Finite Group Databasebeta

← #29725119

Subgroups — #29725119

Order 15616.

638 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454679C211yesmaximal, characteristic, complemented
2#21447286C211yesmaximal, characteristic, complemented
3#21447279C211yesmaximal, characteristic, complemented
4#21447096C211yesmaximal, characteristic, complemented
5#21447089C211yesmaximal, characteristic
6#21446724C211yesmaximal, characteristic, complemented
7#21446607C211yesmaximal, characteristic, complemented
8#21446315C211yesmaximal, characteristic, complemented
9#21446308C211yesmaximal, characteristic
10#21445944C211yesmaximal, characteristic, complemented
11#21445931C211yesmaximal, characteristic, complemented
12#21445928C211yesmaximal, characteristic, complemented
13#21445873C211yesmaximal, characteristic, complemented
14#21445619C211yesmaximal, characteristic, complemented
15C2×C61×#471C211yesmaximal, characteristic
16#16840240C2211yescharacteristic
17#16840237C2211yescharacteristic
18#16839863—22no—
19SD16×Dic61—22no—
20#16839810C2211yescharacteristic
21#16839810C2211yescharacteristic
22C2×#14519713C2211yescharacteristic
23#16839800C2211yescharacteristic
24#16839779C2211yescharacteristic
25#16839779C2211yescharacteristic
26#16839771C2211yescharacteristic
27#16839771C2211yescharacteristic
28#16839767C2211yescharacteristic
29#16839767C2211yescharacteristic
30#16839746C2211yescharacteristic
31C2×#14519696C2211yescharacteristic
32C2×#14519694C2211yescharacteristic
33#16839736C2211yescharacteristic
34#16839717C2211yescharacteristic
35#16839717C2211yescharacteristic
36#16839534C2211yescharacteristic
37#16839534C2211yescharacteristic
38#16839534C2211yescharacteristic
39#16839534C2211yescharacteristic
40#16839533C2211yescharacteristic
41#16839533C2211yescharacteristic
42#16839533C2211yescharacteristic
43#16839533C2211yescharacteristic
44#16839519C2211yescharacteristic
45#16839519C2211yescharacteristic
46C2×#137×C61C2211yescharacteristic
47C2×AGL(1,ℤ/8)×C61C2211yescharacteristic
48C22×SD16×C61C2211yescharacteristic
49C2×#131×C61C2211yescharacteristic
50C61×#471—22no—
51C61×#471—22no—
52C2×#108×C61C2211yescharacteristic
53C2×#101×C61C2211yescharacteristic
54C2×#100×C61C2211yescharacteristic
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2311yescharacteristic
58C2×#11758194C2311yescharacteristic
59Q8×Dic61—22no—
60Q8×Dic61—41no—
61D4×Dic61—22no—
62D4×Dic61—41no—
63#14519736—22no—
64#14519736C2311yescharacteristic
65#14519736C2311yescharacteristic
66#14519736C2311yescharacteristic
67#14519736C2311yescharacteristic
68C2×C22⋊Dic61—22no—
69#14519713—21no—
70#14519713—21no—
71#14519712—41no—
72#14519712—41no—
73#14519711—21no—
74#14519709—21no—
75#14519704—41no—
76#14519704—41no—
77#14519701—22no—
78#14519701—41no—
79#14519698—22no—
80#14519698—41no—
81#14519696—21no—
82#14519696—41no—
83#14519694—21no—
84#14519694—41no—
85#14519692—22no—
86#14519687—21no—
87#14519687—21no—
88C8×Dic61—21no—
89C8×Dic61—21no—
90C2×#41×C61C2311yescharacteristic
91C22×Q8×C61C2311yescharacteristic
92C22×D4×C61C2311yescharacteristic
93#137×C61—22no—
94#137×C61—41no—
95AGL(1,ℤ/8)×C61—22no—
96AGL(1,ℤ/8)×C61—41no—
97C2×Q16×C61—21no—
98C2×SD16×C61—21no—
99C2×SD16×C61—21no—
100C2×SD16×C61—21no—
101C2×SD16×C61—21no—
102C2×SD16×C61—21no—
103C2×SD16×C61—21no—
104C2×SD16×C61D412yes—
105C2×SD16×C61D412yes—
106C2×D8×C61—21no—
107#131×C61—21no—
108#131×C61—21no—
109#131×C61—21no—
110#131×C61—21no—
111C2×M16×C61—21no—
112C2×M16×C61C2311yescharacteristic
113C2×M16×C61C2311yescharacteristic
114C2×M16×C61C2311yescharacteristic
115C22×C8×C61—21no—
116C22×C8×C61C2311yescharacteristic
117#108×C61—21no—
118#108×C61—21no—
119#101×C61—22no—
120#100×C61—22no—
121C24×C61—21no—
122#41×C61—21no—
123#41×C61—21no—
124#41×C61—21no—
125#41×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61#4111yescharacteristic
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61C2×D412yes—
140C2×D4×C61#4111yescharacteristic
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—42no—
145Q16×C61—42no—
146SD16×C61—22no—
147SD16×C61—22no—
148SD16×C61—22no—
149SD16×C61—22no—
150SD16×C61—42no—
151SD16×C61—42no—
152D8×C61—42no—
153M16×C61—21no—
154M16×C61—21no—
155M16×C61—21no—
156M16×C61—21no—
157M16×C61—21no—
158M16×C61—21no—
159M16×C61—41no—
160M16×C61—41no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—41no—
165C2×C8×C61—41no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61#4111yescharacteristic
169C2×C8×C61#4111yescharacteristic
170C2×C8×C61#4111yescharacteristic
171C2×C8×C61#4111yescharacteristic
172C22⋊Dic61—21no—
173C22⋊Dic61—21no—
174C22⋊Dic61—21no—
175C22⋊Dic61—42no—
176C22⋊Dic61#4112yes—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—42no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—42no—
187#11758194—21no—
188#11758194—21no—
189#11758194—21no—
190#11758194—21no—
191#11758194—21no—
192#11758194—21no—
193#11758194—21no—
194#11758194—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C23×C61—21no—
200C23×C61—22no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—161no—
204C23×C61C2×#4111yescharacteristic
205Q8×C61—22no—
206Q8×C61—21no—
207Q8×C61—22no—
208Q8×C61—21no—
209Q8×C61—41no—
210Q8×C61—41no—
211D4×C61—22no—
212D4×C61—21no—
213D4×C61—22no—
214D4×C61—21no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—41no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×Dic61—22no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—162no—
236C8×C61—41no—
237C8×C61—41no—
238C8×C61—41no—
239C8×C61#12311yescharacteristic
240C8×C61#12311yescharacteristic
241C8×C61#12311yescharacteristic
242C8×C61#12311yescharacteristic
243#92418—41no—
244#92418—41no—
245#92418—41no—
246#92418—41no—
247#24987—611nomaximal, Sylow
248C22×C61—41no—
249C22×C61—41no—
250C22×C61—42no—
251C22×C61—82no—
252C22×C61—81no—
253C22×C61—161no—
254C22×C61—161no—
255C22×C61#52911yescharacteristic
256C22×C61#54611yescharacteristic
257C22×C61#55511yescharacteristic
258C4×C61—41no—
259C4×C61—42no—
260C4×C61C2×#12311yescharacteristic
261C4×C61#54011yescharacteristic
262C4×C61#54611yescharacteristic
263C4×C61#55311yescharacteristic
264Dic61—42no—
265Dic61—81no—
266Dic61—81no—
267Dic61—81no—
268C2×#471D6111yescharacteristic
269#2949—611no—
270#2082—611no—
271#2081—611no—
272#2049—611no—
273#2048—611no—
274#1977—611no—
275#1953—611no—
276#1905—611no—
277#1904—611no—
278#1833—611no—
279#1830—611no—
280#1829—611no—
281#1814—611no—
282#1757—611no—
283C2×C61—21no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61—81no—
287C2×C61—82no—
288C2×C61#242011yescharacteristic
289C2×#137C2×D6111yescharacteristic
290C2×AGL(1,ℤ/8)C2×D6111yescharacteristic
291C22×SD16C2×D6111yescharacteristic
292C2×#131C2×D6111yescharacteristic
293#519—611no—
294#518—611no—
295#471—22no—
296#471—22no—
297#440—1222no—
298C4×SD16—1222no—
299#430—611no—
300#430—611no—
301C2×#108—611no—
302C2×#108C2×D6111yescharacteristic
303#427—611no—
304#421—611no—
305#421—611no—
306#419—611no—
307#419—611no—
308#418—611no—
309#418—611no—
310#413—611no—
311C2×#101—611no—
312C2×#101C2×D6111yescharacteristic
313C2×#100—611no—
314C2×#100C2×D6111yescharacteristic
315#410—611no—
316#405—611no—
317#405—611no—
318#344—611no—
319#344—611no—
320#344—611no—
321#344—611no—
322#344—611no—
323#344—611no—
324#344—611no—
325#344—611no—
326H(3,ℤ/4)—611no—
327H(3,ℤ/4)—611no—
328C61#2498711yescharacteristic, complemented, Sylow
329C2×#41C22×D6111yescharacteristic
330C22×Q8C22×D6111yescharacteristic
331C22×D4C22×D6111yescharacteristic
332#137—22no—
333#137—41no—
334AGL(1,ℤ/8)—22no—
335AGL(1,ℤ/8)—41no—
336C2×Q16—21no—
337C2×SD16—21no—
338C2×SD16—21no—
339C2×SD16—21no—
340C2×SD16—21no—
341C2×SD16—21no—
342C2×SD16—21no—
343C2×SD16C22⋊D6112yes—
344C2×SD16C22⋊D6112yes—
345C2×D8—21no—
346#131—21no—
347#131—21no—
348#131—21no—
349#131—21no—
350C2×M16—21no—
351C2×M16—611no—
352C2×M16—611no—
353C2×M16—611no—
354C2×M16—611no—
355C2×M16C22×D6111yescharacteristic
356C2×M16C22×D6111yescharacteristic
357C2×M16C22×D6111yescharacteristic
358C22×C8—21no—
359C22×C8C22×D6111yescharacteristic
360C4×Q8—1222no—
361C4×Q8—2441no—
362C4×D4—1222no—
363C4×D4—2441no—
364#117—611no—
365#117—611no—
366#117—611no—
367#117—611no—
368#117—1222no—
369C2×C22⋊C4—1222no—
370#108—21no—
371#108—21no—
372#108—1221no—
373#108—1221no—
374#108—2441no—
375#108—2441no—
376#107—1221no—
377#106—1221no—
378#104—2441no—
379#104—2441no—
380#103—1222no—
381#103—2441no—
382#102—1222no—
383#102—2441no—
384#101—22no—
385#101—1221no—
386#101—2441no—
387#100—22no—
388#100—1221no—
389#100—2441no—
390#99—1222no—
391#97—1221no—
392#97—1221no—
393C4×C8—1221no—
394C4×C8—1221no—
395C24—21no—
396#41—21no—
397#41—21no—
398#41—21no—
399#41—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8#1175822411yescharacteristic
406C2×Q8C2×C22⋊D6112yes—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4#1175822411yescharacteristic
414C2×D4C2×C22⋊D6112yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—2442no—
418C22×C4C23×D6111yescharacteristic
419Q16—42no—
420SD16—22no—
421SD16—22no—
422SD16—22no—
423SD16—22no—
424SD16—42no—
425SD16—42no—
426D8—42no—
427M16—21no—
428M16—21no—
429M16—21no—
430M16—21no—
431M16—21no—
432M16—21no—
433M16—41no—
434M16—41no—
435M16—1221no—
436M16—1221no—
437M16—1221no—
438M16—1221no—
439M16—1221no—
440M16—1221no—
441M16—1221no—
442M16—1221no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8#1175822411yescharacteristic
453C2×C8#1175822411yescharacteristic
454C2×C8#1175822411yescharacteristic
455C2×C8#1175822411yescharacteristic
456C2×C8C2×C22⋊D6111yescharacteristic
457C2×C8C2×C22⋊D6111yescharacteristic
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—2442no—
463C22⋊C4—612no—
464C22⋊C4—1221no—
465C22⋊C4—1221no—
466C22⋊C4—1221no—
467C22⋊C4—2442no—
468C42—1221no—
469C42—1221no—
470C42—1221no—
471C42—1221no—
472C42—2442no—
473C23—21no—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—161no—
478C23#41×D6111yescharacteristic
479Q8—21no—
480Q8—22no—
481Q8—21no—
482Q8—22no—
483Q8—41no—
484Q8—41no—
485D4—21no—
486D4—22no—
487D4—21no—
488D4—22no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—41no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—1222no—
500C2×C4—9762no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4C22×C22⋊D6111yescharacteristic
505C2×C4C2×#1175822411yescharacteristic
506C2×C4C2×#1175822411yescharacteristic
507C2×C4C2×#1175822411yescharacteristic
508C2×C4#1451982211yescharacteristic
509C2×C4#1451982311yescharacteristic
510C8—41no—
511C8—41no—
512C8—41no—
513C8—2441no—
514C8—2441no—
515C8—2441no—
516C8—2441no—
517C8#1451975811yescharacteristic
518C8#1451975811yescharacteristic
519C8#1451975811yescharacteristic
520C8#1451975811yescharacteristic
521C22—41no—
522C22—41no—
523C22—42no—
524C22—82no—
525C22—81no—
526C22—161no—
527C22—161no—
528C22#1684029311yescharacteristic
529C22#1684040011yescharacteristic
530C22#1684047611yescharacteristic
531C4—41no—
532C4—42no—
533C4—4881no—
534C4—4881no—
535C4—4881no—
536C4—2442no—
537C4C2×#1451975811yescharacteristic
538C4#1684036511yescharacteristic
539C4#1684040611yescharacteristic
540C4#1684046411yescharacteristic
541C2—21no—
542C2—21no—
543C2—21no—
544C2—82no—
545C2—81no—
546C2#2144930711yescharacteristic
subgroupG/H
centerC2#21449307
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#471C2
socleC2×C61#2420
prime radicalC2×AGL(1,ℤ/8)×C61C22
hypercenterC2×#471D61
hypocenterC61#24987
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#471

Complements

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

NH ≅ G/Ntype
1. #21454679C2semidirect
2. #21447286C2semidirect
3. #21447279C2semidirect
4. #21447096C2semidirect
6. #21446724C2semidirect
7. #21446607C2semidirect
8. #21446315C2semidirect
10. #21445944C2semidirect
11. #21445931C2semidirect
12. #21445928C2semidirect
13. #21445873C2semidirect
14. #21445619C2semidirect
328. C61#24987semidirect

Maximal subgroups

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

subgroupindex
#214546792
#214472862
#214472792
#214470962
#214470892
#214467242
#214466072
#214463152
#214463082
#214459442
#214459312
#214459282
#214458732
#214456192
C2×C61×#4712
#2498761