Finite Group Databasebeta

← #195532

Subgroups — #195532

Order 512.

1193 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic
2#60237C211yesmaximal, characteristic, complemented
3#21815C211yesmaximal, characteristic, complemented
4#17544C211yesmaximal, characteristic, complemented
5#17495C211yesmaximal, characteristic, complemented
6#9053C218yesmaximal, complemented
7#7902C211yesmaximal, characteristic, complemented
8#7886C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3245—22no—
12#3226C412yescomplemented
13#3226C412yescomplemented
14#3205C412yescomplemented
15#3205C412yescomplemented
16#3186—22no—
17#3169C2214yes—
18#3100—22no—
19#3100—22no—
20#2981—22no—
21#2980—22no—
22#2974—22no—
23#2973—22no—
24#2935C2214yescomplemented
25#2932—22no—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C2×#431C2211yescharacteristic
30#2880—22no—
31C4×#117C2211yescharacteristic
32#1980C2214yescomplemented
33#1962—22no—
34#1789—22no—
35#1507—22no—
36#1505—22no—
37#1502C2214yescomplemented
38#1496—22no—
39#1490—22no—
40#1480C2214yescomplemented
41#1279C2214yescomplemented
42#1278C2214yescomplemented
43C22×#41—21no—
44C22×#41—21no—
45C2×#131—24no—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532C2×C412yes—
57#532C2×C412yes—
58#532D412yes—
59#532D412yes—
60#532C2312yes—
61#532C2312yes—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#127C2×C411yescharacteristic
65C2×#127C2×C411yescharacteristic
66C2×#124—21no—
67C2×#124—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121D412yes—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4D412yes—
76C2×#117—21no—
77C22×C42C2311yescharacteristic
78#500C2×C414yes—
79#493C2×C414yes—
80#486C2×C414yes—
81#486C2×C414yes—
82#480—24no—
83#480—24no—
84#474—24no—
85#474—24no—
86#466—28no—
87#459—24no—
88#459—24no—
89#447—24no—
90#447—24no—
91C4×D8—28no—
92#436—24no—
93#435—28no—
94#435—28no—
95#435—28no—
96C8×D4—24no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432D412yes—
104#432D412yes—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431C2×C412yes—
111#431C2×C412yes—
112#431C2312yes—
113#423—21no—
114#423D412yes—
115C2×C4⋊C8—21no—
116C2×C4⋊C8C2312yes—
117C2×#104—28no—
118#408—28no—
119#408—24no—
120#407—24no—
121#407—21no—
122#407D412yes—
123C2×C22⋊C8—24no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C4×M16—21no—
127C4×M16C2312yes—
128C2×#97—21no—
129C2×C4×C8C2312yes—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2312yes—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134#376—21no—
135C43—21no—
136#327—28no—
137D4⋊C8—28no—
138C2×#41—22no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—22no—
144C2×#41—41no—
145C2×#41—41no—
146C2×#41—84no—
147C2×#41—84no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—41no—
157C22×D4C22⋊C412yes—
158C23×C4—22no—
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C2×D8—28no—
164#131—88no—
165C2×M16—28no—
166C2×M16—41no—
167C2×M16C2×D414yes—
168C22×C8—22no—
169C22×C8—28no—
170C22×C8—22no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8C22⋊C412yes—
177C4⋊Q8C22×C412yes—
178C4⋊Q8C22×C412yes—
179#127—21no—
180#127—21no—
181#127—21no—
182#127—21no—
183#127—21no—
184#127C22⋊C412yes—
185#127C22×C412yes—
186#127C22×C412yes—
187#124—22no—
188#124—22no—
189#124—22no—
190#124—22no—
191#124—22no—
192#124—22no—
193#124—41no—
194#124—41no—
195#121—22no—
196#121—22no—
197#121—22no—
198#121—22no—
199#121—22no—
200#121—22no—
201#121—22no—
202#121—22no—
203#121—22no—
204#121—22no—
205#121—41no—
206#121—41no—
207#121—41no—
208#121—41no—
209#121C22⋊C414yes—
210#121C2×D414yes—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4C22⋊C414yes—
226C4×D4C2×D414yes—
227#117—44no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4C22⋊C412yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C42—22no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42#4111yescharacteristic
251C2×C42#4111yescharacteristic
252C2×C42C2411yescharacteristic
253#107—28no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—41no—
261C4⋊C8C2×D414yes—
262#104—88no—
263#102—28no—
264#102—28no—
265C22⋊C8—28no—
266C22⋊C8—28no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—41no—
275#97—22no—
276#97—22no—
277#97—22no—
278#97—41no—
279C4×C8—22no—
280C4×C8—22no—
281C4×C8—22no—
282C4×C8C22×C414yes—
283#95—22no—
284#95—41no—
285C24—22no—
286C24—41no—
287C24#12111yescharacteristic
288#41—48no—
289#41—48no—
290#41—84no—
291#41—84no—
292#41—84no—
293#41—84no—
294#41—84no—
295#41—84no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—41no—
300C2×Q8—82no—
301C2×Q8—82no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—82no—
323C2×D4—84no—
324C2×D4—84no—
325C2×D4C2×C22⋊C414yes—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4—81no—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4#12011yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12311yescharacteristic
352C22×C4#12311yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4111yescharacteristic
356D8—88no—
357M16—28no—
358M16—88no—
359M16—82no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—44no—
364C2×C8—42no—
365C2×C8—82no—
366C2×C8—88no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4C2×C22⋊C414yes—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—44no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C42—44no—
387C42—44no—
388C42—82no—
389C42C2×C22⋊C411yescharacteristic
390C42C2×C22⋊C411yescharacteristic
391C42C4×D411yescharacteristic
392C42C4×D411yescharacteristic
393C42C4×D411yescharacteristic
394C42C4×D411yescharacteristic
395C42#12011yescharacteristic
396C42#12011yescharacteristic
397C42#12111yescharacteristic
398C42#12111yescharacteristic
399C42#12311yescharacteristic
400C42#12311yescharacteristic
401C42C23×C411yescharacteristic
402C42C22×D411yescharacteristic
403C42C22×D411yescharacteristic
404C42C2×#4111yescharacteristic
405C23—22no—
406C23—24no—
407C23—41no—
408C23—41no—
409C23—44no—
410C23—42no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23#38611yescharacteristic
417C23#38611yescharacteristic
418C23C2×#12111yescharacteristic
419Q8—44no—
420Q8—44no—
421Q8—82no—
422Q8—82no—
423D4—44no—
424D4—44no—
425D4—48no—
426D4—44no—
427D4—44no—
428D4—44no—
429D4—44no—
430D4—82no—
431D4—82no—
432D4—82no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—84no—
437D4—84no—
438D4—84no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—48no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—84no—
461C2×C4—84no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—84no—
467C2×C4#38611yescharacteristic
468C2×C4#38611yescharacteristic
469C2×C4#38611yescharacteristic
470C2×C4#38611yescharacteristic
471C2×C4#38611yescharacteristic
472C2×C4#38611yescharacteristic
473C2×C4C22×C22⋊C411yescharacteristic
474C2×C4C2×C4×D411yescharacteristic
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C2×#12011yescharacteristic
477C2×C4C2×#12111yescharacteristic
478C2×C4C2×#12311yescharacteristic
479C2×C4#57614yes—
480C8—44no—
481C8—44no—
482C8—44no—
483C8—82no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—44no—
493C22—44no—
494C22—44no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—164no—
500C22—164no—
501C22C2×#38611yescharacteristic
502C22#304112yes—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—44no—
513C4—84no—
514C4—82no—
515C4#294812yes—
516C4#295612yes—
517C4#300812yes—
518C2—22no—
519C2—41no—
520C2—41no—
521C2—84no—
522C2—162no—
523C2#2091811yescharacteristic
524C2#2093411yescharacteristic
525C2#2094711yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#60940C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60237C2semidirect
3. #21815C2semidirect
4. #17544C2semidirect
5. #17495C2semidirect
6. #9053C2semidirect
7. #7902C2semidirect
8. #7886C2semidirect
12. #3226C4semidirect
13. #3226C4semidirect
14. #3205C4semidirect
15. #3205C4semidirect
24. #2935C22semidirect
32. #1980C22semidirect
37. #1502C22semidirect
40. #1480C22semidirect
41. #1279C22semidirect
42. #1278C22semidirect

Maximal subgroups

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

subgroupindex
#609402
#602372
#218152
#175442
#174952
#90532
#79022
#78862