Finite Group Databasebeta

← #29580346

Subgroups — #29580346

Order 15616.

808 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 85 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#21456621C211yesmaximal, characteristic, complemented
2#21451543C211yesmaximal, characteristic
3#21446805C211yesmaximal, characteristic, complemented
4#21445763C211yesmaximal, characteristic, complemented
5#21444336C211yesmaximal, characteristic, complemented
6#21443726C211yesmaximal, characteristic, complemented
7C61×#3007C211yesmaximal, characteristic, complemented
8C2×#14519751—21no—
9C2×#14519751C2211yescharacteristic
10#16840095—21no—
11#16840075—21no—
12#16840010—22no—
13#16839912—21no—
14#16839901—21no—
15C2×#14519704—22no—
16#16839726C2211yescharacteristic
17#16839673—21no—
18#16839670—22no—
19#16839670—21no—
20#16839622—21no—
21#16839622C2211yescharacteristic
22C4×C22⋊Dic61C2211yescharacteristic
23#16839529—22no—
24#16839497—22no—
25C22×#41×C61C2211yescharacteristic
26C2×#137×C61—21no—
27C2×C22⋊Q8×C61C2211yescharacteristic
28C61×#452—22no—
29C61×#451—21no—
30C61×#450—21no—
31C61×#417—21no—
32C61×#407C2211yescharacteristic
33C22×D244D411yescharacteristic
34C22×C22⋊D61—22no—
35C22×C22⋊D61—21no—
36C2×#11758201—22no—
37C2×#11758194D412yescomplemented
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41#14519751—21no—
42#14519751—44no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—22no—
47C2×C22⋊Dic61D412yes—
48C2×#11758198—22no—
49C2×#11758198—21no—
50C2×C4×Dic61—21no—
51#14519710—22no—
52#14519704—44no—
53#14519701—22no—
54#14519697—22no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57#14519684—22no—
58#14519684—21no—
59#14519684—21no—
60C2×#41×C61—21no—
61C2×#41×C61—21no—
62C2×#41×C61—21no—
63C2×#41×C61—21no—
64C2×#41×C61—44no—
65C22×Q8×C61D411yescharacteristic
66C22×Q8×C61D411yescharacteristic, complemented
67C22×D4×C61—21no—
68C22×D4×C61D411yescharacteristic
69C23×C4×C61—21no—
70C23×C4×C61C2311yescharacteristic
71#137×C61—44no—
72C2×Q16×C61—22no—
73C2×SD16×C61—22no—
74C2×M16×C61—21no—
75C22⋊Q8×C61—21no—
76C22⋊Q8×C61—21no—
77C22⋊Q8×C61—21no—
78C22⋊Q8×C61—21no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61D411yescharacteristic, complemented
81C2×C22⋊C4×C61—21no—
82#103×C61—22no—
83#102×C61—22no—
84C22⋊C8×C61—21no—
85C22⋊C8×C61D412yescomplemented
86C24×C61—21no—
87C24×C61C2×D411yescharacteristic
88C23×D61—21no—
89#41×C61—24no—
90#41×C61—24no—
91#41×C61—42no—
92#41×C61—42no—
93#41×C61—42no—
94#41×C61—42no—
95#41×C61—42no—
96#41×C61—42no—
97C2×Q8×C61—21no—
98C2×Q8×C61—21no—
99C2×Q8×C61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61—42no—
102C2×Q8×C61—42no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—42no—
106C2×D4×C61—42no—
107C2×D4×C61—42no—
108C2×D4×C61—41no—
109C2×D4×C61—41no—
110C22×C4×C61—24no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—42no—
115C22×C4×C61—42no—
116C22×C4×C61—41no—
117C22×C4×C61—41no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×C22⋊D61—44no—
125C2×C22⋊D61—41no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—82no—
128C2×C22⋊D61—82no—
129C2×C22⋊D61—82no—
130C22×Dic61—21no—
131C22×Dic61—22no—
132C22×Dic61—41no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—42no—
136Q16×C61—44no—
137SD16×C61—44no—
138M16×C61—42no—
139C2×C8×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C22⋊C4×C61—41no—
145C22⋊C4×C61—41no—
146C22⋊Dic61—22no—
147C22⋊Dic61—41no—
148C22⋊Dic61—42no—
149C22⋊Dic61—82no—
150#11758201—82no—
151#11758198—44no—
152#11758198—41no—
153#11758198—41no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—42no—
157C4×Dic61—42no—
158#11758194—42no—
159C2×#92418—22no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—22no—
163C23×C61—41no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61#12011yescharacteristic
172C23×C61#12111yescharacteristic
173C23×C61#12711yescharacteristic
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179Q8×C61—22no—
180Q8×C61—22no—
181Q8×C61—41no—
182Q8×C61—41no—
183Q8×C61—42no—
184Q8×C61—42no—
185D4×C61—22no—
186D4×C61—22no—
187D4×C61—41no—
188D4×C61—41no—
189D4×C61—44no—
190D4×C61—44no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—44no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—44no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—81no—
205C2×C4×C61—81no—
206C2×C4×C61#12011yescharacteristic
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61#12111yescharacteristic
209C2×C4×C61#12111yescharacteristic
210C22⋊D61—82no—
211C22⋊D61—82no—
212C22⋊D61—88no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—41no—
218C2×Dic61—42no—
219C2×Dic61—81no—
220C2×Dic61—81no—
221D244—42no—
222D244—81no—
223C8×C61—42no—
224#92418—42no—
225#9934—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61—81no—
243C22×C61#39211yescharacteristic
244C22×C61#45511yescharacteristic
245C22×C61#45611yescharacteristic
246C2×D61—82no—
247C2×D61—82no—
248C2×D61—81no—
249C2×D61—81no—
250C2×D61—81no—
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—42no—
255C4×C61—81no—
256C4×C61—81no—
257C4×C61B(4,𝔽2)12yes—
258Dic61—82no—
259Dic61—82no—
260Dic61—81no—
261#3193—611no—
262#3007D6111yescharacteristic, complemented
263#2615—611no—
264#1996—611no—
265#1789—611no—
266#1507—611no—
267#1305—611no—
268C2×C61—21no—
269C2×C61—21no—
270C2×C61—41no—
271C2×C61—41no—
272C2×C61—82no—
273C2×C61#200711yescharacteristic
274C2×C61#200811yescharacteristic
275C2×C61#202111yescharacteristic
276D61—161no—
277C22×#41C2×D6111yescharacteristic
278C2×#137—21no—
279C2×C22⋊Q8C2×D6111yescharacteristic
280C2×#121—611no—
281C2×#121—1221no—
282#484—1221no—
283#480—1221no—
284#468—1222no—
285#452—22no—
286#451—21no—
287#450—21no—
288#450—1221no—
289#448—1221no—
290C2×#104—1222no—
291#417—21no—
292#407—611no—
293#407C2×D6111yescharacteristic
294#393—1221no—
295#392—1222no—
296#392—1221no—
297#380—611no—
298#380—1221no—
299C4×C22⋊C4—611no—
300#342—1222no—
301#327—1222no—
302C61#993411yescharacteristic, complemented, Sylow
303C2×#41—21no—
304C2×#41—21no—
305C2×#41—21no—
306C2×#41—21no—
307C2×#41—44no—
308C22×Q8C22⋊D6111yescharacteristic
309C22×Q8C22⋊D6111yescharacteristic, complemented
310C22×D4—21no—
311C22×D4—611no—
312C22×D4—1222no—
313C22×D4—1221no—
314C22×D4C22⋊D6111yescharacteristic
315C23×C4—21no—
316C23×C4C22×D6111yescharacteristic
317#137—44no—
318C2×Q16—22no—
319C2×SD16—22no—
320C2×SD16—1222no—
321C2×M16—21no—
322C2×M16—612no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—21no—
326C22⋊Q8—21no—
327#121—1221no—
328#121—1221no—
329#121—1221no—
330#121—1221no—
331#121—2444no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4C22⋊D6111yescharacteristic, complemented
336C2×C22⋊C4—21no—
337C2×C22⋊C4—612no—
338C2×C22⋊C4—1221no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1221no—
342C2×C42—1221no—
343#107—1222no—
344#104—2444no—
345#103—22no—
346#103—1222no—
347#102—22no—
348#102—1222no—
349C22⋊C8—21no—
350C22⋊C8—1221no—
351C22⋊C8—1221no—
352C22⋊C8C22⋊D6112yescomplemented
353#95—1222no—
354#95—1221no—
355#95—1221no—
356C24—21no—
357C24—1221no—
358C24D4×D6111yescharacteristic
359#41—24no—
360#41—24no—
361#41—42no—
362#41—42no—
363#41—42no—
364#41—42no—
365#41—42no—
366#41—42no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—1221no—
381C2×D4—1221no—
382C2×D4—4882no—
383C2×D4—4882no—
384C2×D4—4882no—
385C2×D4—2442no—
386C2×D4—2444no—
387C2×D4—2441no—
388C2×D4—2441no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—1222no—
398C22×C4—1221no—
399C22×C4—2441no—
400C22×C4D4×D6111yescharacteristic
401C22×C4D4×D6111yescharacteristic
402C22×C4#1175822411yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406Q16—44no—
407SD16—44no—
408SD16—4882no—
409M16—42no—
410M16—2442no—
411C2×C8—22no—
412C2×C8—1222no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—1221no—
418C4⋊C4—1221no—
419C4⋊C4—2442no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—1222no—
423C22⋊C4—4882no—
424C22⋊C4—2444no—
425C22⋊C4—2441no—
426C22⋊C4—2441no—
427C22⋊C4—2441no—
428C22⋊C4—2442no—
429C42—2442no—
430C23—21no—
431C23—21no—
432C23—22no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—2442no—
442C23—2442no—
443C23—2441no—
444C23—2441no—
445C23—2441no—
446C23#1451974511yescharacteristic
447C23#1451974911yescharacteristic
448C23#1451977311yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—41no—
452Q8—41no—
453Q8—42no—
454Q8—42no—
455D4—22no—
456D4—22no—
457D4—41no—
458D4—41no—
459D4—44no—
460D4—44no—
461D4—4881no—
462D4—4882no—
463D4—4882no—
464D4—4888no—
465D4—2442no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—4881no—
482C2×C4—4881no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4—2441no—
489C2×C4#1451974511yescharacteristic
490C2×C4#1451974611yescharacteristic
491C2×C4#1451974811yescharacteristic
492C2×C4#1451974911yescharacteristic
493C8—42no—
494C8—2442no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—4882no—
513C22—4882no—
514C22—4881no—
515C22—4881no—
516C22—4881no—
517C22#1683967011yescharacteristic
518C22#1683993811yescharacteristic
519C22#1683994311yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—42no—
524C4—81no—
525C4—81no—
526C4—4882no—
527C4—4882no—
528C4—4881no—
529C4#1683994812yes—
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—82no—
535C2—9761no—
536C2#2144686411yescharacteristic
537C2#2144686911yescharacteristic
538C2#2144694211yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#3007C2
socleC22×C61#392
prime radical#21451543C2
hypercenter#3007D61
hypocenterC61#9934
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3007

Complements

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

NH ≅ G/Ntype
1. #21456621C2semidirect
3. #21446805C2semidirect
4. #21445763C2semidirect
5. #21444336C2semidirect
6. #21443726C2semidirect
7. C61×#3007C2semidirect
37. C2×#11758194D4semidirect
66. C22×Q8×C61D4semidirect
80. C2×C4⋊C4×C61D4semidirect
85. C22⋊C8×C61D4semidirect
262. #3007D61semidirect
302. C61#9934semidirect
309. C22×Q8C22⋊D61semidirect
335. C2×C4⋊C4C22⋊D61semidirect
352. C22⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214566212
#214515432
#214468052
#214457632
#214443362
#214437262
C61×#30072
#993461