Finite Group Databasebeta

← #6346156

Subgroups — #6346156

Order 512.

1628 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61436C211yesmaximal, characteristic
2#60328C211yesmaximal, characteristic, complemented
3#60308C211yesmaximal, characteristic, complemented
4#18091C213yesmaximal, complemented
5#18061C213yesmaximal, complemented
6#18025C213yesmaximal, complemented
7#18022C213yesmaximal, complemented
8#17975C213yesmaximal, complemented
9#17940C213yesmaximal, complemented
10#17917C213yesmaximal, complemented
11#17863C213yesmaximal, complemented
12#17837C211yesmaximal, characteristic, complemented
13#17520C211yesmaximal, characteristic, complemented
14#17520C211yesmaximal, characteristic, complemented
15#17320C211yesmaximal, characteristic, complemented
16#3483C2211yescharacteristic
17#3483C2211yescharacteristic
18#3391C411yescharacteristic
19Q8×Q16C411yescharacteristic
20#3362C411yescharacteristic
21#3362C411yescharacteristic
22#3341C413yes—
23#3305C413yes—
24#3305C413yes—
25#3287C413yes—
26#3169C2213yes—
27#3106C2213yes—
28#3106C2213yes—
29#2991—21no—
30#2991C2213yes—
31#2991C2211yescharacteristic
32#2991C2211yescharacteristic
33#2990C2213yes—
34#2988—21no—
35#2981—23no—
36#2979—23no—
37#2979C2213yes—
38#2979C2213yes—
39#2971—21no—
40#2965C2213yes—
41C8×#41—21no—
42C8×#41C2213yes—
43C8×#41C2211yescharacteristic
44C8×#41C2211yescharacteristic
45C4×#135C2213yes—
46#2935—23no—
47#2931—23no—
48#2931C2213yes—
49#2931C2213yes—
50C4×#131C2211yescharacteristic
51#1594C2216yes—
52#1592C2216yes—
53#1584C2213yes—
54#1584C2211yescharacteristic
55#1583C2213yes—
56#1583C2213yes—
57#1582C2213yes—
58#1582C2213yes—
59#1581C2216yes—
60#1581C2213yes—
61#1581C2216yes—
62#1581C2216yes—
63#1581C2213yes—
64#1580C2216yes—
65SD16⋊C8C2216yes—
66#1574C2213yes—
67#1574C2213yes—
68#1567C2213yes—
69#1564C2213yes—
70#1492C2213yes—
71#1492C2213yes—
72#1489C2213yes—
73#1489C2213yes—
74#1484C2213yes—
75#1484C2213yes—
76#1476C2213yes—
77#1476C2213yes—
78#1452C2213yes—
79#1451C2213yes—
80#1451C2213yes—
81C2×#322C2213yes—
82C2×#135C2×C413yes—
83C2×#131—21no—
84C2×#131—23no—
85C2×#131—23no—
86Q82C2×C411yescharacteristic
87Q82C2×C411yescharacteristic
88#555—23no—
89#555—23no—
90#555C2×C413yes—
91#555C2×C413yes—
92#550—21no—
93#550—21no—
94#550C2×C413yes—
95#550C2×C413yes—
96#550C2×C411yescharacteristic
97#550C2×C411yescharacteristic
98#532—23no—
99#532—23no—
100#532C2313yes—
101#532C2313yes—
102C4×#41—21no—
103C4×#41—21no—
104C4×#41C2313yes—
105C4×#41C2313yes—
106C4×#41C2311yescharacteristic
107#500C2×C413yes—
108#493C2×C413yes—
109#486C2×C413yes—
110#486C2×C413yes—
111#482C2×C416yes—
112#482C2×C416yes—
113#477C2×C413yes—
114#477C2×C413yes—
115#476C2×C413yes—
116#476C2×C413yes—
117#466C2×C416yes—
118#466C2×C416yes—
119#460C2×C413yes—
120#460C2×C413yes—
121#459C2×C413yes—
122#459C2×C413yes—
123#449C2×C416yes—
124#449C2×C416yes—
125#446C2311yescharacteristic
126#446C2313yes—
127C8×Q8—21no—
128C8×Q8—21no—
129C8×Q8C2×C411yescharacteristic
130C8×Q8C2×C411yescharacteristic
131C8×Q8D411yescharacteristic
132C8×Q8D411yescharacteristic
133C8×Q8C2313yes—
134C8×Q8C2311yescharacteristic
135C4×Q16C2×C413yes—
136C4×Q16C2311yescharacteristic
137C4×SD16C2×C413yes—
138C4×SD16C2×C413yes—
139C4×SD16C2313yes—
140C4×SD16C2313yes—
141C4×D8C2×C413yes—
142C4×D8C2316yes—
143C4×D8C2313yes—
144#436—23no—
145#436—23no—
146#436—23no—
147#436—23no—
148#436—26no—
149#436—23no—
150#436—23no—
151#436C2313yes—
152#436C2316yes—
153#436C2313yes—
154#435—23no—
155#435—23no—
156#435—26no—
157C8×D4—23no—
158C8×D4—23no—
159C8×D4—23no—
160C8×D4—23no—
161C8×D4—23no—
162C8×D4—23no—
163C8×D4C2×C413yes—
164C8×D4C2×C413yes—
165C8×D4D413yes—
166C8×D4D413yes—
167C8×D4C2316yes—
168C8×D4C2313yes—
169C8×D4C2313yes—
170#432—23no—
171#431—23no—
172#431—23no—
173#431—23no—
174#431C2313yes—
175#431C2313yes—
176#431C2313yes—
177#431C2313yes—
178#427C2×C413yes—
179#423—23no—
180#423C2313yes—
181#423C2313yes—
182C2×C4⋊C8—23no—
183#416C2×C413yes—
184#416C2×C413yes—
185#408—23no—
186#408—23no—
187#408—26no—
188#408—26no—
189#405—23no—
190C4×M16—23no—
191C4×M16C2313yes—
192C2×C4×C8—23no—
193C2×C4×C8C2313yes—
194C2×C4×C8C2313yes—
195C2×C4×C8C2313yes—
196#335C2311yescharacteristic
197#335C2313yes—
198#335C2316yes—
199#334C2313yes—
200#334C2313yes—
201Q8⋊C8C2313yes—
202Q8⋊C8C2311yescharacteristic
203Q8⋊C8C2313yes—
204Q8⋊C8C2311yescharacteristic
205D4⋊C8C2316yes—
206D4⋊C8C2313yes—
207D4⋊C8C2316yes—
208D4⋊C8C2313yes—
209D4⋊C8C2313yes—
210D4⋊C8C2313yes—
211#322C2316yes—
212#322C2311yescharacteristic
213#322C2313yes—
214#322C2313yes—
215#322C2313yes—
216C2×#41—21no—
217C2×#41—21no—
218C2×#41—23no—
219C2×#41—23no—
220C2×#41C22×C413yes—
221C2×#41C22×C413yes—
222C2×#41#4111yescharacteristic
223#135—46no—
224C2×Q16C22×C411yescharacteristic
225C2×SD16C22×C413yes—
226C2×SD16C22×C413yes—
227C2×D8C22×C413yes—
228C2×D8C22×C416yes—
229#131—44no—
230#131—86no—
231#131—86no—
232C2×M16—23no—
233C2×M16—23no—
234C2×M16—23no—
235C2×M16—23no—
236C2×M16—26no—
237C2×M16—23no—
238C22×C8—23no—
239C22×C8—23no—
240C22×C8—26no—
241C22×C8—23no—
242C22×C8C22×C413yes—
243C22×C8C22×C413yes—
244C22×C8C2×D413yes—
245C22×C8C2×D413yes—
246C4⋊Q8—23no—
247C4⋊Q8—23no—
248C4⋊Q8C22×C413yes—
249C4⋊Q8C22×C413yes—
250#127—23no—
251#127—23no—
252#127C22×C413yes—
253#127C22×C413yes—
254#124—26no—
255#124—26no—
256#124—23no—
257#124—23no—
258#121—23no—
259#121—23no—
260#121—23no—
261#121—23no—
262#121—23no—
263#121—23no—
264#121—26no—
265#121—26no—
266#121C22×C416yes—
267#121C22×C416yes—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8C22×C413yes—
271C4×Q8C22×C413yes—
272C4×Q8C22×C411yescharacteristic
273C4×Q8C2×D411yescharacteristic
274C4×Q8C2411yescharacteristic
275C4×Q8C2411yescharacteristic
276C4×D4—23no—
277C4×D4—23no—
278C4×D4—23no—
279C4×D4—23no—
280C4×D4—23no—
281C4×D4—23no—
282C4×D4—23no—
283C4×D4—23no—
284C4×D4C22×C413yes—
285C4×D4C22×C413yes—
286C4×D4C22×C413yes—
287C4×D4C2×D413yes—
288C4×D4C2413yes—
289C4×D4C2413yes—
290#117—23no—
291#117—23no—
292#117—23no—
293#117—23no—
294#117C22×C413yes—
295#117C22×C413yes—
296#117#4113yes—
297C2×C42—23no—
298C2×C42—23no—
299C2×C42C2413yes—
300#107C22×C411yescharacteristic
301#107C22×C413yes—
302#107C22×C416yes—
303#106C22×C413yes—
304#106C22×C413yes—
305C4⋊C8—23no—
306C4⋊C8—23no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—23no—
310C4⋊C8—23no—
311C4⋊C8C22×C413yes—
312C4⋊C8C22×C413yes—
313C4⋊C8C2×D413yes—
314C4⋊C8C2×D413yes—
315C4⋊C8C2411yescharacteristic
316C4⋊C8C2411yescharacteristic
317C4⋊C8C2413yes—
318C4⋊C8C2413yes—
319#103C22×C411yescharacteristic
320#103C22×C411yescharacteristic
321#103C22×C413yes—
322#103C22×C413yes—
323#102C22×C413yes—
324#102C22×C413yes—
325#102C22×C413yes—
326#102C22×C413yes—
327#102C22×C416yes—
328#102C22×C416yes—
329C22⋊C8—23no—
330C22⋊C8—23no—
331C22⋊C8—23no—
332C22⋊C8—23no—
333C22⋊C8—26no—
334C22⋊C8—23no—
335C22⋊C8—23no—
336C22⋊C8—26no—
337C22⋊C8C22×C416yes—
338C22⋊C8C2×D416yes—
339#97—21no—
340#97—23no—
341#97—23no—
342C4×C8—21no—
343C4×C8—23no—
344C4×C8—23no—
345C4×C8—23no—
346C4×C8—23no—
347C4×C8C22×C413yes—
348C4×C8C22×C413yes—
349C4×C8C2×D413yes—
350C4×C8C2×D413yes—
351C4×C8C2411yescharacteristic
352C4×C8C2413yes—
353C4×C8C2413yes—
354C4×C8C2411yescharacteristic
355C4×C8C2411yescharacteristic
356C4×C8C2413yes—
357#41—24no—
358#41—26no—
359#41—43no—
360#41—43no—
361#41—43no—
362#41—44no—
363#41—44no—
364#41—83no—
365#41—83no—
366#41—83no—
367#41—83no—
368C2×Q8—23no—
369C2×Q8—23no—
370C2×Q8C4×D411yescharacteristic
371C2×Q8C23×C411yescharacteristic
372C2×Q8C23×C411yescharacteristic
373C2×D4—23no—
374C2×D4—23no—
375C2×D4—26no—
376C2×D4—26no—
377C2×D4—23no—
378C2×D4—23no—
379C2×D4—23no—
380C2×D4—23no—
381C2×D4C4×D413yes—
382C2×D4C23×C413yes—
383C2×D4C23×C413yes—
384C22×C4—23no—
385C22×C4—23no—
386C22×C4—23no—
387C22×C4—23no—
388C22×C4—23no—
389C22×C4—23no—
390C22×C4C23×C413yes—
391C22×C4C22×D413yes—
392C22×C4C2×#4113yes—
393Q16—22no—
394SD16—26no—
395SD16—43no—
396D8—46no—
397D8—43no—
398M16—46no—
399M16—46no—
400M16—83no—
401M16—86no—
402M16—83no—
403M16—83no—
404C2×C8—26no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—23no—
408C2×C8—23no—
409C2×C8—21no—
410C2×C8—23no—
411C2×C8—43no—
412C2×C8—43no—
413C2×C8—43no—
414C2×C8—46no—
415C2×C8—46no—
416C2×C8—83no—
417C2×C8—86no—
418C2×C8C23×C411yescharacteristic
419C2×C8C23×C411yescharacteristic
420C2×C8C23×C413yes—
421C2×C8C23×C413yes—
422C2×C8C22×D413yes—
423C2×C8C22×D411yescharacteristic
424C2×C8C22×D413yes—
425C2×C8C22×D411yescharacteristic
426C4⋊C4—23no—
427C4⋊C4—23no—
428C4⋊C4—23no—
429C4⋊C4—23no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4C4×D413yes—
433C4⋊C4C4×D413yes—
434C4⋊C4C23×C411yescharacteristic
435C4⋊C4C23×C411yescharacteristic
436C4⋊C4C23×C413yes—
437C4⋊C4C23×C413yes—
438C22⋊C4—23no—
439C22⋊C4—23no—
440C22⋊C4—26no—
441C22⋊C4—26no—
442C22⋊C4—23no—
443C22⋊C4—23no—
444C22⋊C4—23no—
445C22⋊C4—23no—
446C22⋊C4C4×D416yes—
447C42—23no—
448C42—23no—
449C42—23no—
450C42—23no—
451C42—21no—
452C42—21no—
453C42C23×C413yes—
454C42C22×D413yes—
455C42C2×#4113yes—
456C42C2511yescharacteristic
457C23—23no—
458C23—23no—
459C23C2×C4×D413yes—
460Q8—21no—
461Q8—21no—
462Q8—21no—
463Q8—83no—
464Q8—83no—
465Q8#44414yes—
466Q8#56812yes—
467D4—26no—
468D4—23no—
469D4—23no—
470D4—23no—
471D4—23no—
472D4—46no—
473D4—46no—
474D4—46no—
475D4—83no—
476D4—83no—
477D4—83no—
478D4—86no—
479D4—86no—
480C2×C4—21no—
481C2×C4—23no—
482C2×C4—26no—
483C2×C4—23no—
484C2×C4—23no—
485C2×C4—23no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—23no—
490C2×C4—23no—
491C2×C4—46no—
492C2×C4—43no—
493C2×C4—43no—
494C2×C4—46no—
495C2×C4—46no—
496C2×C4—43no—
497C2×C4—43no—
498C2×C4—83no—
499C2×C4C2×C4×D413yes—
500C2×C4C2×C4×D413yes—
501C2×C4C2×C4×D413yes—
502C2×C4C24×C411yescharacteristic
503C2×C4C23×D411yescharacteristic
504C2×C4C22×#4111yescharacteristic
505C8—21no—
506C8—23no—
507C8—21no—
508C8—23no—
509C8—21no—
510C8—23no—
511C8—23no—
512C8—41no—
513C8—43no—
514C8—41no—
515C8—41no—
516C8—43no—
517C8—43no—
518C8#57612yes—
519C22—23no—
520C22—23no—
521C22—23no—
522C22—43no—
523C22—43no—
524C22—43no—
525C22—43no—
526C22—43no—
527C22—43no—
528C22C22×C4×D411yescharacteristic
529C4—23no—
530C4—21no—
531C4—23no—
532C4—22no—
533C4—43no—
534C4—43no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4C2×#44416yes—
539C4C2×#56811yescharacteristic
540C4#357311yescharacteristic
541C4C2×#57611yescharacteristic
542C4#358411yescharacteristic
543C2—43no—
544C2—83no—
545C2—83no—
546C2#6014611yescharacteristic
547C2C4×#57611yescharacteristic
548C2C22×#44411yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#61436C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C8×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60328C2semidirect
3. #60308C2semidirect
4. #18091C2semidirect
5. #18061C2semidirect
6. #18025C2semidirect
7. #18022C2semidirect
8. #17975C2semidirect
9. #17940C2semidirect
10. #17917C2semidirect
11. #17863C2semidirect
12. #17837C2semidirect
13. #17520C2semidirect
14. #17520C2semidirect
15. #17320C2semidirect

Maximal subgroups

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

subgroupindex
#614362
#603282
#603082
#180912
#180612
#180252
#180222
#179752
#179402
#179172
#178632
#178372
#175202
#173202