Finite Group Databasebeta

← #29731015

Subgroups — #29731015

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×#16840023C212yesmaximal, complemented
3C2×#16839967C211yesmaximal, characteristic, complemented
4#21454563C211yesmaximal, characteristic, complemented
5#21447116C212yesmaximal, complemented
6#21446933C212yesmaximal, complemented
7#21446662C211yesmaximal, characteristic, complemented
8#21446592C211yesmaximal, characteristic, complemented
9#21446332C212yesmaximal, complemented
10C4×#14519694C211yesmaximal, characteristic, complemented
11C61×#3036C211yesmaximal, characteristic
12C2×#14519807C2212yes—
13C2×#14519807C2212yes—
14C22×#11758200C2212yes—
15C2×#14519784C2212yes—
16C2×#14519777C2211yescharacteristic
17#16840157—22no—
18#16840141—21no—
19#16840134—21no—
20#16840023—28no—
21#16839967—21no—
22#16839967—22no—
23#16839967—21no—
24#16839840—21no—
25#16839834—22no—
26D4×#92418—21no—
27C2×#14519712C2212yes—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519700C2212yes—
30C2×#14519695C2212yes—
31C2×#14519695C2212yes—
32C2×#14519694C2211yescharacteristic
33C2×#14519694C2211yescharacteristic
34#16839729C2212yes—
35C4×#11758194C2211yescharacteristic
36C4×#11758194C2211yescharacteristic
37#16839531C2211yescharacteristic
38#16839531C2211yescharacteristic
39#16839526C2212yes—
40C2×AGL(1,ℤ/8)×C61C2212yes—
41C61×#532C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#463—22no—
44C61×#461—21no—
45C61×#459—21no—
46C61×#423C2211yescharacteristic
47C61×#417C2212yes—
48C22×Dic122C2312yes—
49#14519807—24no—
50#14519807—24no—
51#14519807—42no—
52#14519807—42no—
53C2×#11758202—22no—
54C2×#11758202—22no—
55C2×#11758200—24no—
56C2×#11758200—22no—
57C2×#11758200—22no—
58C2×#11758200—22no—
59C2×#11758200—22no—
60C2×#11758200—42no—
61#14519784—24no—
62#14519784—42no—
63C2×#11758194—22no—
64C2×#11758194C2311yescharacteristic
65C2×#11758194C2311yescharacteristic
66C2×#11758194C2311yescharacteristic
67C2×#11758194C2311yescharacteristic
68C22×#92418—22no—
69C22×#92418C2312yes—
70#14519777—21no—
71#14519777—21no—
72#14519777—21no—
73#14519777—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76#14519712—24no—
77C4⋊#92418D411yescharacteristic
78C4⋊#92418D412yes—
79C4⋊#92418D411yescharacteristic
80#14519700—22no—
81#14519700—22no—
82#14519695—24no—
83#14519695—42no—
84#14519694—22no—
85#14519694D414yescomplemented
86C22⋊#92418—22no—
87C22⋊#92418—22no—
88#14519688—21no—
89#14519688—21no—
90C4×#92418—21no—
91C4×#92418—21no—
92C2×#41×C61C2312yes—
93C22×D4×C61C2311yescharacteristic
94C23×C4×C61—21no—
95AGL(1,ℤ/8)×C61—24no—
96AGL(1,ℤ/8)×C61—42no—
97C2×SD16×C61—22no—
98C2×D8×C61—22no—
99C2×M16×C61C2312yes—
100C4⋊Q8×C61D411yescharacteristic
101#127×C61D411yescharacteristic
102#124×C61D412yes—
103#121×C61—21no—
104#121×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61—21no—
108C4×D4×C61—21no—
109C4×D4×C61—21no—
110C4×D4×C61—21no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C42×C61C2311yescharacteristic
114C4⋊C8×C61—21no—
115C4⋊C8×C61—21no—
116#103×C61—22no—
117#102×C61—22no—
118C24×C61—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×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61C2×D412yes—
133C2×D4×C61C2×D412yes—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61—41no—
137C22×C4×C61—41no—
138C22×C4×C61—82no—
139C22×C4×C61#4111yescharacteristic
140C22×C4×C61#4111yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—22no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149SD16×C61—44no—
150D8×C61—44no—
151M16×C61—22no—
152M16×C61—22no—
153C2×C8×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—41no—
161C22⋊C4×C61—41no—
162C42×C61C2×D411yescharacteristic
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166#11758202—44no—
167#11758202—44no—
168#11758200—44no—
169#11758200—44no—
170#11758200—44no—
171#11758200—44no—
172C4⋊Dic61—41no—
173C4⋊Dic61—41no—
174C4⋊Dic61—41no—
175C4⋊Dic61—41no—
176#11758194—21no—
177#11758194—21no—
178#11758194—21no—
179#11758194—21no—
180#11758194—21no—
181#11758194—44no—
182#11758194C2×D412yes—
183#11758194C2×D412yes—
184#11758194#4112yes—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—21no—
191C2×#92418—44no—
192C2×#92418C2×D412yes—
193C2×#92418C2×D412yes—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—81no—
199C23×C61—81no—
200C23×C61C2×#4111yescharacteristic
201Q8×C61—22no—
202Q8×C61—22no—
203D4×C61—22no—
204D4×C61—22no—
205D4×C61—22no—
206D4×C61—22no—
207D4×C61—42no—
208D4×C61—42no—
209D4×C61—41no—
210D4×C61—41no—
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—41no—
219C2×C4×C61—82no—
220C2×C4×C61—82no—
221C2×C4×C61—81no—
222C2×C4×C61#12111yescharacteristic
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61#12111yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235Dic122—42no—
236Dic122—42no—
237Dic122—42no—
238Dic122—42no—
239C8×C61—42no—
240#92418—22no—
241#92418—22no—
242#92418—22no—
243#92418—22no—
244#92418—22no—
245#92418—41no—
246#25405—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—82no—
254C22×C61—82no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61C2×#12111yescharacteristic
258C22×C61#53211yescharacteristic
259C22×C61#54011yescharacteristic
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—42no—
264C4×C61—81no—
265C4×C61#47112yescomplemented
266C4×C61D4211yescharacteristic
267C4×C61D4211yescharacteristic
268C4×C61#54611yescharacteristic
269C4×C61#55011yescharacteristic
270Dic61—82no—
271#3152—611no—
272C2×#471—612no—
273#3036D6111yescharacteristic
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—82no—
286C2×C61#242811yescharacteristic
287C2×C61C2×#47111yescharacteristic
288C2×C61#306911yescharacteristic
289C2×#137—612no—
290C2×#137—612no—
291C2×AGL(1,ℤ/8)C2×D6112yes—
292C22×SD16—612no—
293C2×#131—612no—
294#532C2×D6111yescharacteristic
295C2×C4⋊Q8—611no—
296C2×C4×D4C2×D6111yescharacteristic
297#497—1222no—
298#494—1221no—
299#492—1221no—
300#471—1228no—
301#463—22no—
302#461—21no—
303#461—1221no—
304#461—1222no—
305#461—1221no—
306#459—21no—
307#436—1221no—
308#435—1222no—
309C8×D4—1221no—
310C2×#108—612no—
311#423C2×D6111yescharacteristic
312C2×C4⋊C8—611no—
313#417C2×D6112yes—
314C2×#103—612no—
315C2×#101—612no—
316C2×#101—612no—
317C2×#100—611no—
318C2×#100—611no—
319#408—612no—
320C4×M16—611no—
321C4×M16—611no—
322#343—611no—
323#343—611no—
324#341—612no—
325C61#2540511yescharacteristic, complemented, Sylow
326C2×#41C22×D6112yes—
327C22×Q8—612no—
328C22×D4C22×D6111yescharacteristic
329C23×C4—21no—
330#137—1224no—
331#137—1224no—
332#137—2442no—
333#137—2442no—
334AGL(1,ℤ/8)—24no—
335AGL(1,ℤ/8)—42no—
336C2×Q16—1222no—
337C2×Q16—1222no—
338C2×SD16—22no—
339C2×SD16—1222no—
340C2×SD16—1222no—
341C2×SD16—1224no—
342C2×SD16—1222no—
343C2×SD16—1222no—
344C2×SD16—2442no—
345C2×D8—22no—
346#131—1224no—
347#131—2442no—
348C2×M16—611no—
349C2×M16—611no—
350C2×M16—611no—
351C2×M16—611no—
352C2×M16—1222no—
353C2×M16C22×D6112yes—
354C22×C8—612no—
355C22×C8—1222no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8—1221no—
360C4⋊Q8C22⋊D6111yescharacteristic
361#127C22⋊D6111yescharacteristic
362#124C22⋊D6112yes—
363#121—21no—
364#121—21no—
365C4×D4—21no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4—21no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C22⋊C4—21no—
375C2×C42C22×D6111yescharacteristic
376#108—1224no—
377C4⋊C8—21no—
378C4⋊C8—21no—
379C4⋊C8—611no—
380C4⋊C8—612no—
381C4⋊C8—611no—
382#103—22no—
383#103—1222no—
384#103—1222no—
385#102—22no—
386#101—1224no—
387#101—2442no—
388#100—614no—
389#100—1222no—
390C22⋊C8—1222no—
391C22⋊C8—1222no—
392#97—1221no—
393#97—1221no—
394C4×C8—1221no—
395C4×C8—1221no—
396C24—21no—
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×C22⋊D6112yes—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4D4×D6112yes—
417C2×D4C2×C22⋊D6112yes—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—82no—
423C22×C4—1222no—
424C22×C4#1175822411yescharacteristic
425C22×C4#1175822411yescharacteristic
426C22×C4C23×D6111yescharacteristic
427Q16—2444no—
428Q16—2444no—
429SD16—44no—
430SD16—2444no—
431SD16—2444no—
432SD16—2444no—
433SD16—2444no—
434D8—44no—
435M16—22no—
436M16—22no—
437M16—612no—
438M16—612no—
439M16—612no—
440M16—1221no—
441M16—1221no—
442M16—1221no—
443M16—1221no—
444M16—1221no—
445M16—2444no—
446C2×C8—22no—
447C2×C8—612no—
448C2×C8—612no—
449C2×C8—1221no—
450C2×C8—1222no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1222no—
454C2×C8—1221no—
455C2×C8—2444no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—2441no—
461C4⋊C4—2441no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C42D4×D6111yescharacteristic
469C42D4×D6111yescharacteristic
470C42C2×C22⋊D6111yescharacteristic
471C42C2×C22⋊D6111yescharacteristic
472C23—22no—
473C23—22no—
474C23—41no—
475C23—41no—
476C23—81no—
477C23—81no—
478C23C2×#1175822411yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—2442no—
482Q8—2442no—
483Q8—2442no—
484Q8—2442no—
485D4—22no—
486D4—22no—
487D4—22no—
488D4—22no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—41no—
493D4—41no—
494D4—81no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
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×C4#41×D6111yescharacteristic
515C2×C4#1451982211yescharacteristic
516C2×C4#1451982311yescharacteristic
517C8—42no—
518C8—1222no—
519C8—1222no—
520C8—1222no—
521C8—1222no—
522C8—1222no—
523C8—2441no—
524C22—21no—
525C22—21no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—82no—
531C22—82no—
532C22—81no—
533C22—81no—
534C22C2×#1451974911yescharacteristic
535C22#1684030711yescharacteristic
536C22#1684036511yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—42no—
541C4—81no—
542C4—4882no—
543C4#1684002312yescomplemented
544C4D4×C22⋊D6111yescharacteristic
545C4#1684039611yescharacteristic
546C4#1684040011yescharacteristic
547C4#1684043011yescharacteristic
548C2—21no—
549C2—21no—
550C2—82no—
551C2—82no—
552C2#2144940911yescharacteristic
553C2C2×#1684002311yescharacteristic
554C2#2145555811yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3036C2
socleC22×C61C2×#121
hypercenter#3036D61
hypocenterC61#25405
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #21456279C2semidirect
2. C2×#16840023C2semidirect
3. C2×#16839967C2semidirect
4. #21454563C2semidirect
5. #21447116C2semidirect
6. #21446933C2semidirect
7. #21446662C2semidirect
8. #21446592C2semidirect
9. #21446332C2semidirect
10. C4×#14519694C2semidirect
85. #14519694D4semidirect
265. C4×C61#471semidirect
325. C61#25405semidirect
543. C4#16840023semidirect

Maximal subgroups

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

subgroupindex
#214562792
C2×#168400232
C2×#168399672
#214545632
#214471162
#214469332
#214466622
#214465922
#214463322
C4×#145196942
C61×#30362
#2540561