Finite Group Databasebeta

← #29731007

Subgroups — #29731007

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
1#21456279C211yesmaximal, characteristic, complemented
2C2×#16840025C212yesmaximal, complemented
3#21455279C211yesmaximal, characteristic
4C2×#16839968C211yesmaximal, characteristic, complemented
5#21447117C212yesmaximal, complemented
6#21446934C212yesmaximal, complemented
7#21446663C211yesmaximal, characteristic, complemented
8#21446595C211yesmaximal, characteristic, complemented
9#21446334C212yesmaximal, complemented
10C4×#14519693C211yesmaximal, characteristic, complemented
11C61×#2935C211yesmaximal, characteristic, complemented
12C2×#14519807C2212yes—
13C2×#14519807C2212yes—
14C2×#14519802C2212yes—
15C22×#11758190C2212yes—
16C2×#14519777C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18#16840157—22no—
19#16840141—21no—
20#16840134—21no—
21#16840025—28no—
22#16839974—22no—
23#16839968—21no—
24#16839968—22no—
25#16839968—21no—
26#16839964—21no—
27#16839954—21no—
28C2×#14519713C2212yes—
29#16839786C2211yescharacteristic
30#16839763C2212yes—
31C2×#14519702C2212yes—
32C2×#14519695C2212yes—
33C2×#14519695C2212yes—
34C2×#14519693C2211yescharacteristic
35C2×#14519693C2211yescharacteristic
36C4×#11758194C2211yescharacteristic
37#16839530C2211yescharacteristic
38#16839530C2211yescharacteristic
39#16839527C2212yes—
40C2×#131×C61C2212yes—
41C61×#532C2211yescharacteristic
42C61×#436—21no—
43C61×#435—22no—
44C8×D4×C61—21no—
45C2×C4⋊C8×C61C2211yescharacteristic
46C61×#408C2212yes—
47C4×M16×C61C2211yescharacteristic
48C22×Dic122C2312yes—
49C22×D244C2311yescharacteristic
50C22×C4×D61—21no—
51#14519807—24no—
52#14519807—24no—
53#14519807—42no—
54#14519807—42no—
55#14519802—24no—
56#14519802—42no—
57C2×#11758202—22no—
58C2×#11758202—22no—
59C2×#11758190—24no—
60C2×#11758190—22no—
61C2×#11758190—22no—
62C2×#11758190—42no—
63C2×#11758200—22no—
64C2×#11758200—22no—
65C2×#11758201—22no—
66C2×#11758199—22no—
67C2×#11758194C2312yes—
68C2×#11758194C2311yescharacteristic
69C2×#11758194C2311yescharacteristic
70#14519777—21no—
71#14519777—21no—
72#14519777—21no—
73#14519777—21no—
74C4×D244—21no—
75C4×D244—21no—
76C4×D244—21no—
77C4×D244—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758198—21no—
82#14519713—24no—
83C4⋊#92418—21no—
84C4⋊#92418—21no—
85#14519702—22no—
86#14519702—22no—
87#14519701—22no—
88#14519698—22no—
89#14519695—24no—
90#14519695—42no—
91#14519693—22no—
92#14519693D414yescomplemented
93#14519688—21no—
94C4×#92418—21no—
95C2×#41×C61C2312yes—
96#131×C61—24no—
97#131×C61—42no—
98C2×M16×C61—22no—
99C2×M16×C61C2311yescharacteristic
100C2×M16×C61C2311yescharacteristic
101C22×C8×C61—22no—
102C22×C8×C61C2312yes—
103C4⋊Q8×C61D411yescharacteristic
104#127×C61D411yescharacteristic
105#124×C61D412yes—
106#121×C61—21no—
107#121×C61—21no—
108C4×D4×C61—21no—
109C4×D4×C61—21no—
110C2×C42×C61C2311yescharacteristic
111C4⋊C8×C61D411yescharacteristic
112C4⋊C8×C61D412yes—
113C4⋊C8×C61D411yescharacteristic
114C22⋊C8×C61—22no—
115C22⋊C8×C61—22no—
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×C61C2×D412yes—
124C2×D4×C61—22no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—22no—
129C22×C4×C61#4111yescharacteristic
130C22×C4×C61#4111yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—22no—
133C22×Dic61—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244C2×D412yes—
140C2×C4×D61—41no—
141C2×C4×D61—41no—
142C2×C4×D61—82no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149M16×C61—21no—
150M16×C61—21no—
151M16×C61—21no—
152M16×C61—44no—
153M16×C61C2×D412yes—
154C2×C8×C61—21no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—21no—
158C2×C8×C61—44no—
159C2×C8×C61C2×D412yes—
160C2×C8×C61C2×D412yes—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2×D411yescharacteristic
168C42×C61C2×D411yescharacteristic
169#11758202—44no—
170#11758202—44no—
171#11758201—44no—
172#11758200—44no—
173#11758200—44no—
174#11758199—44no—
175#11758198—41no—
176#11758198—41no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—41no—
180C4⋊Dic61—41no—
181C4⋊Dic61—41no—
182C4⋊Dic61—41no—
183#11758194—21no—
184#11758194—21no—
185#11758194—22no—
186#11758194—22no—
187#11758194C2×D412yes—
188#11758194#4112yes—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—22no—
192#11758190—44no—
193#11758190—44no—
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—
203D4×C61—22no—
204D4×C61—22no—
205D4×C61—42no—
206D4×C61—42no—
207D4×C61—42no—
208D4×C61—81no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61#12111yescharacteristic
215C2×C4×C61#12111yescharacteristic
216C2×C4×C61#12111yescharacteristic
217C2×C4×C61#12111yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C2×C4×C61H(5,𝔽2)11yescharacteristic
224C2×Dic61—41no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—81no—
229D244—22no—
230D244—22no—
231D244—41no—
232D244—41no—
233C4×D61—82no—
234C4×D61—82no—
235Dic122—42no—
236Dic122—42no—
237Dic122—42no—
238Dic122—42no—
239C8×C61—22no—
240C8×C61—22no—
241C8×C61—22no—
242C8×C61—41no—
243#92418—22no—
244#92418—22no—
245#92418—42no—
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#47112yescomplemented
265C4×C61D4211yescharacteristic
266C4×C61D4211yescharacteristic
267C4×C61#54611yescharacteristic
268C4×C61#55011yescharacteristic
269Dic61—82no—
270Dic61—81no—
271#3152—611no—
272C2×#471—612no—
273#3036—611no—
274C2×#461—611no—
275#2935D6111yescharacteristic, complemented
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×#137—612no—
290C2×#137—612no—
291C2×AGL(1,ℤ/8)—612no—
292C22×SD16—612no—
293C2×#131C2×D6112yes—
294#532C2×D6111yescharacteristic
295C2×C4⋊Q8—611no—
296C2×C4×D4—611no—
297#497—1222no—
298#494—1221no—
299#492—1221no—
300#471—1228no—
301#463—1222no—
302#461—1221no—
303#461—1221no—
304#461—1222no—
305#461—1221no—
306#459—1221no—
307#436—21no—
308#435—22no—
309C8×D4—21no—
310C2×#108—612no—
311#423—611no—
312C2×C4⋊C8C2×D6111yescharacteristic
313#417—612no—
314C2×#103—612no—
315C2×#101—612no—
316C2×#101—612no—
317C2×#100—611no—
318C2×#100—611no—
319#408C2×D6112yes—
320C4×M16—611no—
321C4×M16C2×D6111yescharacteristic
322#343—611no—
323#343—611no—
324#341—612no—
325C61#2540511yescharacteristic, complemented, Sylow
326C2×#41C22×D6112yes—
327C22×Q8—612no—
328C22×D4—611no—
329C23×C4—1221no—
330#137—1224no—
331#137—1224no—
332#137—2442no—
333#137—2442no—
334AGL(1,ℤ/8)—1224no—
335AGL(1,ℤ/8)—2442no—
336C2×Q16—1222no—
337C2×Q16—1222no—
338C2×SD16—1224no—
339C2×SD16—1222no—
340C2×SD16—1222no—
341C2×SD16—1222no—
342C2×SD16—1222no—
343C2×SD16—1222no—
344C2×SD16—2442no—
345C2×D8—1222no—
346#131—24no—
347#131—42no—
348C2×M16—22no—
349C2×M16—611no—
350C2×M16—611no—
351C2×M16—612no—
352C2×M16C22×D6111yescharacteristic
353C2×M16C22×D6111yescharacteristic
354C22×C8—22no—
355C22×C8C22×D6112yes—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8—1221no—
360C4⋊Q8D24411yescharacteristic
361#127D24411yescharacteristic
362#124D24412yes—
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—611no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C22⋊C4—1221no—
375C2×C42C22×D6111yescharacteristic
376#108—1224no—
377C4⋊C8—1221no—
378C4⋊C8—1221no—
379C4⋊C8C22⋊D6111yescharacteristic
380C4⋊C8C22⋊D6112yes—
381C4⋊C8C22⋊D6111yescharacteristic
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—22no—
391C22⋊C8—22no—
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—1222no—
402C2×Q8—1222no—
403C2×Q8—1222no—
404C2×Q8—1222no—
405C2×Q8—1222no—
406C2×Q8—1222no—
407C2×Q8C2×D24412yes—
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×D24412yes—
418C22×C4—22no—
419C22×C4—1221no—
420C22×C4—1222no—
421C22×C4—4882no—
422C22×C4—2441no—
423C22×C4—2441no—
424C22×C4#1175822211yescharacteristic
425C22×C4#1175822211yescharacteristic
426C22×C4C23×D6111yescharacteristic
427Q16—2444no—
428Q16—2444no—
429SD16—2444no—
430SD16—2444no—
431SD16—2444no—
432SD16—2444no—
433SD16—2444no—
434D8—2444no—
435M16—21no—
436M16—21no—
437M16—21no—
438M16—44no—
439M16—612no—
440M16—612no—
441M16—1221no—
442M16—1221no—
443M16—1222no—
444M16—1222no—
445M16D4×D6112yes—
446C2×C8—22no—
447C2×C8—21no—
448C2×C8—22no—
449C2×C8—21no—
450C2×C8—44no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1222no—
454C2×C8C2×C22⋊D6112yes—
455C2×C8C2×C22⋊D6112yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—2441no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—2441no—
467C22⋊C4—2441no—
468C42C2×D24411yescharacteristic
469C42C2×D24411yescharacteristic
470C42C2×C22⋊D6111yescharacteristic
471C42C2×C22⋊D6111yescharacteristic
472C23—22no—
473C23—1222no—
474C23—4881no—
475C23—4881no—
476C23—2441no—
477C23—2441no—
478C23C2×#1175822211yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—2442no—
482Q8—2442no—
483Q8—2442no—
484Q8—2442no—
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—4882no—
501C2×C4—4882no—
502C2×C4—4881no—
503C2×C4—2441no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4#1451975111yescharacteristic
508C2×C4#1451975111yescharacteristic
509C2×C4#1451975111yescharacteristic
510C2×C4#1451975111yescharacteristic
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C22×C22⋊D6111yescharacteristic
513C2×C4C22×D24411yescharacteristic
514C2×C4#41×D6111yescharacteristic
515C2×C4#1451982211yescharacteristic
516C2×C4#1451982311yescharacteristic
517C8—22no—
518C8—22no—
519C8—22no—
520C8—41no—
521C8—1222no—
522C8—1222no—
523C8—2442no—
524C22—21no—
525C22—21no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—4882no—
530C22—4882no—
531C22—4881no—
532C22—4881no—
533C22—2442no—
534C22C2×#1451975111yescharacteristic
535C22#1684030911yescharacteristic
536C22#1684036311yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—42no—
541C4—4881no—
542C4—4882no—
543C4#1684002512yescomplemented
544C4D4×C22⋊D6111yescharacteristic
545C4D4×D24411yescharacteristic
546C4#1684040711yescharacteristic
547C4#1684043111yescharacteristic
548C2—21no—
549C2—21no—
550C2—82no—
551C2—4882no—
552C2#2144940111yescharacteristic
553C2C2×#1684002511yescharacteristic
554C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2935C2
socleC22×C61C2×#121
prime radical#21455279C2
hypercenter#2935D61
hypocenterC61#25405
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
1. #21456279C2semidirect
2. C2×#16840025C2semidirect
4. C2×#16839968C2semidirect
5. #21447117C2semidirect
6. #21446934C2semidirect
7. #21446663C2semidirect
8. #21446595C2semidirect
9. #21446334C2semidirect
10. C4×#14519693C2semidirect
11. C61×#2935C2semidirect
92. #14519693D4semidirect
264. C4×C61#471semidirect
275. #2935D61semidirect
325. C61#25405semidirect
543. C4#16840025semidirect

Maximal subgroups

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

subgroupindex
#214562792
C2×#168400252
#214552792
C2×#168399682
#214471172
#214469342
#214466632
#214465952
#214463342
C4×#145196932
C61×#29352
#2540561