Finite Group Databasebeta

← #6468646

Subgroups — #6468646

Order 512.

1626 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35332C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#29359C212yesmaximal, complemented
4#29358C212yesmaximal, complemented
5#29342C214yesmaximal, complemented
6#29223C212yesmaximal, complemented
7#28754C212yesmaximal, complemented
8#27979C212yesmaximal, complemented
9#25908C212yesmaximal, complemented
10#23827C214yesmaximal, complemented
11#21655C212yesmaximal, complemented
12C4×#489C212yesmaximal, complemented
13#21399C212yesmaximal
14#21398C212yesmaximal, complemented
15#20465C211yesmaximal, characteristic, complemented
16C2×#541C2211yescharacteristic
17C2×#540C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#489C2212yes—
20C2×#489C2212yes—
21C2×#433C2212yes—
22#2859C2211yescharacteristic
23#2858C2211yescharacteristic
24#2850C2212yes—
25#2642C2211yescharacteristic
26#2641C2211yescharacteristic
27#2620C2212yes—
28#2559—22no—
29#2559—22no—
30#2558—22no—
31#2558—22no—
32#2369—22no—
33#2365—22no—
34#2345—22no—
35Q8×C22⋊C4—22no—
36#2331C2212yes—
37#2331C2212yes—
38#2326C2212yes—
39#2316C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×#124C2212yes—
43#2285C2211yescharacteristic
44#2274C2212yes—
45C4×#117C2211yescharacteristic
46#2100C2214yes—
47#2099C2214yes—
48#2074C2214yes—
49#2073C2214yes—
50#2042C2214yes—
51#2042C2214yes—
52#2024C2214yes—
53#2023C2214yes—
54#1993C2212yes—
55#1993C2212yes—
56#1989C2214yes—
57#1989C2214yes—
58#1982C2214yes—
59#1981C2214yes—
60#1959C2212yes—
61#1959C2212yes—
62#1936C2214yes—
63#1935C2214yes—
64#1934C2214yes—
65#1933C2214yes—
66#1875C2214yes—
67#1874C2214yes—
68#1873C2214yes—
69#1872C2214yes—
70#1836C2212yes—
71#1836C2212yes—
72#1768C2212yes—
73#1767C2212yes—
74#1763C2214yes—
75#1762C2214yes—
76C4×#103C2214yes—
77C4×#102C2214yes—
78#1750C2212yes—
79#1749C2212yes—
80#1746C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83C23×Q8—21no—
84#541—28no—
85#540—28no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×#127C2311yescharacteristic
93C2×#126—22no—
94C2×#125—22no—
95C2×#125C2311yescharacteristic
96C2×#125C2311yescharacteristic
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×#123—22no—
100C2×C22⋊Q8C2312yes—
101C2×#121—22no—
102C2×#121C2312yes—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8C2312yes—
106C2×C4×D4C2312yes—
107C2×#117C2312yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C22×C4⋊C4—21no—
111C22×C4⋊C4C2312yes—
112C22×C42C2311yescharacteristic
113#489—28no—
114#489—28no—
115#433—28no—
116C2×C4⋊C8C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×C22⋊C8C2314yes—
126C2×#97C2312yes—
127C2×#97C2312yes—
128C2×#97C2312yes—
129C2×#97C2312yes—
130C2×C4×C8C2312yes—
131C2×C4×C8C2312yes—
132#400—22no—
133#400—22no—
134#396—22no—
135#396—22no—
136#394—22no—
137#394—22no—
138#394—22no—
139#394—22no—
140#391—22no—
141#391—22no—
142#391—24no—
143#391—24no—
144#391C2312yes—
145#391C2312yes—
146#390—22no—
147#390—22no—
148#390C2312yes—
149#390C2312yes—
150#389—24no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389C2314yes—
156#389C2314yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#387—24no—
164#387—24no—
165#386—22no—
166#386—22no—
167#385—22no—
168#385—22no—
169#384C2312yes—
170#384C2312yes—
171#382C2312yes—
172#382C2312yes—
173#380—21no—
174#380—21no—
175#380—24no—
176C4×C4⋊C4—24no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4D412yes—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4C2314yes—
184C4×C4⋊C4C2314yes—
185C4×C22⋊C4—24no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4D412yes—
189C4×C22⋊C4D412yes—
190#376C2311yescharacteristic
191#376C2311yescharacteristic
192C43C2311yescharacteristic
193C43C2311yescharacteristic
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#336C2314yes—
203#336C2314yes—
204C22×Q8—28no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—84no—
208C22×Q8C2412yes—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2411yescharacteristic
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—28no—
220C4⋊Q8—28no—
221C4⋊Q8—28no—
222C4⋊Q8—24no—
223C4⋊Q8—24no—
224C4⋊Q8#4118yes—
225#127—24no—
226#126—48no—
227#125—24no—
228#125—24no—
229#125—48no—
230#124—24no—
231#124—24no—
232#124—44no—
233#123—48no—
234C22⋊Q8—28no—
235#121—28no—
236#121—48no—
237C4×Q8—28no—
238C4×Q8—84no—
239C4×Q8—84no—
240C4×D4—28no—
241#117—42no—
242#117—44no—
243#117#4118yes—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2414yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D412yes—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297C4⋊C8—28no—
298#103—28no—
299#103—28no—
300#103—44no—
301#103—44no—
302#102—28no—
303#102—28no—
304#102—44no—
305#102—44no—
306C22⋊C8—28no—
307#97—24no—
308#97—24no—
309#97—24no—
310#97—42no—
311C4×C8—24no—
312C4×C8—42no—
313#95—24no—
314#95—24no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95C2×D412yes—
326#95C2×D412yes—
327#95C2×D412yes—
328#95C2×D412yes—
329C24—22no—
330C24C22×D411yescharacteristic
331C2×Q8—28no—
332C2×Q8—28no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—84no—
340C2×Q8—816no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—82no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—82no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—84no—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4114yes—
377C22×C4C2511yescharacteristic
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4#12418yes—
405C4⋊C4C2×#4118yes—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4#12418yes—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—44no—
430C42—44no—
431C42—84no—
432C42—84no—
433C42C2×#4114yes—
434C42C2×#4114yes—
435C42C2×#4114yes—
436C23—22no—
437C23—22no—
438C23—21no—
439C23—21no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23C2×#12412yes—
448C23C23×D411yescharacteristic
449Q8—416no—
450Q8—44no—
451Q8—44no—
452Q8—88no—
453D4—44no—
454D4—44no—
455D4—164no—
456D4—164no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—28no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—28no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—88no—
494C2×C4—164no—
495C2×C4#52512yes—
496C2×C4#52512yes—
497C2×C4C2×#12414yes—
498C2×C4C2×#12414yes—
499C2×C4C2×#12412yes—
500C2×C4C22×#4112yes—
501C2×C4C22×#4112yes—
502C2×C4#58512yes—
503C8—44no—
504C8—44no—
505C22—22no—
506C22—24no—
507C22—42no—
508C22—41no—
509C22—41no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22C2×#52511yescharacteristic
518C22C22×#12411yescharacteristic
519C22#344911yescharacteristic
520C22#358512yes—
521C22#358612yes—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—24no—
526C4—24no—
527C4—42no—
528C4—42no—
529C4—41no—
530C4—48no—
531C4—41no—
532C4—84no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4#348014yes—
537C2—42no—
538C2—162no—
539C2#3495511yescharacteristic
540C2#6056411yescharacteristic
541C2#6056611yescharacteristic
542C2#6062512yes—
543C2#6102112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. #29359C2semidirect
4. #29358C2semidirect
5. #29342C2semidirect
6. #29223C2semidirect
7. #28754C2semidirect
8. #27979C2semidirect
9. #25908C2semidirect
10. #23827C2semidirect
11. #21655C2semidirect
12. C4×#489C2semidirect
14. #21398C2semidirect
15. #20465C2semidirect

Maximal subgroups

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

subgroupindex
#353322
#342162
#293592
#293582
#293422
#292232
#287542
#279792
#259082
#238272
#216552
C4×#4892
#213992
#213982
#204652