Finite Group Databasebeta

← #29578062

Subgroups — #29578062

Order 15616.

742 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 103 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#21448586C211yesmaximal, characteristic, complemented
2#21445763C211yesmaximal, characteristic, complemented
3#21445737C211yesmaximal, characteristic, complemented
4#21444348C211yesmaximal, characteristic, complemented
5#21443728C211yesmaximal, characteristic, complemented
6#21443673C211yesmaximal, characteristic, complemented
7C61×#3006C211yesmaximal, characteristic
8C2×#14519704—21no—
9C2×#14519704—21no—
10C2×#14519692—21no—
11C2×#14519692—21no—
12#16839726C2211yescharacteristic
13#16839665—21no—
14#16839655—21no—
15#16839648—21no—
16#16839642—21no—
17#16839642—21no—
18#16839625C2211yescharacteristic
19#16839625C2211yescharacteristic
20C22⋊C4×Dic61—21no—
21C4×C22⋊Dic61C2211yescharacteristic
22#16839496—22no—
23C22×#41×C61C2211yescharacteristic
24C2×AGL(1,ℤ/8)×C61—21no—
25C2×#121×C61C2211yescharacteristic
26C61×#449—22no—
27C61×#448C412yescomplemented
28C61×#447C412yescomplemented
29C61×#417—21no—
30C61×#407C2211yescharacteristic
31C23×Dic61—21no—
32C2×#11758194—21no—
33C2×C4⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61D411yescharacteristic
40C2×C22⋊Dic61D411yescharacteristic
41C2×C4×Dic61—21no—
42C2×C4×Dic61D411yescharacteristic
43C2×C4×Dic61D411yescharacteristic
44#14519704—42no—
45#14519704—42no—
46#14519692—42no—
47#14519692—42no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C2×#41×C61—21no—
58C2×#41×C61—21no—
59C2×#41×C61—21no—
60C2×#41×C61—21no—
61C2×#41×C61—42no—
62C2×#41×C61—42no—
63C22×Q8×C61C2×C411yescharacteristic, complemented
64C22×D4×C61—21no—
65C22×D4×C61—21no—
66C22×D4×C61C2×C411yescharacteristic, complemented
67C22×D4×C61D411yescharacteristic
68C23×C4×C61—21no—
69C23×C4×C61C2311yescharacteristic
70AGL(1,ℤ/8)×C61—42no—
71AGL(1,ℤ/8)×C61—42no—
72C2×SD16×C61—22no—
73C2×D8×C61—22no—
74C2×M16×C61—21no—
75#121×C61—21no—
76#121×C61—21no—
77#121×C61C2×C412yes—
78#121×C61D412yes—
79C2×C4⋊C4×C61D411yescharacteristic
80C2×C22⋊C4×C61—21no—
81#103×C61—22no—
82#102×C61—22no—
83C22⋊C8×C61—21no—
84C22⋊C8×C61C2×C412yescomplemented
85C24×C61—21no—
86C24×C61—21no—
87C24×C61C2×D411yescharacteristic
88#41×C61—22no—
89#41×C61—22no—
90#41×C61—22no—
91#41×C61—22no—
92#41×C61—41no—
93#41×C61—41no—
94#41×C61—41no—
95#41×C61—41no—
96#41×C61—42no—
97#41×C61—42no—
98#41×C61—42no—
99#41×C61—42no—
100C2×Q8×C61—21no—
101C2×Q8×C61—21no—
102C2×Q8×C61—42no—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61—42no—
113C2×D4×C61—41no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×D4×C61—81no—
118C2×D4×C61—81no—
119C2×D4×C61C22⋊C412yes—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—41no—
123C22×C4×C61—41no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—41no—
127C22×C4×C61—41no—
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61#4111yescharacteristic
133C22×C4×C61#4111yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—42no—
139C22×Dic61—41no—
140C22×Dic61—81no—
141C22×Dic61—81no—
142SD16×C61—44no—
143D8×C61—44no—
144M16×C61—41no—
145M16×C61—41no—
146C2×C8×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61C22⋊C412yes—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C22⋊Dic61—42no—
152C22⋊Dic61—42no—
153C22⋊Dic61—41no—
154C22⋊Dic61—41no—
155C22⋊Dic61—41no—
156C22⋊Dic61—41no—
157C22⋊Dic61—81no—
158C22⋊Dic61—81no—
159C4⋊Dic61—81no—
160C4×Dic61—41no—
161C4×Dic61—41no—
162C4×Dic61—81no—
163#11758194—41no—
164#11758194—41no—
165C2×#92418—41no—
166C23×C61—21no—
167C23×C61—21no—
168C23×C61—22no—
169C23×C61—22no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—81no—
181C23×C61—81no—
182C23×C61C4×D411yescharacteristic
183C23×C61#12011yescharacteristic
184C23×C61#12111yescharacteristic
185Q8×C61—22no—
186Q8×C61—22no—
187Q8×C61—41no—
188Q8×C61—41no—
189D4×C61—22no—
190D4×C61—22no—
191D4×C61—42no—
192D4×C61—41no—
193D4×C61—41no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61—81no—
199D4×C61—82no—
200D4×C61—82no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61C2×C22⋊C411yescharacteristic
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61#12311yescharacteristic
223C2×Dic61—22no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—84no—
236C2×Dic61—81no—
237C2×Dic61—81no—
238C2×Dic61—81no—
239C8×C61—42no—
240#92418—81no—
241#9523—611nomaximal, Sylow
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—82no—
261C22×C61—82no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61#38611yescharacteristic
265C22×C61B(4,𝔽2)11yescharacteristic
266C22×C61B(4,𝔽2)11yescharacteristic
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—42no—
272C4×C61—42no—
273Dic61—42no—
274Dic61—81no—
275Dic61—81no—
276Dic61—81no—
277Dic61—161no—
278#3006D6111yescharacteristic
279#2333—611no—
280#1789—611no—
281#1782—611no—
282#1510—611no—
283#1306—611no—
284#1284—611no—
285C2×C61—21no—
286C2×C61—21no—
287C2×C61—41no—
288C2×C61—41no—
289C2×C61—82no—
290C2×C61—82no—
291C2×C61#188911yescharacteristic
292C2×C61#190411yescharacteristic
293C2×C61#190611yescharacteristic
294C22×#41C2×D6111yescharacteristic
295C2×AGL(1,ℤ/8)—21no—
296C2×#121C2×D6111yescharacteristic
297#449—22no—
298#448Dic6112yescomplemented
299#447Dic6112yescomplemented
300C2×#104—1221no—
301C2×#104—1221no—
302#417—21no—
303C2×#99—1221no—
304C2×#99—1221no—
305#407—611no—
306#407C2×D6111yescharacteristic
307#390—1221no—
308#388—1221no—
309#387—1221no—
310#386—1221no—
311#386—1221no—
312#381—611no—
313#381—611no—
314C4×C22⋊C4—611no—
315C4×C22⋊C4—1221no—
316#327—1222no—
317C61#952311yescharacteristic, complemented, Sylow
318C2×#41—21no—
319C2×#41—21no—
320C2×#41—21no—
321C2×#41—21no—
322C2×#41—42no—
323C2×#41—42no—
324C22×Q8C2×Dic6111yescharacteristic, complemented
325C22×D4—21no—
326C22×D4—21no—
327C22×D4C2×Dic6111yescharacteristic, complemented
328C22×D4C22⋊D6111yescharacteristic
329C23×C4—21no—
330C23×C4—1221no—
331C23×C4C22×D6111yescharacteristic
332AGL(1,ℤ/8)—42no—
333AGL(1,ℤ/8)—42no—
334C2×SD16—22no—
335C2×D8—22no—
336C2×M16—21no—
337C2×M16—1221no—
338#121—21no—
339#121—21no—
340#121C2×Dic6112yes—
341#121C22⋊D6112yes—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4C22⋊D6111yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—611no—
346C2×C22⋊C4—611no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C22⋊C4—1221no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42—1221no—
355#104—2442no—
356#104—2442no—
357#103—22no—
358#102—22no—
359#99—2442no—
360#99—2442no—
361C22⋊C8—21no—
362C22⋊C8—1221no—
363C22⋊C8—1221no—
364C22⋊C8C2×Dic6112yescomplemented
365#95—1221no—
366#95—1221no—
367#95—1221no—
368#95—1221no—
369#95—1221no—
370#95—1221no—
371#95—1221no—
372C24—21no—
373C24—21no—
374C24D4×D6111yescharacteristic
375#41—22no—
376#41—22no—
377#41—22no—
378#41—22no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—42no—
384#41—42no—
385#41—42no—
386#41—42no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—42no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—42no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—81no—
405C2×D4—81no—
406C2×D4C22⋊Dic6112yes—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4—1221no—
418C22×C4—1221no—
419C22×C4—4881no—
420C22×C4—4881no—
421C22×C4—2442no—
422C22×C4—2441no—
423C22×C4D4×D6111yescharacteristic
424C22×C4#1175822411yescharacteristic
425C22×C4#1175822411yescharacteristic
426C22×C4C22×Dic6111yescharacteristic
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6111yescharacteristic
429SD16—44no—
430D8—44no—
431M16—41no—
432M16—41no—
433M16—2441no—
434M16—2441no—
435C2×C8—22no—
436C2×C8—2441no—
437C4⋊C4—21no—
438C4⋊C4—4881no—
439C4⋊C4C22⋊Dic6112yes—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—4881no—
443C22⋊C4—4881no—
444C22⋊C4—2442no—
445C22⋊C4—2442no—
446C22⋊C4—2441no—
447C22⋊C4—2441no—
448C22⋊C4—2441no—
449C22⋊C4—2441no—
450C42—4881no—
451C42—2441no—
452C42—2441no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—81no—
468C23—81no—
469C23D4×Dic6111yescharacteristic
470C23#1451974511yescharacteristic
471C23#1451974911yescharacteristic
472Q8—22no—
473Q8—22no—
474Q8—41no—
475Q8—41no—
476D4—22no—
477D4—22no—
478D4—42no—
479D4—41no—
480D4—41no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—81no—
486D4—82no—
487D4—82no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—1222no—
507C2×C4—4881no—
508C2×C4—4881no—
509C2×C4—4881no—
510C2×C4—4884no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4C2×C22⋊Dic6111yescharacteristic
523C2×C4D4×Dic6111yescharacteristic
524C2×C4#1451974911yescharacteristic
525C2×C4#1451975811yescharacteristic
526C8—42no—
527C8—4881no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—82no—
547C22—82no—
548C22—81no—
549C22—81no—
550C22#1683964211yescharacteristic
551C22#1683994811yescharacteristic
552C22#1683994811yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—42no—
558C4—42no—
559C4—9761no—
560C4—4881no—
561C4—4881no—
562C4—4881no—
563C4—2442no—
564C2—21no—
565C2—21no—
566C2—41no—
567C2—41no—
568C2—82no—
569C2—82no—
570C2#2144622811yescharacteristic
571C2#2144630811yescharacteristic
572C2#2144632211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#3006C2
socleC22×C61#386
cosocleC23×C4×C61C23
hypercenter#3006D61
hypocenterC61#9523
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
1. #21448586C2semidirect
2. #21445763C2semidirect
3. #21445737C2semidirect
4. #21444348C2semidirect
5. #21443728C2semidirect
6. #21443673C2semidirect
27. C61×#448C4semidirect
28. C61×#447C4semidirect
63. C22×Q8×C61C2×C4semidirect
66. C22×D4×C61C2×C4semidirect
84. C22⋊C8×C61C2×C4semidirect
298. #448Dic61semidirect
299. #447Dic61semidirect
317. C61#9523semidirect
324. C22×Q8C2×Dic61semidirect
327. C22×D4C2×Dic61semidirect
364. C22⋊C8C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214485862
#214457632
#214457372
#214443482
#214437282
#214436732
C61×#30062
#952361