Finite Group Databasebeta

← #6806145

Subgroups — #6806145

Order 512.

1518 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36602C211yesmaximal, characteristic
2Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
3#33230C211yesmaximal, characteristic
4#31640C214yesmaximal, complemented
5#31638C214yesmaximal
6#30216C212yesmaximal, complemented
7#29225C211yesmaximal, characteristic, complemented
8#29148C212yesmaximal, complemented
9#28619C212yesmaximal, complemented
10#28447C211yesmaximal, characteristic
11#25942C214yesmaximal, complemented
12#25336C214yesmaximal, complemented
13#22273C212yesmaximal, complemented
14#21657C211yesmaximal, characteristic, complemented
15#21346C211yesmaximal, characteristic, complemented
16C2×Q82C2212yes—
17C2×#497C2212yes—
18C2×#491C2211yescharacteristic
19C2×#489C2214yes—
20C2×#464C2214yes—
21#2868C2212yes—
22#2868C2211yescharacteristic
23#2861C2212yes—
24#2858C2211yescharacteristic
25#2597—22no—
26#2593—22no—
27#2592C2212yes—
28#2567C2214yes—
29#2567C2212yes—
30#2551—22no—
31#2551—22no—
32#2551—22no—
33#2458—22no—
34#2446C2212yes—
35#2445C2212yes—
36#2374—22no—
37#2367C2212yes—
38Q8×C4⋊C4—22no—
39C4×C4⋊Q8C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×#125C2211yescharacteristic
42C4×#124C2212yes—
43C42×Q8C2211yescharacteristic
44#2100C2214yes—
45#2072C2214yes—
46#2071C2214yes—
47#2056C2214yes—
48#2055C2214yes—
49#2024C2214yes—
50#2024C2214yes—
51#2023C2214yes—
52#2023C2214yes—
53#2016C2212yes—
54#2014C2212yes—
55#1992C2211yescharacteristic
56#1992C2212yes—
57#1988C2214yes—
58#1985C2212yes—
59#1960C2211yescharacteristic
60#1959C2212yes—
61#1959C2212yes—
62#1959C2212yes—
63#1957C2212yes—
64#1956C2212yes—
65#1940C2212yes—
66#1933C2214yes—
67#1916C2214yes—
68#1880C2214yes—
69#1880C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1878C2214yes—
73#1877C2214yes—
74#1836C2212yes—
75#1836C2211yescharacteristic
76#1803C2214yes—
77#1802C2214yes—
78#1767C2211yescharacteristic
79#1767C2212yes—
80#1763C2214yes—
81#1745C2212yes—
82C4×#97C2211yescharacteristic
83C22×Q16C2314yes—
84C22×SD16C2314yes—
85Q82—48no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2314yes—
95C2×#127C2311yescharacteristic
96C2×#125C2311yescharacteristic
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×#123—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8D412yes—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2314yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×D4C2312yes—
107#497—28no—
108#491—24no—
109#489—48no—
110#464—48no—
111C2×C4⋊C8C2312yes—
112C2×C4⋊C8C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#97C2311yescharacteristic
122C2×#97C2312yes—
123C2×#97C2312yes—
124C2×#97C2311yescharacteristic
125C2×C4×C8C2312yes—
126#400—22no—
127#398—22no—
128#397—22no—
129#395—22no—
130#395—24no—
131#395—22no—
132#395—24no—
133#394—22no—
134#394—22no—
135#393—22no—
136#391C2314yes—
137#391C2312yes—
138#391C2312yes—
139#391C2312yes—
140#391C2312yes—
141#390—22no—
142#390—22no—
143#390C2312yes—
144#390C2312yes—
145#390C2312yes—
146#389—24no—
147#389—24no—
148#389—24no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389C2312yes—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#386—22no—
167#385—22no—
168#384—22no—
169#384—22no—
170#384—22no—
171#384C2312yes—
172#384C2311yescharacteristic
173#384C2314yes—
174#384C2312yes—
175#382C2311yescharacteristic
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D412yes—
181C4×C4⋊C4D412yes—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2311yescharacteristic
185C4×C4⋊C4C2312yes—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C43C2311yescharacteristic
190C43C2312yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2312yes—
199#336C2314yes—
200#336C2314yes—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8C2×D411yescharacteristic
204C22×Q8C2412yes—
205C22×Q8C2412yes—
206C22×D4—22no—
207C22×D4C2412yes—
208C23×C4—22no—
209C2×Q16—28no—
210C2×Q16—44no—
211C2×Q16—44no—
212C2×SD16—28no—
213C2×SD16—44no—
214C2×SD16—44no—
215C22×C8C2414yes—
216C22×C8C2412yes—
217C22×C8C2412yes—
218C4⋊Q8—28no—
219C4⋊Q8—22no—
220C4⋊Q8—44no—
221C4⋊Q8—44no—
222C4⋊Q8—42no—
223C4⋊Q8—416no—
224C4⋊Q8—48no—
225C4⋊Q8—84no—
226C4⋊Q8—84no—
227C4⋊Q8—84no—
228C4⋊Q8#4114yes—
229#127—42no—
230#125—22no—
231#125—22no—
232#124—24no—
233#124—28no—
234#124—24no—
235#123—48no—
236C4×Q8—28no—
237C4×Q8—24no—
238C4×Q8—28no—
239C4×Q8—44no—
240C4×Q8—44no—
241C4×Q8—44no—
242C4×Q8—48no—
243C4×D4—44no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
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—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2412yes—
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⋊C4—22no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2412yes—
290C2×C42C2412yes—
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C4⋊C8—24no—
295C4⋊C8—44no—
296#103—28no—
297#103—44no—
298#103—44no—
299#103—44no—
300#103—44no—
301#102—44no—
302#102—44no—
303#102—44no—
304#97—22no—
305#97—24no—
306#97—24no—
307#97—22no—
308C4×C8—42no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95C2×D411yescharacteristic
322#95C2×D412yes—
323#95C2×D411yescharacteristic
324#95C2×D412yes—
325C24—22no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—48no—
336C2×Q8—44no—
337C2×Q8—84no—
338C2×Q8—84no—
339C2×Q8—84no—
340C2×Q8C2×#4114yes—
341C2×D4—22no—
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—24no—
345C2×D4—44no—
346C2×D4—82no—
347C2×D4—82no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—44no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365C22×C4C2511yescharacteristic
366Q16—88no—
367SD16—88no—
368C2×C8—22no—
369C2×C8—24no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—24no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—28no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4#13718yes—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C42—24no—
415C42—24no—
416C42—24no—
417C42—24no—
418C42—22no—
419C42—24no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—48no—
424C42—44no—
425C42—82no—
426C42—82no—
427C42C2×#4112yes—
428C42H(5,𝔽2)14yes—
429C422−1+412yes—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23C23×D411yescharacteristic
438Q8—28no—
439Q8—28no—
440Q8—44no—
441Q8—44no—
442Q8—44no—
443Q8—164no—
444Q8—168no—
445D4—44no—
446D4—44no—
447D4—164no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—164no—
482C2×C4#53512yes—
483C2×C4#53512yes—
484C2×C4#54814yes—
485C2×C4C2×#13714yes—
486C2×C4C2×#13714yes—
487C2×C4C2×#13714yes—
488C2×C4C22×#4111yescharacteristic
489C2×C4C2×H(5,𝔽2)12yes—
490C2×C4C2×2−1+411yescharacteristic
491C2×C4#58512yes—
492C8—42no—
493C8—44no—
494C8—42no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22C2×#53511yescharacteristic
503C22C2×#54811yescharacteristic
504C22#350011yescharacteristic
505C22C22×#13712yes—
506C22#358512yes—
507C4—24no—
508C4—24no—
509C4—22no—
510C4—24no—
511C4—24no—
512C4—22no—
513C4—44no—
514C4—42no—
515C4—42no—
516C4—82no—
517C4—82no—
518C4—82no—
519C4—84no—
520C4—82no—
521C4#351412yes—
522C4#355412yes—
523C2—162no—
524C2#3917111yescharacteristic
525C2#6080011yescharacteristic
526C2#6083011yescharacteristic
527C2#6131912yes—
528C2#6258212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Q8C2semidirect
4. #31640C2semidirect
6. #30216C2semidirect
7. #29225C2semidirect
8. #29148C2semidirect
9. #28619C2semidirect
11. #25942C2semidirect
12. #25336C2semidirect
13. #22273C2semidirect
14. #21657C2semidirect
15. #21346C2semidirect

Maximal subgroups

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

subgroupindex
#366022
Q8×C4⋊Q82
#332302
#316402
#316382
#302162
#292252
#291482
#286192
#284472
#259422
#253362
#222732
#216572
#213462