Finite Group Databasebeta

← #6798002

Subgroups — #6798002

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#36151C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#31956C218yesmaximal, complemented
5#29845C212yesmaximal, complemented
6#28719C211yesmaximal, characteristic, complemented
7#28512C211yesmaximal, characteristic, complemented
8#27948C212yesmaximal, complemented
9#25600C212yesmaximal, complemented
10#24831C214yesmaximal, complemented
11#24134C214yesmaximal
12#21860C212yesmaximal, complemented
13#21691C211yesmaximal, characteristic, complemented
14#21265C211yesmaximal, characteristic
15C2×#554C2212yes—
16C2×#533C2212yes—
17C2×#494C2211yescharacteristic
18C2×#487C2212yes—
19C2×#475C2214yes—
20C2×#441C2214yes—
21C2×#431C2212yes—
22#2868C2211yescharacteristic
23#2862C2211yescharacteristic
24#2857C2212yes—
25#2857C2212yes—
26#2567C2212yes—
27#2552—22no—
28#2536C2212yes—
29#2502C2214yes—
30#2489—22no—
31#2453—22no—
32#2440—22no—
33#2440—22no—
34#2421C2214yes—
35#2415—22no—
36#2362—22no—
37#2351C2212yes—
38D4×C4⋊C4—22no—
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C2×#384C2212yes—
42C42×D4C2211yescharacteristic
43#2096C2214yes—
44#2084C2218yes—
45#2074C2214yes—
46#2066C2218yes—
47#2035C2218yes—
48#2001C2218yes—
49#1993C2214yes—
50#1990C2212yes—
51#1990C2212yes—
52#1990C2212yes—
53#1971C2214yes—
54#1956C2212yes—
55#1956C2211yescharacteristic
56#1956C2211yescharacteristic
57#1955C2212yes—
58#1940C2212yes—
59#1924C2214yes—
60#1878C2218yes—
61#1872C2218yes—
62#1853C2218yes—
63#1844C2211yescharacteristic
64#1844C2212yes—
65#1836C2214yes—
66#1805C2214yes—
67#1803C2218yes—
68C4×C4⋊C8C2211yescharacteristic
69C4×C22⋊C8C2212yes—
70C4×#97C2211yescharacteristic
71C22×Q16C2314yes—
72#554—48no—
73#533—28no—
74#533—44no—
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2311yescharacteristic
78C2×C4⋊Q8C2312yes—
79C2×C4⋊Q8C2312yes—
80C2×#125—22no—
81C2×#125—22no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8C2314yes—
86C2×C22⋊Q8C2314yes—
87C2×C4×Q8—22no—
88C2×C4×Q8C2312yes—
89C2×C4×Q8C2312yes—
90C2×C4×Q8C2314yes—
91C2×C4×D4—22no—
92C2×C4×D4D412yes—
93C2×C4×D4C2311yescharacteristic
94C2×#117—22no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4C2314yes—
98C22×C42C2312yes—
99#494—24no—
100#487—44no—
101#475—48no—
102#441—48no—
103#431—44no—
104C2×#106C2318yes—
105C2×#106C2314yes—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2314yes—
108C2×#103C2318yes—
109C2×#103C2314yes—
110C2×#103C2318yes—
111C2×#103C2314yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114C2×#97C2312yes—
115C2×#97C2312yes—
116C2×C4×C8C2312yes—
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#400—24no—
120#398—22no—
121#398—24no—
122#398—24no—
123#397—24no—
124#395—22no—
125#395—22no—
126#394—22no—
127#391—22no—
128#391—22no—
129#391—22no—
130#391C2312yes—
131#391C2312yes—
132#391C2314yes—
133#389—24no—
134#389—24no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389C2314yes—
141#388—24no—
142#388—24no—
143#387—24no—
144#385—24no—
145#385—24no—
146#384—22no—
147#384—22no—
148#384—22no—
149#384C2311yescharacteristic
150#384C2311yescharacteristic
151#384C2314yes—
152#384C2312yes—
153#384C2312yes—
154#382—22no—
155#382C2312yes—
156#382C2312yes—
157#380—22no—
158#380—22no—
159#380—22no—
160#380C2314yes—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4D412yes—
165C4×C4⋊C4C2311yescharacteristic
166C4×C22⋊C4D412yes—
167C4×C22⋊C4D412yes—
168C4×C22⋊C4C2312yes—
169C43C2311yescharacteristic
170#340C2318yes—
171#340C2318yes—
172#340C2318yes—
173#336C2312yes—
174#336C2314yes—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8C2412yes—
178C22×Q8C2412yes—
179C22×D4C2×D411yescharacteristic
180C23×C4—24no—
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2412yes—
184C2×Q16—28no—
185C2×Q16—28no—
186C2×Q16—44no—
187C22×C8C2414yes—
188C22×C8C2412yes—
189C22×C8C2412yes—
190C4⋊Q8—24no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—24no—
194C4⋊Q8—28no—
195C4⋊Q8—22no—
196C4⋊Q8—84no—
197C4⋊Q8#4114yes—
198#125—48no—
199#125—48no—
200C22⋊Q8—28no—
201C22⋊Q8—416no—
202C22⋊Q8—44no—
203C22⋊Q8—48no—
204C22⋊Q8—48no—
205C22⋊Q8—48no—
206C4×Q8—28no—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8—44no—
210C4×Q8—48no—
211C4×D4—28no—
212C4×D4—24no—
213C4×D4—48no—
214#117—48no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—84no—
235C2×C4⋊C4—84no—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2414yes—
240C2×C4⋊C4C2414yes—
241C2×C4⋊C4C2412yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4C2×D412yes—
246C2×C22⋊C4C2×D412yes—
247C2×C22⋊C4C2×D412yes—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2412yes—
258C2×C42C2411yescharacteristic
259C2×C42C2411yescharacteristic
260C2×C42C2411yescharacteristic
261#106—44no—
262#106—48no—
263C4⋊C8—24no—
264C4⋊C8—44no—
265#103—44no—
266#103—44no—
267#103—48no—
268#103—48no—
269C22⋊C8—28no—
270C22⋊C8—44no—
271#97—24no—
272#97—24no—
273C4×C8—22no—
274C4×C8—22no—
275C4×C8—24no—
276#95—22no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—22no—
285#95C2×D411yescharacteristic
286#95C2×D411yescharacteristic
287#95C2×D412yes—
288#95C2×D412yes—
289C24C22×D412yes—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—44no—
298C2×Q8—44no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—84no—
302C2×Q8C2×#4114yes—
303C2×D4—24no—
304C2×D4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—48no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4#13718yes—
324C22×C4C22×D412yes—
325C22×C4C22×D412yes—
326C22×C4C22×D412yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C2511yescharacteristic
332Q16—416no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—24no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—22no—
340C2×C8—24no—
341C2×C82−1+414yes—
342C4⋊C4—22no—
343C4⋊C4—28no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—28no—
350C4⋊C4—48no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—48no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C22⋊C4—48no—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—24no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—48no—
381C42—44no—
382C42—44no—
383C42C2×#4112yes—
384C42H(5,𝔽2)14yes—
385C42H(5,𝔽2)12yes—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23C2×#13714yes—
393C23C23×D411yescharacteristic
394Q8—28no—
395Q8—44no—
396Q8—44no—
397Q8—44no—
398Q8—88no—
399Q8—164no—
400D4—28no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—216no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—48no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—168no—
435C2×C4#53512yes—
436C2×C4#53512yes—
437C2×C4#54614yes—
438C2×C4C2×#13714yes—
439C2×C4C2×#13714yes—
440C2×C4#57714yes—
441C2×C4C22×#4111yescharacteristic
442C2×C4C2×H(5,𝔽2)12yes—
443C2×C4C2×H(5,𝔽2)11yescharacteristic
444C2×C4C2×2−1+412yes—
445C8—24no—
446C8—42no—
447C8—44no—
448C22—24no—
449C22—24no—
450C22—24no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22C2×#53511yescharacteristic
456C22C2×#54611yescharacteristic
457C22#349911yescharacteristic
458C22C22×#13712yes—
459C22C2×#57712yes—
460C4—22no—
461C4—22no—
462C4—24no—
463C4—42no—
464C4—42no—
465C4—44no—
466C4—42no—
467C4—82no—
468C4—82no—
469C4—82no—
470C4—84no—
471C4—84no—
472C4#331618yes—
473C4#352012yes—
474C4#354812yes—
475C2—44no—
476C2#3895111yescharacteristic
477C2#6080011yescharacteristic
478C2#6081811yescharacteristic
479C2C2×#331612yes—
480C2#6247812yes—
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. #33293C2semidirect
4. #31956C2semidirect
5. #29845C2semidirect
6. #28719C2semidirect
7. #28512C2semidirect
8. #27948C2semidirect
9. #25600C2semidirect
10. #24831C2semidirect
12. #21860C2semidirect
13. #21691C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#361512
#332932
#319562
#298452
#287192
#285122
#279482
#256002
#248312
#241342
#218602
#216912
#212652