Finite Group Databasebeta

← #6468563

Subgroups — #6468563

Order 512.

1432 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic
3#29327C212yesmaximal, complemented
4#29326C212yesmaximal, complemented
5#29308C214yesmaximal
6#29213C212yesmaximal, complemented
7#28695C212yesmaximal, complemented
8#27955C212yesmaximal
9#25819C212yesmaximal
10#23845C212yesmaximal, complemented
11#23844C212yesmaximal, complemented
12#21731C212yesmaximal, complemented
13C4×#501C212yesmaximal, complemented
14#21367C214yesmaximal
15#20465C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C2×#501C2212yes—
18C2×#501C2212yes—
19C2×#433C2212yes—
20#2868C2211yescharacteristic
21#2862C2211yescharacteristic
22#2857C2212yes—
23#2647C2212yes—
24#2628C2212yes—
25#2511—22no—
26#2492—22no—
27#2451—22no—
28#2423—22no—
29#2370—22no—
30#2370—22no—
31#2345—22no—
32#2345—22no—
33#2331C2212yes—
34#2330C2211yescharacteristic
35#2330C2211yescharacteristic
36#2330C2211yescharacteristic
37#2330C2211yescharacteristic
38#2316C2212yes—
39#2314C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×#125C2212yes—
42C4×#125C2212yes—
43#2285C2211yescharacteristic
44#2274C2212yes—
45C4×#117C2211yescharacteristic
46#2085C2214yes—
47#2085C2214yes—
48#2084C2214yes—
49#2083C2214yes—
50#2059C2214yes—
51#2059C2214yes—
52#2058C2214yes—
53#2057C2214yes—
54#1993C2212yes—
55#1993C2212yes—
56#1992C2212yes—
57#1992C2212yes—
58#1949C2214yes—
59#1949C2214yes—
60#1948C2214yes—
61#1947C2214yes—
62#1944C2214yes—
63#1943C2214yes—
64#1942C2214yes—
65#1941C2214yes—
66#1854C2214yes—
67#1854C2214yes—
68#1853C2214yes—
69#1852C2214yes—
70#1847C2212yes—
71#1847C2212yes—
72#1776C2214yes—
73#1776C2214yes—
74C4×#107C2214yes—
75C4×#106C2214yes—
76#1768C2212yes—
77#1767C2212yes—
78#1750C2212yes—
79#1749C2212yes—
80#1746C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#125—28no—
87C2×#125—22no—
88C2×#125C2311yescharacteristic
89C2×#125C2311yescharacteristic
90C2×#125C2312yes—
91C2×C22⋊Q8—22no—
92C2×C4×Q8—22no—
93C2×#117C2312yes—
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C22×C4⋊C4—22no—
97C22×C4⋊C4C2312yes—
98C22×C42C2311yescharacteristic
99#501—28no—
100#501—28no—
101#433Q8116yescomplemented
102C2×#107C2314yes—
103C2×#107C2314yes—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#106C2314yes—
107C2×#106C2314yes—
108C2×#106C2314yes—
109C2×#106C2314yes—
110C2×C4⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112C2×#97C2312yes—
113C2×#97C2312yes—
114C2×#97C2312yes—
115C2×#97C2312yes—
116C2×C4×C8C2312yes—
117C2×C4×C8C2312yes—
118#400—22no—
119#400—24no—
120#399—22no—
121#398—24no—
122#395—22no—
123#393—22no—
124#391—22no—
125#391—22no—
126#391—22no—
127#391—22no—
128#391—24no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389C2312yes—
138#389C2312yes—
139#389C2312yes—
140#389C2312yes—
141#389C2314yes—
142#389C2314yes—
143#388—24no—
144#387—24no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—24no—
154#384—22no—
155#384C2312yes—
156#384C2312yes—
157#384C2311yescharacteristic
158#384C2311yescharacteristic
159#382C2312yes—
160#382C2312yes—
161#382C2312yes—
162#380—22no—
163#380—22no—
164#380—22no—
165#380—22no—
166#380C2314yes—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2314yes—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4D412yes—
185C4×C22⋊C4D412yes—
186#376C2311yescharacteristic
187#376C2311yescharacteristic
188C43C2311yescharacteristic
189C43C2311yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C22×Q8—22no—
201C22×Q8—22no—
202C23×C4—22no—
203C23×C4C2×D412yes—
204C23×C4C2411yescharacteristic
205C22×C8C2414yes—
206C22×C8C2414yes—
207C4⋊Q8—24no—
208C4⋊Q8—24no—
209C4⋊Q8—44no—
210C4⋊Q8—84no—
211#125—216no—
212#125—24no—
213#125—24no—
214#125—24no—
215#125#4118yes—
216C22⋊Q8—84no—
217C4×Q8—84no—
218#117—42no—
219#117—44no—
220#117C2×Q818yes—
221C2×C4⋊C4—216no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4C2×D411yescharacteristic
254C2×C22⋊C4C2×D411yescharacteristic
255C2×C22⋊C4C2×D412yes—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274#107—28no—
275#107—28no—
276#107—44no—
277#107—44no—
278#106—28no—
279#106—28no—
280#106—44no—
281#106—44no—
282C4⋊C8C2×Q8116yes—
283C22⋊C8C2×Q8116yes—
284#97—24no—
285#97—24no—
286#97—42no—
287#97C2×Q818yes—
288C4×C8—42no—
289C4×C8C2×Q818yes—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—24no—
306#95—22no—
307#95C2×D412yes—
308#95C2×D412yes—
309#95C2×D412yes—
310#95C2×D412yes—
311C24C22×D411yescharacteristic
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—84no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—28no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D412yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D412yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×Q814yes—
341C22×C4C2511yescharacteristic
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8C22×Q818yes—
347C2×C8C22×Q818yes—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—28no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C4⋊C4C4⋊Q818yes—
377C4⋊C4C2×#4114yes—
378C4⋊C4C2×#4114yes—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—44no—
382C22⋊C4—84no—
383C22⋊C4C4⋊Q818yes—
384C42—24no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—44no—
393C42—82no—
394C42—82no—
395C42—82no—
396C42—82no—
397C42—84no—
398C42C22×Q814yes—
399C42C22×Q814yes—
400C42C2×#4114yes—
401C23—22no—
402C23—22no—
403C23—21no—
404C23—21no—
405C23C2×C4⋊Q812yes—
406C23C23×D411yescharacteristic
407Q8—88no—
408Q8—164no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—28no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—84no—
448C2×C4—88no—
449C2×C4#52514yes—
450C2×C4C2×C4⋊Q814yes—
451C2×C4C2×C4⋊Q814yes—
452C2×C4C2×C4⋊Q812yes—
453C2×C4C23×Q812yes—
454C2×C4C22×#4112yes—
455C2×C4#58512yes—
456C8—28no—
457C8—28no—
458C22—22no—
459C22—24no—
460C22—41no—
461C22—41no—
462C22—42no—
463C22C2×#52511yescharacteristic
464C22C22×C4⋊Q811yescharacteristic
465C22#344911yescharacteristic
466C22#358512yes—
467C22#358612yes—
468C4—22no—
469C4—24no—
470C4—24no—
471C4—24no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—42no—
477C4—44no—
478C4—44no—
479C4—82no—
480C4—84no—
481C4—82no—
482C4—84no—
483C4Q8×#4114yes—
484C2—42no—
485C2#3495311yescharacteristic
486C2#6056712yes—
487C2#6066812yes—
488C2#6101911yescharacteristic
489C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #29327C2semidirect
4. #29326C2semidirect
6. #29213C2semidirect
7. #28695C2semidirect
10. #23845C2semidirect
11. #23844C2semidirect
12. #21731C2semidirect
13. C4×#501C2semidirect
101. #433Q8semidirect

Maximal subgroups

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

subgroupindex
#352832
#342172
#293272
#293262
#293082
#292132
#286952
#279552
#258192
#238452
#238442
#217312
C4×#5012
#213672
#204652