Finite Group Databasebeta

← #29733306

Subgroups — #29733306

Order 15616.

1226 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840124C211yesmaximal, characteristic, complemented
2C2×#16839977C211yesmaximal, characteristic, complemented
3#21450758C211yesmaximal, characteristic, complemented
4#21446822C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446772C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446141C212yesmaximal, complemented
9#21446131C212yesmaximal, complemented
10#21445644C211yesmaximal, characteristic, complemented
11C2×C61×#555C211yesmaximal, characteristic, complemented
12C22×#11758200C2211yescharacteristic
13C22×#11758199C2211yescharacteristic
14C2×#14519765C2211yescharacteristic
15C2×#14519756C2212yes—
16C2×#14519751C2212yes—
17#16840124—24no—
18#16839977—24no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519700C2212yes—
21C2×#14519700C2211yescharacteristic
22C2×#14519698C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2211yescharacteristic
25C2×C22⋊#92418C2212yes—
26C2×C22⋊#92418C2212yes—
27C2×#14519688C2211yescharacteristic
28#16839697—21no—
29#16839670—21no—
30#16839662—22no—
31C4×C4⋊Dic61C2211yescharacteristic
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C61×#555—24no—
36C61×#555—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#124×C61—22no—
39C2×#121×C61—21no—
40C2×#121×C61C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C2×#117×C61—21no—
44C2×#117×C61C2212yes—
45C22×Dic122C2311yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C22⋊D61—22no—
48C2×#11758200—22no—
49C2×#11758200—42no—
50C2×#11758199—22no—
51C2×#11758199—42no—
52C22×#92418C2312yes—
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55#14519765—22no—
56#14519765—22no—
57#14519756—24no—
58#14519756—42no—
59#14519751—24no—
60#14519751—42no—
61C2×C4⋊Dic61C2312yes—
62C2×#11758196—22no—
63C2×C22⋊Dic61—22no—
64C2×#11758198—22no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×C4×Dic61—21no—
68C4⋊#92418—22no—
69#14519700—22no—
70#14519700—24no—
71#14519698—24no—
72#14519697—22no—
73#14519697—24no—
74C22⋊#92418—42no—
75C22⋊#92418—42no—
76#14519688—22no—
77#14519684—21no—
78C22×Q8×C61—21no—
79C22×D4×C61—22no—
80C22×D4×C61D411yescharacteristic
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61—21no—
83C23×C4×C61C2312yes—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61D414yescomplemented
86#124×C61—216no—
87#121×C61—28no—
88#121×C61—24no—
89#121×C61—24no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61D414yescomplemented
94#117×C61—28no—
95#117×C61—24no—
96#117×C61—24no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61C2312yes—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61D412yes—
105C2×C42×C61—22no—
106C2×C42×C61C2311yescharacteristic
107C24×C61—21no—
108C24×C61C2×D412yes—
109C23×D61—21no—
110C2×Q8×C61—24no—
111C2×Q8×C61—81no—
112C2×Q8×C61—81no—
113C2×D4×C61—28no—
114C2×D4×C61—22no—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—42no—
119C2×D4×C61—82no—
120C2×D4×C61—82no—
121C2×D4×C61C2×D412yes—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—24no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—44no—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—44no—
137C2×C22⋊D61—44no—
138C2×C22⋊D61—82no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—22no—
144C2×D244—22no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C2×D412yes—
154C4⋊C4×C61C2×D414yes—
155C22⋊C4×C61—28no—
156C22⋊C4×C61—28no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—24no—
159C42×C61—28no—
160C42×C61C2×D412yes—
161C42×C61C2×D412yes—
162C22⋊Dic61—44no—
163#11758200—44no—
164#11758199—44no—
165#11758198—44no—
166#11758198—42no—
167#11758198—42no—
168C4⋊Dic61—22no—
169C4⋊Dic61#4114yes—
170#11758196—44no—
171C4×Dic61—81no—
172C2×#92418—22no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61C22×D411yescharacteristic
192C22×D61—41no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—41no—
196C22×D61—41no—
197Q8×C61—84no—
198D4×C61—22no—
199D4×C61—22no—
200D4×C61—44no—
201D4×C61—42no—
202D4×C61—88no—
203C2×C4×C61—22no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—48no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61#12012yes—
220C2×C4×C61#12012yes—
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4112yes—
224C2×C4×C61H(5,𝔽2)12yes—
225C22⋊D61—88no—
226C2×Dic61—24no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231D244—41no—
232D244—41no—
233D244—42no—
234Dic122—41no—
235Dic122—41no—
236Dic122—42no—
237#92418—41no—
238#92418—41no—
239#92418—42no—
240#25578—611nomaximal, Sylow
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—82no—
252C22×C61—82no—
253C22×C61C2×#12011yescharacteristic
254C22×C61#54612yes—
255C22×C61#57512yes—
256C22×C61#57611yescharacteristic
257C22×C61#57711yescharacteristic
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—82no—
262C2×D61—82no—
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—42no—
266C4×C61—42no—
267C4×C61—44no—
268C4×C61#53211yescharacteristic
269C4×C61#53511yescharacteristic
270C4×C61#55012yes—
271Dic61—44no—
272Dic61—81no—
273C2×#555D6111yescharacteristic, complemented
274C2×#489—611no—
275C2×#463—611no—
276#2540—611no—
277#2000—612no—
278#1999—612no—
279#1988—611no—
280#1955—611no—
281#1875—612no—
282#1873—612no—
283#1762—611no—
284C2×C61—82no—
285C2×C61—82no—
286C2×C61#243111yescharacteristic
287C2×C61#300511yescharacteristic
288C2×C61#301211yescharacteristic
289C2×C61#331912yes—
290C2×C61#332112yes—
291D61—161no—
292C22×SD16—611no—
293C22×D8—611no—
294#555—24no—
295#555—24no—
296C2×C4⋊Q8C2×D6111yescharacteristic
297C2×#124—22no—
298C2×#124—611no—
299C2×C22⋊Q8—612no—
300C2×#121—21no—
301C2×#121—612no—
302C2×#121C2×D6112yes—
303C2×C4×D4C2×D6111yescharacteristic
304C2×C4×D4C2×D6111yescharacteristic
305C2×#117—21no—
306C2×#117C2×D6112yes—
307#489—1224no—
308#463—1224no—
309C2×C4⋊C8—611no—
310C2×#103—612no—
311C2×#103—611no—
312C2×#102—612no—
313C2×#102—612no—
314C2×#102—611no—
315C2×C22⋊C8—612no—
316C2×C22⋊C8—612no—
317C2×#97—611no—
318#399—1221no—
319#392—1221no—
320#390—1222no—
321C4×C4⋊C4—611no—
322#340—612no—
323#340—612no—
324#340—612no—
325C61#2557811yescharacteristic, complemented, Sylow
326C22×Q8—21no—
327C22×Q8—611no—
328C22×D4—22no—
329C22×D4—611no—
330C22×D4—1222no—
331C22×D4C22⋊D6111yescharacteristic
332C22×D4C22×D6111yescharacteristic
333C23×C4—21no—
334C23×C4C22×D6112yes—
335C2×SD16—1222no—
336C2×SD16—2442no—
337C2×D8—1222no—
338C2×D8—2442no—
339C22×C8—611no—
340C22×C8—611no—
341C22×C8—612no—
342C4⋊Q8—22no—
343C4⋊Q8C22⋊D6114yescomplemented
344#124—216no—
345#124—1222no—
346#124—1222no—
347C22⋊Q8—1224no—
348C22⋊Q8—2442no—
349#121—28no—
350#121—24no—
351#121—24no—
352#121—1224no—
353#121—2442no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4C22⋊D6114yescomplemented
358#117—28no—
359#117—24no—
360#117—24no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4C22⋊D6111yescharacteristic
365C2×C4⋊C4C22×D6111yescharacteristic
366C2×C4⋊C4C22×D6112yes—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—1222no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4C22⋊D6112yes—
375C2×C42—22no—
376C2×C42—1221no—
377C2×C42C22×D6111yescharacteristic
378C4⋊C8—1222no—
379#103—1222no—
380#103—1224no—
381#102—1224no—
382#102—1222no—
383#102—1224no—
384C22⋊C8—2442no—
385C22⋊C8—2442no—
386#97—1222no—
387#95—1221no—
388C24—21no—
389C24—1221no—
390C24C2×C22⋊D6112yes—
391C2×Q8—24no—
392C2×Q8—81no—
393C2×Q8—81no—
394C2×Q8—1221no—
395C2×Q8—1221no—
396C2×Q8—1222no—
397C2×Q8—1222no—
398C2×D4—28no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—1222no—
409C2×D4—1222no—
410C2×D4—4882no—
411C2×D4—2444no—
412C2×D4—2444no—
413C2×D4C2×C22⋊D6112yes—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—44no—
424C22×C4—1222no—
425C22×C4—1221no—
426C22×C4C2×C22⋊D6112yes—
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430SD16—2444no—
431D8—2444no—
432C2×C8—1222no—
433C2×C8—1221no—
434C2×C8—1221no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—614no—
444C4⋊C4—1222no—
445C4⋊C4—2444no—
446C4⋊C4C2×C22⋊D6112yes—
447C4⋊C4C2×C22⋊D6114yes—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—2444no—
453C22⋊C4—2444no—
454C22⋊C4—2442no—
455C22⋊C4—2442no—
456C42—28no—
457C42—4881no—
458C42C2×C22⋊D6112yes—
459C42C2×C22⋊D6112yes—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—42no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—2441no—
473C23—2442no—
474C23—2441no—
475C23—2441no—
476C23—2442no—
477C23C22×C22⋊D6111yescharacteristic
478Q8—84no—
479Q8—2441no—
480Q8—2441no—
481Q8—2442no—
482D4—22no—
483D4—22no—
484D4—44no—
485D4—42no—
486D4—88no—
487D4—4888no—
488D4—2441no—
489D4—2441no—
490D4—2442no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—1224no—
508C2×C4—2441no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4#1451974612yes—
513C2×C4#1451974612yes—
514C2×C4C22×C22⋊D6111yescharacteristic
515C2×C4C22×C22⋊D6111yescharacteristic
516C2×C4#41×D6112yes—
517C2×C4#1451982312yes—
518C8—2441no—
519C8—2441no—
520C8—2442no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—82no—
532C22—82no—
533C22—4881no—
534C22—4882no—
535C22—4882no—
536C22—4881no—
537C22—4881no—
538C22C2×#1451974611yescharacteristic
539C22#1684040012yes—
540C22#1684056012yes—
541C22#1684056511yescharacteristic
542C22#1684057311yescharacteristic
543C4—22no—
544C4—22no—
545C4—44no—
546C4—42no—
547C4—42no—
548C4—4881no—
549C4—2444no—
550C4#1684030811yescharacteristic
551C4#1684032611yescharacteristic
552C4#1684043112yes—
553C2—82no—
554C2—82no—
555C2—9761no—
556C2#2144943411yescharacteristic
557C2#2145506111yescharacteristic
558C2#2145512311yescharacteristic
559C2#2145796912yes—
560C2#2145799912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#555C2
socleC23×C61C22×D4
hypercenterC2×#555D61
hypocenterC61#25578
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×#16840124C2semidirect
2. C2×#16839977C2semidirect
3. #21450758C2semidirect
4. #21446822C2semidirect
5. #21446815C2semidirect
6. #21446772C2semidirect
7. #21446618C2semidirect
8. #21446141C2semidirect
9. #21446131C2semidirect
10. #21445644C2semidirect
11. C2×C61×#555C2semidirect
85. C4⋊Q8×C61D4semidirect
93. C4×D4×C61D4semidirect
273. C2×#555D61semidirect
325. C61#25578semidirect
343. C4⋊Q8C22⋊D61semidirect
357. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401242
C2×#168399772
#214507582
#214468222
#214468152
#214467722
#214466182
#214461412
#214461312
#214456442
C2×C61×#5552
#2557861