Finite Group Databasebeta

← #29729885

Subgroups — #29729885

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×#16839957C211yesmaximal, characteristic
3#21448883C211yesmaximal, characteristic, complemented
4#21447233C212yesmaximal, complemented
5#21447129C212yesmaximal, complemented
6#21446768C211yesmaximal, characteristic, complemented
7#21446538C211yesmaximal, characteristic, complemented
8#21446170C212yesmaximal, complemented
9#21445826C212yesmaximal, complemented
10C4×#14519698C211yesmaximal, characteristic, complemented
11C61×#2563C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×C4×D244C2212yes—
15C2×#14519739—21no—
16#16840110—24no—
17#16839957—24no—
18C2×C4⋊#92418C2211yescharacteristic
19C2×C4⋊#92418C2212yes—
20C2×#14519698C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519697C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25C2×C4×#92418C2211yescharacteristic
26#16839664C2212yes—
27#16839663—21no—
28#16839640—22no—
29C4×C4⋊Dic61C2211yescharacteristic
30#16839524C2211yescharacteristic
31#16839524C2211yescharacteristic
32#16839524C2212yes—
33#16839524C2212yes—
34#16839517C2212yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#393—22no—
38C61×#390C2212yes—
39C61×#382C2212yes—
40C61×#380—21no—
41C61×#380—21no—
42C4×C4⋊C4×C61C2211yescharacteristic
43C22×D244—21no—
44C22×D244—21no—
45C22×D244C2312yes—
46C22×C4×D61—22no—
47C2×#11758199—24no—
48C2×#11758199—24no—
49C2×#11758199—42no—
50C22×#92418C2311yescharacteristic
51C22×#92418C2311yescharacteristic
52C22×#92418C2312yes—
53#14519774—22no—
54#14519774D414yes—
55C4×D244—24no—
56C4×D244—42no—
57#14519739—44no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758198—22no—
61C2×#11758198—22no—
62C2×#11758198—22no—
63C2×C4×Dic61—21no—
64C4⋊#92418—22no—
65C4⋊#92418—42no—
66#14519698—22no—
67#14519698—22no—
68#14519697—24no—
69#14519697—24no—
70#14519697—24no—
71C4×#92418—22no—
72#14519684—21no—
73C22×Q8×C61—21no—
74C22×D4×C61C2311yescharacteristic
75C23×C4×C61—21no—
76C4⋊Q8×C61—22no—
77C4⋊Q8×C61—22no—
78C4×D4×C61—22no—
79C4×D4×C61—22no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61D411yescharacteristic
82C2×C4⋊C4×C61D411yescharacteristic
83C2×C4⋊C4×C61C2312yes—
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C22⋊C4×C61—21no—
86C2×C22⋊C4×C61—22no—
87C2×C22⋊C4×C61—22no—
88C2×C42×C61C2312yes—
89C2×C42×C61C2311yescharacteristic
90#95×C61—22no—
91#95×C61—22no—
92#95×C61D412yes—
93C24×C61—21no—
94C23×D61—22no—
95C2×Q8×C61—42no—
96C2×Q8×C61—81no—
97C2×Q8×C61—81no—
98C2×D4×C61—21no—
99C2×D4×C61—21no—
100C2×D4×C61—21no—
101C2×D4×C61—21no—
102C2×D4×C61—21no—
103C2×D4×C61#4112yes—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—42no—
107C22×C4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61C2×D412yes—
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C22×Dic61—21no—
115C22×Dic61—21no—
116C2×D244—22no—
117C2×D244—22no—
118C2×D244—22no—
119C2×D244—22no—
120C2×D244—24no—
121C2×D244—22no—
122C2×D244—24no—
123C2×D244—42no—
124C2×D244—42no—
125C2×D244—81no—
126C2×D244—81no—
127C2×D244C2×D414yes—
128C2×C4×D61—44no—
129C2×C4×D61—44no—
130C2×C4×D61—82no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—42no—
138C22⋊C4×C61—42no—
139C22⋊C4×C61—82no—
140C22⋊C4×C61—82no—
141C42×C61—22no—
142C42×C61—22no—
143C42×C61C2×D412yes—
144C42×C61#4112yes—
145#11758199—48no—
146#11758198—44no—
147#11758198—44no—
148#11758198—82no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61#4112yes—
153C4×Dic61—42no—
154C2×#92418—21no—
155C2×#92418—21no—
156C2×#92418—22no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418—22no—
160C2×#92418—21no—
161C2×#92418—22no—
162C2×#92418—21no—
163C23×C61—22no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61C22×D411yescharacteristic
171C22×D61—24no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178Q8×C61—84no—
179D4×C61—22no—
180D4×C61—22no—
181D4×C61—41no—
182D4×C61—41no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—84no—
198C2×C4×C61#12112yes—
199C2×C4×C61#12112yes—
200C2×C4×C61#13512yes—
201C2×C4×C61AGL(1,ℤ/8)12yes—
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61H(5,𝔽2)11yescharacteristic
207C2×C4×C612−1+411yescharacteristic
208C2×Dic61—22no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214D244—24no—
215D244—24no—
216D244—42no—
217D244—42no—
218D244—48no—
219D244—84no—
220C4×D61—88no—
221#92418—41no—
222#92418—42no—
223#92418—41no—
224#25321—611nomaximal, Sylow
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—81no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61—81no—
232C22×C61C2×#12111yescharacteristic
233C22×C61#54611yescharacteristic
234C22×C61#54811yescharacteristic
235C22×C61C2×#13511yescharacteristic
236C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C61#57611yescharacteristic
238C22×C61#57611yescharacteristic
239C2×D61—44no—
240C2×D61—44no—
241C2×D61—44no—
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—82no—
245C2×D61—82no—
246C4×C61—22no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61—81no—
250C4×C61—82no—
251C4×C61#46314yes—
252C4×C61#52911yescharacteristic
253C4×C61#54211yescharacteristic
254C4×C61#54611yescharacteristic
255C4×C61#54711yescharacteristic
256Dic61—42no—
257Dic61—81no—
258C2×#486—611no—
259C2×#459—611no—
260#2563D6111yescharacteristic, complemented
261#2366—611no—
262#2073—612no—
263#2054—612no—
264#1987—611no—
265#1939—611no—
266#1879—612no—
267#1804—612no—
268C4×#102—611no—
269C2×C61—82no—
270C2×C61#242711yescharacteristic
271C2×C61C2×#46311yescharacteristic
272C2×C61#304111yescharacteristic
273C2×C61#330511yescharacteristic
274C2×C61#332511yescharacteristic
275C2×C61#336111yescharacteristic
276C2×C61#336611yescharacteristic
277D61—84no—
278C22×D8—612no—
279C2×C4⋊Q8C2×D6111yescharacteristic
280C2×#127—611no—
281C2×C4×D4—612no—
282C2×C4×D4—1221no—
283C2×C4×D4C2×D6111yescharacteristic
284#486—1224no—
285#459—1224no—
286C2×C4⋊C8—611no—
287C2×C4⋊C8—612no—
288C2×#102—611no—
289C2×#102—611no—
290C2×#102—612no—
291C2×#102—612no—
292C2×#102—612no—
293C2×C4×C8—611no—
294#393—22no—
295#390—612no—
296#390—1221no—
297#390C2×D6112yes—
298#386—1222no—
299#382C2×D6112yes—
300#380—21no—
301#380—21no—
302C4×C4⋊C4—611no—
303C4×C4⋊C4C2×D6111yescharacteristic
304#340—612no—
305#340—611no—
306#340—612no—
307#340—611no—
308#336—612no—
309C61#2532111yescharacteristic, complemented, Sylow
310C22×Q8—21no—
311C22×D4—612no—
312C22×D4—1221no—
313C22×D4—1221no—
314C22×D4C22×D6111yescharacteristic
315C23×C4—21no—
316C23×C4—1222no—
317C2×D8—1224no—
318C2×D8—1224no—
319C2×D8—2442no—
320C22×C8—612no—
321C22×C8—611no—
322C22×C8—611no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325#127—614no—
326#127—1222no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—1224no—
330C4×D4—2442no—
331C4×D4—2444no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4C22⋊D6111yescharacteristic
336C2×C4⋊C4C22⋊D6111yescharacteristic
337C2×C4⋊C4C22×D6111yescharacteristic
338C2×C4⋊C4C22×D6112yes—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—1222no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1222no—
345C2×C42—1221no—
346C2×C42C22×D6112yes—
347C2×C42C22×D6111yescharacteristic
348C4⋊C8—1222no—
349C4⋊C8—2442no—
350#102—1224no—
351#102—1224no—
352#102—1224no—
353#102—1222no—
354#102—1222no—
355C4×C8—1222no—
356#95—22no—
357#95—22no—
358#95—1221no—
359#95C22⋊D6112yes—
360C24—21no—
361C24—1222no—
362C2×Q8—42no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—614no—
371C2×D4—1222no—
372C2×D4—1222no—
373C2×D4—1222no—
374C2×D4—1222no—
375C2×D4—1222no—
376C2×D4—1224no—
377C2×D4—1224no—
378C2×D4—4881no—
379C2×D4—4881no—
380C2×D4—2442no—
381C2×D4—2442no—
382C2×D4#1175822612yes—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4—4882no—
391C22×C4—2444no—
392C22×C4—2444no—
393C22×C4C2×C22⋊D6112yes—
394C22×C4C2×C22⋊D6112yes—
395C22×C4C2×C22⋊D6111yescharacteristic
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C23×D6111yescharacteristic
398D8—2448no—
399C2×C8—1221no—
400C2×C8—1221no—
401C2×C8—1222no—
402C2×C8—1221no—
403C2×C8—1221no—
404C2×C8—1222no—
405C2×C8—1221no—
406C2×C8—1222no—
407C2×C8—1221no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—612no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4—1221no—
419C22⋊C4—42no—
420C22⋊C4—82no—
421C22⋊C4—82no—
422C22⋊C4—4882no—
423C22⋊C4—2444no—
424C22⋊C4—2444no—
425C42—22no—
426C42—22no—
427C42—2442no—
428C42D4×D6112yes—
429C42#1175822412yes—
430C23—22no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—1224no—
438C23—2442no—
439C23—2442no—
440C23—2442no—
441C23—2442no—
442C23—2442no—
443C23—2442no—
444C23C22×C22⋊D6111yescharacteristic
445Q8—84no—
446D4—22no—
447D4—22no—
448D4—41no—
449D4—41no—
450D4—1224no—
451D4—1224no—
452D4—4884no—
453D4—2448no—
454D4—2442no—
455D4—2442no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—84no—
471C2×C4—1222no—
472C2×C4—4888no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4#1451974812yes—
479C2×C4#1451974812yes—
480C2×C4#1451979712yes—
481C2×C4#1451980212yes—
482C2×C4C2×D4×D6111yescharacteristic
483C2×C4#41×D6111yescharacteristic
484C2×C4C2×#1175822411yescharacteristic
485C2×C4C2×#1175822611yescharacteristic
486C2×C4#1451982311yescharacteristic
487C2×C4#1451982511yescharacteristic
488C8—2441no—
489C8—2442no—
490C8—2441no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—4882no—
499C22—4882no—
500C22—4882no—
501C22—4882no—
502C22—2444no—
503C22—2444no—
504C22—2444no—
505C22C2×#1451974811yescharacteristic
506C22#1684040011yescharacteristic
507C22#1684042311yescharacteristic
508C22C2×#1451979711yescharacteristic
509C22C2×#1451980211yescharacteristic
510C22#1684056511yescharacteristic
511C22#1684056511yescharacteristic
512C4—22no—
513C4—42no—
514C4—42no—
515C4—81no—
516C4—82no—
517C4—4881no—
518C4—2442no—
519C4#1683997414yes—
520C4#1684029811yescharacteristic
521C4#1684038511yescharacteristic
522C4#1684040311yescharacteristic
523C4#1684041411yescharacteristic
524C2—82no—
525C2—4884no—
526C2#2144939211yescharacteristic
527C2C2×#1683997411yescharacteristic
528C2#2145531811yescharacteristic
529C2#2145776311yescharacteristic
530C2#2145805511yescharacteristic
531C2#2145847911yescharacteristic
532C2#2145852611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2563C2
socleC23×C61C22×D4
prime radicalC2×#16839957C2
hypercenter#2563D61
hypocenterC61#25321
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2563

Complements

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

NH ≅ G/Ntype
1. C2×#16840110C2semidirect
3. #21448883C2semidirect
4. #21447233C2semidirect
5. #21447129C2semidirect
6. #21446768C2semidirect
7. #21446538C2semidirect
8. #21446170C2semidirect
9. #21445826C2semidirect
10. C4×#14519698C2semidirect
11. C61×#2563C2semidirect
260. #2563D61semidirect
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×#168399572
#214488832
#214472332
#214471292
#214467682
#214465382
#214461702
#214458262
C4×#145196982
C61×#25632
#2532161