Finite Group Databasebeta

← #6806147

Subgroups — #6806147

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36602C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic, complemented
3#33230C211yesmaximal, characteristic
4#31637C214yesmaximal
5#31635C214yesmaximal, complemented
6#30217C212yesmaximal, complemented
7#29226C211yesmaximal, characteristic, complemented
8#29146C212yesmaximal, complemented
9#28619C212yesmaximal, complemented
10#28447C211yesmaximal, characteristic
11#25941C214yesmaximal, complemented
12#25337C214yesmaximal, complemented
13#22273C212yesmaximal, complemented
14#21657C211yesmaximal, characteristic, complemented
15#21346C211yesmaximal, characteristic, complemented
16C2×#557C2212yes—
17C2×#497C2212yes—
18C2×#491C2211yescharacteristic
19C2×#489C2214yes—
20C2×#464C2214yes—
21#2868C2211yescharacteristic
22#2861C2212yes—
23#2859C2212yes—
24#2858C2211yescharacteristic
25#2597—22no—
26#2593—22no—
27#2592C2212yes—
28#2568C2212yes—
29#2565C2214yes—
30#2546—22no—
31#2506—22no—
32#2506—22no—
33#2458—22no—
34#2446C2212yes—
35#2445C2212yes—
36#2374—22no—
37#2367C2212yes—
38#2365—22no—
39C4×C4⋊Q8C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×#125C2211yescharacteristic
42C4×#124C2212yes—
43C42×Q8C2211yescharacteristic
44#2099C2214yes—
45#2074C2214yes—
46#2073C2214yes—
47#2056C2214yes—
48#2055C2214yes—
49#2026C2214yes—
50#2026C2214yes—
51#2025C2214yes—
52#2025C2214yes—
53#2016C2212yes—
54#2014C2212yes—
55#1992C2211yescharacteristic
56#1992C2212yes—
57#1987C2214yes—
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#1805C2214yes—
77#1804C2214yes—
78#1767C2211yescharacteristic
79#1767C2212yes—
80#1763C2214yes—
81#1745C2212yes—
82C4×#97C2211yescharacteristic
83C22×Q16C2314yes—
84C22×SD16C2314yes—
85#557—48no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×#127C2311yescharacteristic
92C2×#125—24no—
93C2×#125C2311yescharacteristic
94C2×#125C2314yes—
95C2×#124C2312yes—
96C2×#124C2312yes—
97C2×#123—22no—
98C2×C4×Q8—22no—
99C2×C4×Q8D412yes—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2314yes—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×D4C2312yes—
105#497—28no—
106#491—24no—
107#489—48no—
108#464—48no—
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#97C2311yescharacteristic
120C2×#97C2312yes—
121C2×#97C2312yes—
122C2×#97C2311yescharacteristic
123C2×C4×C8C2312yes—
124#400—22no—
125#400—22no—
126#400—22no—
127#398—24no—
128#398—24no—
129#398—22no—
130#397—22no—
131#394—22no—
132#394—22no—
133#393—22no—
134#391—24no—
135#391C2312yes—
136#391C2314yes—
137#391C2312yes—
138#391C2312yes—
139#391C2312yes—
140#390—22no—
141#390—22no—
142#390C2312yes—
143#390C2312yes—
144#390C2312yes—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
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—24no—
166#387—24no—
167#387—22no—
168#386—22no—
169#385—22no—
170#384—22no—
171#384C2311yescharacteristic
172#384C2312yes—
173#384C2312yes—
174#382C2311yescharacteristic
175#382C2314yes—
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⋊C4C2312yes—
185C4×C4⋊C4C2311yescharacteristic
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—22no—
202C22×Q8C2×D411yescharacteristic
203C22×Q8C2412yes—
204C22×Q8C2412yes—
205C22×D4—22no—
206C22×D4C2412yes—
207C23×C4—22no—
208C2×Q16—28no—
209C2×Q16—44no—
210C2×Q16—44no—
211C2×SD16—28no—
212C2×SD16—44no—
213C2×SD16—44no—
214C22×C8C2414yes—
215C22×C8C2412yes—
216C22×C8C2412yes—
217C4⋊Q8—28no—
218C4⋊Q8—22no—
219C4⋊Q8—48no—
220C4⋊Q8—42no—
221C4⋊Q8—48no—
222C4⋊Q8#4114yes—
223#127—42no—
224#125—22no—
225#125—22no—
226#125—44no—
227#125—44no—
228#125—416no—
229#124—24no—
230#124—28no—
231#124—24no—
232#123—48no—
233C4×Q8—24no—
234C4×Q8—28no—
235C4×Q8—44no—
236C4×Q8—44no—
237C4×Q8—44no—
238C4×Q8—44no—
239C4×Q8—48no—
240C4×D4—44no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
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×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2412yes—
286C2×C42C2412yes—
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C4⋊C8—24no—
291C4⋊C8—44no—
292#103—28no—
293#103—44no—
294#103—44no—
295#103—44no—
296#103—44no—
297#102—44no—
298#102—44no—
299#102—44no—
300#97—22no—
301#97—24no—
302#97—24no—
303#97—22no—
304C4×C8—42no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95C2×D411yescharacteristic
320#95C2×D412yes—
321#95C2×D412yes—
322#95C2×D411yescharacteristic
323C24—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—42no—
333C2×Q8—48no—
334C2×Q8—44no—
335C2×Q8C2×#4114yes—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—44no—
341C2×D4—82no—
342C2×D4—82no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—44no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4C22×D412yes—
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C22×D412yes—
360C22×C4C2511yescharacteristic
361Q16—88no—
362SD16—88no—
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—24no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—24no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—48no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C42—24no—
409C42—24no—
410C42—22no—
411C42—24no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—48no—
418C42—44no—
419C42—82no—
420C42—82no—
421C42C2×#4112yes—
422C42H(5,𝔽2)14yes—
423C422−1+412yes—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23C23×D411yescharacteristic
432Q8—28no—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—44no—
437Q8—88no—
438D4—44no—
439D4—44no—
440D4—164no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—164no—
477C2×C4#53512yes—
478C2×C4#53512yes—
479C2×C4#54814yes—
480C2×C4#57514yes—
481C2×C4C22×#4111yescharacteristic
482C2×C4C2×H(5,𝔽2)12yes—
483C2×C4C2×2−1+411yescharacteristic
484C2×C4#58512yes—
485C8—42no—
486C8—44no—
487C8—42no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22C2×#53511yescharacteristic
496C22C2×#54811yescharacteristic
497C22#350011yescharacteristic
498C22C2×#57512yes—
499C22#358612yes—
500C4—24no—
501C4—22no—
502C4—24no—
503C4—22no—
504C4—44no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—82no—
510C4—82no—
511C4—84no—
512C4—82no—
513C4—82no—
514C4#351412yes—
515C4#355412yes—
516C2—162no—
517C2#3917111yescharacteristic
518C2#6080111yescharacteristic
519C2#6083111yescharacteristic
520C2#6132012yes—
521C2#6258512yes—
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. #36560C2semidirect
5. #31635C2semidirect
6. #30217C2semidirect
7. #29226C2semidirect
8. #29146C2semidirect
9. #28619C2semidirect
11. #25941C2semidirect
12. #25337C2semidirect
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
#365602
#332302
#316372
#316352
#302172
#292262
#291462
#286192
#284472
#259412
#253372
#222732
#216572
#213462