Finite Group Databasebeta

← #29714081

Subgroups — #29714081

Order 15616.

1140 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840049C211yesmaximal, characteristic, complemented
2C2×#16839860C211yesmaximal, characteristic, complemented
3#21450720C211yesmaximal, characteristic
4#21447012C212yesmaximal, complemented
5#21446820C212yesmaximal, complemented
6#21446793C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446141C212yesmaximal, complemented
9#21446018C212yesmaximal, complemented
10#21445834C211yesmaximal, characteristic, complemented
11C2×C61×#556C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13C2×#14519751C2212yes—
14C2×C4×D244C2211yescharacteristic
15#16840049—24no—
16#16839860—24no—
17C2×#14519710C2211yescharacteristic
18C2×#14519708C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519701C2212yes—
22C2×#14519697C2211yescharacteristic
23C2×#14519697C2212yes—
24C2×C22⋊#92418C2212yes—
25C2×C22⋊#92418C2212yes—
26C2×#14519688C2211yescharacteristic
27#16839688—21no—
28#16839670—21no—
29#16839638—22no—
30#16839628C2211yescharacteristic
31#16839622C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35C61×#556—24no—
36C61×#556—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#126×C61—22no—
39C2×#124×C61—21no—
40C2×C22⋊Q8×C61C2212yes—
41C2×C4×Q8×C61—21no—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×#117×C61C2212yes—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—22no—
47C22×C4×D61—21no—
48C2×#11758201—22no—
49C2×#11758201—22no—
50C2×#11758201—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2312yes—
53C22×#92418C2311yescharacteristic
54#14519751—24no—
55#14519751—42no—
56C4×D244—22no—
57C4×D244—22no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×C22⋊Dic61—22no—
61C2×#11758198—21no—
62C2×#11758198—22no—
63#14519710—22no—
64#14519708—24no—
65C4⋊#92418—22no—
66#14519701—22no—
67#14519701—42no—
68#14519697—24no—
69#14519697—22no—
70C22⋊#92418—24no—
71C22⋊#92418—42no—
72#14519688—22no—
73#14519684—22no—
74#14519684—21no—
75#14519684—21no—
76C22×Q8×C61—21no—
77C22×Q8×C61C2311yescharacteristic
78C22×D4×C61D411yescharacteristic
79C23×C4×C61C2312yes—
80C4⋊Q8×C61—22no—
81C4⋊Q8×C61D414yescomplemented
82#126×C61—216no—
83#124×C61—28no—
84C22⋊Q8×C61—24no—
85C22⋊Q8×C61—24no—
86C4×Q8×C61—28no—
87C4×Q8×C61—22no—
88C4×Q8×C61D414yescomplemented
89C4×D4×C61—22no—
90C4×D4×C61—22no—
91#117×C61—24no—
92#117×C61—24no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61D411yescharacteristic
97C2×C4⋊C4×C61C2312yes—
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61D412yes—
102C2×C42×C61—22no—
103C2×C42×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C24×C61C2×D412yes—
106C23×D61—21no—
107C2×Q8×C61—24no—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—44no—
114C2×Q8×C61C2×D412yes—
115C2×D4×C61—22no—
116C2×D4×C61—42no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—82no—
133C22×Dic61—21no—
134C22×Dic61—22no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244#4112yes—
141C2×C4×D61—42no—
142C2×C4×D61—82no—
143C4⋊C4×C61—28no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C2×D412yes—
150C4⋊C4×C61C2×D414yes—
151C22⋊C4×C61—28no—
152C22⋊C4×C61—28no—
153C22⋊C4×C61—24no—
154C42×C61—28no—
155C42×C61—24no—
156C42×C61C2×D412yes—
157C42×C61C2×D412yes—
158C22⋊Dic61—82no—
159#11758201—28no—
160#11758198—44no—
161#11758198—42no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—21no—
172C2×#92418—22no—
173C2×#92418—21no—
174C2×#92418#4112yes—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—42no—
187C22×D61—42no—
188Q8×C61—22no—
189Q8×C61—22no—
190Q8×C61—48no—
191Q8×C61—41no—
192Q8×C61—41no—
193D4×C61—44no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—44no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61#12012yes—
211C2×C4×C61#12012yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61H(5,𝔽2)11yescharacteristic
217C2×C4×C612−1+411yescharacteristic
218C22⋊D61—88no—
219C2×Dic61—41no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224D244—22no—
225D244—22no—
226D244—41no—
227D244—41no—
228C4×D61—84no—
229#92418—22no—
230#92418—41no—
231#92418—42no—
232#24131—611nomaximal, Sylow
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61C2×#12011yescharacteristic
241C22×C61#54611yescharacteristic
242C22×C61#54711yescharacteristic
243C22×C61#57512yes—
244C22×C61#57712yes—
245C2×D61—42no—
246C2×D61—44no—
247C2×D61—82no—
248C2×D61—81no—
249C2×D61—81no—
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—42no—
253C4×C61—42no—
254C4×C61—44no—
255C4×C61—42no—
256C4×C61#52511yescharacteristic
257C4×C61#53711yescharacteristic
258C4×C61#54811yescharacteristic
259C4×C61#55011yescharacteristic
260Dic61—81no—
261Dic61—82no—
262C2×#556D6111yescharacteristic, complemented
263C2×#476—611no—
264C2×#440—611no—
265#2535—611no—
266#2035—612no—
267#2000—612no—
268#1993—611no—
269#1955—611no—
270#1875—612no—
271#1854—612no—
272#1805—611no—
273C2×C61—82no—
274C2×C61#240711yescharacteristic
275C2×C61#300511yescharacteristic
276C2×C61#301011yescharacteristic
277C2×C61#331112yes—
278C2×C61#335012yes—
279D61—82no—
280C22×SD16—611no—
281#556—24no—
282#556—24no—
283C2×C4⋊Q8C2×D6111yescharacteristic
284C2×#126—22no—
285C2×#124—21no—
286C2×C22⋊Q8C2×D6112yes—
287C2×#121—612no—
288C2×C4×Q8—21no—
289C2×C4×Q8C2×D6111yescharacteristic
290C2×C4×D4—611no—
291C2×C4×D4C2×D6111yescharacteristic
292C2×#117C2×D6112yes—
293#476—1224no—
294#440—1224no—
295C2×#107—611no—
296C2×#106—612no—
297C2×C4⋊C8—611no—
298C2×#103—611no—
299C2×#103—612no—
300C2×#102—611no—
301C2×#102—612no—
302C2×C22⋊C8—612no—
303C2×C22⋊C8—612no—
304C2×#97—611no—
305#397—1221no—
306#392—1221no—
307#386—1222no—
308#382—611no—
309#380—612no—
310#340—612no—
311#340—612no—
312#340—612no—
313C61#2413111yescharacteristic, complemented, Sylow
314C22×Q8—21no—
315C22×Q8C22×D6111yescharacteristic
316C22×D4—611no—
317C22×D4—1222no—
318C22×D4C22⋊D6111yescharacteristic
319C23×C4—1221no—
320C23×C4C22×D6112yes—
321C2×SD16—1222no—
322C2×SD16—1222no—
323C2×SD16—1222no—
324C22×C8—611no—
325C22×C8—611no—
326C22×C8—612no—
327C4⋊Q8—22no—
328C4⋊Q8C22⋊D6114yescomplemented
329#126—216no—
330#124—28no—
331C22⋊Q8—24no—
332C22⋊Q8—24no—
333#121—1224no—
334#121—2442no—
335C4×Q8—28no—
336C4×Q8—22no—
337C4×Q8C22⋊D6114yescomplemented
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—1222no—
341C4×D4—1222no—
342#117—24no—
343#117—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4C22⋊D6111yescharacteristic
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C4⋊C4C22×D6112yes—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—1222no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1222no—
357C2×C22⋊C4C22⋊D6112yes—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42C22×D6111yescharacteristic
361#107—1222no—
362#106—1224no—
363C4⋊C8—1222no—
364#103—1222no—
365#103—2442no—
366#102—1222no—
367#102—1224no—
368C22⋊C8—1224no—
369C22⋊C8—2442no—
370#97—1222no—
371#95—1222no—
372#95—1221no—
373#95—1221no—
374C24—1221no—
375C24C2×C22⋊D6112yes—
376C2×Q8—21no—
377C2×Q8—24no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—44no—
383C2×Q8C2×C22⋊D6112yes—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—612no—
387C2×D4—1221no—
388C2×D4—1221no—
389C2×D4—1221no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—4882no—
393C2×D4—2444no—
394C2×D4—2444no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—1222no—
405C22×C4—1221no—
406C22×C4—4882no—
407C22×C4—2442no—
408C22×C4C2×C22⋊D6112yes—
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C2×C22⋊D6111yescharacteristic
411C22×C4C23×D6111yescharacteristic
412SD16—1228no—
413C2×C8—612no—
414C2×C8—1221no—
415C2×C8—1221no—
416C2×C8—1221no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—1221no—
420C2×C8—1222no—
421C2×C8—1221no—
422C4⋊C4—28no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1222no—
431C4⋊C4—1222no—
432C4⋊C4C2×C22⋊D6114yes—
433C4⋊C4C2×C22⋊D6112yes—
434C22⋊C4—28no—
435C22⋊C4—28no—
436C22⋊C4—24no—
437C22⋊C4—4882no—
438C22⋊C4—2444no—
439C22⋊C4—2442no—
440C42—28no—
441C42—24no—
442C42C2×C22⋊D6112yes—
443C42C2×C22⋊D6112yes—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—1222no—
452C23—2441no—
453C23—2442no—
454C23—2442no—
455C23—2441no—
456C23C22×C22⋊D6111yescharacteristic
457Q8—22no—
458Q8—22no—
459Q8—48no—
460Q8—41no—
461Q8—41no—
462D4—44no—
463D4—1222no—
464D4—1222no—
465D4—4888no—
466D4—2441no—
467D4—2441no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—4884no—
485C2×C4—2441no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4#1451974612yes—
491C2×C4#1451974612yes—
492C2×C4C22×C22⋊D6111yescharacteristic
493C2×C4C22×C22⋊D6111yescharacteristic
494C2×C4#41×D6111yescharacteristic
495C2×C4#41×D6111yescharacteristic
496C2×C4#1451982311yescharacteristic
497C2×C4#1451982411yescharacteristic
498C8—1222no—
499C8—2441no—
500C8—2442no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—4881no—
509C22—4882no—
510C22—4881no—
511C22—2444no—
512C22—2442no—
513C22C2×#1451974611yescharacteristic
514C22#1684040011yescharacteristic
515C22#1684041111yescharacteristic
516C22#1684056012yes—
517C22#1684057012yes—
518C4—22no—
519C4—22no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—44no—
524C4—4881no—
525C4—4882no—
526C4#1684026911yescharacteristic
527C4#1684034011yescharacteristic
528C4#1684041911yescharacteristic
529C4#1684043111yescharacteristic
530C2—82no—
531C2—4882no—
532C2#2144915811yescharacteristic
533C2#2145506111yescharacteristic
534C2#2145510511yescharacteristic
535C2#2145784912yes—
536C2#2145836412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#556C2
socleC23×C61C22×D4
prime radical#21450720C2
hypercenterC2×#556D61
hypocenterC61#24131
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×#16840049C2semidirect
2. C2×#16839860C2semidirect
4. #21447012C2semidirect
5. #21446820C2semidirect
6. #21446793C2semidirect
7. #21446618C2semidirect
8. #21446141C2semidirect
9. #21446018C2semidirect
10. #21445834C2semidirect
11. C2×C61×#556C2semidirect
81. C4⋊Q8×C61D4semidirect
88. C4×Q8×C61D4semidirect
262. C2×#556D61semidirect
313. C61#24131semidirect
328. C4⋊Q8C22⋊D61semidirect
337. C4×Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
C2×#168398602
#214507202
#214470122
#214468202
#214467932
#214466182
#214461412
#214460182
#214458342
C2×C61×#5562
#2413161