Finite Group Databasebeta

← #164675

Subgroups — #164675

Order 512.

1039 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36404C211yesmaximal, characteristic
2#22469C211yesmaximal, characteristic, complemented
3#20684C211yesmaximal, characteristic, complemented
4#7721C212yesmaximal, complemented
5#7687C211yesmaximal, characteristic, complemented
6#7667C211yesmaximal, characteristic, complemented
7#7655C212yesmaximal, complemented
8#7645C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic, complemented
10#7578C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic, complemented
12C2×#547C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C8×D4—21no—
15C2×#408—21no—
16#2857C2211yescharacteristic
17#2855C2212yes—
18#2568C411yescharacteristic
19#2538C411yescharacteristic
20#2470—22no—
21#2457—22no—
22#2420—21no—
23#2415C412yes—
24#2407—21no—
25#2407C412yes—
26#2360—21no—
27#2354C411yescharacteristic
28#2352—21no—
29C4×#121C2211yescharacteristic
30#2297C2212yes—
31#1984—21no—
32#1869—22no—
33#1818—22no—
34#1817C2212yes—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2211yescharacteristic
38C4×C22⋊C8C2211yescharacteristic
39#1729—21no—
40#1728C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×#41C2×C412yes—
54C22×#41D411yescharacteristic
55C22×M16—21no—
56C23×C8—21no—
57C23×C8—22no—
58#547—44no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#123—22no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8C2×C412yes—
63C2×C22⋊Q8C2×C412yes—
64C2×#121—21no—
65C2×#121C2×C411yescharacteristic
66C4×#41—44no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C412yes—
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D412yes—
75C2×#117C2×C412yes—
76C2×#117D412yes—
77C2×#117D411yescharacteristic
78C22×C4⋊C4—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2312yes—
81C8×D4—44no—
82C2×C4⋊C8—21no—
83#408—44no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90#400—22no—
91#397—22no—
92#396—22no—
93#394—22no—
94#394—22no—
95#392—22no—
96#391—21no—
97#391C2×C412yes—
98#391C2×C412yes—
99#391C2312yes—
100#390—21no—
101#390C2×C412yes—
102#390C2311yescharacteristic
103#389—22no—
104#389—21no—
105#389—21no—
106#389C2×C412yes—
107#389C2311yescharacteristic
108#388—22no—
109#388—22no—
110#388—21no—
111#387—22no—
112#387—22no—
113#386—22no—
114#386—22no—
115#386—21no—
116#386—22no—
117#385—22no—
118#385—22no—
119#384C2×C411yescharacteristic
120#382—21no—
121#381—22no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380C2×C412yes—
126#380C2×C411yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C22⋊C4—22no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—21no—
136#336—21no—
137#336—22no—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—24no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41—44no—
151C2×#41—82no—
152C22×Q8C22⋊C411yescharacteristic
153C22×Q8C22×C412yes—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4C22⋊C412yes—
157C22×D4C22⋊C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—22no—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C2×M16—42no—
168C2×M16—42no—
169C2×M16—42no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—44no—
177C22×C8—42no—
178C22×C8—42no—
179C22×C8—44no—
180C22×C8—42no—
181C22×C8—82no—
182C4⋊Q8—42no—
183#123—48no—
184C22⋊Q8—44no—
185C22⋊Q8—44no—
186C22⋊Q8—82no—
187#121—22no—
188#121—41no—
189#121—44no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—42no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—42no—
198C4×D4—42no—
199#117—44no—
200#117—42no—
201#117—82no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C412yes—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—42no—
236C2×C42—44no—
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2411yescharacteristic
244C4⋊C8—42no—
245C22⋊C8—42no—
246C22⋊C8—44no—
247C22⋊C8—44no—
248C22⋊C8—44no—
249C22⋊C8—81no—
250C4×C8—42no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—21no—
262C24—21no—
263C24C2×C22⋊C412yes—
264C24C2×C22⋊C411yescharacteristic
265#41—48no—
266#41—48no—
267#41—88no—
268#41—88no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—42no—
288C2×D4—81no—
289C2×D4—82no—
290C2×D4#9914yes—
291C2×D4#13112yes—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—82no—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4#13512yes—
332C22×C4#13512yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336M16—84no—
337C2×C8—22no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—88no—
356C2×C8—84no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—81no—
374C4⋊C4—82no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—42no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C42—22no—
392C42—22no—
393C42—41no—
394C42—42no—
395C42—41no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—82no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—22no—
405C23—22no—
406C23—21no—
407C23—22no—
408C23—21no—
409C23—22no—
410C23—22no—
411C23—21no—
412C23—22no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23C2×#9914yes—
420C23#41612yes—
421C23C22×C22⋊C411yescharacteristic
422Q8—44no—
423Q8—82no—
424Q8—82no—
425D4—28no—
426D4—42no—
427D4—42no—
428D4—48no—
429D4—84no—
430D4—88no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—88no—
480C2×C4—84no—
481C2×C4C2×#9912yes—
482C2×C4#41612yes—
483C2×C4#42112yes—
484C2×C4C2×#13111yescharacteristic
485C2×C4#56811yescharacteristic
486C2×C4C2×#13511yescharacteristic
487C2×C4C2×#13511yescharacteristic
488C2×C4#57711yescharacteristic
489C2×C4#57811yescharacteristic
490C8—42no—
491C8—81no—
492C8—81no—
493C8—81no—
494C8—82no—
495C8—82no—
496C22—24no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—24no—
501C22—24no—
502C22—41no—
503C22—41no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—41no—
508C22—42no—
509C22—41no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22#288011yescharacteristic
518C22C22×#9911yescharacteristic
519C22#288811yescharacteristic
520C22C2×#41611yescharacteristic
521C22#289811yescharacteristic
522C22C2×#42111yescharacteristic
523C22#290111yescharacteristic
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—42no—
534C4—41no—
535C4—82no—
536C4—82no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4#151614yes—
541C2—44no—
542C2—42no—
543C2—82no—
544C2#1741111yescharacteristic
545C2#1750811yescharacteristic
546C2#1752311yescharacteristic
547C2C2×#151611yescharacteristic
548C2#1760811yescharacteristic
549C2#1768411yescharacteristic
550C2#1772611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36404C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22469C2semidirect
3. #20684C2semidirect
4. #7721C2semidirect
5. #7687C2semidirect
6. #7667C2semidirect
7. #7655C2semidirect
8. #7645C2semidirect
9. #7616C2semidirect
10. #7578C2semidirect
11. #7565C2semidirect

Maximal subgroups

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

subgroupindex
#364042
#224692
#206842
#77212
#76872
#76672
#76552
#76452
#76162
#75782
#75652