Finite Group Databasebeta

← #198834

Subgroups — #198834

Order 512.

1072 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C211yesmaximal, characteristic
2#60237C211yesmaximal, characteristic
3#17596C211yesmaximal, characteristic, complemented
4#17589C211yesmaximal, characteristic
5#17491C211yesmaximal, characteristic, complemented
6#9279C218yesmaximal, complemented
7#7966C211yesmaximal, characteristic, complemented
8#7857C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3228—22no—
12#3228—22no—
13#3228C414yescomplemented
14#3228C2214yes—
15C2×#500C411yescharacteristic, complemented
16C2×#498C411yescharacteristic, complemented
17C2×#486C411yescharacteristic, complemented
18C2×#486C411yescharacteristic, complemented
19C2×#482—22no—
20C2×#478—22no—
21#2981C2214yes—
22#2980—22no—
23#2974—22no—
24#2973—22no—
25#2935—22no—
26#2932—22no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2880—22no—
32#2283C2211yescharacteristic
33#1520—22no—
34#1517—22no—
35#1515—22no—
36#1510—22no—
37#1506C2214yescomplemented
38#1505C2214yes—
39#1497C2214yescomplemented
40#1487—22no—
41#1485—22no—
42#1287C2214yescomplemented
43#1274C2214yescomplemented
44C22×#41—21no—
45C2×#131—24no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533C2×C412yes—
53#533C2312yes—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532C2×C412yes—
59#532D412yes—
60#532D412yes—
61#532C2312yes—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×#127C2×C411yescharacteristic
66C2×#125—21no—
67C2×#124—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×#121—21no—
71C2×#121D412yes—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×D4—21no—
75C2×C4×D4D412yes—
76C2×#117—21no—
77C22×C42C2311yescharacteristic
78#500—22no—
79#500C2×C414yes—
80#498—22no—
81#498C2×C414yes—
82#486—22no—
83#486—22no—
84#486C2×C414yes—
85#486C2×C414yes—
86#482—28no—
87#482—28no—
88#478—28no—
89#478—28no—
90#436—24no—
91#435—28no—
92#435—28no—
93#435—28no—
94C8×D4—24no—
95#432—21no—
96#432—21no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432D412yes—
101#432C2312yes—
102#432C2312yes—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431C2×C412yes—
108#431D412yes—
109#431C2312yes—
110#431C2312yes—
111C2×#107—22no—
112C2×#106—22no—
113#423—21no—
114#423—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×#102—22no—
118C2×#102—22no—
119#408—24no—
120#408—28no—
121#407—24no—
122#407—21no—
123#407—21no—
124C2×C22⋊C8—24no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8D412yes—
127C4×M16—21no—
128C4×M16D412yes—
129C2×#97—21no—
130C2×C4×C8C2×C411yescharacteristic
131C2×C4×C8C2×C411yescharacteristic
132#383—21no—
133#382—21no—
134#381—21no—
135#380—21no—
136C4×C4⋊C4—21no—
137C4×C22⋊C4—21no—
138#376C2312yes—
139#328—28no—
140#327—28no—
141Q8⋊C8—28no—
142C2×#41—22no—
143C2×#41—22no—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41—84no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C22⋊C412yes—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4#4111yescharacteristic
158#131—88no—
159C2×M16—22no—
160C2×M16—28no—
161C2×M16—41no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—28no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8C22⋊C412yes—
175C4⋊Q8C22×C412yes—
176C4⋊Q8C22×C412yes—
177C4⋊Q8C22×C412yes—
178#127—21no—
179#127—21no—
180#127C22⋊C412yes—
181#127C22×C412yes—
182#125—22no—
183#125—22no—
184#125—41no—
185#125—41no—
186#124—22no—
187#124—22no—
188#124—22no—
189#124—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—41no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198#121—22no—
199#121—22no—
200#121—22no—
201#121—22no—
202#121—22no—
203#121—22no—
204#121C22⋊C414yes—
205#121C2×D414yes—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4C22⋊C414yes—
221C4×D4C2×D414yes—
222#117—22no—
223#117—22no—
224#117—41no—
225#117—41no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4C22⋊C412yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C42—22no—
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—
254#106—28no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—41no—
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—41no—
274C22⋊C8C2×D414yes—
275#97—22no—
276#97—22no—
277#97—22no—
278#97—22no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—41no—
282C4×C8C22×C412yes—
283C4×C8C22×C412yes—
284C4×C8C2×D414yes—
285#95—22no—
286#95—22no—
287#95—41no—
288#95—41no—
289C24—22no—
290C24#12111yescharacteristic
291#41—48no—
292#41—84no—
293#41—84no—
294#41—84no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—82no—
302C2×Q8—82no—
303C2×Q8—82no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—44no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4—84no—
315C2×D4C2×C22⋊C414yes—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—84no—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12111yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12311yescharacteristic
340C22×C4#12311yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4111yescharacteristic
344M16—44no—
345M16—88no—
346M16—82no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—44no—
354C2×C8—42no—
355C2×C8—88no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—82no—
372C4⋊C4C2×C22⋊C414yes—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—82no—
380C42—44no—
381C42—42no—
382C42—42no—
383C42C2×C22⋊C411yescharacteristic
384C42C2×C22⋊C411yescharacteristic
385C42C4×D411yescharacteristic
386C42C4×D411yescharacteristic
387C42C4×D411yescharacteristic
388C42C4×D411yescharacteristic
389C42#12011yescharacteristic
390C42#12011yescharacteristic
391C42#12111yescharacteristic
392C42#12111yescharacteristic
393C42#12311yescharacteristic
394C42#12311yescharacteristic
395C42C23×C411yescharacteristic
396C42C22×D411yescharacteristic
397C42C22×D411yescharacteristic
398C42C2×#4111yescharacteristic
399C23—22no—
400C23—24no—
401C23—41no—
402C23—41no—
403C23—44no—
404C23—82no—
405C23—82no—
406C23#38611yescharacteristic
407C23#38611yescharacteristic
408C23C2×#12111yescharacteristic
409Q8—44no—
410Q8—44no—
411Q8—82no—
412Q8—82no—
413Q8—82no—
414Q8—82no—
415D4—44no—
416D4—48no—
417D4—44no—
418D4—44no—
419D4—82no—
420D4—82no—
421D4—82no—
422D4—84no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—48no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—84no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4#38611yescharacteristic
453C2×C4#38611yescharacteristic
454C2×C4#38611yescharacteristic
455C2×C4#38611yescharacteristic
456C2×C4#38611yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4C22×C22⋊C411yescharacteristic
459C2×C4C2×C4×D411yescharacteristic
460C2×C4C2×C4×D411yescharacteristic
461C2×C4C2×#12011yescharacteristic
462C2×C4C2×#12111yescharacteristic
463C2×C4C2×#12311yescharacteristic
464C8—42no—
465C8—42no—
466C8—44no—
467C8—44no—
468C8—82no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22—82no—
481C22—164no—
482C22#190912yes—
483C22C2×#38611yescharacteristic
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—44no—
495C4—82no—
496C4—82no—
497C4—162no—
498C4#190412yes—
499C4#295712yes—
500C2—22no—
501C2—41no—
502C2—41no—
503C2—84no—
504C2#2091911yescharacteristic
505C2C2×#190411yescharacteristic
506C2C2×#190911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17596C2semidirect
5. #17491C2semidirect
6. #9279C2semidirect
7. #7966C2semidirect
8. #7857C2semidirect
13. #3228C4semidirect
15. C2×#500C4semidirect
16. C2×#498C4semidirect
17. C2×#486C4semidirect
18. C2×#486C4semidirect
37. #1506C22semidirect
39. #1497C22semidirect
42. #1287C22semidirect
43. #1274C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32282
#602372
#175962
#175892
#174912
#92792
#79662
#78572