Finite Group Databasebeta

← #29729890

Subgroups — #29729890

Order 15616.

978 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×#16840110C211yesmaximal, characteristic, complemented
2C2×#16839958C211yesmaximal, characteristic
3#21450993C211yesmaximal, characteristic, complemented
4#21448883C211yesmaximal, characteristic, complemented
5#21447236C212yesmaximal, complemented
6#21447131C212yesmaximal, complemented
7#21446535C211yesmaximal, characteristic, complemented
8#21446168C212yesmaximal, complemented
9#21445828C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic, complemented
11C61×#1987C211yesmaximal, characteristic, complemented
12C22×D488C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×C4×D244—21no—
15C2×C4×D244C2211yescharacteristic
16C2×#14519739C2212yes—
17#16840110—24no—
18#16839958—24no—
19C2×#14519706C2212yes—
20C2×#14519699C2212yes—
21C2×#14519697C2211yescharacteristic
22C2×#14519697C2211yescharacteristic
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25C2×C4×#92418C2211yescharacteristic
26#16839673—22no—
27#16839664—21no—
28#16839663C2212yes—
29#16839663C2212yes—
30#16839640—22no—
31#16839627C2212yes—
32#16839619—21no—
33#16839619—21no—
34C4×C4⋊Dic61C2211yescharacteristic
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839516C2212yes—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2211yescharacteristic
40C4×C4⋊C4×C61C2211yescharacteristic
41C61×#340C2212yes—
42C61×#340C2212yes—
43C22×D244—21no—
44C22×D244—21no—
45C22×D244C2312yes—
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61—22no—
49C2×D488—24no—
50C2×D488—24no—
51C2×D488—42no—
52C22×#92418C2311yescharacteristic
53C22×#92418C2311yescharacteristic
54#14519774—22no—
55#14519774D414yes—
56C4×D244—22no—
57C4×D244—22no—
58C4×D244—44no—
59#14519739—24no—
60#14519739—42no—
61C2×C4⋊Dic61D411yescharacteristic
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758198—21no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×C4×Dic61C2312yes—
71#14519706—42no—
72#14519699—24no—
73#14519697—24no—
74#14519697—24no—
75#14519697—22no—
76#14519697—22no—
77C4×#92418—22no—
78#14519684—22no—
79#14519684—22no—
80#14519684D412yes—
81C22×Q8×C61—21no—
82C22×C8×C61C2312yes—
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61—22no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C42×C61—21no—
90C2×C42×C61C2311yescharacteristic
91C4⋊C8×C61—22no—
92#95×C61—21no—
93C23×D61—22no—
94C23×D61—21no—
95C2×Q8×C61—42no—
96C2×Q8×C61—81no—
97C2×Q8×C61—81no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—22no—
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2411yescharacteristic
104C22×Dic61—21no—
105C22×Dic61C2×D412yes—
106C22×Dic61C2×D412yes—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—22no—
110C2×D244—21no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—21no—
114C2×D244—22no—
115C2×D244—24no—
116C2×D244—22no—
117C2×D244—21no—
118C2×D244—24no—
119C2×D244—42no—
120C2×D244—42no—
121C2×D244—81no—
122C2×D244—81no—
123C2×D244C2×D414yes—
124C2×D244#4112yes—
125C2×C4×D61—44no—
126C2×C4×D61—44no—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×C4×D61—82no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—42no—
140C4⋊C4×C61#4112yes—
141C42×C61—42no—
142C42×C61C2×D412yes—
143C42×C61#4112yes—
144#11758198—44no—
145#11758198—44no—
146#11758198—42no—
147#11758198—82no—
148#11758198—82no—
149#11758198—82no—
150C4⋊Dic61—22no—
151C4⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4×Dic61—22no—
155C4×Dic61—22no—
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—21no—
162D488—48no—
163C23×C61C22×D411yescharacteristic
164C22×D61—24no—
165C22×D61—22no—
166C22×D61—41no—
167C22×D61—42no—
168C22×D61—41no—
169C22×D61—42no—
170C22×D61—41no—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—42no—
177C22×D61—42no—
178Q8×C61—84no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—22no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61#12112yes—
191C2×C4×C61#12112yes—
192C2×C4×C61#13512yes—
193C2×C4×C61AGL(1,ℤ/8)12yes—
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—22no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209D244—22no—
210D244—22no—
211D244—24no—
212D244—24no—
213D244—42no—
214D244—41no—
215D244—42no—
216D244—41no—
217D244—48no—
218D244—84no—
219C4×D61—88no—
220C4×D61—84no—
221C8×C61—42no—
222#92418—41no—
223#92418—41no—
224#25321—611nomaximal, Sylow
225C22×C61C2×#12111yescharacteristic
226C22×C61#54611yescharacteristic
227C22×C61#54811yescharacteristic
228C22×C61C2×#13511yescharacteristic
229C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
230C22×C61#57611yescharacteristic
231C22×C61#57611yescharacteristic
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—44no—
236C2×D61—44no—
237C2×D61—44no—
238C2×D61—82no—
239C2×D61—82no—
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—82no—
243C2×D61—81no—
244C2×D61—82no—
245C2×D61—81no—
246C4×C61—22no—
247C4×C61—42no—
248C4×C61—81no—
249C4×C61—82no—
250C4×C61#46314yes—
251C4×C61#52911yescharacteristic
252C4×C61#54211yescharacteristic
253C4×C61#54611yescharacteristic
254C4×C61#54711yescharacteristic
255Dic61—42no—
256Dic61—42no—
257Dic61—81no—
258C2×#486—611no—
259C2×#459—611no—
260#2563—611no—
261#2366—611no—
262#2073—612no—
263#2054—612no—
264#1987D6111yescharacteristic, complemented
265#1939—611no—
266#1879—612no—
267#1804—612no—
268C4×#102—611no—
269C2×C61#242711yescharacteristic
270C2×C61C2×#46311yescharacteristic
271C2×C61#304111yescharacteristic
272C2×C61#330511yescharacteristic
273C2×C61#332511yescharacteristic
274C2×C61#336111yescharacteristic
275C2×C61#336611yescharacteristic
276D61—82no—
277D61—84no—
278C22×D8—612no—
279C2×C4⋊Q8C2×D6111yescharacteristic
280C2×#127—611no—
281C2×C4×D4—612no—
282C2×C4×D4—611no—
283C2×C4×D4—1221no—
284#486—1224no—
285#459—1224no—
286C2×C4⋊C8—612no—
287C2×C4⋊C8C2×D6111yescharacteristic
288C2×#102—612no—
289C2×#102—611no—
290C2×#102—612no—
291C2×#102—612no—
292C2×#102—611no—
293C2×C4×C8—611no—
294#393—1222no—
295#390—612no—
296#390—612no—
297#390—1221no—
298#386—1222no—
299#382—612no—
300#380—1221no—
301#380—1221no—
302C4×C4⋊C4—611no—
303C4×C4⋊C4C2×D6111yescharacteristic
304#340—611no—
305#340—611no—
306#340C2×D6112yes—
307#340C2×D6112yes—
308#336—612no—
309C61#2532111yescharacteristic, complemented, Sylow
310C22×Q8—21no—
311C22×D4—612no—
312C22×D4—611no—
313C22×D4—1221no—
314C22×D4—1221no—
315C23×C4—1222no—
316C23×C4—1221no—
317C2×D8—1224no—
318C2×D8—1224no—
319C2×D8—2442no—
320C22×C8—611no—
321C22×C8—611no—
322C22×C8C22×D6112yes—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325#127—614no—
326#127—1222no—
327C4×D4—1224no—
328C4×D4—1222no—
329C4×D4—1222no—
330C4×D4—2444no—
331C4×D4—2442no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4C22×D6112yes—
337C2×C4⋊C4C22×D6111yescharacteristic
338C2×C4⋊C4C22×D6111yescharacteristic
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1222no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1221no—
345C2×C42—21no—
346C2×C42—612no—
347C2×C42C22×D6111yescharacteristic
348C4⋊C8—22no—
349C4⋊C8—2442no—
350#102—1222no—
351#102—1224no—
352#102—1224no—
353#102—1224no—
354#102—1222no—
355C4×C8—1222no—
356#95—21no—
357#95—612no—
358#95—1222no—
359#95—1222no—
360C24—1221no—
361C24—1222no—
362C2×Q8—42no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×D4—614no—
366C2×D4—612no—
367C2×D4—1221no—
368C2×D4—1222no—
369C2×D4—1221no—
370C2×D4—1222no—
371C2×D4—1221no—
372C2×D4—1222no—
373C2×D4—1222no—
374C2×D4—1221no—
375C2×D4—1222no—
376C2×D4—1221no—
377C2×D4—1224no—
378C2×D4—1224no—
379C2×D4—4881no—
380C2×D4—4881no—
381C2×D4—2442no—
382C2×D4—2442no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—612no—
387C22×C4—612no—
388C22×C4—1221no—
389C22×C4—4882no—
390C22×C4—2444no—
391C22×C4—2444no—
392C22×C4—2442no—
393C22×C4—2442no—
394C22×C4—2442no—
395C22×C4D4×D6111yescharacteristic
396C22×C4D4×D6111yescharacteristic
397C22×C4C23×D6111yescharacteristic
398D8—2448no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—1221no—
403C2×C8—1221no—
404C2×C8—1221no—
405C2×C8—1221no—
406C2×C8—1221no—
407C2×C8—1221no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—42no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4#1175822612yes—
419C22⋊C4—4882no—
420C22⋊C4—4882no—
421C22⋊C4—4882no—
422C22⋊C4—2444no—
423C22⋊C4—2444no—
424C22⋊C4—2442no—
425C42—42no—
426C42—1222no—
427C42—1222no—
428C42#1175822212yes—
429C42D4×D6112yes—
430C23—1224no—
431C23—1222no—
432C23—2441no—
433C23—2441no—
434C23—2442no—
435C23—2442no—
436C23—2441no—
437C23—2441no—
438C23—2441no—
439C23—2442no—
440C23—2442no—
441C23—2442no—
442C23—2441no—
443C23—2442no—
444C23C2×D4×D6111yescharacteristic
445Q8—84no—
446D4—1222no—
447D4—1222no—
448D4—1224no—
449D4—1224no—
450D4—4884no—
451D4—2448no—
452D4—2442no—
453D4—2441no—
454D4—2442no—
455D4—2441no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—1222no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—4888no—
474C2×C4—4884no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4#1451975012yes—
479C2×C4#1451975012yes—
480C2×C4#1451980012yes—
481C2×C4#1451980412yes—
482C2×C4C2×D4×D6111yescharacteristic
483C2×C4#41×D6111yescharacteristic
484C2×C4C2×#1175822211yescharacteristic
485C2×C4C2×#1175822611yescharacteristic
486C2×C4#1451982311yescharacteristic
487C2×C4#1451982511yescharacteristic
488C8—42no—
489C8—2441no—
490C8—2441no—
491C22—4882no—
492C22—4882no—
493C22—4881no—
494C22—4882no—
495C22—4882no—
496C22—4881no—
497C22—4881no—
498C22—4881no—
499C22—2442no—
500C22—2442no—
501C22—2444no—
502C22—2442no—
503C22—2444no—
504C22—2444no—
505C22C2×#1451975011yescharacteristic
506C22#1684040311yescharacteristic
507C22#1684042211yescharacteristic
508C22C2×#1451980011yescharacteristic
509C22C2×#1451980411yescharacteristic
510C22#1684056611yescharacteristic
511C22#1684056611yescharacteristic
512C4—22no—
513C4—42no—
514C4—81no—
515C4—82no—
516C4—4881no—
517C4—2442no—
518C4—2442no—
519C4#1683998014yes—
520C4#1684029911yescharacteristic
521C4#1684038411yescharacteristic
522C4#1684040511yescharacteristic
523C4#1684041511yescharacteristic
524C2—4884no—
525C2—4882no—
526C2#2144939711yescharacteristic
527C2C2×#1683998011yescharacteristic
528C2#2145532111yescharacteristic
529C2#2145776611yescharacteristic
530C2#2145805811yescharacteristic
531C2#2145847811yescharacteristic
532C2#2145852511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1987C2
socleC23×C61C22×D4
prime radicalC2×#16839958C2
hypercenter#1987D61
hypocenterC61#25321
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
1. C2×#16840110C2semidirect
3. #21450993C2semidirect
4. #21448883C2semidirect
5. #21447236C2semidirect
6. #21447131C2semidirect
7. #21446535C2semidirect
8. #21446168C2semidirect
9. #21445828C2semidirect
10. C4×#14519697C2semidirect
11. C61×#1987C2semidirect
264. #1987D61semidirect
309. C61#25321semidirect

Maximal subgroups

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

subgroupindex
C2×#168401102
C2×#168399582
#214509932
#214488832
#214472362
#214471312
#214465352
#214461682
#214458282
C4×#145196972
C61×#19872
#2532161