Finite Group Databasebeta

← #230097

Subgroups — #230097

Order 512.

593 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23180C211yesmaximal, characteristic
2#22746C211yesmaximal, characteristic, complemented
3#19114C211yesmaximal, characteristic
4#18004C211yesmaximal, characteristic, complemented
5#17997C211yesmaximal, characteristic, complemented
6#9860C211yesmaximal, characteristic
7#9632C211yesmaximal, characteristic, complemented
8#9592C211yesmaximal, characteristic, complemented
9#9590C211yesmaximal, characteristic, complemented
10#8992C211yesmaximal, characteristic, complemented
11#8992C211yesmaximal, characteristic, complemented
12#8932C211yesmaximal, characteristic, complemented
13#8930C211yesmaximal, characteristic, complemented
14#8297C211yesmaximal, characteristic, complemented
15#8205C211yesmaximal, characteristic, complemented
16#3147C2211yescharacteristic
17#3103C2211yescharacteristic
18#3103C2211yescharacteristic
19Q8×M16C2211yescharacteristic
20Q8×M16C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22C2×#444—21no—
23#2935C2211yescharacteristic
24#2935C2211yescharacteristic
25#1980C2211yescharacteristic
26#1980C2211yescharacteristic
27#1975—21no—
28#1950—21no—
29#1950—21no—
30#1950C2211yescharacteristic
31#1950C2211yescharacteristic
32#1876—21no—
33#1876—21no—
34#1837—21no—
35#1837C2211yescharacteristic
36C4×#108—21no—
37C4×#108C2211yescharacteristic
38#1770C2211yescharacteristic
39C4×#103C2211yescharacteristic
40#1670C411yescharacteristic, complemented
41#1670C411yescharacteristic, complemented
42#1668C411yescharacteristic, complemented
43#1668C411yescharacteristic, complemented
44#1666C411yescharacteristic, complemented
45#1666C411yescharacteristic, complemented
46#1666C411yescharacteristic, complemented
47#1666C411yescharacteristic, complemented
48#1592—21no—
49#1590—21no—
50#1588—21no—
51#1585—21no—
52#1583—21no—
53#1582—21no—
54SD16⋊C8—21no—
55C8×SD16—21no—
56#1571C2211yescharacteristic
57#1570C2211yescharacteristic
58#1564C2211yescharacteristic
59#1540C2211yescharacteristic
60#1540C2211yescharacteristic
61#1492C2211yescharacteristic
62#1492C2211yescharacteristic
63#1492C2211yescharacteristic
64#1492C2211yescharacteristic
65C2×Q8⋊C8C2211yescharacteristic
66#1451C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1279C2211yescharacteristic
72#1278C2211yescharacteristic
73#1278C2211yescharacteristic
74#1277C2211yescharacteristic
75C2×#135—21no—
76C2×#135—21no—
77C2×#131—21no—
78C2×#131—21no—
79#532C2311yescharacteristic
80#532C2311yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82#495—21no—
83#495—21no—
84#494C2×C411yescharacteristic
85#493C2×C411yescharacteristic
86#492C2×C411yescharacteristic
87#492C2×C411yescharacteristic
88#481—21no—
89#481—21no—
90#475C2×C411yescharacteristic
91#475C2×C411yescharacteristic
92#475C2×C411yescharacteristic
93#475C2×C411yescharacteristic
94#460C2×C411yescharacteristic
95#460C2×C411yescharacteristic
96#460C2×C411yescharacteristic
97#460C2×C411yescharacteristic
98#448—21no—
99#448—21no—
100#446—21no—
101#446—21no—
102#446—41no—
103#446—41no—
104C8×Q8—21no—
105C8×Q8—21no—
106C8×Q8—21no—
107C8×Q8—21no—
108C8×Q8C2×C411yescharacteristic
109C8×Q8C2×C411yescharacteristic
110C8×Q8C2×C411yescharacteristic
111C8×Q8C2×C411yescharacteristic
112#444—44no—
113C4×SD16—21no—
114C4×SD16—21no—
115C4×SD16—21no—
116C4×SD16—21no—
117#436—21no—
118#436—21no—
119#435—41no—
120#435—41no—
121C8×D4—21no—
122C8×D4—21no—
123#431C2311yescharacteristic
124#431C2311yescharacteristic
125C2×#108—21no—
126C2×#108—21no—
127C2×#108D411yescharacteristic
128C2×#108D411yescharacteristic
129#423—21no—
130#423—21no—
131#423C2311yescharacteristic
132#423C2311yescharacteristic
133C2×C4⋊C8—21no—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×#104—21no—
137C2×#104—21no—
138C2×#104—21no—
139C2×#104—21no—
140C2×#103—21no—
141#408—21no—
142#408—21no—
143C4×M16—21no—
144C4×M16—21no—
145C4×M16C2311yescharacteristic
146C4×M16C2311yescharacteristic
147C2×#97—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C4×C4⋊C4C2311yescharacteristic
153#341—21no—
154#341—21no—
155#341—21no—
156#341—21no—
157#341D411yescharacteristic
158#341D411yescharacteristic
159#340—21no—
160#336—21no—
161#335—21no—
162#335—21no—
163#335C2×C411yescharacteristic
164#335C2×C411yescharacteristic
165#334—21no—
166#334—21no—
167#334C2×C411yescharacteristic
168#334C2×C411yescharacteristic
169#332C2×C411yescharacteristic
170#332C2×C411yescharacteristic
171#332C2×C411yescharacteristic
172#332C2×C411yescharacteristic
173#331C2×C411yescharacteristic
174#331C2×C411yescharacteristic
175#331C2×C411yescharacteristic
176#331C2×C411yescharacteristic
177Q8⋊C8D411yescharacteristic
178Q8⋊C8D411yescharacteristic
179Q8⋊C8D411yescharacteristic
180Q8⋊C8D411yescharacteristic
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—41no—
186D4⋊C8—41no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191#322—21no—
192C82—21no—
193C2×#41—21no—
194C2×#41—21no—
195C22×Q8#4111yescharacteristic
196#135—42no—
197#135—42no—
198#135—81no—
199#135—81no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×SD16—21no—
203C2×SD16—21no—
204C2×SD16—21no—
205C2×SD16—21no—
206C2×SD16—21no—
207C2×SD16—21no—
208C2×SD16—21no—
209C2×SD16—21no—
210C2×D8—21no—
211C2×D8—21no—
212#131—42no—
213#131—42no—
214#131—81no—
215#131—81no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—41no—
219C2×M16—41no—
220C2×M16C2×D411yescharacteristic
221C2×M16C2×D411yescharacteristic
222C2×M16C2×D411yescharacteristic
223C2×M16C2×D411yescharacteristic
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—41no—
229C22×C8—41no—
230C22×C8C2×D411yescharacteristic
231C22×C8C2×D411yescharacteristic
232C4⋊Q8C22×C411yescharacteristic
233C4⋊Q8C22×C411yescharacteristic
234#127C22×C411yescharacteristic
235#127C22×C411yescharacteristic
236#124—21no—
237#124—21no—
238#121—21no—
239#121—21no—
240#121—21no—
241#121—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8C22×C411yescharacteristic
245C4×Q8C22×C411yescharacteristic
246C4×Q8C2×D411yescharacteristic
247C4×Q8C2×D411yescharacteristic
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4#4111yescharacteristic
255C2×C42—21no—
256C2×C42—21no—
257C2×C42C2411yescharacteristic
258#108—22no—
259#108—41no—
260#108—42no—
261#108—42no—
262#106—21no—
263#106—21no—
264#106—21no—
265#106—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—41no—
277C4⋊C8—41no—
278C4⋊C8—41no—
279C4⋊C8—41no—
280C4⋊C8C22×C411yescharacteristic
281C4⋊C8C22×C411yescharacteristic
282C4⋊C8C22×C411yescharacteristic
283C4⋊C8C22×C411yescharacteristic
284C4⋊C8C22×C411yescharacteristic
285C4⋊C8C22×C411yescharacteristic
286C4⋊C8C2×D411yescharacteristic
287C4⋊C8C2×D411yescharacteristic
288#104—42no—
289#104—42no—
290#104—81no—
291#104—81no—
292#103—21no—
293#103—21no—
294#103—21no—
295#103—21no—
296#103—42no—
297#102—21no—
298#102—21no—
299#102—21no—
300#102—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—41no—
304C22⋊C8—41no—
305C22⋊C8—41no—
306C22⋊C8—41no—
307#97—21no—
308#97—21no—
309#97—41no—
310#97—41no—
311#97—42no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—41no—
324C4×C8—41no—
325C4×C8C22×C411yescharacteristic
326C4×C8C22×C411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8#4111yescharacteristic
330C4×C8#4111yescharacteristic
331#95—21no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340C2×Q8—21no—
341C2×Q8—22no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—41no—
345C2×Q8C4×D411yescharacteristic
346C2×Q8C4×D411yescharacteristic
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—41no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4C22×D411yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C42−1+411yescharacteristic
363Q16—42no—
364Q16—81no—
365SD16—42no—
366SD16—42no—
367SD16—42no—
368SD16—42no—
369SD16—81no—
370SD16—81no—
371SD16—81no—
372SD16—81no—
373D8—42no—
374D8—81no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—82no—
387M16—82no—
388M16AGL(1,ℤ/8)12yes—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—42no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—82no—
418C2×C8—82no—
419C2×C8C4×D411yescharacteristic
420C2×C8C4×D411yescharacteristic
421C2×C8C4×D411yescharacteristic
422C2×C8C4×D411yescharacteristic
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4C4×D411yescharacteristic
435C4⋊C4C4×D411yescharacteristic
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—41no—
445C42—41no—
446C42C23×C411yescharacteristic
447C42C22×D411yescharacteristic
448C42C2×#4111yescharacteristic
449C42H(5,𝔽2)11yescharacteristic
450C23—41no—
451C23—41no—
452C23#54811yescharacteristic
453Q8—21no—
454Q8—21no—
455Q8—21no—
456Q8—42no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461Q8#44412yescomplemented
462D4—41no—
463D4—41no—
464D4—41no—
465D4—41no—
466D4—41no—
467D4—41no—
468D4—41no—
469D4—41no—
470D4—81no—
471D4—81no—
472D4—81no—
473D4—81no—
474D4—81no—
475D4—81no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4C2×C4×D411yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4#52011yescharacteristic
497C2×C4D4211yescharacteristic
498C2×C4#54711yescharacteristic
499C2×C4#55311yescharacteristic
500C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
501C2×C4#57511yescharacteristic
502C2×C4#57711yescharacteristic
503C2×C4#57711yescharacteristic
504C8—22no—
505C8—21no—
506C8—21no—
507C8—21no—
508C8—21no—
509C8—22no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—81no—
523C8—81no—
524C8—81no—
525C22—21no—
526C22—21no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22#236111yescharacteristic
534C22#336811yescharacteristic
535C22#337111yescharacteristic
536C4—21no—
537C4—21no—
538C4—21no—
539C4—22no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—81no—
544C4—81no—
545C4—161no—
546C4C4×AGL(1,ℤ/8)11yescharacteristic
547C4#294911yescharacteristic
548C4C2×#44411yescharacteristic
549C4#295711yescharacteristic
550C4#326511yescharacteristic
551C4#327111yescharacteristic
552C2—21no—
553C2—21no—
554C2—161no—
555C2—161no—
556C2#2301011yescharacteristic
557C2#2301911yescharacteristic
558C2#2304211yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#3147C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22746C2semidirect
4. #18004C2semidirect
5. #17997C2semidirect
7. #9632C2semidirect
8. #9592C2semidirect
9. #9590C2semidirect
10. #8992C2semidirect
11. #8992C2semidirect
12. #8932C2semidirect
13. #8930C2semidirect
14. #8297C2semidirect
15. #8205C2semidirect
40. #1670C4semidirect
41. #1670C4semidirect
42. #1668C4semidirect
43. #1668C4semidirect
44. #1666C4semidirect
45. #1666C4semidirect
46. #1666C4semidirect
47. #1666C4semidirect
461. Q8#444semidirect

Maximal subgroups

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

subgroupindex
#231802
#227462
#191142
#180042
#179972
#98602
#96322
#95922
#95902
#89922
#89322
#89302
#82972
#82052