Finite Group Databasebeta

← #6798472

Subgroups — #6798472

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36165C211yesmaximal, characteristic
2#36151C211yesmaximal, characteristic
3#33296C211yesmaximal, characteristic, complemented
4#31360C214yesmaximal, complemented
5#31358C214yesmaximal, complemented
6#29883C212yesmaximal, complemented
7#29150C211yesmaximal, characteristic, complemented
8#28695C211yesmaximal, characteristic, complemented
9#27969C212yesmaximal, complemented
10#26123C214yesmaximal, complemented
11#25677C212yesmaximal, complemented
12#24979C214yesmaximal
13#21866C212yesmaximal, complemented
14#21657C211yesmaximal, characteristic, complemented
15#21271C211yesmaximal, characteristic
16C2×#533C2212yes—
17C2×#501C2212yes—
18C2×#491C2211yescharacteristic
19C2×#432C2212yes—
20#2862C2211yescharacteristic
21#2859C2211yescharacteristic
22#2857C2212yes—
23#2851C2212yes—
24#2571C2212yes—
25#2536C2212yes—
26#2508—22no—
27#2505—22no—
28#2505—22no—
29#2489—22no—
30#2453—22no—
31#2442C2212yes—
32#2442C2212yes—
33#2421C2214yes—
34#2415—22no—
35#2362—22no—
36#2351C2212yes—
37#2349—22no—
38D4×C4⋊C4C2212yes—
39C4×C4⋊Q8C2211yescharacteristic
40C4×#125C2211yescharacteristic
41C2×#382C2212yes—
42C42×D4C2211yescharacteristic
43#2101C2214yes—
44#2101C2214yes—
45#2088C2214yes—
46#2087C2214yes—
47#2085C2214yes—
48#2077C2214yes—
49#2075C2214yes—
50#2036C2214yes—
51#2035C2214yes—
52#2026C2212yes—
53#2026C2212yes—
54#1993C2212yes—
55#1993C2212yes—
56#1993C2212yes—
57#1992C2211yescharacteristic
58#1991C2212yes—
59#1991C2212yes—
60#1960C2212yes—
61#1960C2211yescharacteristic
62#1955C2212yes—
63#1949C2214yes—
64#1940C2212yes—
65#1933C2214yes—
66#1923C2214yes—
67#1872C2214yes—
68#1872C2214yes—
69#1865C2214yes—
70#1863C2214yes—
71#1836C2214yes—
72#1836C2211yescharacteristic
73#1836C2212yes—
74#1824C2214yes—
75#1823C2214yes—
76#1805C2214yes—
77#1805C2214yes—
78#1803C2214yes—
79#1767C2211yescharacteristic
80#1763C2214yes—
81#1749C2212yes—
82C4×#97C2211yescharacteristic
83#533—28no—
84#533—44no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×#125—22no—
88C2×#125C2311yescharacteristic
89C2×#125C2311yescharacteristic
90C2×#125C2311yescharacteristic
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8C2314yes—
94C2×C4×Q8—22no—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2314yes—
97C2×C4×D4—22no—
98C2×C4×D4D412yes—
99C2×C4×D4C2311yescharacteristic
100C2×#117—22no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C42C2312yes—
104#501—44no—
105#491—24no—
106#432—44no—
107C2×#107C2314yes—
108C2×#106C2314yes—
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2314yes—
117C2×#97C2312yes—
118C2×#97C2311yescharacteristic
119C2×#97C2312yes—
120C2×#97C2311yescharacteristic
121C2×C4×C8C2312yes—
122#400—22no—
123#400—22no—
124#400—24no—
125#400—22no—
126#400—22no—
127#398—22no—
128#398—22no—
129#398—22no—
130#397—24no—
131#394—22no—
132#391—22no—
133#391—22no—
134#391—22no—
135#391C2312yes—
136#389—22no—
137#389—22no—
138#389—24no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389C2312yes—
144#389C2312yes—
145#389C2312yes—
146#389C2314yes—
147#388—24no—
148#388—24no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—24no—
153#385—22no—
154#385—22no—
155#385—22no—
156#385—22no—
157#385—22no—
158#385—22no—
159#384C2311yescharacteristic
160#384C2312yes—
161#382—22no—
162#382—22no—
163#382—22no—
164#382—22no—
165#382C2311yescharacteristic
166#382C2312yes—
167#382C2312yes—
168#382C2312yes—
169#382C2312yes—
170#382C2312yes—
171#380—22no—
172#380—22no—
173#380—22no—
174#380C2312yes—
175#380C2314yes—
176#380C2312yes—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D412yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2311yescharacteristic
183C4×C22⋊C4D412yes—
184C4×C22⋊C4D412yes—
185C4×C22⋊C4C2312yes—
186C2×#95—22no—
187C2×#95—22no—
188C43C2311yescharacteristic
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2312yes—
200#336C2314yes—
201C22×Q8—22no—
202C22×Q8C2412yes—
203C22×D4C2×D411yescharacteristic
204C23×C4—22no—
205C23×C4—22no—
206C23×C4C2×D412yes—
207C23×C4C2×D412yes—
208C23×C4C2412yes—
209C22×C8C2412yes—
210C22×C8C2414yes—
211C22×C8C2412yes—
212C4⋊Q8—24no—
213C4⋊Q8—22no—
214C4⋊Q8—24no—
215C4⋊Q8#4114yes—
216#125—22no—
217#125—22no—
218#125—42no—
219#125—42no—
220#125—48no—
221C22⋊Q8—28no—
222C22⋊Q8—44no—
223C22⋊Q8—48no—
224C22⋊Q8—48no—
225C4×Q8—28no—
226C4×Q8—28no—
227C4×Q8—44no—
228C4×Q8—48no—
229C4×D4—24no—
230C4×D4—44no—
231C4×D4—48no—
232#117—48no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4C2×D412yes—
265C2×C22⋊C4C2×D412yes—
266C2×C22⋊C4C2×D412yes—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2412yes—
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280#107—44no—
281#106—44no—
282C4⋊C8—24no—
283C4⋊C8—44no—
284#103—28no—
285#103—44no—
286#103—44no—
287#103—44no—
288C22⋊C8—44no—
289C22⋊C8—44no—
290#97—22no—
291#97—22no—
292#97—24no—
293#97—24no—
294C4×C8—42no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—22no—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—22no—
314#95C2×D411yescharacteristic
315#95C2×D411yescharacteristic
316#95C2×D412yes—
317#95C2×D412yes—
318C24C22×D412yes—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8C2×#4114yes—
326C2×D4—24no—
327C2×D4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4C22×D412yes—
352C22×C4C22×D412yes—
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D412yes—
358C22×C4C2511yescharacteristic
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—24no—
368C4⋊C4—22no—
369C4⋊C4—28no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—42no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—42no—
398C22⋊C4—48no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—48no—
402C42—24no—
403C42—24no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—42no—
408C42—42no—
409C42—48no—
410C42—44no—
411C42—44no—
412C42C2×#4112yes—
413C42H(5,𝔽2)12yes—
414C422−1+414yes—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23#57514yes—
422C23C23×D411yescharacteristic
423Q8—28no—
424Q8—44no—
425Q8—88no—
426D4—28no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—48no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—84no—
467C2×C4—164no—
468C2×C4—164no—
469C2×C4#53612yes—
470C2×C4#53612yes—
471C2×C4#54614yes—
472C2×C4C22×#4111yescharacteristic
473C2×C4C2×H(5,𝔽2)11yescharacteristic
474C2×C4C2×2−1+412yes—
475C2×C4#58512yes—
476C8—42no—
477C8—42no—
478C8—44no—
479C22—24no—
480C22—24no—
481C22—24no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22C2×#53611yescharacteristic
487C22C2×#54611yescharacteristic
488C22#350011yescharacteristic
489C22C2×#57512yes—
490C22#358612yes—
491C4—22no—
492C4—22no—
493C4—24no—
494C4—42no—
495C4—44no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4—82no—
503C4—82no—
504C4—84no—
505C4#351512yes—
506C4#354712yes—
507C2—44no—
508C2#3895511yescharacteristic
509C2#6086111yescharacteristic
510C2#6088411yescharacteristic
511C2#6115812yes—
512C2#6258512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33296C2semidirect
4. #31360C2semidirect
5. #31358C2semidirect
6. #29883C2semidirect
7. #29150C2semidirect
8. #28695C2semidirect
9. #27969C2semidirect
10. #26123C2semidirect
11. #25677C2semidirect
13. #21866C2semidirect
14. #21657C2semidirect

Maximal subgroups

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

subgroupindex
#361652
#361512
#332962
#313602
#313582
#298832
#291502
#286952
#279692
#261232
#256772
#249792
#218662
#216572
#212712