Finite Group Databasebeta

← #29731009

Subgroups — #29731009

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#21456276C211yesmaximal, characteristic, complemented
2C2×#16840022C212yesmaximal, complemented
3#21455283C211yesmaximal, characteristic
4#21454562C211yesmaximal, characteristic, complemented
5#21447119C212yesmaximal, complemented
6#21446931C212yesmaximal, complemented
7#21446664C211yesmaximal, characteristic, complemented
8#21446595C211yesmaximal, characteristic, complemented
9#21446337C212yesmaximal, complemented
10C4×#14519694C211yesmaximal, characteristic, complemented
11C2×C61×#461C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519808C2212yes—
14C2×#14519803C2212yes—
15C2×#14519783C2212yes—
16#16840312C2211yescharacteristic
17#16840154—22no—
18#16840141—21no—
19#16840136—21no—
20#16840022—28no—
21#16839977—22no—
22#16839967—21no—
23#16839957—21no—
24#16839839—21no—
25#16839832—22no—
26#16839827—21no—
27C2×#14519713C2212yes—
28#16839786C2211yescharacteristic
29#16839761C2212yes—
30C2×#14519696C2212yes—
31C2×#14519696C2212yes—
32C2×#14519694C2211yescharacteristic
33C2×#14519694C2211yescharacteristic
34#16839728C2212yes—
35C4×#11758194C2211yescharacteristic
36C4×#11758194C2211yescharacteristic
37#16839531C2211yescharacteristic
38#16839531C2211yescharacteristic
39#16839527C2212yes—
40C22×SD16×C61C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#461—21no—
44C61×#461—22no—
45C61×#461—21no—
46C2×C4⋊C8×C61C2211yescharacteristic
47C2×#103×C61C2212yes—
48C2×#11758222C2312yes—
49#14519808—24no—
50#14519808—24no—
51#14519808—42no—
52#14519808—42no—
53#14519803—24no—
54#14519803—42no—
55C2×#11758202—22no—
56C2×#11758202—22no—
57C2×#11758200—22no—
58C2×#11758201—22no—
59C2×#11758201—22no—
60C2×#11758199—22no—
61#14519783—24no—
62#14519783—42no—
63C2×#11758194C2311yescharacteristic
64C2×#11758194C2312yes—
65C2×#11758194C2311yescharacteristic
66C2×#11758194C2311yescharacteristic
67C2×#11758194C2311yescharacteristic
68C2×#11758189—22no—
69C2×C8×D61—22no—
70#14519777D411yescharacteristic
71#14519774D411yescharacteristic
72#14519765D412yes—
73#14519751—21no—
74#14519751—21no—
75C4×D244—21no—
76C4×D244—21no—
77#14519713—24no—
78C4⋊#92418—21no—
79C4⋊#92418—21no—
80#14519700—22no—
81#14519697—22no—
82#14519696—24no—
83#14519696—42no—
84#14519694—22no—
85#14519694D414yescomplemented
86#14519689—22no—
87#14519689—22no—
88#14519688—21no—
89#14519688—21no—
90C4×#92418—21no—
91C4×#92418—21no—
92C22×Q8×C61C2312yes—
93C22×D4×C61C2311yescharacteristic
94C23×C4×C61—21no—
95C2×SD16×C61—24no—
96C2×SD16×C61—22no—
97C2×SD16×C61—22no—
98C2×SD16×C61—42no—
99C22×C8×C61C2312yes—
100C4⋊Q8×C61—21no—
101C4⋊Q8×C61—21no—
102C4⋊Q8×C61—21no—
103C4⋊Q8×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C42×C61C2311yescharacteristic
113C4⋊C8×C61D411yescharacteristic
114C4⋊C8×C61D412yes—
115C4⋊C8×C61D411yescharacteristic
116#103×C61—22no—
117#103×C61—22no—
118C24×C61—21no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61C2×D412yes—
131C22×C4×C61—22no—
132C22×C4×C61—21no—
133C22×C4×C61—41no—
134C22×C4×C61—41no—
135C22×C4×C61—82no—
136C22×C4×C61#4111yescharacteristic
137C22×C4×C61#4111yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C2×C22⋊D61—22no—
140#11758222—22no—
141#11758222—22no—
142#11758222—22no—
143#11758222—22no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244C2×D412yes—
147C2×C4×D61—22no—
148C2×Dic122C2×D412yes—
149SD16×C61—44no—
150SD16×C61—44no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61C2×D412yes—
154C2×C8×C61C2×D412yes—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—41no—
158C4⋊C4×C61—41no—
159C4⋊C4×C61—41no—
160C4⋊C4×C61—41no—
161C22⋊C4×C61—41no—
162C22⋊C4×C61—41no—
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167#11758202—44no—
168#11758202—44no—
169#11758201—44no—
170#11758201—44no—
171#11758200—44no—
172#11758199—44no—
173#11758198—22no—
174#11758198—22no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177#11758194—21no—
178#11758194—21no—
179#11758194—21no—
180#11758194—22no—
181#11758194—21no—
182#11758194—21no—
183#11758194—22no—
184#11758194C2×D412yes—
185#11758194C2×D412yes—
186#11758194#4112yes—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—22no—
192#11758189—44no—
193C8×D61—44no—
194C23×C61—22no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—81no—
198C23×C61—81no—
199C23×C61C2×#4111yescharacteristic
200C22×D61—22no—
201Q8×C61—42no—
202Q8×C61—42no—
203Q8×C61—42no—
204Q8×C61—42no—
205D4×C61—22no—
206D4×C61—22no—
207D4×C61—41no—
208D4×C61—41no—
209C2×C4×C61—22no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—82no—
216C2×C4×C61—82no—
217C2×C4×C61—81no—
218C2×C4×C61#12111yescharacteristic
219C2×C4×C61#12111yescharacteristic
220C2×C4×C61#12111yescharacteristic
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C22⋊D61—42no—
229C22⋊D61—42no—
230C2×Dic61—22no—
231D244—22no—
232D244—22no—
233D244—42no—
234D244—81no—
235C4×D61—42no—
236C4×D61—42no—
237Dic122—22no—
238Dic122—22no—
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—82no—
251C22×C61—82no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61C2×#12111yescharacteristic
255C22×C61#53211yescharacteristic
256C22×C61#54011yescharacteristic
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—82no—
264C4×C61—81no—
265C4×C61#47112yescomplemented
266C4×C61D4211yescharacteristic
267C4×C61D4211yescharacteristic
268C4×C61#54611yescharacteristic
269C4×C61#55011yescharacteristic
270Dic61—42no—
271#3152—611no—
272C2×#471—612no—
273#3036—611no—
274C2×#461D6111yescharacteristic, complemented
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×#137—612no—
290C2×#137—612no—
291C2×AGL(1,ℤ/8)—612no—
292C22×SD16C2×D6112yes—
293C2×#131—612no—
294#532—611no—
295C2×C4⋊Q8C2×D6111yescharacteristic
296C2×C4×D4C2×D6111yescharacteristic
297#497—1222no—
298#494—1221no—
299#492—1221no—
300#471—1228no—
301#463—1222no—
302#461—21no—
303#461—22no—
304#461—21no—
305#461—1221no—
306#459—1221no—
307#436—1221no—
308#435—1222no—
309C8×D4—1221no—
310C2×#108—612no—
311#423—611no—
312C2×C4⋊C8C2×D6111yescharacteristic
313#417—612no—
314C2×#103C2×D6112yes—
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×#41—612no—
327C22×Q8C22×D6112yes—
328C22×D4C22×D6111yescharacteristic
329C23×C4—21no—
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—24no—
339C2×SD16—22no—
340C2×SD16—22no—
341C2×SD16—42no—
342C2×SD16—1222no—
343C2×SD16—1222no—
344C2×SD16—1222no—
345C2×D8—1222no—
346#131—1224no—
347#131—2442no—
348C2×M16—611no—
349C2×M16—611no—
350C2×M16—611no—
351C2×M16—612no—
352C2×M16—611no—
353C2×M16—1222no—
354C22×C8—1222no—
355C22×C8C22×D6112yes—
356C4⋊Q8—21no—
357C4⋊Q8—21no—
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8—611no—
361#127—611no—
362#124—612no—
363#121—1221no—
364#121—1221no—
365C4×D4—21no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—1221no—
370C4×D4—1221no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C22⋊C4—21no—
375C2×C42C22×D6111yescharacteristic
376#108—1224no—
377C4⋊C8—1221no—
378C4⋊C8—1221no—
379C4⋊C8C22⋊D6111yescharacteristic
380C4⋊C8C22⋊D6112yes—
381C4⋊C8C22⋊D6111yescharacteristic
382#103—22no—
383#103—22no—
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—1221no—
393#97—1221no—
394C4×C8—1221no—
395C4×C8—1221no—
396C24—21no—
397#41—1222no—
398#41—1222no—
399#41—1222no—
400#41—1222no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—612no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—612no—
414C2×D4—1222no—
415C2×D4—1221no—
416C2×D4—1221no—
417C2×D4D4×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—44no—
431SD16—2444no—
432SD16—2444no—
433SD16—2444no—
434D8—2444no—
435M16—612no—
436M16—612no—
437M16—612no—
438M16—1221no—
439M16—1221no—
440M16—1221no—
441M16—1222no—
442M16—1221no—
443M16—1221no—
444M16—1222no—
445M16—2444no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1222no—
452C2×C8—1221no—
453C2×C8—2444no—
454C2×C8C2×C22⋊D6112yes—
455C2×C8C2×C22⋊D6112yes—
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—41no—
465C22⋊C4—41no—
466C22⋊C4—1222no—
467C22⋊C4—1222no—
468C42D4×D6111yescharacteristic
469C42D4×D6111yescharacteristic
470C42C2×C22⋊D6111yescharacteristic
471C42C2×C22⋊D6111yescharacteristic
472C23—22no—
473C23—41no—
474C23—41no—
475C23—81no—
476C23—81no—
477C23—1222no—
478C23C2×#1175822411yescharacteristic
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—1222no—
484Q8—1222no—
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489D4—1222no—
490D4—1222no—
491D4—4881no—
492D4—2442no—
493D4—2442no—
494D4—2442no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—1222no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4#1451974811yescharacteristic
508C2×C4#1451974811yescharacteristic
509C2×C4#1451974811yescharacteristic
510C2×C4#1451974811yescharacteristic
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C2×D4×D6111yescharacteristic
513C2×C4C22×C22⋊D6111yescharacteristic
514C2×C4#41×D6111yescharacteristic
515C2×C4#1451982211yescharacteristic
516C2×C4#1451982311yescharacteristic
517C8—22no—
518C8—22no—
519C8—1222no—
520C8—1222no—
521C8—1222no—
522C8—2442no—
523C8—2441no—
524C22—21no—
525C22—21no—
526C22—42no—
527C22—82no—
528C22—82no—
529C22—81no—
530C22—81no—
531C22—2442no—
532C22—2442no—
533C22—2442no—
534C22C2×#1451974811yescharacteristic
535C22#1684030711yescharacteristic
536C22#1684036511yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—81no—
541C4—82no—
542C4—2442no—
543C4#1684002212yescomplemented
544C4D4×C22⋊D6111yescharacteristic
545C4#1684039611yescharacteristic
546C4#1684040911yescharacteristic
547C4#1684043111yescharacteristic
548C2—21no—
549C2—21no—
550C2—82no—
551C2—4882no—
552C2#2144940311yescharacteristic
553C2C2×#1684002211yescharacteristic
554C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#461C2
socleC22×C61C2×#121
prime radical#21455283C2
hypercenterC2×#461D61
hypocenterC61#25405
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. #21456276C2semidirect
2. C2×#16840022C2semidirect
4. #21454562C2semidirect
5. #21447119C2semidirect
6. #21446931C2semidirect
7. #21446664C2semidirect
8. #21446595C2semidirect
9. #21446337C2semidirect
10. C4×#14519694C2semidirect
11. C2×C61×#461C2semidirect
85. #14519694D4semidirect
265. C4×C61#471semidirect
274. C2×#461D61semidirect
325. C61#25405semidirect
543. C4#16840022semidirect

Maximal subgroups

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

subgroupindex
#214562762
C2×#168400222
#214552832
#214545622
#214471192
#214469312
#214466642
#214465952
#214463372
C4×#145196942
C2×C61×#4612
#2540561