×#14519699C2212yes—
31C2×#14519698C2212yes—
32C2×#14519697C2212yes—
33C2×#14519689C2212yes—
34C2×C8×Dic61C2212yes—
35#16839631C2212yes—
36#16839626—21no—
37#16839621—21no—
38#16839619C2212yes—
39C4×#11758196—21no—
40C4×C4⋊Dic61C2211yescharacteristic
41C2×#124×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C2×#102×C61C2212yes—
45C4×C4⋊C4×C61C2211yescharacteristic
46C2×#11758226—22no—
47C2×#11758226—42no—
48C2×#11758226—42no—
49C2×#11758226C2×C414yes—
50C2×#11758224—22no—
51C2×#11758224—42no—
52C2×#11758224—42no—
53C2×#11758224C2×C414yes—
54C22×Dic122C2311yescharacteristic
55C22×D244—21no—
56C22×D244C2311yescharacteristic
57C22×C22⋊D61—21no—
58C22×C22⋊D61—21no—
59C22×C22⋊D61—21no—
60C23×Dic61—21no—
61C22×C4×D61—21no—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C22×#92418C2312yes—
66#14519775—22no—
67#14519775C2×C414yes—
68#14519773—22no—
69#14519773C2×C414yes—
70#14519765—22no—
71#14519765C2×C414yes—
72#14519750—28no—
73#14519750—28no—
74#14519737—28no—
75#14519737—28no—
76#14519734—44no—
77C2×C4⋊Dic61C2312yes—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—22no—
81C2×#11758196—22no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—22no—
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—
99C8×Dic61—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×D61C2×D411yescharacteristic
119C2×Q8×C61—21no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61C22×C412yes—
123C2×D4×C61—21no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61C22×C412yes—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—24no—
133C2×C22⋊D61—24no—
134C2×C22⋊D61—24no—
135C2×C22⋊D61—44no—
136C2×C22⋊D61—44no—
137C2×C22⋊D61—42no—
138C2×C22⋊D61—42no—
139C22×Dic61—21no—
140C22×Dic61—24no—
141C22×Dic61—22no—
142C22×Dic61—44no—
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146#11758226—48no—
147#11758226—44no—
148#11758226—44no—
149#11758224—48no—
150#11758224—44no—
151#11758224—44no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—24no—
155C2×D244—21no—
156C2×D244—44no—
157C2×D244C22×C412yes—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—24no—
164C2×C4×D61—44no—
165C2×C4×D61C22⋊C414yes—
166C2×C4×D61C22⋊C414yes—
167C2×C4×D61C22×C412yes—
168C2×C4×D61C2×D412yes—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—21no—
172C2×Dic122C22×C412yes—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—24no—
180C42×C61—42no—
181C42×C61C22×C412yes—
182C42×C61C2×D412yes—
183#11758198—24no—
184#11758198—24no—
185#11758198—44no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188#11758196—24no—
189#11758196—24no—
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—42no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61C22×D411yescharacteristic
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—22no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61C2×C22⋊C412yes—
216Q8×C61—24no—
217Q8×C61—41no—
218Q8×C61—41no—
219D4×C61—24no—
220D4×C61—41no—
221D4×C61—41no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61C2×C22⋊C412yes—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C23×C411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61H(5,𝔽2)12yes—
235C2×C4×C61H(5,𝔽2)12yes—
236C22⋊D61—48no—
237C22⋊D61—48no—
238C22⋊D61—48no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—48no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×C22⋊C412yes—
249C2×Dic61#13514yes—
250C2×Dic61AGL(1,ℤ/8)12yes—
251C2×Dic61#13712yes—
252D244—24no—
253D244—48no—
254D244—41no—
255D244—41no—
256C4×D61—24no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—28no—
260C4×D61—28no—
261C4×D61—48no—
262Dic122—24no—
263Dic122—41no—
264Dic122—41no—
265C8×C61—42no—
266#92418—42no—
267#21805—611nomaximal, Sylow
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—82no—
272C22×C61—82no—
273C22×C61C22×C22⋊C411yescharacteristic
274C22×C61#53512yes—
275C22×C61C2×#13512yes—
276C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—82no—
287C2×D61—82no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—42no—
291C4×C61—82no—
292C4×C61#51812yes—
293C4×C61#53511yescharacteristic
294C4×C61#53511yescharacteristic
295Dic61—22no—
296Dic61—42no—
297Dic61—82no—
298Dic61#41614yes—
299Dic61#41714yes—
300C2×#532—611no—
301C2×#431—611no—
302#2285—611no—
303#1983—611no—
304#1983—611no—
305#1983—611no—
306#1983D6111yescharacteristic, complemented
307#1957—612no—
308#1956—612no—
309#1788—612no—
310#1788—612no—
311C2×C61—82no—
312C2×C61#230911yescharacteristic
313C2×C61C2×#41611yescharacteristic
314C2×C61C2×#41711yescharacteristic
315C2×C61#320712yes—
316C2×C61#321012yes—
317D61—42no—
318D61—82no—
319C22×#41—611no—
320C22×#41—611no—
321#532—618no—
322#532—1224no—
323C2×C4⋊Q8—611no—
324C2×#127—611no—
325C2×#124—611no—
326C2×#124C2×D6111yescharacteristic
327C2×#121—1221no—
328C2×#121—1221no—
329C2×C4×D4—1221no—
330C2×C4×D4—1221no—
331C2×#117—1221no—
332C22×C42—611no—
333#431—1224no—
334C2×C4⋊C8—611no—
335C2×C4⋊C8C2×D6111yescharacteristic
336C2×#103—612no—
337C2×#103—612no—
338C2×#103—612no—
339C2×#103C2×D6112yes—
340C2×#102—612no—
341C2×#102—612no—
342C2×#102—612no—
343C2×#102C2×D6112yes—
344C2×C22⋊C8—612no—
345C2×C4×C8—612no—
346#384—612no—
347#382—1221no—
348#380—612no—
349#380—1221no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—1221no—
352C4×C4⋊C4C2×D6111yescharacteristic
353C61#2180511yescharacteristic, complemented, Sylow
354C2×#41—614no—
355C2×#41—614no—
356C2×#41—1222no—
357C2×#41—1222no—
358C2×#41—2442no—
359C2×#41—2442no—
360C2×#41—2442no—
361C2×#41—2442no—
362C22×Q8—611no—
363C22×Q8C22×D6111yescharacteristic
364C22×D4—611no—
365C22×D4—1221no—
366C22×D4—1221no—
367C22×D4—1221no—
368C22×D4—1221no—
369C22×D4C22×D6111yescharacteristic
370C23×C4—611no—
371C23×C4—611no—
372C23×C4—611no—
373C23×C4—1221no—
374C23×C4—1221no—
375C22×C8—612no—
376C22×C8C22×D6112yes—
377C4⋊Q8—614no—
378C4⋊Q8—1222no—
379#127—614no—
380#127—1222no—
381#124—22no—
382#124—614no—
383#124—1222no—
384#124C4×D6114yes—
385#121—1228no—
386#121—1228no—
387C4×D4—1228no—
388C4×D4—1228no—
389#117—2444no—
390C2×C4⋊C4—612no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4C22×D6112yes—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1222no—
401C2×C42—21no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—614no—
406C2×C42—614no—
407C2×C42—1221no—
408C2×C42C22×D6111yescharacteristic
409C4⋊C8—22no—
410C4⋊C8—1222no—
411#103—42no—
412#103—2442no—
413#103—2442no—
414#103—2442no—
415#102—42no—
416#102—2442no—
417#102—2442no—
418#102—2442no—
419C22⋊C8—1224no—
420C4×C8—1224no—
421#95—21no—
422#95—1222no—
423#95—1221no—
424C24—21no—
425C24—611no—
426C24—1221no—
427#41—2448no—
428#41—2444no—
429#41—2444no—
430#41—2448no—
431#41—2444no—
432#41—2444no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—612no—
437C2×Q8—1221no—
438C2×Q8—1222no—
439C2×Q8—1222no—
440C2×Q8C2×C4×D6112yes—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—612no—
445C2×D4—1224no—
446C2×D4—1224no—
447C2×D4—1224no—
448C2×D4—1224no—
449C2×D4—1221no—
450C2×D4—1222no—
451C2×D4—1222no—
452C2×D4—2444no—
453C2×D4—2444no—
454C2×D4—2442no—
455C2×D4—2442no—
456C2×D4—2444no—
457C2×D4C2×C4×D6112yes—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—612no—
464C22×C4—612no—
465C22×C4—614no—
466C22×C4—614no—
467C22×C4—1222no—
468C22×C4—1224no—
469C22×C4—1222no—
470C22×C4—1224no—
471C22×C4—1224no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—2444no—
477C22×C4—2444no—
478C22×C4D4×D6111yescharacteristic
479C22×C4D4×D6111yescharacteristic
480C22×C4C23×D6111yescharacteristic
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—1222no—
485C2×C8—1222no—
486C2×C8—1222no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—1224no—
490C4⋊C4—1224no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—2444no—
494C4⋊C4—2444no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—1224no—
498C22⋊C4—1224no—
499C22⋊C4—2444no—
500C42—42no—
501C42—612no—
502C42—612no—
503C42—614no—
504C42—614no—
505C42—2442no—
506C42C2×C4×D6112yes—
507C42D4×D6112yes—
508C23—22no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—612no—
514C23—1222no—
515C23—1221no—
516C23—1221no—
517C23—1222no—
518C23—1222no—
519C23—2442no—
520C23—2441no—
521C23—2441no—
522C23—2442no—
523C23C2×D4×D6111yescharacteristic
524Q8—24no—
525Q8—41no—
526Q8—41no—
527Q8—1224no—
528Q8—2441no—
529Q8—2441no—
530D4—24no—
531D4—41no—
532D4—41no—
533D4—1224no—
534D4—2448no—
535D4—2448no—
536D4—2448no—
537D4—2448no—
538D4—2441no—
539D4—2441no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—614no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—1228no—
555C2×C4—1228no—
556C2×C4—1222no—
557C2×C4—1224no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—2448no—
562C2×C4—2448no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4C22⋊C4×D6112yes—
568C2×C4C22⋊C4×D6112yes—
569C2×C4C22×C4×D6111yescharacteristic
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4#1451982212yes—
572C2×C4#1451982312yes—
573C8—42no—
574C8—2442no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—82no—
579C22—82no—
580C22—1222no—
581C22—1222no—
582C22—1222no—
583C22—4882no—
584C22—4882no—
585C22—2442no—
586C22—2442no—
587C22—2442no—
588C22—2442no—
589C22—2442no—
590C22C2×C22⋊C4×D6111yescharacteristic
591C22#1684032812yes—
592C22#135×D6112yes—
593C22AGL(1,ℤ/8)×D6111yescharacteristic
594C22#137×D6111yescharacteristic
595C4—22no—
596C4—22no—
597C4—42no—
598C4—82no—
599C4—614no—
600C4—614no—
601C4—1222no—
602C4—4882no—
603C4—2442no—
604C4#1684023412yes—
605C4#1684032511yescharacteristic
606C4#1684033011yescharacteristic
607C2—82no—
608C2—4882no—
609C2—2442no—
610C2#2144839811yescharacteristic
611C2#416×D6111yescharacteristic
612C2#417×D6111yescharacteristic
613C2#2145675112yes—
614C2#2145677412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1983C2
socleC23×C61C22×D4
prime radicalC2×#16840311C2
cosocleC22×C4×C61C24
hypercenter#1983D61
hypocenterC61#21805
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×#16839814C2semidirect
3. #21448223C2semidirect
4. #21446752C2semidirect
5. #21446751C2semidirect
6. #21446749C2semidirect
7. #21446623C2semidirect
8. #21446619C2semidirect
9. #21445758C2semidirect
10. #21445756C2semidirect
11. C61×#1983C2semidirect
15. #16840311C4semidirect
306. #1983D61semidirect
353. C61#21805semidirect

Maximal subgroups

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

subgroupindex
C2×#168403112
C2×#168398142
#214482232
#214467522
#214467512
#214467492
#214466232
#214466192
#214457582
#214457562
C61×#19832
#2180561