Finite Group Databasebeta

← #29729512

Subgroups — #29729512

Order 15616.

954 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839958C211yesmaximal, characteristic
2C2×#16839886C211yesmaximal, characteristic, complemented
3#21450821C211yesmaximal, characteristic, complemented
4#21450742C211yesmaximal, characteristic, complemented
5#21447202C212yesmaximal, complemented
6#21447131C212yesmaximal, complemented
7#21446683C212yesmaximal, complemented
8#21446624C211yesmaximal, characteristic, complemented
9#21446503C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic, complemented
11C61×#1947C211yesmaximal, characteristic, complemented
12C22×D488C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519765—21no—
15C2×#14519757—21no—
16C2×C4×Dic122C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18#16839958—24no—
19#16839886—24no—
20C2×#14519706C2212yes—
21C2×#14519699C2212yes—
22C2×#14519697C2211yescharacteristic
23C2×#14519697C2211yescharacteristic
24C2×#14519697C2212yes—
25C2×#14519697C2212yes—
26C2×C4×#92418C2211yescharacteristic
27C2×C8×Dic61C2212yes—
28#16839684—22no—
29#16839680—22no—
30#16839664—21no—
31#16839663C2212yes—
32#16839663C2212yes—
33#16839663C2212yes—
34#16839661C2212yes—
35#16839619—21no—
36#16839524C2211yescharacteristic
37#16839524C2211yescharacteristic
38C2×#107×C61C2212yes—
39C2×C4⋊C8×C61C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C22×Dic122D411yescharacteristic
44C22×D244—21no—
45C22×D244—21no—
46C22×D244C2312yes—
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C2×D488—24no—
50C2×D488—24no—
51C2×D488—42no—
52C22×#92418C2311yescharacteristic
53C22×#92418C2311yescharacteristic
54#14519774—22no—
55#14519774—22no—
56#14519765—44no—
57#14519757—44no—
58C4×Dic122—22no—
59C4×Dic122—22no—
60C4×D244—22no—
61C4×D244—22no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758196D412yes—
66C2×#11758198—21no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×C4×Dic61C2312yes—
74#14519706—24no—
75#14519699—24no—
76#14519697—22no—
77#14519697—22no—
78#14519697—24no—
79#14519697—42no—
80C4×#92418—22no—
81C8×Dic61—24no—
82#14519684—22no—
83C22×C8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2312yes—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C42×C61—21no—
90C2×C42×C61C2311yescharacteristic
91#107×C61—42no—
92C4⋊C8×C61D414yes—
93#95×C61—21no—
94C23×D61—21no—
95C23×D61—22no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—22no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2411yescharacteristic
102C22×Dic61—21no—
103C22×Dic61C2×D412yes—
104C22×Dic61C2×D412yes—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—22no—
114C2×D244—21no—
115C2×D244—22no—
116C2×D244—42no—
117C2×D244—42no—
118C2×D244—42no—
119C2×D244—42no—
120C2×D244—81no—
121C2×D244—81no—
122C2×D244#4112yes—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×C4×D61—42no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122D814yescomplemented
129C2×C8×C61—22no—
130C2×C8×C61—22no—
131C2×C8×C61C2×D414yes—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61—81no—
140C42×C61—42no—
141C42×C61C2×D412yes—
142C42×C61#4112yes—
143#11758198—42no—
144#11758198—44no—
145#11758198—44no—
146#11758198—44no—
147#11758198—82no—
148#11758198—82no—
149#11758198—82no—
150C4⋊Dic61—22no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—44no—
154#11758196—24no—
155C4×Dic61—22no—
156C4×Dic61—22no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418#4112yes—
163D488—48no—
164C23×C61C22×D411yescharacteristic
165C22×D61—22no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—41no—
190C2×C4×C61#12112yes—
191C2×C4×C61#12112yes—
192C2×C4×C61C2×D812yes—
193C2×C4×C61#13512yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×C4×C61H(5,𝔽2)11yescharacteristic
199C2×C4×C612−1+411yescharacteristic
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61C2×D814yes—
209D244—22no—
210D244—22no—
211D244—42no—
212D244—41no—
213D244—42no—
214D244—42no—
215D244—41no—
216D244—42no—
217D244—84no—
218D244—84no—
219C4×D61—84no—
220Dic122—28no—
221C8×C61—24no—
222#92418—22no—
223#92418—41no—
224#25292—611nomaximal, Sylow
225C22×C61C2×#12111yescharacteristic
226C22×C61#54811yescharacteristic
227C22×C61#55011yescharacteristic
228C22×C61C22×D811yescharacteristic
229C22×C61C2×#13511yescharacteristic
230C22×C61#57611yescharacteristic
231C22×C61#57711yescharacteristic
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—82no—
245C2×D61—82no—
246C4×C61—22no—
247C4×C61—81no—
248C4×C61—82no—
249C4×C61—81no—
250C4×C61#46614yes—
251C4×C61#52911yescharacteristic
252C4×C61#54211yescharacteristic
253C4×C61#54611yescharacteristic
254C4×C61#54811yescharacteristic
255Dic61—24no—
256Dic61—42no—
257Dic61—81no—
258C2×#459—611no—
259C2×C8×Q8—611no—
260#2548—611no—
261#2538—611no—
262#2067—612no—
263#2054—612no—
264#1970—612no—
265#1957—611no—
266#1947D6111yescharacteristic, complemented
267#1934—612no—
268C4×#102—611no—
269C2×C61#242611yescharacteristic
270C2×C61C2×#46611yescharacteristic
271C2×C61#306211yescharacteristic
272C2×C61#335811yescharacteristic
273C2×C61#336111yescharacteristic
274C2×C61#339111yescharacteristic
275C2×C61#339411yescharacteristic
276D61—82no—
277D61—162no—
278C22×D8—612no—
279C2×#127—611no—
280C2×#124—1221no—
281C2×C22⋊Q8—1221no—
282C2×C4×Q8—611no—
283C2×C4×D4—611no—
284#459—1224no—
285C8×Q8—1224no—
286C2×#107C2×D6112yes—
287C2×C4⋊C8—612no—
288C2×C4⋊C8C2×D6111yescharacteristic
289C2×#102—612no—
290C2×#102—611no—
291C2×#102—612no—
292C2×#102—611no—
293C2×#102—612no—
294C2×C4×C8—612no—
295C2×C4×C8—611no—
296#396—1222no—
297#394—1222no—
298#390—612no—
299#390—612no—
300#390—612no—
301#390—1221no—
302#389—612no—
303#384C2×D6111yescharacteristic
304#380—1221no—
305C4×C4⋊C4C2×D6111yescharacteristic
306#340—611no—
307#340—611no—
308#340C2×D6112yes—
309C61#2529211yescharacteristic, complemented, Sylow
310C22×Q8—611no—
311C22×D4—611no—
312C22×D4—612no—
313C22×D4—1221no—
314C22×D4—1221no—
315C23×C4—1221no—
316C2×D8—1224no—
317C2×D8—1224no—
318C2×D8—2442no—
319C22×C8—611no—
320C22×C8—611no—
321C22×C8C22×D6112yes—
322#127—1222no—
323#127—1222no—
324#124—2444no—
325C22⋊Q8—2444no—
326C4×Q8—1222no—
327C4×Q8—1222no—
328C4×D4—1222no—
329C4×D4—1222no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—1222no—
336C2×C4⋊C4C22×D6111yescharacteristic
337C2×C4⋊C4C22×D6111yescharacteristic
338C2×C4⋊C4C22×D6112yes—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1222no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1222no—
345C2×C22⋊C4—1221no—
346C2×C42—21no—
347C2×C42—612no—
348C2×C42C22×D6111yescharacteristic
349#107—42no—
350C4⋊C8—1224no—
351C4⋊C8D24414yes—
352#102—1222no—
353#102—1224no—
354#102—1222no—
355#102—1224no—
356#102—2442no—
357C4×C8—1224no—
358C4×C8—1222no—
359#95—21no—
360#95—1222no—
361C24—1221no—
362C24—1222no—
363C2×Q8—614no—
364C2×Q8—1222no—
365C2×Q8—1222no—
366C2×D4—612no—
367C2×D4—1221no—
368C2×D4—1222no—
369C2×D4—1221no—
370C2×D4—1222no—
371C2×D4—1222no—
372C2×D4—1221no—
373C2×D4—1222no—
374C2×D4—1221no—
375C2×D4—1222no—
376C2×D4—1221no—
377C2×D4—1222no—
378C2×D4—4881no—
379C2×D4—4881no—
380C2×D4—2442no—
381C2×D4—2442no—
382C2×D4—2442no—
383C2×D4—2442no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—612no—
388C22×C4—612no—
389C22×C4—1221no—
390C22×C4—2442no—
391C22×C4—2442no—
392C22×C4—2442no—
393C22×C4D4×D6111yescharacteristic
394C22×C4D4×D6111yescharacteristic
395C22×C4C23×D6111yescharacteristic
396D8—2448no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—612no—
400C2×C8—1221no—
401C2×C8—1221no—
402C2×C8—1221no—
403C2×C8—1221no—
404C2×C8—1221no—
405C2×C8C2×D24414yes—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—42no—
413C4⋊C4—81no—
414C4⋊C4—1224no—
415C4⋊C4—1222no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4—2444no—
419C22⋊C4—4882no—
420C22⋊C4—4882no—
421C22⋊C4—4882no—
422C22⋊C4—2444no—
423C22⋊C4—2444no—
424C22⋊C4—2444no—
425C22⋊C4—2442no—
426C42—42no—
427C42—1222no—
428C42—1222no—
429C42C2×D24412yes—
430C42#1175822612yes—
431C23—1222no—
432C23—2442no—
433C23—2442no—
434C23—2442no—
435C23—2442no—
436C23—2442no—
437C23—2442no—
438C23—2442no—
439C23—2441no—
440C23—2441no—
441C23—2441no—
442C23—2441no—
443C23—2441no—
444C23—2441no—
445C23C2×D4×D6111yescharacteristic
446Q8—1228no—
447D4—1222no—
448D4—1222no—
449D4—4884no—
450D4—4884no—
451D4—2442no—
452D4—2442no—
453D4—2441no—
454D4—2442no—
455D4—2442no—
456D4—2441no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—614no—
469C2×C4—1222no—
470C2×C4—1222no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—4884no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4#1451975212yes—
479C2×C4#1451975212yes—
480C2×C4D8×D6112yes—
481C2×C4#1451980012yes—
482C2×C4C22×D24411yescharacteristic
483C2×C4#41×D6111yescharacteristic
484C2×C4#41×D6111yescharacteristic
485C2×C4C2×#1175822611yescharacteristic
486C2×C4#1451982211yescharacteristic
487C2×C4#1451982511yescharacteristic
488C8—24no—
489C8—1222no—
490C8—2441no—
491C22—4882no—
492C22—4882no—
493C22—4882no—
494C22—4882no—
495C22—4882no—
496C22—4882no—
497C22—4882no—
498C22—4881no—
499C22—4881no—
500C22—4881no—
501C22—4881no—
502C22—2442no—
503C22—2442no—
504C22—2442no—
505C22C2×#1451975211yescharacteristic
506C22#1684042211yescharacteristic
507C22#1684043011yescharacteristic
508C22C2×D8×D6111yescharacteristic
509C22C2×#1451980011yescharacteristic
510C22#1684056611yescharacteristic
511C22#1684057111yescharacteristic
512C4—22no—
513C4—81no—
514C4—81no—
515C4—82no—
516C4—1224no—
517C4—4881no—
518C4—2442no—
519C4#1683999914yes—
520C4#1684029711yescharacteristic
521C4#1684038611yescharacteristic
522C4#1684040711yescharacteristic
523C4#1684042411yescharacteristic
524C2—9762no—
525C2—4882no—
526C2#2144938311yescharacteristic
527C2C2×#1683999911yescharacteristic
528C2#2145550011yescharacteristic
529C2#2145845311yescharacteristic
530C2#2145847811yescharacteristic
531C2#2145879011yescharacteristic
532C2#2145881411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1947C2
socleC23×C61C22×D4
prime radicalC2×#16839958C2
hypercenter#1947D61
hypocenterC61#25292
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
2. C2×#16839886C2semidirect
3. #21450821C2semidirect
4. #21450742C2semidirect
5. #21447202C2semidirect
6. #21447131C2semidirect
7. #21446683C2semidirect
8. #21446624C2semidirect
9. #21446503C2semidirect
10. C4×#14519697C2semidirect
11. C61×#1947C2semidirect
128. C2×Dic122D8semidirect
266. #1947D61semidirect
309. C61#25292semidirect

Maximal subgroups

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

subgroupindex
C2×#168399582
C2×#168398862
#214508212
#214507422
#214472022
#214471312
#214466832
#214466242
#214465032
C4×#145196972
C61×#19472
#2529261