Finite Group Databasebeta

← #29729850

Subgroups — #29729850

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×#16840108C211yesmaximal, characteristic, complemented
2C2×#16839954C211yesmaximal, characteristic
3#21450768C211yesmaximal, characteristic, complemented
4#21447222C212yesmaximal, complemented
5#21447129C212yesmaximal, complemented
6#21446771C211yesmaximal, characteristic, complemented
7#21446538C211yesmaximal, characteristic, complemented
8#21446167C212yesmaximal, complemented
9#21445819C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic, complemented
11C61×#2366C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519757—21no—
15C2×#14519757—21no—
16C2×C4×D244C2211yescharacteristic
17#16840108—24no—
18#16839954—24no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×C4⋊#92418C2212yes—
21C2×#14519698C2212yes—
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24C2×#14519697C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×C4×#92418C2211yescharacteristic
27#16839699—22no—
28#16839664C2212yes—
29#16839632C2212yes—
30#16839524C2211yescharacteristic
31#16839524C2211yescharacteristic
32#16839524C2212yes—
33#16839524C2212yes—
34#16839517C2212yes—
35C2×#127×C61C2211yescharacteristic
36C2×C4×D4×C61—21no—
37C2×C4×D4×C61C2212yes—
38C61×#390—21no—
39C61×#390C2212yes—
40C61×#386—22no—
41C4×C4⋊C4×C61C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C22×Dic122—21no—
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C2×#11758199—24no—
47C2×#11758199—24no—
48C2×#11758199—42no—
49C22×#92418C2311yescharacteristic
50C22×#92418C2311yescharacteristic
51C22×#92418C2312yes—
52#14519777—22no—
53#14519777—22no—
54#14519757—44no—
55#14519757—44no—
56C4×D244—22no—
57C4×D244—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2312yes—
63C2×#11758198—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C4⋊#92418—22no—
67C4⋊#92418—42no—
68#14519698—24no—
69#14519698—24no—
70#14519698—24no—
71#14519697—22no—
72#14519697—22no—
73C4×#92418—22no—
74C22×D4×C61—21no—
75C22×D4×C61—21no—
76C22×D4×C61C2312yes—
77C23×C4×C61—22no—
78#127×C61—22no—
79#127×C61D414yes—
80C4×D4×C61—24no—
81C4×D4×C61—42no—
82C4×D4×C61—44no—
83C2×C4⋊C4×C61D411yescharacteristic
84C2×C4⋊C4×C61D411yescharacteristic
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C22⋊C4×C61—22no—
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61—22no—
90C2×C42×C61—21no—
91C2×C42×C61C2312yes—
92C2×C42×C61C2311yescharacteristic
93#95×C61—21no—
94#95×C61D412yes—
95C24×C61—22no—
96C23×D61—21no—
97C2×D4×C61—24no—
98C2×D4×C61—24no—
99C2×D4×C61—22no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—22no—
103C2×D4×C61—22no—
104C2×D4×C61—42no—
105C2×D4×C61—42no—
106C2×D4×C61—81no—
107C2×D4×C61—81no—
108C2×D4×C61C2×D414yes—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—44no—
112C22×C4×C61—44no—
113C22×C4×C61—82no—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—22no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244#4112yes—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×Dic122—42no—
131C2×Dic122—42no—
132C2×Dic122—42no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61D814yescomplemented
138C4⋊C4×C61#4112yes—
139C22⋊C4×C61—44no—
140C22⋊C4×C61—44no—
141C22⋊C4×C61—82no—
142C42×C61—22no—
143C42×C61—22no—
144C42×C61—42no—
145C42×C61C2×D412yes—
146C42×C61#4112yes—
147#11758199—48no—
148#11758198—44no—
149#11758198—42no—
150#11758198—44no—
151C4⋊Dic61—21no—
152C4⋊Dic61—22no—
153C4⋊Dic61—21no—
154C4⋊Dic61—22no—
155C4⋊Dic61—44no—
156C4⋊Dic61—42no—
157C4⋊Dic61—44no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—22no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—22no—
164C2×#92418—21no—
165C2×#92418—22no—
166C2×#92418—21no—
167C23×C61—24no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61C22×D411yescharacteristic
175C22×D61—22no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182D4×C61—24no—
183D4×C61—24no—
184D4×C61—42no—
185D4×C61—42no—
186D4×C61—48no—
187D4×C61—84no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—88no—
202C2×C4×C61#12112yes—
203C2×C4×C61#12112yes—
204C2×C4×C61C2×D814yes—
205C2×C4×C61C2×D812yes—
206C2×C4×C61AGL(1,ℤ/8)12yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61H(5,𝔽2)11yescharacteristic
212C2×C4×C612−1+411yescharacteristic
213C2×Dic61—42no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—42no—
217C2×Dic61—41no—
218C2×Dic61—42no—
219D244—22no—
220D244—22no—
221D244—41no—
222D244—41no—
223C4×D61—84no—
224Dic122—84no—
225#92418—41no—
226#92418—42no—
227#92418—41no—
228#25318—611nomaximal, Sylow
229C22×C61—44no—
230C22×C61—44no—
231C22×C61—44no—
232C22×C61—82no—
233C22×C61—82no—
234C22×C61—82no—
235C22×C61—82no—
236C22×C61C2×#12111yescharacteristic
237C22×C61#54611yescharacteristic
238C22×C61#54811yescharacteristic
239C22×C61C22×D811yescharacteristic
240C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
241C22×C61#57711yescharacteristic
242C22×C61#57711yescharacteristic
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—81no—
247C2×D61—81no—
248C2×D61—81no—
249C2×D61—81no—
250C4×C61—24no—
251C4×C61—22no—
252C4×C61—42no—
253C4×C61—42no—
254C4×C61—81no—
255C4×C61#45914yes—
256C4×C61#52911yescharacteristic
257C4×C61#54211yescharacteristic
258C4×C61#54611yescharacteristic
259C4×C61#54711yescharacteristic
260Dic61—82no—
261Dic61—81no—
262C2×#486—611no—
263C2×#459—611no—
264#2542—611no—
265#2366D6111yescharacteristic, complemented
266#2071—612no—
267#2054—612no—
268#1988—611no—
269#1939—611no—
270#1879—612no—
271#1802—612no—
272C4×#102—611no—
273C2×C61—84no—
274C2×C61#242711yescharacteristic
275C2×C61C2×#45911yescharacteristic
276C2×C61#304211yescharacteristic
277C2×C61#329411yescharacteristic
278C2×C61#333211yescharacteristic
279C2×C61#335811yescharacteristic
280C2×C61#337311yescharacteristic
281D61—82no—
282C22×D8—612no—
283C2×C4⋊Q8—611no—
284C2×#127C2×D6111yescharacteristic
285C2×C22⋊Q8—1221no—
286C2×C22⋊Q8—1221no—
287C2×C4×D4—21no—
288C2×C4×D4—611no—
289C2×C4×D4C2×D6112yes—
290#486—1224no—
291#459—1224no—
292C2×C4⋊C8—611no—
293C2×C4⋊C8—612no—
294C2×#102—612no—
295C2×#102—612no—
296C2×#102—612no—
297C2×#102—611no—
298C2×#102—611no—
299C2×C4×C8—611no—
300#399—1222no—
301#390—21no—
302#390—612no—
303#390C2×D6112yes—
304#386—22no—
305#384—612no—
306C4×C4⋊C4C2×D6111yescharacteristic
307C4×C4⋊C4C2×D6111yescharacteristic
308#340—612no—
309#340—611no—
310#340—612no—
311#340—611no—
312#336—612no—
313C61#2531811yescharacteristic, complemented, Sylow
314C22×Q8—1221no—
315C22×D4—21no—
316C22×D4—21no—
317C22×D4—611no—
318C22×D4C22×D6112yes—
319C23×C4—22no—
320C23×C4—1221no—
321C2×D8—1224no—
322C2×D8—1224no—
323C2×D8—2442no—
324C22×C8—612no—
325C22×C8—611no—
326C22×C8—611no—
327C4⋊Q8—1222no—
328C4⋊Q8—1222no—
329#127—22no—
330#127D24414yes—
331C22⋊Q8—2444no—
332C22⋊Q8—2444no—
333C4×D4—24no—
334C4×D4—42no—
335C4×D4—44no—
336C4×D4—1222no—
337C4×D4—1222no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—612no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4C22⋊D6111yescharacteristic
344C2×C4⋊C4C22⋊D6111yescharacteristic
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C4⋊C4C22×D6111yescharacteristic
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1222no—
352C2×C22⋊C4—1222no—
353C2×C42—21no—
354C2×C42C22×D6112yes—
355C2×C42C22×D6111yescharacteristic
356C4⋊C8—1222no—
357C4⋊C8—2442no—
358#102—1222no—
359#102—1224no—
360#102—1224no—
361#102—1224no—
362#102—1222no—
363C4×C8—1222no—
364#95—21no—
365#95C22⋊D6112yes—
366C24—22no—
367C24—1221no—
368C2×Q8—2442no—
369C2×Q8—2442no—
370C2×Q8—2442no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4—612no—
383C2×D4—1221no—
384C2×D4—1221no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4C2×D24414yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—82no—
394C22×C4—1222no—
395C22×C4—1221no—
396C22×C4—2442no—
397C22×C4—2442no—
398C22×C4—2442no—
399C22×C4C2×C22⋊D6112yes—
400C22×C4C2×C22⋊D6111yescharacteristic
401C22×C4C2×C22⋊D6112yes—
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404D8—2448no—
405C2×C8—1221no—
406C2×C8—1221no—
407C2×C8—1222no—
408C2×C8—1221no—
409C2×C8—1222no—
410C2×C8—1221no—
411C2×C8—1221no—
412C2×C8—1222no—
413C2×C8—1221no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—1221no—
419C4⋊C4—1222no—
420C4⋊C4—1221no—
421C4⋊C4—1222no—
422C4⋊C4—2444no—
423C4⋊C4—2442no—
424C4⋊C4—2444no—
425C4⋊C4#1175819914yescomplemented
426C4⋊C4#1175822412yes—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—82no—
430C22⋊C4—2444no—
431C22⋊C4—2442no—
432C22⋊C4—2444no—
433C42—22no—
434C42—22no—
435C42—42no—
436C42C2×D24412yes—
437C42#1175822212yes—
438C23—24no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—1222no—
446C23—2441no—
447C23—2441no—
448C23—2441no—
449C23—2441no—
450C23—2441no—
451C23—2441no—
452C23C22×C22⋊D6111yescharacteristic
453Q8—4884no—
454D4—24no—
455D4—24no—
456D4—48no—
457D4—42no—
458D4—42no—
459D4—84no—
460D4—1222no—
461D4—1222no—
462D4—2441no—
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—22no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—88no—
478C2×C4—4884no—
479C2×C4—2441no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2441no—
483C2×C4—2442no—
484C2×C4—2441no—
485C2×C4#1451975112yes—
486C2×C4#1451975112yes—
487C2×C4C2×#1175819914yes—
488C2×C4C2×#1175819912yes—
489C2×C4#1451980212yes—
490C2×C4C22×D24411yescharacteristic
491C2×C4#41×D6111yescharacteristic
492C2×C4C2×#1175822411yescharacteristic
493C2×C4C2×#1175822211yescharacteristic
494C2×C4#1451982211yescharacteristic
495C2×C4#1451982411yescharacteristic
496C8—2441no—
497C8—2442no—
498C8—2441no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—4881no—
507C22—4881no—
508C22—4881no—
509C22—4881no—
510C22—2442no—
511C22—2442no—
512C22—2442no—
513C22C2×#1451975111yescharacteristic
514C22#1684040611yescharacteristic
515C22#1684041911yescharacteristic
516C22C22×#1175819911yescharacteristic
517C22C2×#1451980211yescharacteristic
518C22#1684057311yescharacteristic
519C22#1684057311yescharacteristic
520C4—24no—
521C4—22no—
522C4—42no—
523C4—42no—
524C4—81no—
525C4—4881no—
526C4—4882no—
527C4#1683995714yes—
528C4#1684029411yescharacteristic
529C4#1684037711yescharacteristic
530C4#1684040711yescharacteristic
531C4#1684041611yescharacteristic
532C2—84no—
533C2—4882no—
534C2#2144939011yescharacteristic
535C2C2×#1683995711yescharacteristic
536C2#2145533111yescharacteristic
537C2#2145763211yescharacteristic
538C2#2145814611yescharacteristic
539C2#2145845011yescharacteristic
540C2#2145860711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2366C2
socleC23×C61C22×D4
prime radicalC2×#16839954C2
hypercenter#2366D61
hypocenterC61#25318
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2366

Complements

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

NH ≅ G/Ntype
1. C2×#16840108C2semidirect
3. #21450768C2semidirect
4. #21447222C2semidirect
5. #21447129C2semidirect
6. #21446771C2semidirect
7. #21446538C2semidirect
8. #21446167C2semidirect
9. #21445819C2semidirect
10. C4×#14519697C2semidirect
11. C61×#2366C2semidirect
137. C4⋊C4×C61D8semidirect
265. #2366D61semidirect
313. C61#25318semidirect
425. C4⋊C4#11758199semidirect

Maximal subgroups

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

subgroupindex
C2×#168401082
C2×#168399542
#214507682
#214472222
#214471292
#214467712
#214465382
#214461672
#214458192
C4×#145196972
C61×#23662
#2531861