Finite Group Databasebeta

← #6654119

Subgroups — #6654119

Order 512.

1320 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2#34948C211yesmaximal, characteristic
3#33108C212yesmaximal
4#31595C214yesmaximal, complemented
5#31144C214yesmaximal
6#30834C214yesmaximal, complemented
7#30649C214yesmaximal
8#29923C212yesmaximal
9#29321C212yesmaximal, complemented
10#29257C212yesmaximal, complemented
11#28755C212yesmaximal, complemented
12#28287C212yesmaximal, complemented
13#21926C211yesmaximal, characteristic
14C2×#491C2214yes—
15C2×#483C2214yes—
16C2×#479C2214yes—
17#2868C2211yescharacteristic
18#2859C2211yescharacteristic
19#2857C2212yes—
20#2647—21no—
21#2629—21no—
22#2571C2212yes—
23#2570—22no—
24#2569—22no—
25#2552C2212yes—
26#2507—22no—
27#2497—22no—
28#2446C2214yes—
29#2442C2212yes—
30#2442C2212yes—
31#2429C2214yes—
32#2402—21no—
33#2402—21no—
34#2330—21no—
35#2314—21no—
36C4×C4⋊Q8C2212yes—
37C4×#125C2212yes—
38C4×#125C2212yes—
39#2296—21no—
40#2285—21no—
41#2285C2211yescharacteristic
42C4×#117C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44#2097C2214yes—
45#2095C2214yes—
46#2085C2214yes—
47#2072C2214yes—
48#2068C2214yes—
49#2066C2214yes—
50#2059C2214yes—
51#2043C2214yes—
52#2043C2214yes—
53#2036C2214yes—
54#2036C2214yes—
55#2026C2214yes—
56#1981C2212yes—
57#1981C2212yes—
58#1981C2212yes—
59#1981C2212yes—
60#1960C2214yes—
61#1948C2212yes—
62#1948C2212yes—
63#1947C2212yes—
64#1947C2212yes—
65#1944C2214yes—
66#1943C2214yes—
67#1927C2214yes—
68#1927C2214yes—
69#1924C2214yes—
70#1917C2214yes—
71#1872C2214yes—
72#1865C2214yes—
73#1863C2214yes—
74#1842C2211yescharacteristic
75#1831C2211yescharacteristic
76#1805C2214yes—
77#1797C2212yes—
78#1767C2212yes—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2314yes—
81C2×#125—21no—
82C2×#125C2312yes—
83C2×#125C2312yes—
84C2×C22⋊Q8C2314yes—
85C2×C4×Q8C2314yes—
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C2×#117D411yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92#491—48no—
93#483—48no—
94#479—48no—
95C2×#107C2314yes—
96C2×#107C2314yes—
97C2×#106C2314yes—
98C2×#106C2314yes—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2314yes—
102C2×#103C2314yes—
103C2×#103C2314yes—
104C2×#103C2314yes—
105C2×#103C2314yes—
106C2×#103C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×C22⋊C8C2314yes—
111C2×#97C2314yes—
112#400—22no—
113#400—22no—
114#400—22no—
115#400—22no—
116#400—24no—
117#398—24no—
118#398—22no—
119#398—22no—
120#398—22no—
121#398—22no—
122#398—24no—
123#396—24no—
124#391C2314yes—
125#391C2314yes—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—24no—
134#389C2312yes—
135#389C2312yes—
136#389C2312yes—
137#389C2312yes—
138#389C2314yes—
139#388—24no—
140#387—24no—
141#387—22no—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—24no—
146#385—22no—
147#385—22no—
148#384—22no—
149#384—21no—
150#384—21no—
151#384—22no—
152#384C2312yes—
153#384C2312yes—
154#382—22no—
155#382—21no—
156#382—21no—
157#382C2312yes—
158#382C2312yes—
159#380—21no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380—22no—
164#380C2311yescharacteristic
165#380C2311yescharacteristic
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2311yescharacteristic
179C4×C4⋊C4C2311yescharacteristic
180C4×C4⋊C4C2312yes—
181C4×C22⋊C4D412yes—
182C4×C22⋊C4C2312yes—
183#376D411yescharacteristic
184#376D411yescharacteristic
185C2×#95—21no—
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195C22×Q8C2414yes—
196C23×C4—22no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2411yescharacteristic
200C22×C8C2414yes—
201C22×C8C2414yes—
202C4⋊Q8—24no—
203C4⋊Q8—24no—
204C4⋊Q8—44no—
205C4⋊Q8—44no—
206#125—24no—
207#125—24no—
208#125—42no—
209#125—42no—
210#125—44no—
211C22⋊Q8—44no—
212C22⋊Q8—44no—
213C4×Q8—44no—
214C4×Q8—44no—
215#117—24no—
216#117—42no—
217#117—44no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2414yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4C2×D412yes—
251C2×C22⋊C4C2×D412yes—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—24no—
259C2×C42—44no—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2412yes—
269C2×C42C2412yes—
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272#107—44no—
273#107—44no—
274#106—28no—
275#106—44no—
276C4⋊C8—24no—
277C4⋊C8—24no—
278C4⋊C8—44no—
279#103—28no—
280#103—28no—
281#103—44no—
282#103—44no—
283#103—44no—
284#103—44no—
285#103—44no—
286#103—44no—
287C22⋊C8—44no—
288#97—44no—
289#95—22no—
290#95—21no—
291#95—21no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—21no—
297#95—21no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95C2×D412yes—
309#95C2×D412yes—
310#95C2×D412yes—
311#95C2×D412yes—
312C24C22×D411yescharacteristic
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—82no—
336C22×C4#13714yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D412yes—
339C22×C4C22×D412yes—
340C22×C4C22×D412yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C2511yescharacteristic
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—48no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4—84no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C4⋊C4H(5,𝔽2)14yes—
385C22⋊C4—24no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—84no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—21no—
399C42—21no—
400C42—22no—
401C42—22no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—44no—
406C42—42no—
407C42—44no—
408C42—84no—
409C422−1+412yes—
410C422−1+412yes—
411C23—22no—
412C23—21no—
413C23—22no—
414C23—21no—
415C23C2×#13712yes—
416C23C23×D411yescharacteristic
417Q8—44no—
418Q8—44no—
419Q8—44no—
420Q8—44no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—28no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—82no—
463C2×C4—88no—
464C2×C4#53612yes—
465C2×C4#53612yes—
466C2×C4C2×#13712yes—
467C2×C4C2×#13712yes—
468C2×C4#57512yes—
469C2×C4C2×H(5,𝔽2)12yes—
470C2×C4C2×2−1+411yescharacteristic
471C2×C4C2×2−1+411yescharacteristic
472C2×C4#58512yes—
473C8—44no—
474C8—44no—
475C22—24no—
476C22—22no—
477C22—42no—
478C22—41no—
479C22—41no—
480C22C2×#53611yescharacteristic
481C22#348812yes—
482C22C22×#13711yescharacteristic
483C22C2×#57511yescharacteristic
484C22#358612yes—
485C4—22no—
486C4—22no—
487C4—42no—
488C4—42no—
489C4—44no—
490C4—42no—
491C4—42no—
492C4—41no—
493C4—41no—
494C4—82no—
495C4—84no—
496C4—82no—
497C4—82no—
498C4—84no—
499C4#323614yes—
500C4#354811yescharacteristic
501C4#355412yes—
502C4#355711yescharacteristic
503C2—42no—
504C2#3727411yescharacteristic
505C2C2×#323611yescharacteristic
506C2#6086911yescharacteristic
507C2#6196012yes—
508C2#6196912yes—
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
4. #31595C2semidirect
6. #30834C2semidirect
9. #29321C2semidirect
10. #29257C2semidirect
11. #28755C2semidirect
12. #28287C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#349482
#331082
#315952
#311442
#308342
#306492
#299232
#293212
#292572
#287552
#282872
#219262