Finite Group Databasebeta

← #29779746

Subgroups — #29779746

Order 15616.

878 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454180C211yesmaximal, characteristic, complemented
2C4×#14519773C211yesmaximal, characteristic
3#21447385C212yesmaximal, complemented
4#21447052C212yesmaximal
5#21446514C212yesmaximal, complemented
6#21446513C212yesmaximal, complemented
7#21445820C211yesmaximal, characteristic, complemented
8#21445819C211yesmaximal, characteristic, complemented
9#21445816C211yesmaximal, characteristic, complemented
10#21445674C211yesmaximal, characteristic, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C2×#14519773C2211yescharacteristic
13C2×#14519768—21no—
14C2×C4×D244C2211yescharacteristic
15C2×C4×C22⋊D61—21no—
16C2×C4×C22⋊D61—21no—
17C2×C4⋊#92418C2211yescharacteristic
18C2×#14519699C2212yes—
19C2×#14519698C2212yes—
20C2×#14519697C2212yes—
21C2×#14519687C2212yes—
22#16839665C2211yescharacteristic
23#16839663C2211yescharacteristic
24#16839662—21no—
25#16839662—21no—
26#16839661—21no—
27#16839661C2212yes—
28#16839658—21no—
29#16839658C2212yes—
30#16839632C2211yescharacteristic
31#16839631C2211yescharacteristic
32#16839627—21no—
33C42×Dic61C2211yescharacteristic
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839516C2212yes—
38C2×C4×D4×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2211yescharacteristic
40C2×#102×C61C2212yes—
41C61×#384C2211yescharacteristic
42C61×#340C2212yes—
43C22×D244C2311yescharacteristic
44C22×C22⋊D61—21no—
45C22×C22⋊D61—21no—
46C22×C4×D61—21no—
47C22×#92418C2312yes—
48#14519773—22no—
49#14519773—22no—
50#14519768—44no—
51C4×D244—22no—
52C4×D244—22no—
53C4×C22⋊D61—44no—
54C4×C22⋊D61—44no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×#11758196—21no—
61C2×#11758196—21no—
62C2×#11758196—22no—
63C2×C22⋊Dic61—21no—
64C2×C22⋊Dic61—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×#11758198—21no—
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61D411yescharacteristic
70C2×C4×Dic61D411yescharacteristic
71C2×C4×Dic61D411yescharacteristic
72C2×C4×Dic61C2311yescharacteristic
73C2×C4×Dic61C2311yescharacteristic
74C4⋊#92418—22no—
75#14519699—24no—
76#14519698—24no—
77#14519697—42no—
78#14519687—42no—
79#14519684—22no—
80#14519684—22no—
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61—21no—
83C22×C8×C61C2312yes—
84C4×D4×C61—22no—
85C4×D4×C61—22no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2312yes—
90C2×C22⋊C4×C61—21no—
91C2×C42×C61C2311yescharacteristic
92C4⋊C8×C61—22no—
93#102×C61—42no—
94C24×C61—21no—
95C23×D61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—21no—
98C2×D4×C61—21no—
99C2×D4×C61—21no—
100C2×D4×C61—21no—
101C2×D4×C61#4112yes—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61—42no—
105C22×C4×C61—42no—
106C22×C4×C61—42no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2411yescharacteristic
110C2×C22⋊D61—24no—
111C2×C22⋊D61—24no—
112C2×C22⋊D61—42no—
113C2×C22⋊D61—42no—
114C2×C22⋊D61—42no—
115C2×C22⋊D61—42no—
116C22×Dic61—21no—
117C22×Dic61—22no—
118C22×Dic61C2×D411yescharacteristic
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244#4112yes—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×C4×D61—42no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61—42no—
140C22⋊C4×C61—42no—
141C42×C61#4112yes—
142C42×C61#4112yes—
143C22⋊Dic61—42no—
144C22⋊Dic61—42no—
145#11758198—42no—
146#11758198—42no—
147#11758198—42no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—42no—
153C4⋊Dic61—42no—
154C4⋊Dic61—44no—
155#11758196—42no—
156#11758196—42no—
157#11758196—44no—
158C4×Dic61—22no—
159C4×Dic61—22no—
160C4×Dic61—21no—
161C4×Dic61—21no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164C4×Dic61—22no—
165C4×Dic61—22no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—22no—
169C23×C61—22no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61C22×D411yescharacteristic
177C22×D61—22no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184D4×C61—22no—
185D4×C61—22no—
186D4×C61—41no—
187D4×C61—41no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—84no—
197C2×C4×C61#12412yes—
198C2×C4×C61#12412yes—
199C2×C4×C61AGL(1,ℤ/8)12yes—
200C2×C4×C61AGL(1,ℤ/8)12yes—
201C2×C4×C61C2×#4111yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61H(5,𝔽2)11yescharacteristic
206C2×C4×C612−1+411yescharacteristic
207C22⋊D61—48no—
208C22⋊D61—48no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—42no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—42no—
225C2×Dic61AGL(1,ℤ/8)12yes—
226C2×Dic61#13712yes—
227D244—22no—
228D244—22no—
229D244—41no—
230D244—41no—
231C4×D61—84no—
232C8×C61—42no—
233#92418—42no—
234#29130—611nomaximal, Sylow
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61#52511yescharacteristic
243C22×C61C2×#12411yescharacteristic
244C22×C61#53711yescharacteristic
245C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
247C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C61C2×#13711yescharacteristic
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—81no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—81no—
256C4×C61—22no—
257C4×C61—81no—
258C4×C61—82no—
259C4×C61#49014yes—
260C4×C61#55211yescharacteristic
261C4×C61#55211yescharacteristic
262C4×C61#55311yescharacteristic
263C4×C61#55311yescharacteristic
264Dic61—22no—
265Dic61—22no—
266Dic61—41no—
267Dic61—41no—
268Dic61—82no—
269Dic61—81no—
270#2859—611no—
271C4×#127—611no—
272#2100—612no—
273#2041—612no—
274#1936—612no—
275#1936—612no—
276#1802—611no—
277#1802—611no—
278#1802—611no—
279#1802D6111yescharacteristic, complemented
280#1767—611no—
281C2×C61—82no—
282C2×C61#256211yescharacteristic
283C2×C61#310911yescharacteristic
284C2×C61#311011yescharacteristic
285C2×C61C2×#49011yescharacteristic
286C2×C61#314111yescharacteristic
287C2×C61#325311yescharacteristic
288C2×C61#325411yescharacteristic
289D61—82no—
290C2×#127—611no—
291C2×#125—1221no—
292C2×C4×D4—611no—
293C2×C4×D4—1221no—
294C2×C4×D4—1221no—
295C2×C4×D4C2×D6111yescharacteristic
296C2×C4⋊C8—611no—
297C2×C4⋊C8C2×D6111yescharacteristic
298C2×#102—612no—
299C2×#102—612no—
300C2×#102—612no—
301C2×#102C2×D6112yes—
302C2×#97—612no—
303#390—611no—
304#390—611no—
305#390—1221no—
306#390—1221no—
307#389—612no—
308#389—612no—
309#389—1221no—
310#389—1221no—
311#384—611no—
312#384—611no—
313#384C2×D6111yescharacteristic
314#382—1221no—
315C43—611no—
316#340—612no—
317#340—612no—
318#340—612no—
319#340C2×D6112yes—
320#336—612no—
321C61#2913011yescharacteristic, complemented, Sylow
322C22×D4—611no—
323C22×D4—1221no—
324C22×D4—1221no—
325C22×D4C22×D6111yescharacteristic
326C23×C4—21no—
327C23×C4—1221no—
328C22×C8—612no—
329C22×C8C22×D6112yes—
330#127—1222no—
331#127—1222no—
332#125—2444no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—1222no—
336C4×D4—1222no—
337C4×D4—2444no—
338C4×D4—2444no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—611no—
342C2×C4⋊C4—612no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4C22×D6111yescharacteristic
350C2×C4⋊C4C22×D6112yes—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C42—611no—
358C2×C42—611no—
359C2×C42—611no—
360C2×C42—611no—
361C2×C42—611no—
362C2×C42—611no—
363C2×C42C22×D6111yescharacteristic
364C4⋊C8—22no—
365C4⋊C8—1222no—
366#102—42no—
367#102—1224no—
368#102—1224no—
369#102—2442no—
370#97—2442no—
371#95—1222no—
372#95—1222no—
373C24—21no—
374C24—1221no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—612no—
381C2×D4—1224no—
382C2×D4—1224no—
383C2×D4—1221no—
384C2×D4—1221no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—2442no—
389C2×D4—2442no—
390C2×D4—2442no—
391C2×D4—2442no—
392C2×D4#1175822612yes—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—1222no—
403C22×C4—1221no—
404C22×C4—2442no—
405C22×C4—2442no—
406C22×C4—2442no—
407C22×C4D4×D6111yescharacteristic
408C22×C4D4×D6111yescharacteristic
409C22×C4C23×D6111yescharacteristic
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—1222no—
414C2×C8—1222no—
415C2×C8—1222no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—1222no—
423C4⋊C4—1221no—
424C4⋊C4—1221no—
425C4⋊C4—1222no—
426C4⋊C4—2442no—
427C4⋊C4—2444no—
428C4⋊C4—2442no—
429C4⋊C4—2442no—
430C4⋊C4—2444no—
431C4⋊C4—2442no—
432C22⋊C4—42no—
433C22⋊C4—2442no—
434C22⋊C4—2442no—
435C22⋊C4—2442no—
436C22⋊C4—2442no—
437C22⋊C4—2442no—
438C42—1222no—
439C42—1222no—
440C42—1221no—
441C42—1221no—
442C42—1221no—
443C42—1221no—
444C42—1222no—
445C42—1222no—
446C42#1175822212yes—
447C42#1175822412yes—
448C23—22no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—1222no—
456C23—2441no—
457C23—2441no—
458C23—2441no—
459C23—2441no—
460C23—2441no—
461C23—2441no—
462C23C2×D4×D6111yescharacteristic
463D4—22no—
464D4—22no—
465D4—41no—
466D4—41no—
467D4—1222no—
468D4—1222no—
469D4—2448no—
470D4—2448no—
471D4—2441no—
472D4—2441no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—84no—
482C2×C4—612no—
483C2×C4—612no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1221no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—1221no—
490C2×C4—1221no—
491C2×C4—1221no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—4884no—
495C2×C4—2442no—
496C2×C4—2441no—
497C2×C4—2442no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2442no—
501C2×C4#1451976312yes—
502C2×C4#1451976312yes—
503C2×C4#1451980112yes—
504C2×C4#1451980412yes—
505C2×C4#41×D6111yescharacteristic
506C2×C4C2×#1175822411yescharacteristic
507C2×C4C2×#1175822211yescharacteristic
508C2×C4C2×#1175822611yescharacteristic
509C2×C4#1451982211yescharacteristic
510C2×C4#1451982411yescharacteristic
511C8—42no—
512C8—2442no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—4881no—
521C22—4881no—
522C22—4881no—
523C22—4881no—
524C22—2442no—
525C22—2442no—
526C22—2442no—
527C22#1684027511yescharacteristic
528C22C2×#1451976311yescharacteristic
529C22#1684034111yescharacteristic
530C22AGL(1,ℤ/8)×D6111yescharacteristic
531C22C2×#1451980111yescharacteristic
532C22C2×#1451980411yescharacteristic
533C22#137×D6111yescharacteristic
534C4—22no—
535C4—81no—
536C4—82no—
537C4—1222no—
538C4—1222no—
539C4—4881no—
540C4—4882no—
541C4—2441no—
542C4—2441no—
543C4#1684012514yes—
544C4#1684045111yescharacteristic
545C4#1684045411yescharacteristic
546C4#1684045911yescharacteristic
547C4#1684046011yescharacteristic
548C2—82no—
549C2—4882no—
550C2#2145098411yescharacteristic
551C2#2145591211yescharacteristic
552C2#2145592011yescharacteristic
553C2C2×#1684012511yescharacteristic
554C2#2145621011yescharacteristic
555C2#2145719811yescharacteristic
556C2#2145720311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC2×#14519773C22
hypercenter#1802D61
hypocenterC61#29130
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. #21454180C2semidirect
3. #21447385C2semidirect
5. #21446514C2semidirect
6. #21446513C2semidirect
7. #21445820C2semidirect
8. #21445819C2semidirect
9. #21445816C2semidirect
10. #21445674C2semidirect
11. C61×#1802C2semidirect
279. #1802D61semidirect
321. C61#29130semidirect

Maximal subgroups

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

subgroupindex
#214541802
C4×#145197732
#214473852
#214470522
#214465142
#214465132
#214458202
#214458192
#214458162
#214456742
C61×#18022
#2913061