Finite Group Databasebeta

← #164157

Subgroups — #164157

Order 512.

1217 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36274C211yesmaximal, characteristic
2#22469C211yesmaximal, characteristic
3#20336C211yesmaximal, characteristic
4#7687C211yesmaximal, characteristic, complemented
5#7681C212yesmaximal, complemented
6#7667C211yesmaximal, characteristic, complemented
7#7655C212yesmaximal, complemented
8#7617C211yesmaximal, characteristic, complemented
9#7609C212yesmaximal, complemented
10#7566C211yesmaximal, characteristic, complemented
11C2×#1272C212yesmaximal, complemented
12C22×#131—21no—
13C2×D4×Q8C411yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×C8×D4—21no—
16#2857C2211yescharacteristic
17#2855C2212yes—
18#2582C411yescharacteristic
19#2568C411yescharacteristic
20#2451—21no—
21#2450—22no—
22#2439—21no—
23#2438—22no—
24#2417C412yes—
25#2409C412yes—
26#2349—21no—
27D4×C4⋊C4C411yescharacteristic
28#2339—21no—
29C4×C22⋊Q8C2211yescharacteristic
30C2×#391C2212yes—
31#1984—21no—
32#1869—22no—
33#1817C2212yes—
34#1797C2211yescharacteristic
35C4×C22⋊C8C2211yescharacteristic
36#1729—22no—
37#1729—21no—
38#1728C2212yes—
39#1728C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2211yescharacteristic
47#1272C2212yes—
48#1272C2212yes—
49#1272C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2212yes—
52#1271C2211yescharacteristic
53C22×#41D411yescharacteristic
54C23×Q8C2×C412yes—
55C2×#131—28no—
56C2×#131—44no—
57C2×#131—44no—
58C22×M16—22no—
59C22×M16—21no—
60C23×C8—22no—
61C23×C8—21no—
62C23×C8—22no—
63D4×Q8—44no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×C22⋊Q8C2×C412yes—
69C2×C22⋊Q8C2×C412yes—
70C4×#41—44no—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C412yes—
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D412yes—
80C2×#117D412yes—
81C2×#117D411yescharacteristic
82C22×C4⋊C4C2×C412yes—
83C22×C42C2311yescharacteristic
84C22×C42C2312yes—
85C8×D4—44no—
86C2×C4⋊C8—21no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C4×C8—21no—
91#400—22no—
92#398—22no—
93#398—22no—
94#396—22no—
95#395—22no—
96#393—22no—
97#391—21no—
98#391—21no—
99#391—22no—
100#391—22no—
101#391C2×C412yes—
102#391C2×C412yes—
103#391C2×C412yes—
104#391C2312yes—
105#391C2311yescharacteristic
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389C2×C412yes—
111#389C2311yescharacteristic
112#387—22no—
113#387—22no—
114#387—21no—
115#386—22no—
116#386—22no—
117#385—22no—
118#385—21no—
119#385—22no—
120#385—22no—
121#384C2×C411yescharacteristic
122#382—21no—
123#381—22no—
124#380—21no—
125#380—21no—
126#380—22no—
127#380C2×C412yes—
128#380C2×C411yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C22⋊C4—21no—
132C2×#95—22no—
133C2×#95—22no—
134#336—22no—
135#336—22no—
136#336—21no—
137#336—22no—
138#336—21no—
139#336—22no—
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144#336C2312yes—
145#336C2311yescharacteristic
146C2×#41—24no—
147C2×#41—24no—
148C2×#41SD1614yes—
149C2×#41Q1614yes—
150C22×Q8—22no—
151C22×Q8—24no—
152C22×Q8—42no—
153C22×Q8—44no—
154C22×Q8—82no—
155C22×Q8C22⋊C411yescharacteristic
156C22×Q8C22×C412yes—
157C22×Q8C22×C411yescharacteristic
158C22×D4C22⋊C412yes—
159C22×D4C22⋊C411yescharacteristic
160C23×C4—22no—
161C23×C4—22no—
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168#131—432no—
169C2×M16—28no—
170C2×M16—24no—
171C2×M16—44no—
172C2×M16—42no—
173C2×M16—44no—
174C2×M16—42no—
175C22×C8—28no—
176C22×C8—24no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—44no—
184C22×C8—44no—
185C22×C8—42no—
186C22×C8—44no—
187C22×C8—42no—
188C22×C8—44no—
189C22×C8—82no—
190C4⋊Q8—42no—
191C4⋊Q8—44no—
192C22⋊Q8—22no—
193C22⋊Q8—48no—
194C22⋊Q8—41no—
195C22⋊Q8—44no—
196C22⋊Q8—82no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206#117—44no—
207#117—42no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C22⋊C412yes—
235C2×C22⋊C4C22⋊C412yes—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—42no—
239C2×C42—44no—
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2411yescharacteristic
247C4⋊C8—42no—
248C22⋊C8—42no—
249C22⋊C8—44no—
250C22⋊C8—81no—
251C4×C8—42no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—21no—
257#95—21no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—22no—
266#95—21no—
267C24C2×C22⋊C411yescharacteristic
268C24C2×C22⋊C412yes—
269#41—216no—
270#41—216no—
271C2×Q8—21no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—82no—
283C2×Q8—88no—
284C2×Q8—88no—
285C2×Q8#10314yes—
286C2×Q8#13112yescomplemented
287C2×D4—24no—
288C2×D4—24no—
289C2×D4#9914yes—
290C2×D4#10318yes—
291C2×D4#10314yes—
292C2×D4#10414yes—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
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—44no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×SD1614yes—
332C22×C4C2×SD1612yes—
333C22×C4C2×Q1614yes—
334C22×C4C2×Q1612yes—
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338M16—416no—
339M16—48no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—416no—
349C2×C8—48no—
350C2×C8—41no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—88no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—82no—
379C4⋊C4—81no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—81no—
389C22⋊C4—82no—
390C42—22no—
391C42—22no—
392C42—41no—
393C42—42no—
394C42—41no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23C2×#9914yes—
410C23C2×#10312yes—
411C23C2×#10314yes—
412C23C2×#10414yes—
413C23C22×C22⋊C411yescharacteristic
414Q8—28no—
415Q8—42no—
416Q8—42no—
417Q8—82no—
418Q8—82no—
419Q8—88no—
420D4—216no—
421D4#328116yes—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—88no—
473C2×C4—164no—
474C2×C4C2×#9912yes—
475C2×C4C2×#10314yes—
476C2×C4C2×#10312yes—
477C2×C4C2×#10412yes—
478C2×C4C2×#13111yescharacteristic
479C2×C4#56811yescharacteristic
480C2×C4C22×SD1611yescharacteristic
481C2×C4C22×Q1611yescharacteristic
482C2×C4#57711yescharacteristic
483C2×C4#57811yescharacteristic
484C8—42no—
485C8—44no—
486C8—42no—
487C8—42no—
488C8—81no—
489C8—82no—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—24no—
494C22—24no—
495C22—24no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22C2×#32818yes—
504C22#288011yescharacteristic
505C22C22×#9911yescharacteristic
506C22#288811yescharacteristic
507C22C22×#10311yescharacteristic
508C22#289811yescharacteristic
509C22C22×#10411yescharacteristic
510C22#290111yescharacteristic
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—24no—
516C4—21no—
517C4—21no—
518C4—42no—
519C4—41no—
520C4—42no—
521C4—42no—
522C4—41no—
523C4—82no—
524C4—81no—
525C4—81no—
526C4—82no—
527C4—81no—
528C4C2×#32814yes—
529C2—28no—
530C2—42no—
531C2#1741111yescharacteristic
532C2#1750911yescharacteristic
533C2#1751911yescharacteristic
534C2C22×#32811yescharacteristic
535C2#1760811yescharacteristic
536C2#1765211yescharacteristic
537C2#1772211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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
4. #7687C2semidirect
5. #7681C2semidirect
6. #7667C2semidirect
7. #7655C2semidirect
8. #7617C2semidirect
9. #7609C2semidirect
10. #7566C2semidirect
11. C2×#1272C2semidirect
286. C2×Q8#131semidirect

Maximal subgroups

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

subgroupindex
#362742
#224692
#203362
#76872
#76812
#76672
#76552
#76172
#76092
#75662
C2×#12722