Finite Group Databasebeta

← #29702253

Subgroups — #29702253

Order 15616.

1170 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839958C211yesmaximal, characteristic
2C2×C4×D488C212yesmaximal, complemented
3#21448883C211yesmaximal, characteristic, complemented
4C4⋊C4×D244C211yesmaximal, characteristic, complemented
5#21446625C211yesmaximal, characteristic, complemented
6#21446578C211yesmaximal, characteristic, complemented
7#21446502C212yesmaximal, complemented
8#21446166C212yesmaximal, complemented
9#21446069C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
12C22×D488C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×C4×D244C2212yes—
15C2×C4×D244C2212yes—
16C2×C4×D244C2211yescharacteristic
17C2×#14519739—21no—
18C2×C4⋊C4×D61—21no—
19#16839958C418yescomplemented
20C4×D488—28no—
21C2×#14519711C2212yes—
22C2×#14519699C2211yescharacteristic
23C2×#14519699C2212yes—
24C2×#14519699C2212yes—
25C2×#14519699C2212yes—
26C2×#14519699C2211yescharacteristic
27#16839664C2212yes—
28#16839663—21no—
29#16839660C2212yes—
30#16839640—22no—
31#16839636—22no—
32#16839632C2211yescharacteristic
33#16839619—21no—
34C4×C4⋊Dic61C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839525C2212yes—
38C2×C4⋊C8×C61C2211yescharacteristic
39C2×C4×C8×C61C2211yescharacteristic
40C2×C4×C8×C61C2212yes—
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#336C2212yes—
43C22×D244—21no—
44C22×D244—21no—
45C22×D244C2312yes—
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C2×D488—24no—
51C2×D488—24no—
52C2×D488C2×C418yescomplemented
53#14519774—22no—
54#14519774C2×C414yescomplemented
55C4×D244—22no—
56C4×D244—24no—
57C4×D244—24no—
58C4×D244—42no—
59C4×D244—42no—
60C4×D244C2×C414yescomplemented
61#14519739—44no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61C2312yes—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×#11758196—22no—
69C4⋊C4×D61—44no—
70C2×#11758198—21no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×C4×Dic61—21no—
76#14519711—24no—
77#14519699—22no—
78#14519699—24no—
79#14519699—42no—
80#14519699C2×C418yescomplemented
81#14519699D414yescomplemented
82#14519684—21no—
83#14519684—22no—
84C22×C8×C61C2311yescharacteristic
85C22×C8×C61C2312yes—
86C22×C8×C61C2311yescharacteristic
87C2×C4⋊C4×C61D411yescharacteristic
88C2×C4⋊C4×C61D411yescharacteristic
89C2×C42×C61C2312yes—
90C2×C42×C61C2311yescharacteristic
91C4⋊C8×C61C2×C414yescomplemented
92C4×C8×C61—24no—
93C4×C8×C61—22no—
94#95×C61D412yes—
95C23×D61—21no—
96C23×D61—22no—
97C22×C4×C61C2×D412yes—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D412yes—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2411yescharacteristic
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—22no—
105C22×Dic61—21no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—24no—
114C2×D244—22no—
115C2×D244—24no—
116C2×D244—42no—
117C2×D244—42no—
118C2×D244—81no—
119C2×D244—81no—
120C2×D244C22×C414yes—
121C2×D244C22×C412yes—
122C2×D244C2×D412yes—
123C2×D244#4112yes—
124C2×C4×D61—24no—
125C2×C4×D61—42no—
126C2×C4×D61—42no—
127C2×C4×D61—44no—
128C2×C4×D61—44no—
129C2×C4×D61—44no—
130C2×C4×D61—44no—
131C2×C4×D61—82no—
132C2×C4×D61—82no—
133C2×C8×C61—22no—
134C2×C8×C61—21no—
135C2×C8×C61—22no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61C22×C414yes—
141C2×C8×C61C2×D412yes—
142C4⋊C4×C61—22no—
143C4⋊C4×C61D814yescomplemented
144C42×C61—22no—
145C42×C61—22no—
146C42×C61C22×C412yes—
147C42×C61#4112yes—
148#11758198—24no—
149#11758198—44no—
150#11758198—44no—
151#11758198—44no—
152#11758198—82no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61—21no—
157C4⋊Dic61—22no—
158C4⋊Dic61—21no—
159C4⋊Dic61—42no—
160C4⋊Dic61—81no—
161C4⋊Dic61C22×C412yes—
162C4⋊Dic61C2×D412yes—
163#11758196—44no—
164C4×Dic61—42no—
165D488—216no—
166C23×C61C22×D411yescharacteristic
167C22×D61—22no—
168C22×D61—22no—
169C22×D61—22no—
170C22×D61—24no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—42no—
181C2×C4×C61—22no—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—21no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61C4×D412yes—
189C2×C4×C61C4×D412yes—
190C2×C4×C61C2×D814yes—
191C2×C4×C61C2×D812yes—
192C2×C4×C61#13512yes—
193C2×C4×C61C23×C411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61H(5,𝔽2)11yescharacteristic
198C2×C4×C612−1+411yescharacteristic
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—42no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211D244—24no—
212D244—22no—
213D244—24no—
214D244—22no—
215D244—22no—
216D244—42no—
217D244—42no—
218D244—48no—
219D244—84no—
220D244C4×D414yescomplemented
221C4×D61—48no—
222C4×D61—88no—
223C4×D61—88no—
224C8×C61—24no—
225C8×C61—22no—
226C8×C61—41no—
227#22995—611nomaximal, Sylow
228C22×C61C2×C4×D411yescharacteristic
229C22×C61D4211yescharacteristic
230C22×C61#54811yescharacteristic
231C22×C61C22×D811yescharacteristic
232C22×C61C2×#13511yescharacteristic
233C22×C61#57611yescharacteristic
234C22×C61#57711yescharacteristic
235C2×D61—24no—
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—44no—
243C2×D61—44no—
244C2×D61—44no—
245C2×D61—82no—
246C2×D61—82no—
247C2×D61—82no—
248C2×D61—82no—
249C4×C61—22no—
250C4×C61—24no—
251C4×C61—42no—
252C4×C61C4×D814yescomplemented
253C4×C61C2×C4×D411yescharacteristic
254C4×C61#52011yescharacteristic
255C4×C61#54711yescharacteristic
256C4×C61#55311yescharacteristic
257Dic61—42no—
258Dic61—42no—
259Dic61—81no—
260Dic61—82no—
261C2×#459—611no—
262C2×C4×D8—612no—
263#2366—611no—
264D4×C4⋊C4—611no—
265#1957—611no—
266#1947—611no—
267#1934—612no—
268#1879—612no—
269#1862—612no—
270C8×C4⋊C4D6111yescharacteristic, complemented
271C4×#102—611no—
272C2×C61#236111yescharacteristic
273C2×C61C2×C4×D811yescharacteristic
274C2×C61#294611yescharacteristic
275C2×C61D4×D811yescharacteristic
276C2×C61#328411yescharacteristic
277C2×C61#335811yescharacteristic
278C2×C61#336111yescharacteristic
279D61—44no—
280D61—84no—
281C22×D8—612no—
282C2×#127—611no—
283C2×C4×D4—611no—
284C2×C4×D4—612no—
285C2×C4×D4—612no—
286C2×C4×D4—1221no—
287C22×C4⋊C4—1221no—
288#459—618no—
289C4×D8—1228no—
290C2×#107—612no—
291C2×C4⋊C8C2×D6111yescharacteristic
292C2×#102—612no—
293C2×#102—611no—
294C2×#102—611no—
295C2×#102—612no—
296C2×#102—612no—
297C2×C4×C8C2×D6112yes—
298C2×C4×C8C2×D6111yescharacteristic
299#390—612no—
300#390—1221no—
301#389—612no—
302#386—1222no—
303#385—1222no—
304#384—611no—
305#380—1221no—
306C4×C4⋊C4—611no—
307C4×C4⋊C4C2×D6111yescharacteristic
308#340—611no—
309#340—612no—
310#340—611no—
311#336C2×D6112yes—
312C61#2299511yescharacteristic, complemented, Sylow
313C22×D4—611no—
314C22×D4—612no—
315C22×D4—1221no—
316C22×D4—1221no—
317C23×C4—1221no—
318C23×C4—1222no—
319C23×C4—1222no—
320C2×D8—618no—
321C2×D8—1224no—
322C2×D8—1224no—
323C22×C8C22×D6111yescharacteristic
324C22×C8C22×D6112yes—
325C22×C8C22×D6111yescharacteristic
326#127—614no—
327#127—1222no—
328C4×D4—614no—
329C4×D4—1224no—
330C4×D4—1224no—
331C4×D4—1222no—
332C4×D4—2442no—
333C4×D4—2442no—
334C4×D4—2444no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—2444no—
343C2×C4⋊C4D24411yescharacteristic
344C2×C4⋊C4D24411yescharacteristic
345C2×C22⋊C4—1221no—
346C2×C22⋊C4—1222no—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1222no—
350C2×C42—1221no—
351C2×C42C22×D6112yes—
352C2×C42C22×D6111yescharacteristic
353#107—1224no—
354C4⋊C8C4×D6114yescomplemented
355#102—618no—
356#102—614no—
357#102—1222no—
358#102—1224no—
359#102—2442no—
360C4×C8—24no—
361C4×C8—22no—
362#95—1221no—
363#95—1222no—
364#95D24412yes—
365C24—1221no—
366C24—1222no—
367C2×D4—612no—
368C2×D4—614no—
369C2×D4—612no—
370C2×D4—612no—
371C2×D4—1222no—
372C2×D4—1222no—
373C2×D4—1221no—
374C2×D4—1222no—
375C2×D4—1221no—
376C2×D4—1222no—
377C2×D4—1221no—
378C2×D4—1222no—
379C2×D4—1224no—
380C2×D4—1224no—
381C2×D4—4881no—
382C2×D4—4881no—
383C2×D4—2442no—
384C2×D4—2442no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—1224no—
388C22×C4—1221no—
389C22×C4—1222no—
390C22×C4—4882no—
391C22×C4—4882no—
392C22×C4—2444no—
393C22×C4—2444no—
394C22×C4—2444no—
395C22×C4—2444no—
396C22×C4—2442no—
397C22×C4—2442no—
398C22×C4C2×D24412yes—
399C22×C4C2×D24411yescharacteristic
400C22×C4C2×D24412yes—
401C22×C4C2×D24411yescharacteristic
402C22×C4C23×D6111yescharacteristic
403D8—12216no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8C2×C4×D6114yes—
412C2×C8D4×D6112yes—
413C4⋊C4—22no—
414C4⋊C4—612no—
415C4⋊C4—612no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4—1222no—
419C4⋊C4—1221no—
420C4⋊C4—1222no—
421C4⋊C4—1221no—
422C4⋊C4—4881no—
423C4⋊C4—2444no—
424C4⋊C4—2442no—
425C4⋊C4D48814yescomplemented
426C22⋊C4—1224no—
427C22⋊C4—4882no—
428C22⋊C4—2444no—
429C22⋊C4—2444no—
430C22⋊C4—2444no—
431C42—22no—
432C42—22no—
433C42—2442no—
434C42C2×C4×D6112yes—
435C42#1175822212yes—
436C23—1222no—
437C23—1222no—
438C23—1222no—
439C23—1224no—
440C23—2442no—
441C23—2442no—
442C23—2442no—
443C23—2442no—
444C23—2442no—
445C23—2442no—
446C23—2441no—
447C23—2441no—
448C23—2441no—
449C23—2441no—
450C23C22×D24411yescharacteristic
451D4—614no—
452D4—1222no—
453D4—1224no—
454D4—1222no—
455D4—1224no—
456D4—1222no—
457D4—4884no—
458D4—2442no—
459D4—2448no—
460D4—2442no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—4888no—
471C2×C4—4888no—
472C2×C4—2441no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2441no—
476C2×C4—2442no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2448no—
483C2×C4C4×D24412yes—
484C2×C4C4×D24412yes—
485C2×C4C2×D48814yes—
486C2×C4C2×D48812yes—
487C2×C4#1451979912yes—
488C2×C4C22×C4×D6111yescharacteristic
489C2×C4C2×D4×D6111yescharacteristic
490C2×C4#41×D6111yescharacteristic
491C2×C4C2×#1175822211yescharacteristic
492C2×C4#1451982311yescharacteristic
493C2×C4#1451982511yescharacteristic
494C8—24no—
495C8—22no—
496C8—41no—
497C22—1224no—
498C22—4882no—
499C22—4882no—
500C22—4882no—
501C22—4882no—
502C22—2444no—
503C22—2444no—
504C22—2444no—
505C22—2442no—
506C22—2442no—
507C22—2442no—
508C22—2442no—
509C22—2442no—
510C22—2442no—
511C22C2×C4×D24411yescharacteristic
512C22D4×D24411yescharacteristic
513C22#1684042411yescharacteristic
514C22C22×D48811yescharacteristic
515C22C2×#1451979911yescharacteristic
516C22#1684056711yescharacteristic
517C22#1684057411yescharacteristic
518C4—22no—
519C4—24no—
520C4—42no—
521C4—4881no—
522C4—4882no—
523C4—2442no—
524C4—2442no—
525C4C4×D48814yescomplemented
526C4C4×D4×D6111yescharacteristic
527C4#1684024411yescharacteristic
528C4#1684041311yescharacteristic
529C4#1684045611yescharacteristic
530C2—4884no—
531C2—2444no—
532C2#2144883711yescharacteristic
533C2C2×C4×D48811yescharacteristic
534C2#2145465311yescharacteristic
535C2D4×D48811yescharacteristic
536C2#2145755111yescharacteristic
537C2#2145845511yescharacteristic
538C2#2145848111yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC8×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC2×#16839958C2
cosocleC22×C4×C61C24
hypercenterC8×C4⋊C4D61
hypocenterC61#22995
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×D488C2semidirect
3. #21448883C2semidirect
4. C4⋊C4×D244C2semidirect
5. #21446625C2semidirect
6. #21446578C2semidirect
7. #21446502C2semidirect
8. #21446166C2semidirect
9. #21446069C2semidirect
10. C4×#14519699C2semidirect
11. C8×C4⋊C4×C61C2semidirect
19. #16839958C4semidirect
52. C2×D488C2×C4semidirect
54. #14519774C2×C4semidirect
60. C4×D244C2×C4semidirect
80. #14519699C2×C4semidirect
81. #14519699D4semidirect
91. C4⋊C8×C61C2×C4semidirect
143. C4⋊C4×C61D8semidirect
220. D244C4×D4semidirect
252. C4×C61C4×D8semidirect
270. C8×C4⋊C4D61semidirect
312. C61#22995semidirect
354. C4⋊C8C4×D61semidirect
425. C4⋊C4D488semidirect
525. C4C4×D488semidirect

Maximal subgroups

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

subgroupindex
C2×#168399582
C2×C4×D4882
#214488832
C4⋊C4×D2442
#214466252
#214465782
#214465022
#214461662
#214460692
C4×#145196992
C8×C4⋊C4×C612
#2299561