Finite Group Databasebeta

← #205035

Subgroups — #205035

Order 512.

1213 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35238C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic, complemented
3C2×#1505C211yesmaximal, characteristic, complemented
4C2×#1509C211yesmaximal, characteristic, complemented
5C2×#1502C211yesmaximal, characteristic, complemented
6C2×#1500C211yesmaximal, characteristic, complemented
7#13797C211yesmaximal, characteristic, complemented
8#9716C214yesmaximal, complemented
9#9715C214yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C2×#529C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2856C2212yes—
15#2850C2212yes—
16#2662—22no—
17#2567C411yescharacteristic, complemented
18#2566C412yescomplemented
19#2563C411yescharacteristic, complemented
20#2548C411yescharacteristic, complemented
21#2544C412yescomplemented
22#2541C411yescharacteristic, complemented
23#2430—22no—
24#2416—22no—
25#2324—22no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#1940—22no—
30#1939—22no—
31#1797C2214yes—
32#1509—24no—
33#1505—24no—
34#1502—24no—
35#1500—24no—
36C2×#328C2212yes—
37C2×#328C2212yes—
38C2×#327C2212yes—
39C2×#327C2214yes—
40C2×#327C2212yes—
41C2×#327C2214yes—
42C4⋊#95—22no—
43#1434—22no—
44#1434C2214yes—
45C22×#41—21no—
46#532—22no—
47#532—22no—
48#532—22no—
49#532—22no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#127C2×C412yes—
53#529—22no—
54#529—22no—
55#529—22no—
56#529—22no—
57C2×#126—21no—
58C2×#125C2×C411yescharacteristic
59C2×#124—21no—
60C2×#124C2×C411yescharacteristic
61C2×#124C2×C412yes—
62C2×#123—21no—
63C2×C22⋊Q8C2311yescharacteristic
64C2×#121—21no—
65C2×#121—24no—
66C2×#121—22no—
67C2×#121C2312yes—
68C2×#121C2311yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C412yes—
74C2×#117—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#432—22no—
82#432—22no—
83#431—22no—
84#431—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—24no—
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2312yes—
90C2×#97—21no—
91C2×C4×C8—21no—
92#396—24no—
93#395—22no—
94#394—24no—
95#394—22no—
96#393—22no—
97#392—22no—
98#391C2×C412yes—
99#390—22no—
100#390—24no—
101#390C2×C412yes—
102#390C2×C412yes—
103#390C2×C412yes—
104#389—22no—
105#388—24no—
106#386—24no—
107#386—24no—
108#385—24no—
109#384C2×C412yes—
110#382C2×C412yes—
111#380—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4D412yes—
117C4×C4⋊C4D412yes—
118C4×C4⋊C4D414yescomplemented
119#376—22no—
120#376D414yescomplemented
121C2×#95—21no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95D412yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127#336—24no—
128#328—24no—
129#328—24no—
130#327—24no—
131#327—28no—
132#327—28no—
133#327—24no—
134C2×#41—42no—
135C2×#41—42no—
136C2×#41—82no—
137C2×#41—82no—
138C22×Q8—21no—
139C22×Q8C22×C411yescharacteristic
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—24no—
144C22×D4—21no—
145C22×D4C22×C411yescharacteristic
146C22×D4C22×C412yes—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D414yes—
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159#127—22no—
160#127—22no—
161#127—82no—
162#126—42no—
163#126—42no—
164#125—22no—
165#125—22no—
166#124—22no—
167#124—22no—
168#124—42no—
169#124—42no—
170#124—82no—
171#123—42no—
172#123—42no—
173C22⋊Q8—21no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177#121—21no—
178#121—21no—
179#121—22no—
180#121—22no—
181#121—21no—
182#121—21no—
183#121—22no—
184#121—22no—
185#121—48no—
186#121—42no—
187#121—42no—
188#121—84no—
189#121—84no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—42no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—42no—
200#117—42no—
201#117—42no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C414yes—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C42—21no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—44no—
232C2×C42—82no—
233C2×C42—82no—
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D414yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4112yes—
244C2×C42#4112yes—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—84no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252#97—42no—
253C4×C8—42no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—22no—
259#95—84no—
260#95—84no—
261#95C22⋊C414yes—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C2×D414yes—
265#95C2×D414yes—
266C24—22no—
267C24—21no—
268C24C22×D411yescharacteristic
269#41—84no—
270#41—84no—
271#41—88no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—82no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—48no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—44no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—44no—
298C2×D4—82no—
299C2×D4—84no—
300C2×D4—84no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—84no—
330C22×C4C2×C22⋊C414yes—
331C22×C4C2×C22⋊C414yes—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—84no—
369C22⋊C4—48no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—42no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C42—44no—
384C42—42no—
385C42—42no—
386C42—48no—
387C42—42no—
388C42—84no—
389C42C4×D412yes—
390C42C4×D412yes—
391C42C4×D412yes—
392C42C4×D412yes—
393C42#12412yes—
394C42#12412yes—
395C42#12712yes—
396C42#12712yes—
397C23—21no—
398C23—21no—
399C23—22no—
400C23—42no—
401C23—41no—
402C23—41no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—44no—
407C23—42no—
408C23—41no—
409C23—42no—
410C23C22×C22⋊C411yescharacteristic
411C23C2×#12111yescharacteristic
412C23C2×#12111yescharacteristic
413C23#57612yes—
414C23#57712yes—
415Q8—81no—
416Q8—81no—
417Q8—84no—
418D4—81no—
419D4—81no—
420D4—82no—
421D4—82no—
422D4—84no—
423D4—81no—
424D4—82no—
425D4—81no—
426D4—82no—
427D4—82no—
428D4—84no—
429D4—84no—
430D4—168no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—88no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—82no—
472C2×C4—84no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4—82no—
477C2×C4—84no—
478C2×C4#39012yes—
479C2×C4#39012yes—
480C2×C4#39012yes—
481C2×C4#39012yes—
482C2×C4#42114yes—
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C2×#12411yescharacteristic
486C2×C4C2×#12711yescharacteristic
487C8—82no—
488C8—82no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—82no—
498C22—81no—
499C22—82no—
500C22—84no—
501C22—82no—
502C22—84no—
503C22—81no—
504C22—82no—
505C22—82no—
506C22—81no—
507C22C2×#39011yescharacteristic
508C22#288311yescharacteristic
509C22#288411yescharacteristic
510C22#289712yes—
511C22C2×#42112yes—
512C22#304111yescharacteristic
513C22#304311yescharacteristic
514C22#304512yes—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—44no—
523C4—44no—
524C4—84no—
525C4—84no—
526C4—82no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4#196314yes—
531C2—21no—
532C2—22no—
533C2—21no—
534C2—162no—
535C2—161no—
536C2#1759912yes—
537C2#1760212yes—
538C2C2×#196311yescharacteristic
539C2#2121411yescharacteristic
540C2#2126311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35238C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21953C2semidirect
3. C2×#1505C2semidirect
4. C2×#1509C2semidirect
5. C2×#1502C2semidirect
6. C2×#1500C2semidirect
7. #13797C2semidirect
8. #9716C2semidirect
9. #9715C2semidirect
17. #2567C4semidirect
18. #2566C4semidirect
19. #2563C4semidirect
20. #2548C4semidirect
21. #2544C4semidirect
22. #2541C4semidirect
118. C4×C4⋊C4D4semidirect
120. #376D4semidirect

Maximal subgroups

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

subgroupindex
#352382
#219532
C2×#15052
C2×#15092
C2×#15022
C2×#15002
#137972
#97162
#97152