Finite Group Databasebeta

← #29578060

Subgroups — #29578060

Order 15616.

742 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 99 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#21455069C211yesmaximal, characteristic
2#21445764C211yesmaximal, characteristic, complemented
3#21445736C211yesmaximal, characteristic, complemented
4#21444348C211yesmaximal, characteristic, complemented
5#21443727C211yesmaximal, characteristic, complemented
6#21443672C211yesmaximal, characteristic, complemented
7C61×#2333C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C2×#14519803—21no—
10#16839905—22no—
11#16839900C412yescomplemented
12#16839895C412yescomplemented
13C2×#14519705—21no—
14C2×#14519705—21no—
15#16839761—21no—
16C2×#14519691—21no—
17C2×#14519691—21no—
18#16839726C2211yescharacteristic
19#16839726C2211yescharacteristic
20#16839625C2211yescharacteristic
21#16839496—22no—
22C2×#121×C61C2211yescharacteristic
23C61×#390—21no—
24C61×#388—21no—
25C61×#387—21no—
26C61×#386—21no—
27C61×#386—21no—
28C61×#381C2211yescharacteristic
29C4×C22⋊C4×C61—21no—
30C4×C22⋊C4×C61C2211yescharacteristic
31C2×#11758222—21no—
32C2×#11758222—21no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—42no—
36C2×#11758222—42no—
37C22×Dic122C2×C411yescharacteristic, complemented
38C22×D244C2×C411yescharacteristic, complemented
39C22×C22⋊D61—21no—
40C22×C4×D61—21no—
41#14519803—42no—
42#14519803—42no—
43C2×#11758200—22no—
44C2×#11758199—22no—
45C2×#11758194—21no—
46C2×#11758194—21no—
47C2×C22⋊Dic61—21no—
48#14519705—42no—
49#14519705—42no—
50#14519700—22no—
51#14519697—22no—
52#14519691—42no—
53#14519691—42no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57C22⋊#92418C2×C412yescomplemented
58#14519684—21no—
59#14519684—21no—
60C22×D4×C61—21no—
61C22×D4×C61D411yescharacteristic
62C23×C4×C61—21no—
63C23×C4×C61C2311yescharacteristic
64#121×C61—21no—
65#121×C61—21no—
66#121×C61C2×C412yes—
67#121×C61D412yes—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C22⋊C4×C61—21no—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61D411yescharacteristic
76C2×C22⋊C4×C61D411yescharacteristic
77C2×C42×C61—21no—
78C2×C42×C61D411yescharacteristic
79C2×C42×C61D411yescharacteristic
80#95×C61—21no—
81#95×C61—21no—
82#95×C61—21no—
83#95×C61—21no—
84#95×C61—21no—
85C24×C61—21no—
86C24×C61C2×D411yescharacteristic
87C23×D61—21no—
88C2×D4×C61—22no—
89C2×D4×C61—22no—
90C2×D4×C61—21no—
91C2×D4×C61—41no—
92C2×D4×C61—41no—
93C2×D4×C61—81no—
94C2×D4×C61—81no—
95C2×D4×C61C22⋊C412yes—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61—42no—
104C22×C4×C61—81no—
105C22×C4×C61—81no—
106C22×C4×C61C22×C411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×C22⋊D61—22no—
113C2×C22⋊D61—22no—
114C2×C22⋊D61—41no—
115C2×C22⋊D61—41no—
116C2×C22⋊D61—41no—
117C2×C22⋊D61—41no—
118C22×Dic61—21no—
119C22×Dic61—41no—
120#11758222—22no—
121#11758222—22no—
122#11758222—22no—
123#11758222—22no—
124#11758222—42no—
125#11758222—42no—
126#11758222—42no—
127#11758222—42no—
128#11758222—41no—
129#11758222—41no—
130#11758222—41no—
131#11758222—41no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—42no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×C4×D61—41no—
138C2×C4×D61—41no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—42no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—81no—
144C4⋊C4×C61C22⋊C412yes—
145C22⋊C4×C61—22no—
146C22⋊C4×C61—22no—
147C22⋊C4×C61—42no—
148C22⋊C4×C61—42no—
149C22⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C22⋊C4×C61—81no—
152C22⋊C4×C61—81no—
153C42×C61—41no—
154C42×C61—41no—
155C42×C61—81no—
156C22⋊Dic61—41no—
157C22⋊Dic61—41no—
158#11758200—44no—
159#11758199—44no—
160#11758194—41no—
161#11758194—41no—
162#11758194—41no—
163#11758194—41no—
164C2×#92418—22no—
165C2×#92418—41no—
166C23×C61—21no—
167C23×C61—21no—
168C23×C61—22no—
169C23×C61—41no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61C4×D411yescharacteristic
178C23×C61#12011yescharacteristic
179C23×C61#12111yescharacteristic
180C22×D61—22no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—81no—
184C22×D61—81no—
185D4×C61—42no—
186D4×C61—81no—
187D4×C61—82no—
188D4×C61—82no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—84no—
213C2×C4×C61C2×C22⋊C411yescharacteristic
214C2×C4×C61C4×D411yescharacteristic
215C2×C4×C61#12111yescharacteristic
216C2×C4×C61#12311yescharacteristic
217C22⋊D61—42no—
218C22⋊D61—42no—
219C22⋊D61—42no—
220C22⋊D61—42no—
221C2×Dic61—22no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—81no—
225C2×Dic61—81no—
226C2×Dic61—81no—
227D244—22no—
228D244—22no—
229D244—41no—
230D244—41no—
231C4×D61—42no—
232C4×D61—42no—
233C4×D61—42no—
234C4×D61—42no—
235Dic122—22no—
236Dic122—22no—
237Dic122—41no—
238Dic122—41no—
239#92418—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—81no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61—81no—
258C22×C61—81no—
259C22×C61#38611yescharacteristic
260C22×C61B(4,𝔽2)11yescharacteristic
261C22×C61B(4,𝔽2)11yescharacteristic
262C2×D61—42no—
263C2×D61—82no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—82no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61—81no—
276Dic61—42no—
277Dic61—161no—
278#3006—611no—
279#2333D6111yescharacteristic, complemented
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#188911yescharacteristic
291C2×C61#190411yescharacteristic
292C2×C61#190611yescharacteristic
293D61—82no—
294C22×#41—611no—
295C2×AGL(1,ℤ/8)—1221no—
296C2×#121C2×D6111yescharacteristic
297#449—1222no—
298#448—612no—
299#447—612no—
300C2×#104—1221no—
301C2×#104—1221no—
302#417—1221no—
303C2×#99—1221no—
304C2×#99—1221no—
305#407—611no—
306#407—611no—
307#390—21no—
308#388—21no—
309#387—21no—
310#386—21no—
311#386—21no—
312#381—611no—
313#381C2×D6111yescharacteristic
314C4×C22⋊C4—21no—
315C4×C22⋊C4C2×D6111yescharacteristic
316#327—1222no—
317C61#952311yescharacteristic, complemented, Sylow
318C2×#41—1221no—
319C2×#41—1221no—
320C2×#41—1221no—
321C2×#41—1221no—
322C2×#41—2442no—
323C2×#41—2442no—
324C22×Q8—611no—
325C22×D4—21no—
326C22×D4—611no—
327C22×D4—1221no—
328C22×D4C22⋊D6111yescharacteristic
329C23×C4—21no—
330C23×C4—1221no—
331C23×C4C22×D6111yescharacteristic
332AGL(1,ℤ/8)—2442no—
333AGL(1,ℤ/8)—2442no—
334C2×SD16—1222no—
335C2×D8—1222no—
336C2×M16—1221no—
337C2×M16—1221no—
338#121—21no—
339#121—21no—
340#121C4×D6112yes—
341#121D24412yes—
342C2×C4⋊C4—21no—
343C2×C4⋊C4C22⋊D6111yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4C22⋊D6111yescharacteristic
351C2×C22⋊C4C22⋊D6111yescharacteristic
352C2×C42—21no—
353C2×C42C22⋊D6111yescharacteristic
354C2×C42C22⋊D6111yescharacteristic
355#104—2442no—
356#104—2442no—
357#103—1222no—
358#102—1222no—
359#99—2442no—
360#99—2442no—
361C22⋊C8—612no—
362C22⋊C8—1221no—
363C22⋊C8—1221no—
364C22⋊C8—1221no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—1221no—
371#95—1221no—
372C24—21no—
373C24—1221no—
374C24C2×C22⋊D6111yescharacteristic
375#41—1222no—
376#41—1222no—
377#41—1222no—
378#41—1222no—
379#41—2441no—
380#41—2441no—
381#41—2441no—
382#41—2441no—
383#41—2442no—
384#41—2442no—
385#41—2442no—
386#41—2442no—
387C2×Q8—1221no—
388C2×Q8—1221no—
389C2×Q8—2442no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—81no—
396C2×D4—81no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—1221no—
400C2×D4—1221no—
401C2×D4—2442no—
402C2×D4—2441no—
403C2×D4—2441no—
404C2×D4—2441no—
405C2×D4—2441no—
406C2×D4#1175819812yes—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—1221no—
418C22×C4—2442no—
419C22×C4—2442no—
420C22×C4—2441no—
421C22×C4—2441no—
422C22×C4—2441no—
423C22×C4C2×C4×D6111yescharacteristic
424C22×C4C2×D24411yescharacteristic
425C22×C4#1175822211yescharacteristic
426C22×C4#1175822211yescharacteristic
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6111yescharacteristic
429SD16—2444no—
430D8—2444no—
431M16—2441no—
432M16—2441no—
433M16—2441no—
434M16—2441no—
435C2×C8—1222no—
436C2×C8—2441no—
437C4⋊C4—21no—
438C4⋊C4—81no—
439C4⋊C4#1175819812yes—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—81no—
447C22⋊C4—81no—
448C22⋊C4—2441no—
449C22⋊C4—2441no—
450C42—41no—
451C42—41no—
452C42—81no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—1222no—
465C23—4881no—
466C23—4881no—
467C23—2441no—
468C23—2441no—
469C23C4×C22⋊D6111yescharacteristic
470C23#1451974611yescharacteristic
471C23#1451975111yescharacteristic
472Q8—1222no—
473Q8—1222no—
474Q8—2441no—
475Q8—2441no—
476D4—42no—
477D4—81no—
478D4—82no—
479D4—82no—
480D4—1222no—
481D4—1222no—
482D4—2441no—
483D4—2441no—
484D4—2442no—
485D4—2442no—
486D4—2442no—
487D4—2442no—
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—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4—1222no—
513C2×C4—4881no—
514C2×C4—4881no—
515C2×C4—4881no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4C2×#1175819811yescharacteristic
523C2×C4C4×C22⋊D6111yescharacteristic
524C2×C4#1451975111yescharacteristic
525C2×C4#1451976011yescharacteristic
526C8—4881no—
527C8—2442no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—4882no—
546C22—4882no—
547C22—4881no—
548C22—4881no—
549C22—2442no—
550C22#1683964111yescharacteristic
551C22#1683994911yescharacteristic
552C22#1683994911yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—42no—
558C4—42no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—9761no—
563C4—2442no—
564C2—21no—
565C2—21no—
566C2—41no—
567C2—41no—
568C2—82no—
569C2—4882no—
570C2#2144622711yescharacteristic
571C2#2144630611yescharacteristic
572C2#2144632011yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2333C2
socleC22×C61#386
prime radical#21455069C2
cosocleC23×C4×C61C23
hypercenter#2333D61
hypocenterC61#9523
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2333

Complements

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

NH ≅ G/Ntype
2. #21445764C2semidirect
3. #21445736C2semidirect
4. #21444348C2semidirect
5. #21443727C2semidirect
6. #21443672C2semidirect
7. C61×#2333C2semidirect
11. #16839900C4semidirect
12. #16839895C4semidirect
37. C22×Dic122C2×C4semidirect
38. C22×D244C2×C4semidirect
57. C22⋊#92418C2×C4semidirect
279. #2333D61semidirect
317. C61#9523semidirect

Maximal subgroups

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

subgroupindex
#214550692
#214457642
#214457362
#214443482
#214437272
#214436722
C61×#23332
#952361