Finite Group Databasebeta

← #29731031

Subgroups — #29731031

Order 15616.

936 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456269C211yesmaximal, characteristic
2C2×#16840022C212yesmaximal, complemented
3#21455291C211yesmaximal, characteristic, complemented
4#21454562C211yesmaximal, characteristic, complemented
5#21447112C212yesmaximal, complemented
6#21446931C212yesmaximal, complemented
7#21446660C211yesmaximal, characteristic, complemented
8#21446595C211yesmaximal, characteristic, complemented
9#21446330C212yesmaximal, complemented
10C4×#14519696C211yesmaximal, characteristic, complemented
11C2×C61×#460C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519803C2212yes—
14C2×#14519803C2212yes—
15C2×#14519783C2212yes—
16#16840312C2211yescharacteristic
17#16840149—22no—
18#16840138—21no—
19#16840134—21no—
20#16840022—28no—
21#16839984—22no—
22#16839972—21no—
23#16839962—21no—
24#16839839—21no—
25#16839832—22no—
26#16839827—21no—
27C2×#14519713C2212yes—
28#16839786C2211yescharacteristic
29#16839761C2212yes—
30C2×#14519696C2211yescharacteristic
31C2×#14519696C2211yescharacteristic
32C2×#14519694C2212yes—
33C2×#14519694C2212yes—
34#16839728C2212yes—
35C4×#11758194C2211yescharacteristic
36C4×#11758194C2211yescharacteristic
37#16839531C2211yescharacteristic
38#16839531C2211yescharacteristic
39#16839527C2212yes—
40C22×SD16×C61C2212yes—
41C2×#127×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C61×#460—21no—
44C61×#460—22no—
45C61×#460—21no—
46C2×C4⋊C8×C61C2211yescharacteristic
47C2×#102×C61C2212yes—
48C2×#11758222C2312yes—
49#14519808—24no—
50#14519808—42no—
51#14519803—24no—
52#14519803—24no—
53#14519803—42no—
54#14519803—42no—
55C2×#11758202—22no—
56C2×#11758200—22no—
57C2×#11758200—22no—
58C2×#11758201—22no—
59C2×#11758199—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—22no—
83#14519696D414yescomplemented
84#14519694—24no—
85#14519694—42no—
86#14519689—22no—
87#14519689—22no—
88#14519688—21no—
89#14519688—21no—
90C4×#92418—21no—
91C4×#92418—21no—
92C22×Q8×C61C2311yescharacteristic
93C22×D4×C61—21no—
94C22×D4×C61—21no—
95C22×D4×C61C2312yes—
96C2×SD16×C61—24no—
97C2×SD16×C61—22no—
98C2×SD16×C61—22no—
99C2×SD16×C61—42no—
100C22×C8×C61C2312yes—
101#127×C61—21no—
102#127×C61—21no—
103#127×C61—21no—
104#127×C61—21no—
105C4×Q8×C61—21no—
106C4×Q8×C61—21no—
107C4×Q8×C61—21no—
108C4×Q8×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C42×C61—21no—
112C2×C42×C61C2311yescharacteristic
113C4⋊C8×C61D411yescharacteristic
114C4⋊C8×C61D412yes—
115C4⋊C8×C61D411yescharacteristic
116#102×C61—22no—
117#102×C61—22no—
118C24×C61—22no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61C2×D412yes—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—41no—
132C2×D4×C61—41no—
133C2×D4×C61—41no—
134C2×D4×C61—41no—
135C2×D4×C61—82no—
136C2×D4×C61—82no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61#4111yescharacteristic
140C22×C4×C61#4111yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—22no—
143#11758222—22no—
144#11758222—22no—
145#11758222—22no—
146#11758222—22no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244C2×D412yes—
150C2×C4×D61—22no—
151C2×Dic122C2×D412yes—
152SD16×C61—44no—
153SD16×C61—44no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61C2×D412yes—
157C2×C8×C61C2×D412yes—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—41no—
161C4⋊C4×C61—41no—
162C42×C61—41no—
163C42×C61—41no—
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2×D411yescharacteristic
168#11758202—44no—
169#11758201—44no—
170#11758200—44no—
171#11758200—44no—
172#11758199—44no—
173#11758199—44no—
174#11758198—22no—
175#11758198—22no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178#11758194—21no—
179#11758194—21no—
180#11758194—21no—
181#11758194—22no—
182#11758194—21no—
183#11758194—22no—
184#11758194—21no—
185#11758194C2×D412yes—
186#11758194C2×D412yes—
187#11758194#4112yes—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—22no—
193#11758189—44no—
194C8×D61—44no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—82no—
199C23×C61—82no—
200C23×C61C2×#4111yescharacteristic
201C22×D61—22no—
202Q8×C61—22no—
203Q8×C61—22no—
204Q8×C61—41no—
205Q8×C61—41no—
206D4×C61—42no—
207D4×C61—42no—
208D4×C61—42no—
209D4×C61—42no—
210D4×C61—82no—
211D4×C61—82no—
212D4×C61—82no—
213D4×C61—82no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—81no—
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
231C22⋊D61—42no—
232C22⋊D61—42no—
233C2×Dic61—22no—
234D244—22no—
235D244—22no—
236D244—42no—
237D244—81no—
238C4×D61—42no—
239C4×D61—42no—
240Dic122—22no—
241Dic122—22no—
242C8×C61—22no—
243C8×C61—22no—
244#92418—22no—
245#92418—22no—
246#92418—22no—
247#92418—42no—
248#92418—41no—
249#25407—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—82no—
253C22×C61—82no—
254C22×C61—82no—
255C22×C61—162no—
256C22×C61—162no—
257C22×C61C2×#12111yescharacteristic
258C22×C61#53211yescharacteristic
259C22×C61#54011yescharacteristic
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—42no—
267C4×C61—81no—
268C4×C61#47112yescomplemented
269C4×C61D4211yescharacteristic
270C4×C61D4211yescharacteristic
271C4×C61#54611yescharacteristic
272C4×C61#55011yescharacteristic
273Dic61—42no—
274#3151—611no—
275C2×#471—612no—
276#3037—611no—
277C2×#460D6111yescharacteristic, complemented
278#2935—611no—
279#2051—612no—
280#2019—612no—
281#1965—611no—
282#1950—611no—
283#1907—612no—
284C4×#101—611no—
285C2×C61—21no—
286C2×C61—21no—
287C2×C61—162no—
288C2×C61#242811yescharacteristic
289C2×C61C2×#47111yescharacteristic
290C2×C61#306811yescharacteristic
291D61—82no—
292C2×#137—612no—
293C2×AGL(1,ℤ/8)—612no—
294C2×AGL(1,ℤ/8)—612no—
295C22×SD16C2×D6112yes—
296C2×#131—612no—
297#532—611no—
298C2×#127C2×D6111yescharacteristic
299C2×C4×Q8C2×D6111yescharacteristic
300#496—1222no—
301#493—1221no—
302#492—1221no—
303#471—1228no—
304#464—1222no—
305#462—1221no—
306#460—21no—
307#460—22no—
308#460—21no—
309#460—1221no—
310#436—1221no—
311#435—1222no—
312C8×D4—1221no—
313C2×#108—612no—
314#423—611no—
315C2×C4⋊C8C2×D6111yescharacteristic
316#417—612no—
317C2×#102C2×D6112yes—
318C2×#101—611no—
319C2×#101—611no—
320C2×#100—612no—
321C2×#100—612no—
322#408—612no—
323C4×M16—611no—
324C4×M16—611no—
325#343—611no—
326#343—611no—
327#341—612no—
328C61#2540711yescharacteristic, complemented, Sylow
329C2×#41—612no—
330C22×Q8C22×D6111yescharacteristic
331C22×D4—21no—
332C22×D4—21no—
333C22×D4C22×D6112yes—
334#137—1224no—
335#137—2442no—
336AGL(1,ℤ/8)—1224no—
337AGL(1,ℤ/8)—1224no—
338AGL(1,ℤ/8)—2442no—
339AGL(1,ℤ/8)—2442no—
340C2×Q16—1222no—
341C2×SD16—24no—
342C2×SD16—22no—
343C2×SD16—22no—
344C2×SD16—42no—
345C2×SD16—1222no—
346C2×SD16—1222no—
347C2×SD16—1222no—
348C2×D8—1222no—
349C2×D8—1222no—
350#131—1224no—
351#131—2442no—
352C2×M16—611no—
353C2×M16—611no—
354C2×M16—611no—
355C2×M16—612no—
356C2×M16—611no—
357C2×M16—1222no—
358C22×C8—1222no—
359C22×C8C22×D6112yes—
360C4⋊Q8—611no—
361#127—21no—
362#127—21no—
363#127—21no—
364#127—21no—
365#127—611no—
366#124—612no—
367#121—1221no—
368#121—1221no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×D4—1221no—
374C4×D4—1221no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C42—21no—
378C2×C42C22×D6111yescharacteristic
379#108—1224no—
380C4⋊C8—1221no—
381C4⋊C8—1221no—
382C4⋊C8C22⋊D6111yescharacteristic
383C4⋊C8C22⋊D6112yes—
384C4⋊C8C22⋊D6111yescharacteristic
385#103—1222no—
386#102—22no—
387#102—22no—
388#102—1222no—
389#101—614no—
390#101—1222no—
391#100—1224no—
392#100—2442no—
393C22⋊C8—1222no—
394C22⋊C8—1222no—
395#97—1221no—
396#97—1221no—
397C4×C8—1221no—
398C4×C8—1221no—
399C24—22no—
400#41—1222no—
401#41—1222no—
402#41—1222no—
403#41—1222no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—612no—
410C2×Q8D4×D6112yes—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—612no—
424C2×D4—1222no—
425C2×D4—1221no—
426C2×D4—1221no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—1222no—
430C22×C4#1175822411yescharacteristic
431C22×C4#1175822411yescharacteristic
432C22×C4C23×D6111yescharacteristic
433Q16—2444no—
434SD16—44no—
435SD16—44no—
436SD16—2444no—
437SD16—2444no—
438SD16—2444no—
439D8—2444no—
440D8—2444no—
441M16—612no—
442M16—612no—
443M16—612no—
444M16—1221no—
445M16—1221no—
446M16—1221no—
447M16—1222no—
448M16—1221no—
449M16—1221no—
450M16—1222no—
451M16—2444no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1222no—
458C2×C8—1221no—
459C2×C8—2444no—
460C2×C8C2×C22⋊D6112yes—
461C2×C8C2×C22⋊D6112yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C22⋊C4—1222no—
469C22⋊C4—1222no—
470C42—41no—
471C42—41no—
472C42D4×D6111yescharacteristic
473C42D4×D6111yescharacteristic
474C42C2×C22⋊D6111yescharacteristic
475C42C2×C22⋊D6111yescharacteristic
476C23—42no—
477C23—42no—
478C23—42no—
479C23—82no—
480C23—82no—
481C23—1222no—
482C23C2×#1175822411yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—41no—
486Q8—41no—
487Q8—1222no—
488Q8—1222no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—82no—
494D4—82no—
495D4—82no—
496D4—82no—
497D4—1222no—
498D4—1222no—
499D4—4881no—
500D4—2442no—
501D4—2442no—
502D4—2442no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—81no—
510C2×C4—1222no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4#1451974811yescharacteristic
514C2×C4#1451974811yescharacteristic
515C2×C4#1451974811yescharacteristic
516C2×C4#1451974811yescharacteristic
517C2×C4C2×D4×D6111yescharacteristic
518C2×C4C2×D4×D6111yescharacteristic
519C2×C4C22×C22⋊D6111yescharacteristic
520C2×C4#41×D6111yescharacteristic
521C2×C4#1451982211yescharacteristic
522C2×C4#1451982311yescharacteristic
523C8—22no—
524C8—22no—
525C8—1222no—
526C8—1222no—
527C8—1222no—
528C8—2442no—
529C8—2441no—
530C22—21no—
531C22—21no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—162no—
536C22—162no—
537C22—2442no—
538C22—2442no—
539C22—2442no—
540C22C2×#1451974811yescharacteristic
541C22#1684030711yescharacteristic
542C22#1684036511yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—42no—
547C4—81no—
548C4—2442no—
549C4#1684002212yescomplemented
550C4D4×C22⋊D6111yescharacteristic
551C4#1684039611yescharacteristic
552C4#1684040911yescharacteristic
553C4#1684043111yescharacteristic
554C2—21no—
555C2—21no—
556C2—162no—
557C2—4882no—
558C2#2144940311yescharacteristic
559C2C2×#1684002211yescharacteristic
560C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#460C2
socleC22×C61C2×#121
prime radical#21456269C2
hypercenterC2×#460D61
hypocenterC61#25407
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. C2×#16840022C2semidirect
3. #21455291C2semidirect
4. #21454562C2semidirect
5. #21447112C2semidirect
6. #21446931C2semidirect
7. #21446660C2semidirect
8. #21446595C2semidirect
9. #21446330C2semidirect
10. C4×#14519696C2semidirect
11. C2×C61×#460C2semidirect
83. #14519696D4semidirect
268. C4×C61#471semidirect
277. C2×#460D61semidirect
328. C61#25407semidirect
549. C4#16840022semidirect

Maximal subgroups

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

subgroupindex
#214562692
C2×#168400222
#214552912
#214545622
#214471122
#214469312
#214466602
#214465952
#214463302
C4×#145196962
C2×C61×#4602
#2540761