Finite Group Databasebeta

← #29729854

Subgroups — #29729854

Order 15616.

1034 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840109C211yesmaximal, characteristic, complemented
2C2×#16839958C211yesmaximal, characteristic
3#21450768C211yesmaximal, characteristic, complemented
4#21448883C211yesmaximal, characteristic, complemented
5#21447224C212yesmaximal, complemented
6#21447130C212yesmaximal, complemented
7#21446770C211yesmaximal, characteristic, complemented
8#21446166C212yesmaximal, complemented
9#21445820C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C61×#1939C211yesmaximal, characteristic, complemented
12C22×D488C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15C2×#14519757—21no—
16C2×#14519757—21no—
17C2×C4×D244C2211yescharacteristic
18C2×C4×D244C2212yes—
19C2×#14519739—21no—
20#16840109—24no—
21#16839958—24no—
22C2×#14519699C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2212yes—
25C2×#14519699C2212yes—
26C2×#14519699C2212yes—
27#16839699—22no—
28#16839664C2212yes—
29#16839664C2212yes—
30#16839663—21no—
31#16839640—22no—
32#16839632C2212yes—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839525C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839525C2212yes—
37#16839525C2212yes—
38C2×C4⋊C8×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2212yes—
40C2×C4×C8×C61C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#336C2212yes—
43C22×Dic122—21no—
44C22×D244—21no—
45C22×D244—21no—
46C22×D244C2312yes—
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C22×C4×D61—22no—
50C2×D488—24no—
51C2×D488—24no—
52C2×D488—42no—
53#14519777—22no—
54#14519777—22no—
55#14519774—22no—
56#14519774D414yes—
57#14519757—44no—
58#14519757—44no—
59C4×D244—22no—
60C4×D244—22no—
61C4×D244—24no—
62C4×D244—42no—
63#14519739—44no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758198—21no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×C4×Dic61—21no—
78#14519699—24no—
79#14519699—24no—
80#14519699—24no—
81#14519699—22no—
82#14519699—22no—
83#14519684—21no—
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×C61—22no—
92C4⋊C8×C61—42no—
93C4×C8×C61—22no—
94#95×C61D412yes—
95C23×D61—22no—
96C23×D61—21no—
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—21no—
113C2×D244—22no—
114C2×D244—24no—
115C2×D244—22no—
116C2×D244—21no—
117C2×D244—24no—
118C2×D244—42no—
119C2×D244—42no—
120C2×D244—81no—
121C2×D244—81no—
122C2×D244C2×D414yes—
123C2×D244#4112yes—
124C2×C4×D61—44no—
125C2×C4×D61—42no—
126C2×C4×D61—44no—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C4×D61—82no—
130C2×Dic122—42no—
131C2×Dic122—42no—
132C2×Dic122—42no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—22no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—22no—
139C2×C8×C61—21no—
140C2×C8×C61—22no—
141C2×C8×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61D814yescomplemented
144C42×C61—22no—
145C42×C61—22no—
146C42×C61C2×D412yes—
147C42×C61#4112yes—
148#11758198—44no—
149#11758198—44no—
150#11758198—42no—
151#11758198—44no—
152#11758198—44no—
153#11758198—82no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—22no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—22no—
160C4⋊Dic61—21no—
161C4⋊Dic61—44no—
162C4⋊Dic61—42no—
163C4⋊Dic61—44no—
164C4⋊Dic61#4112yes—
165C4×Dic61—42no—
166D488—48no—
167C23×C61C22×D411yescharacteristic
168C22×D61—24no—
169C22×D61—22no—
170C22×D61—41no—
171C22×D61—42no—
172C22×D61—41no—
173C22×D61—42no—
174C22×D61—41no—
175C22×D61—42no—
176C22×D61—41no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—41no—
181C22×D61—41no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61#12112yes—
190C2×C4×C61#12112yes—
191C2×C4×C61C2×D814yes—
192C2×C4×C61C2×D812yes—
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—41no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—41no—
212D244—22no—
213D244—22no—
214D244—24no—
215D244—24no—
216D244—42no—
217D244—41no—
218D244—42no—
219D244—41no—
220D244—48no—
221D244—84no—
222C4×D61—88no—
223C4×D61—84no—
224Dic122—84no—
225C8×C61—42no—
226C8×C61—41no—
227C8×C61—41no—
228#25318—611nomaximal, Sylow
229C22×C61C2×#12111yescharacteristic
230C22×C61#54611yescharacteristic
231C22×C61#54811yescharacteristic
232C22×C61C22×D811yescharacteristic
233C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
234C22×C61#57711yescharacteristic
235C22×C61#57711yescharacteristic
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—44no—
240C2×D61—44no—
241C2×D61—44no—
242C2×D61—82no—
243C2×D61—81no—
244C2×D61—82no—
245C2×D61—82no—
246C2×D61—81no—
247C2×D61—82no—
248C2×D61—81no—
249C2×D61—81no—
250C4×C61—24no—
251C4×C61—22no—
252C4×C61—42no—
253C4×C61#45914yes—
254C4×C61#52911yescharacteristic
255C4×C61#54211yescharacteristic
256C4×C61#54611yescharacteristic
257C4×C61#54711yescharacteristic
258Dic61—42no—
259Dic61—81no—
260Dic61—82no—
261Dic61—81no—
262C2×#486—611no—
263C2×#459—611no—
264#2542—611no—
265#2366—611no—
266#2071—612no—
267#2054—612no—
268#1988—611no—
269#1939D6111yescharacteristic, complemented
270#1879—612no—
271#1802—612no—
272C4×#102—611no—
273C2×C61#242711yescharacteristic
274C2×C61C2×#45911yescharacteristic
275C2×C61#304211yescharacteristic
276C2×C61#329411yescharacteristic
277C2×C61#333211yescharacteristic
278C2×C61#335811yescharacteristic
279C2×C61#337311yescharacteristic
280D61—82no—
281D61—84no—
282C22×D8—612no—
283C2×C4⋊Q8—611no—
284C2×#127—611no—
285C2×C22⋊Q8—1221no—
286C2×C22⋊Q8—1221no—
287C2×C4×D4—611no—
288C2×C4×D4—612no—
289C2×C4×D4—1221no—
290#486—1224no—
291#459—1224no—
292C2×C4⋊C8C2×D6111yescharacteristic
293C2×C4⋊C8C2×D6112yes—
294C2×#102—611no—
295C2×#102—612no—
296C2×#102—612no—
297C2×#102—612no—
298C2×#102—611no—
299C2×C4×C8C2×D6111yescharacteristic
300#399—1222no—
301#390—612no—
302#390—612no—
303#390—1221no—
304#386—1222no—
305#384—612no—
306C4×C4⋊C4—611no—
307C4×C4⋊C4C2×D6111yescharacteristic
308#340—612no—
309#340—611no—
310#340—612no—
311#340—611no—
312#336C2×D6112yes—
313C61#2531811yescharacteristic, complemented, Sylow
314C22×Q8—1221no—
315C22×D4—612no—
316C22×D4—611no—
317C22×D4—1221no—
318C22×D4—1221no—
319C23×C4—1222no—
320C23×C4—1221no—
321C2×D8—1224no—
322C2×D8—1224no—
323C2×D8—2442no—
324C22×C8C22×D6112yes—
325C22×C8C22×D6111yescharacteristic
326C22×C8C22×D6111yescharacteristic
327C4⋊Q8—1222no—
328C4⋊Q8—1222no—
329#127—614no—
330#127—1222no—
331C22⋊Q8—2444no—
332C22⋊Q8—2444no—
333C4×D4—1224no—
334C4×D4—1222no—
335C4×D4—1222no—
336C4×D4—2442no—
337C4×D4—2444no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—612no—
341C2×C4⋊C4—611no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4D24411yescharacteristic
346C2×C4⋊C4D24411yescharacteristic
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1222no—
351C2×C22⋊C4—1222no—
352C2×C22⋊C4—1222no—
353C2×C42—1221no—
354C2×C42C22×D6112yes—
355C2×C42C22×D6111yescharacteristic
356C4⋊C8—22no—
357C4⋊C8—42no—
358#102—1222no—
359#102—1222no—
360#102—1224no—
361#102—1224no—
362#102—1224no—
363C4×C8—22no—
364#95—1221no—
365#95D24412yes—
366C24—1221no—
367C24—1222no—
368C2×Q8—2442no—
369C2×Q8—2442no—
370C2×Q8—2442no—
371C2×D4—614no—
372C2×D4—612no—
373C2×D4—1221no—
374C2×D4—1221no—
375C2×D4—1222no—
376C2×D4—1222no—
377C2×D4—1221no—
378C2×D4—1222no—
379C2×D4—1222no—
380C2×D4—1221no—
381C2×D4—1222no—
382C2×D4—1221no—
383C2×D4—1224no—
384C2×D4—1224no—
385C2×D4—4881no—
386C2×D4—4881no—
387C2×D4—2442no—
388C2×D4—2442no—
389C22×C4—1221no—
390C22×C4—1221no—
391C22×C4—1222no—
392C22×C4—1221no—
393C22×C4—4882no—
394C22×C4—2444no—
395C22×C4—2442no—
396C22×C4—2444no—
397C22×C4—2442no—
398C22×C4—2442no—
399C22×C4C2×D24412yes—
400C22×C4C2×D24411yescharacteristic
401C22×C4C2×D24412yes—
402C22×C4C2×D24411yescharacteristic
403C22×C4C23×D6111yescharacteristic
404D8—2448no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—22no—
411C2×C8—21no—
412C2×C8—22no—
413C2×C8—21no—
414C4⋊C4—22no—
415C4⋊C4—612no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4—1222no—
419C4⋊C4—1221no—
420C4⋊C4—1221no—
421C4⋊C4—1222no—
422C4⋊C4—1221no—
423C4⋊C4—2444no—
424C4⋊C4—2442no—
425C4⋊C4—2444no—
426C4⋊C4D48814yescomplemented
427C22⋊C4—4882no—
428C22⋊C4—2444no—
429C22⋊C4—2444no—
430C22⋊C4—2442no—
431C22⋊C4—2444no—
432C22⋊C4—2444no—
433C42—22no—
434C42—22no—
435C42—2442no—
436C42D4×D6112yes—
437C42#1175822612yes—
438C23—1224no—
439C23—1222no—
440C23—2441no—
441C23—2441no—
442C23—2442no—
443C23—2441no—
444C23—2442no—
445C23—2442no—
446C23—2442no—
447C23—2441no—
448C23—2441no—
449C23—2442no—
450C23—2441no—
451C23—2442no—
452C23C22×D24411yescharacteristic
453Q8—4884no—
454D4—1222no—
455D4—1222no—
456D4—1224no—
457D4—1224no—
458D4—4884no—
459D4—2448no—
460D4—2442no—
461D4—2441no—
462D4—2442no—
463D4—2441no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—1222no—
472C2×C4—4888no—
473C2×C4—4884no—
474C2×C4—2441no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4—2442no—
478C2×C4—2441no—
479C2×C4—2442no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4#1451975212yes—
486C2×C4#1451975212yes—
487C2×C4C2×D48814yes—
488C2×C4C2×D48812yes—
489C2×C4#1451980512yes—
490C2×C4C2×D4×D6111yescharacteristic
491C2×C4#41×D6111yescharacteristic
492C2×C4#41×D6111yescharacteristic
493C2×C4C2×#1175822611yescharacteristic
494C2×C4#1451982211yescharacteristic
495C2×C4#1451982511yescharacteristic
496C8—41no—
497C8—42no—
498C8—41no—
499C22—4882no—
500C22—4881no—
501C22—4882no—
502C22—4882no—
503C22—4881no—
504C22—4881no—
505C22—4882no—
506C22—4881no—
507C22—2442no—
508C22—2444no—
509C22—2442no—
510C22—2444no—
511C22—2442no—
512C22—2444no—
513C22C2×#1451975211yescharacteristic
514C22#1684040711yescharacteristic
515C22#1684042411yescharacteristic
516C22C22×D48811yescharacteristic
517C22C2×#1451980511yescharacteristic
518C22#1684057411yescharacteristic
519C22#1684057411yescharacteristic
520C4—24no—
521C4—22no—
522C4—42no—
523C4—4881no—
524C4—4881no—
525C4—4882no—
526C4—2442no—
527C4#1683995814yes—
528C4#1684029711yescharacteristic
529C4#1684038611yescharacteristic
530C4#1684040911yescharacteristic
531C4#1684041411yescharacteristic
532C2—4884no—
533C2—4882no—
534C2#2144939411yescharacteristic
535C2C2×#1683995811yescharacteristic
536C2#2145533211yescharacteristic
537C2#2145763311yescharacteristic
538C2#2145814711yescharacteristic
539C2#2145845511yescharacteristic
540C2#2145861311yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1939C2
socleC23×C61C22×D4
prime radicalC2×#16839958C2
hypercenter#1939D61
hypocenterC61#25318
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
1. C2×#16840109C2semidirect
3. #21450768C2semidirect
4. #21448883C2semidirect
5. #21447224C2semidirect
6. #21447130C2semidirect
7. #21446770C2semidirect
8. #21446166C2semidirect
9. #21445820C2semidirect
10. C4×#14519699C2semidirect
11. C61×#1939C2semidirect
143. C4⋊C4×C61D8semidirect
269. #1939D61semidirect
313. C61#25318semidirect
426. C4⋊C4D488semidirect

Maximal subgroups

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

subgroupindex
C2×#168401092
C2×#168399582
#214507682
#214488832
#214472242
#214471302
#214467702
#214461662
#214458202
C4×#145196992
C61×#19392
#2531861