Finite Group Databasebeta

← #29731016

Subgroups — #29731016

Order 15616.

916 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840027C212yesmaximal, complemented
2#21455279C211yesmaximal, characteristic
3C2×#16839964C211yesmaximal, characteristic, complemented
4#21454563C211yesmaximal, characteristic, complemented
5#21447115C212yesmaximal, complemented
6#21446930C212yesmaximal, complemented
7#21446664C211yesmaximal, characteristic, complemented
8#21446594C211yesmaximal, characteristic, complemented
9#21446336C212yesmaximal, complemented
10C4×#14519693C211yesmaximal, characteristic, complemented
11C61×#3152C211yesmaximal, characteristic, complemented
12C2×#14519802C2212yes—
13C22×#11758201C2212yes—
14C2×#14519784C2212yes—
15C2×C4×D244C2211yescharacteristic
16#16840027—28no—
17#16839974—22no—
18#16839964—21no—
19#16839964—22no—
20#16839964—21no—
21#16839964—21no—
22#16839954—21no—
23#16839840—21no—
24#16839834—22no—
25D4×#92418—21no—
26C2×#14519712C2212yes—
27#16839786C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29#16839763C2212yes—
30C2×#14519701C2212yes—
31C2×#14519696C2212yes—
32C2×#14519696C2212yes—
33C2×#14519693C2211yescharacteristic
34C2×#14519693C2211yescharacteristic
35#16839729C2212yes—
36C4×#11758194C2211yescharacteristic
37#16839530C2211yescharacteristic
38#16839530C2211yescharacteristic
39#16839526C2212yes—
40C2×#137×C61C2212yes—
41C2×#137×C61C2212yes—
42C61×#532C2211yescharacteristic
43C2×C4⋊Q8×C61C2211yescharacteristic
44C61×#497—22no—
45C61×#494—21no—
46C61×#492—21no—
47C4×M16×C61C2211yescharacteristic
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50#14519802—24no—
51#14519802—42no—
52C2×#11758201—22no—
53C2×#11758201—24no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758201—42no—
57C2×#11758199—22no—
58#14519784—24no—
59#14519784—42no—
60C2×#11758194—22no—
61C2×#11758194C2311yescharacteristic
62C2×#11758194C2312yes—
63C2×#11758194C2311yescharacteristic
64C22×#92418—22no—
65C22×#92418C2312yes—
66C4×D244—21no—
67C4×D244—21no—
68C4×D244—21no—
69C4×D244—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758198—21no—
72#14519712—24no—
73C4⋊#92418—21no—
74C4⋊#92418—21no—
75C4⋊#92418D411yescharacteristic
76C4⋊#92418D412yes—
77C4⋊#92418D411yescharacteristic
78#14519701—22no—
79#14519701—22no—
80#14519701—22no—
81#14519698—22no—
82#14519696—24no—
83#14519696—42no—
84#14519693—22no—
85#14519693D414yescomplemented
86C22⋊#92418—22no—
87C22⋊#92418—22no—
88#14519688—21no—
89C4×#92418—21no—
90C2×#41×C61C2312yes—
91C22×Q8×C61C2312yes—
92#137×C61—24no—
93#137×C61—24no—
94#137×C61—42no—
95#137×C61—42no—
96C2×Q16×C61—22no—
97C2×Q16×C61—22no—
98C2×SD16×C61—22no—
99C2×SD16×C61—22no—
100C2×M16×C61C2311yescharacteristic
101C2×M16×C61C2311yescharacteristic
102C4⋊Q8×C61—21no—
103C4⋊Q8×C61—21no—
104C4⋊Q8×C61—21no—
105C4⋊Q8×C61—21no—
106C4⋊Q8×C61D411yescharacteristic
107#127×C61D411yescharacteristic
108#124×C61D412yes—
109#121×C61—21no—
110#121×C61—21no—
111C4×D4×C61—21no—
112C4×D4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C42×C61C2311yescharacteristic
116#97×C61—21no—
117C4×C8×C61—21no—
118C23×D61—21no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—22no—
122#41×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61C2×D412yes—
130C2×D4×C61—22no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61C2×D412yes—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61#4111yescharacteristic
137C22×C4×C61#4111yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244C2×D412yes—
146C2×C4×D61—41no—
147C2×C4×D61—41no—
148C2×C4×D61—82no—
149Q16×C61—44no—
150Q16×C61—44no—
151SD16×C61—44no—
152SD16×C61—44no—
153M16×C61—21no—
154M16×C61—21no—
155M16×C61C2×D412yes—
156M16×C61#4112yes—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—41no—
162C4⋊C4×C61—41no—
163C4⋊C4×C61—41no—
164C4⋊C4×C61—41no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C42×C61C2×D411yescharacteristic
168C42×C61C2×D411yescharacteristic
169C42×C61C2×D411yescharacteristic
170C42×C61C2×D411yescharacteristic
171#11758201—44no—
172#11758201—44no—
173#11758201—44no—
174#11758199—44no—
175#11758198—41no—
176#11758198—41no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179#11758194—21no—
180#11758194—22no—
181#11758194—21no—
182#11758194—21no—
183#11758194—22no—
184#11758194—44no—
185#11758194C2×D412yes—
186C2×#92418—21no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—21no—
190C2×#92418—22no—
191C2×#92418—44no—
192C2×#92418C2×D412yes—
193C2×#92418C2×D412yes—
194C23×C61—22no—
195C23×C61C2×#4111yescharacteristic
196C22×D61—22no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—81no—
200C22×D61—81no—
201Q8×C61—22no—
202Q8×C61—22no—
203Q8×C61—42no—
204Q8×C61—42no—
205Q8×C61—42no—
206Q8×C61—42no—
207D4×C61—22no—
208D4×C61—22no—
209D4×C61—42no—
210D4×C61—42no—
211D4×C61—42no—
212D4×C61—81no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61#12111yescharacteristic
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×Dic61—41no—
232C2×Dic61—81no—
233D244—22no—
234D244—22no—
235D244—41no—
236D244—41no—
237C4×D61—82no—
238C4×D61—82no—
239C8×C61—22no—
240C8×C61—22no—
241#92418—22no—
242#92418—22no—
243#92418—22no—
244#92418—42no—
245#92418—41no—
246#25405—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61C2×#12111yescharacteristic
253C22×C61#53211yescharacteristic
254C22×C61#54011yescharacteristic
255C2×D61—42no—
256C2×D61—82no—
257C2×D61—81no—
258C2×D61—81no—
259C2×D61—82no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—42no—
264C4×C61—82no—
265C4×C61#47112yescomplemented
266C4×C61D4211yescharacteristic
267C4×C61D4211yescharacteristic
268C4×C61#54611yescharacteristic
269C4×C61#55011yescharacteristic
270Dic61—81no—
271#3152D6111yescharacteristic, complemented
272C2×#471—612no—
273#3036—611no—
274C2×#461—611no—
275#2935—611no—
276#2052—612no—
277#2019—612no—
278#1966—611no—
279#1950—611no—
280#1908—612no—
281C4×#100—611no—
282C2×C61—21no—
283C2×C61—21no—
284C2×C61—82no—
285C2×C61#242811yescharacteristic
286C2×C61C2×#47111yescharacteristic
287C2×C61#306911yescharacteristic
288D61—82no—
289C2×#137C2×D6112yes—
290C2×#137C2×D6112yes—
291C2×AGL(1,ℤ/8)—612no—
292C22×SD16—612no—
293C2×#131—612no—
294#532C2×D6111yescharacteristic
295C2×C4⋊Q8C2×D6111yescharacteristic
296C2×C4×D4—611no—
297#497—22no—
298#494—21no—
299#492—21no—
300#471—1228no—
301#463—1222no—
302#461—1221no—
303#461—1222no—
304#461—1221no—
305#461—1221no—
306#459—1221no—
307#436—1221no—
308#435—1222no—
309C8×D4—1221no—
310C2×#108—612no—
311#423—611no—
312C2×C4⋊C8—611no—
313#417—612no—
314C2×#103—612no—
315C2×#101—612no—
316C2×#101—612no—
317C2×#100—611no—
318C2×#100—611no—
319#408—612no—
320C4×M16—611no—
321C4×M16C2×D6111yescharacteristic
322#343—611no—
323#343—611no—
324#341—612no—
325C61#2540511yescharacteristic, complemented, Sylow
326C2×#41C22×D6112yes—
327C22×Q8C22×D6112yes—
328C22×D4—611no—
329C23×C4—1221no—
330#137—24no—
331#137—24no—
332#137—42no—
333#137—42no—
334AGL(1,ℤ/8)—1224no—
335AGL(1,ℤ/8)—2442no—
336C2×Q16—22no—
337C2×Q16—22no—
338C2×SD16—22no—
339C2×SD16—22no—
340C2×SD16—1222no—
341C2×SD16—1222no—
342C2×SD16—1224no—
343C2×SD16—1222no—
344C2×SD16—2442no—
345C2×D8—1222no—
346#131—1224no—
347#131—2442no—
348C2×M16—611no—
349C2×M16—612no—
350C2×M16—611no—
351C2×M16—1222no—
352C2×M16C22×D6111yescharacteristic
353C2×M16C22×D6111yescharacteristic
354C22×C8—612no—
355C22×C8—1222no—
356C4⋊Q8—21no—
357C4⋊Q8—21no—
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8C22⋊D6111yescharacteristic
361#127C22⋊D6111yescharacteristic
362#124C22⋊D6112yes—
363#121—21no—
364#121—21no—
365C4×D4—21no—
366C4×D4—21no—
367C4×D4—1221no—
368C4×D4—1221no—
369C4×D4—1221no—
370C4×D4—1221no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—611no—
374C2×C22⋊C4—1221no—
375C2×C42C22×D6111yescharacteristic
376#108—1224no—
377C4⋊C8—611no—
378C4⋊C8—612no—
379C4⋊C8—611no—
380C4⋊C8—1221no—
381C4⋊C8—1221no—
382#103—1222no—
383#103—1222no—
384#103—1222no—
385#102—1222no—
386#101—1224no—
387#101—2442no—
388#100—614no—
389#100—1222no—
390C22⋊C8—1222no—
391C22⋊C8—1222no—
392#97—21no—
393#97—1221no—
394C4×C8—21no—
395C4×C8—1221no—
396C24—1221no—
397#41—22no—
398#41—22no—
399#41—22no—
400#41—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8C2×C22⋊D6112yes—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—612no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—1221no—
415C2×D4—1221no—
416C2×D4—1221no—
417C2×D4C2×C22⋊D6112yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—1221no—
421C22×C4—4882no—
422C22×C4—2441no—
423C22×C4—2441no—
424C22×C4#1175822411yescharacteristic
425C22×C4#1175822411yescharacteristic
426C22×C4C23×D6111yescharacteristic
427Q16—44no—
428Q16—44no—
429SD16—44no—
430SD16—44no—
431SD16—2444no—
432SD16—2444no—
433SD16—2444no—
434D8—2444no—
435M16—21no—
436M16—21no—
437M16—612no—
438M16—1221no—
439M16—1222no—
440M16—1221no—
441M16—1221no—
442M16—1222no—
443M16—2444no—
444M16D4×D6112yes—
445M16#1175822412yes—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—612no—
449C2×C8—612no—
450C2×C8—1221no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1221no—
454C2×C8—1222no—
455C2×C8—2444no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—2441no—
467C22⋊C4—2441no—
468C42D4×D6111yescharacteristic
469C42D4×D6111yescharacteristic
470C42C2×C22⋊D6111yescharacteristic
471C42C2×C22⋊D6111yescharacteristic
472C23—22no—
473C23—1222no—
474C23—4881no—
475C23—4881no—
476C23—2441no—
477C23—2441no—
478C23C2×#1175822411yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484Q8—42no—
485D4—22no—
486D4—22no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—81no—
491D4—1222no—
492D4—1222no—
493D4—2441no—
494D4—2441no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—4882no—
504C2×C4—4882no—
505C2×C4—4881no—
506C2×C4—2441no—
507C2×C4#1451974911yescharacteristic
508C2×C4#1451974911yescharacteristic
509C2×C4#1451974911yescharacteristic
510C2×C4#1451974911yescharacteristic
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C2×D4×D6111yescharacteristic
513C2×C4C22×C22⋊D6111yescharacteristic
514C2×C4C2×#1175822411yescharacteristic
515C2×C4#1451982211yescharacteristic
516C2×C4#1451982211yescharacteristic
517C8—22no—
518C8—22no—
519C8—1222no—
520C8—1222no—
521C8—1222no—
522C8—2442no—
523C8—2441no—
524C22—21no—
525C22—21no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—4882no—
530C22—4882no—
531C22—4881no—
532C22—4881no—
533C22—2442no—
534C22C2×#1451974911yescharacteristic
535C22#1684031011yescharacteristic
536C22#1684036611yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—42no—
541C4—82no—
542C4—4881no—
543C4#1684002712yescomplemented
544C4D4×C22⋊D6111yescharacteristic
545C4#1684039711yescharacteristic
546C4#1684040611yescharacteristic
547C4#1684043211yescharacteristic
548C2—21no—
549C2—21no—
550C2—82no—
551C2—4882no—
552C2#2144941011yescharacteristic
553C2C2×#1684002711yescharacteristic
554C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3152C2
socleC22×C61C2×#121
prime radical#21455279C2
hypercenter#3152D61
hypocenterC61#25405
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C2×#16840027C2semidirect
3. C2×#16839964C2semidirect
4. #21454563C2semidirect
5. #21447115C2semidirect
6. #21446930C2semidirect
7. #21446664C2semidirect
8. #21446594C2semidirect
9. #21446336C2semidirect
10. C4×#14519693C2semidirect
11. C61×#3152C2semidirect
85. #14519693D4semidirect
265. C4×C61#471semidirect
271. #3152D61semidirect
325. C61#25405semidirect
543. C4#16840027semidirect

Maximal subgroups

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

subgroupindex
C2×#168400272
#214552792
C2×#168399642
#214545632
#214471152
#214469302
#214466642
#214465942
#214463362
C4×#145196932
C61×#31522
#2540561