Finite Group Databasebeta

← #29580743

Subgroups — #29580743

Order 15616.

686 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455819C211yesmaximal, characteristic, complemented
2#21455283C211yesmaximal, characteristic, complemented
3#21444378C211yesmaximal, characteristic, complemented
4#21444244C211yesmaximal, characteristic, complemented
5#21444210C211yesmaximal, characteristic, complemented
6#21443677C211yesmaximal, characteristic, complemented
7C61×#2535C211yesmaximal, characteristic, complemented
8C2×#14519803—21no—
9#16840312C2211yescharacteristic
10#16840075—21no—
11#16840042—21no—
12#16839977—22no—
13#16839967—21no—
14#16839957—21no—
15#16839957—21no—
16#16839896—21no—
17C4×#11758199—22no—
18#16839815C2211yescharacteristic
19#16839815C2211yescharacteristic
20#16839786C2211yescharacteristic
21#16839761—21no—
22#16839495—22no—
23D4⋊#92418—22no—
24D4⋊#92418—22no—
25C2×#121×C61—21no—
26C2×C4×D4×C61C2211yescharacteristic
27C2×C4×D4×C61C2211yescharacteristic
28C61×#397—21no—
29C61×#392—21no—
30C61×#386—22no—
31C61×#382C2211yescharacteristic
32C61×#380—21no—
33C2×#11758222—21no—
34#14519803—42no—
35#14519803—42no—
36C2×#11758200—22no—
37C2×#11758199—22no—
38C2×#11758199—22no—
39C2×#11758194—21no—
40#14519777D411yescharacteristic, complemented
41#14519774D411yescharacteristic, complemented
42#14519765—21no—
43#14519751—21no—
44#14519751—21no—
45C4×D244—21no—
46C4×D244—21no—
47#14519710—22no—
48C4⋊#92418—21no—
49C4⋊#92418—21no—
50C4⋊#92418—21no—
51C4⋊#92418D412yescomplemented
52#14519700—22no—
53#14519698—22no—
54#14519697—22no—
55#14519697—22no—
56C22⋊#92418—21no—
57C22⋊#92418—21no—
58C4×#92418—21no—
59C4×#92418D412yescomplemented
60C22×D4×C61—22no—
61C22×D4×C61D411yescharacteristic
62C22×D4×C61D411yescharacteristic
63C23×C4×C61—21no—
64C23×C4×C61—21no—
65#121×C61—42no—
66#121×C61—42no—
67C4×D4×C61—21no—
68C4×D4×C61—21no—
69C4×D4×C61—21no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61—21no—
74C4×D4×C61—21no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61D411yescharacteristic
77C2×C4⋊C4×C61D411yescharacteristic
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—21no—
80C2×C22⋊C4×C61—21no—
81C2×C22⋊C4×C61—22no—
82C2×C42×C61C2311yescharacteristic
83#95×C61—22no—
84#95×C61—21no—
85#95×C61D412yes—
86C24×C61—21no—
87C24×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—42no—
96C2×D4×C61—44no—
97C2×D4×C61—44no—
98C2×D4×C61—82no—
99C2×D4×C61D812yescomplemented
100C22×C4×C61—22no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—41no—
104C22×C4×C61—41no—
105C22×C4×C61—41no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61—41no—
109C22×C4×C61—41no—
110C22×C4×C61—41no—
111C22×C4×C61—82no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C2×C22⋊D61—41no—
116#11758222—41no—
117#11758222—41no—
118#11758222—41no—
119#11758222—41no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244—21no—
123C2×C4×D61—41no—
124C2×Dic122—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—41no—
130C4⋊C4×C61—41no—
131C22⋊C4×C61—44no—
132C22⋊C4×C61—41no—
133C22⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C22⋊C4×C61—82no—
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61#4111yescharacteristic
141#11758200—44no—
142#11758199—44no—
143#11758199—44no—
144#11758198—41no—
145#11758198—41no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148#11758194—41no—
149#11758194—41no—
150C2×#92418—22no—
151C2×#92418—22no—
152C2×#92418—41no—
153C23×C61—22no—
154C23×C61—22no—
155C23×C61—41no—
156C23×C61—42no—
157C23×C61—42no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61—41no—
161C23×C61—41no—
162C23×C61—41no—
163C23×C61—41no—
164C23×C61—41no—
165C23×C61#12011yescharacteristic
166C22×D61—41no—
167D4×C61—22no—
168D4×C61—22no—
169D4×C61—22no—
170D4×C61—22no—
171D4×C61—41no—
172D4×C61—41no—
173D4×C61—41no—
174D4×C61—41no—
175D4×C61—88no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—82no—
187C2×C4×C61—82no—
188C2×C4×C61—81no—
189C2×C4×C61—81no—
190C2×C4×C61—81no—
191C2×C4×C61—81no—
192C2×C4×C61#12011yescharacteristic
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12711yescharacteristic
198C2×C4×C61C2×D811yescharacteristic
199C2×C4×C61#13511yescharacteristic
200C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
201C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
202C22⋊D61—81no—
203C22⋊D61—81no—
204C2×Dic61—41no—
205D244—42no—
206D244—41no—
207D244—41no—
208D244—81no—
209C4×D61—81no—
210C4×D61—81no—
211Dic122—41no—
212Dic122—41no—
213#92418—42no—
214#92418—42no—
215#92418—81no—
216#9996—611nomaximal, Sylow
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—42no—
220C22×C61—44no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—42no—
224C22×C61—81no—
225C22×C61—82no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61#39211yescharacteristic
232C22×C61#44711yescharacteristic
233C22×C61#44911yescharacteristic
234C2×D61—81no—
235C2×D61—81no—
236C2×D61—81no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—82no—
242C4×C61—81no—
243C4×C61—81no—
244C4×C61#44711yescharacteristic
245C4×C61#44911yescharacteristic
246C4×C61#46611yescharacteristic
247C4×C61#46911yescharacteristic
248Dic61—81no—
249#3100—611no—
250#3036—611no—
251#2535D6111yescharacteristic, complemented
252#1517—611no—
253#1486—611no—
254#1478—611no—
255#1286—611no—
256C2×C61—21no—
257C2×C61—21no—
258C2×C61—82no—
259C2×C61—82no—
260C2×C61#199911yescharacteristic
261C2×C61#200811yescharacteristic
262C2×C61#201011yescharacteristic
263D61—161no—
264C2×AGL(1,ℤ/8)—1221no—
265#532—611no—
266C2×#121—21no—
267C2×C4×D4C2×D6111yescharacteristic
268C2×C4×D4C2×D6111yescharacteristic
269#480—1221no—
270#474—1221no—
271#463—1222no—
272#461—1221no—
273#459—1221no—
274#459—1221no—
275#447—1221no—
276C4×D8—1222no—
277#431—611no—
278#431—611no—
279#423—611no—
280#417—1221no—
281#397—21no—
282#392—21no—
283#386—22no—
284#382C2×D6111yescharacteristic
285#380—21no—
286#327—1222no—
287D4⋊C8—1222no—
288D4⋊C8—1222no—
289C61#999611yescharacteristic, complemented, Sylow
290C2×#41—1221no—
291C22×D4—22no—
292C22×D4C22⋊D6111yescharacteristic
293C22×D4C22⋊D6111yescharacteristic
294C23×C4—21no—
295C23×C4—21no—
296AGL(1,ℤ/8)—2442no—
297AGL(1,ℤ/8)—2442no—
298C2×SD16—1222no—
299C2×D8—1222no—
300C2×D8—1222no—
301C2×M16—1221no—
302C4⋊Q8—611no—
303#127—611no—
304#124—1221no—
305#121—42no—
306#121—42no—
307#121—1221no—
308#121—1221no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—1221no—
318C4×D4—1221no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4C22⋊D6111yescharacteristic
321C2×C4⋊C4C22⋊D6111yescharacteristic
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C42C22×D6111yescharacteristic
327#107—1222no—
328C4⋊C8—612no—
329C4⋊C8—1221no—
330C4⋊C8—1221no—
331C4⋊C8—1221no—
332#103—1222no—
333#102—1222no—
334#102—1222no—
335#102—1222no—
336C22⋊C8—1221no—
337C22⋊C8—1221no—
338C4×C8—612no—
339C4×C8—1221no—
340#95—22no—
341#95—21no—
342#95C22⋊D6112yes—
343C24—21no—
344C24—21no—
345#41—2441no—
346#41—2441no—
347#41—2441no—
348#41—2441no—
349C2×Q8—1221no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—42no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—82no—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—1221no—
364C2×D4—2441no—
365C2×D4#1175819912yescomplemented
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—82no—
378C22×C4—2441no—
379C22×C4C2×C22⋊D6111yescharacteristic
380C22×C4C2×C22⋊D6111yescharacteristic
381C22×C4C2×C22⋊D6111yescharacteristic
382SD16—2444no—
383D8—2444no—
384D8—2444no—
385M16—2441no—
386M16—2441no—
387C2×C8—1222no—
388C2×C8—1222no—
389C2×C8—2441no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—1221no—
397C4⋊C4—1221no—
398C22⋊C4—44no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—82no—
404C22⋊C4—2441no—
405C22⋊C4—2441no—
406C42D4×D6111yescharacteristic
407C42D4×D6111yescharacteristic
408C42D4×D6111yescharacteristic
409C42#1175822411yescharacteristic
410C23—22no—
411C23—22no—
412C23—41no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—2441no—
423C23#1451974611yescharacteristic
424Q8—2441no—
425Q8—2441no—
426D4—22no—
427D4—22no—
428D4—22no—
429D4—22no—
430D4—41no—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—88no—
435D4—4881no—
436D4—4881no—
437D4—4881no—
438D4—2442no—
439D4—2441no—
440D4—2441no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—4881no—
458C2×C4—4881no—
459C2×C4—2441no—
460C2×C4#1451974511yescharacteristic
461C2×C4#1451974511yescharacteristic
462C2×C4#1451974811yescharacteristic
463C2×C4#1451974911yescharacteristic
464C2×C4#1451974911yescharacteristic
465C2×C4#1451977311yescharacteristic
466C2×C4C2×#1175819911yescharacteristic
467C2×C4#1451979711yescharacteristic
468C2×C4#1451980211yescharacteristic
469C2×C4#1451980311yescharacteristic
470C8—4881no—
471C8—2442no—
472C8—2442no—
473C22—21no—
474C22—21no—
475C22—42no—
476C22—44no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—81no—
481C22—82no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—4881no—
488C22—4881no—
489C22—4881no—
490C22#1683967011yescharacteristic
491C22#1683989611yescharacteristic
492C22#1683990711yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—82no—
498C4—81no—
499C4—81no—
500C4—4881no—
501C4#1683989511yescharacteristic
502C4#1683990511yescharacteristic
503C4#1683999611yescharacteristic
504C4#1684001311yescharacteristic
505C2—21no—
506C2—21no—
507C2—82no—
508C2—82no—
509C2—9761no—
510C2#2144681511yescharacteristic
511C2#2144687211yescharacteristic
512C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2535C2
socleC22×C61#392
hypercenter#2535D61
hypocenterC61#9996
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. #21455819C2semidirect
2. #21455283C2semidirect
3. #21444378C2semidirect
4. #21444244C2semidirect
5. #21444210C2semidirect
6. #21443677C2semidirect
7. C61×#2535C2semidirect
40. #14519777D4semidirect
41. #14519774D4semidirect
51. C4⋊#92418D4semidirect
59. C4×#92418D4semidirect
99. C2×D4×C61D8semidirect
251. #2535D61semidirect
289. C61#9996semidirect
365. C2×D4#11758199semidirect

Maximal subgroups

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

subgroupindex
#214558192
#214552832
#214443782
#214442442
#214442102
#214436772
C61×#25352
#999661