Finite Group Databasebeta

← #6797997

Subgroups — #6797997

Order 512.

1812 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#36149C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#31954C218yesmaximal, complemented
5#29844C212yesmaximal, complemented
6#28719C211yesmaximal, characteristic, complemented
7#28508C211yesmaximal, characteristic, complemented
8#27950C212yesmaximal, complemented
9#25597C212yesmaximal, complemented
10#24833C214yesmaximal, complemented
11#24130C214yesmaximal, complemented
12#21859C212yesmaximal, complemented
13#21692C211yesmaximal, characteristic, complemented
14#21265C211yesmaximal, characteristic, complemented
15C2×#554C2212yes—
16C2×#532C2212yes—
17C2×#492C2211yescharacteristic
18C2×#487C2212yes—
19C2×#474C2214yes—
20C2×#440C2214yes—
21C2×#431C2212yes—
22#2868C2211yescharacteristic
23#2862C2211yescharacteristic
24#2857C2212yes—
25#2857C2212yes—
26#2567C2212yes—
27#2552—22no—
28#2535C2212yes—
29#2502C2214yes—
30#2447—22no—
31#2440—22no—
32#2440—22no—
33#2427—22no—
34#2420C2214yes—
35#2407—22no—
36#2349C2212yes—
37D4×C4⋊C4—22no—
38#2339—22no—
39C4×C4⋊Q8C2211yescharacteristic
40C4×#127C2211yescharacteristic
41C2×#384C2212yes—
42C42×D4C2211yescharacteristic
43#2093C2214yes—
44#2083C2218yes—
45#2074C2214yes—
46#2068C2218yes—
47#2036C2218yes—
48#2000C2218yes—
49#1993C2214yes—
50#1990C2212yes—
51#1990C2212yes—
52#1990C2212yes—
53#1973C2214yes—
54#1957C2212yes—
55#1957C2211yescharacteristic
56#1956C2211yescharacteristic
57#1955C2212yes—
58#1940C2212yes—
59#1924C2214yes—
60#1877C2218yes—
61#1874C2218yes—
62#1852C2218yes—
63#1845C2211yescharacteristic
64#1845C2212yes—
65#1836C2214yes—
66#1805C2214yes—
67#1802C2218yes—
68C4×C4⋊C8C2211yescharacteristic
69C4×C22⋊C8C2212yes—
70C4×#97C2211yescharacteristic
71C22×#41—22no—
72C22×SD16C2314yes—
73#554—48no—
74#532—28no—
75#532—44no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8C2311yescharacteristic
78C2×C4⋊Q8C2312yes—
79C2×C4⋊Q8C2312yes—
80C2×#127C2311yescharacteristic
81C2×#125—22no—
82C2×#124—22no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8C2314yes—
86C2×#121—22no—
87C2×#121C2314yes—
88C2×C4×Q8C2312yes—
89C2×C4×D4—22no—
90C2×C4×D4—22no—
91C2×C4×D4—22no—
92C2×C4×D4D412yes—
93C2×C4×D4C2312yes—
94C2×C4×D4C2314yes—
95C2×C4×D4C2311yescharacteristic
96C22×C4⋊C4—22no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2314yes—
99C22×C42C2312yes—
100#492—24no—
101#487—44no—
102#474—48no—
103#440—48no—
104#431—44no—
105C2×#107C2318yes—
106C2×#107C2314yes—
107C2×C4⋊C8C2312yes—
108C2×C4⋊C8C2314yes—
109C2×#103C2318yes—
110C2×#103C2314yes—
111C2×#102C2314yes—
112C2×#102C2318yes—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×#97C2312yes—
116C2×#97C2312yes—
117C2×C4×C8C2312yes—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#398—24no—
121#398—24no—
122#397—22no—
123#397—24no—
124#395—22no—
125#395—22no—
126#394—24no—
127#392—22no—
128#391—22no—
129#391—22no—
130#391—22no—
131#391C2312yes—
132#391C2314yes—
133#390—22no—
134#390C2312yes—
135#389—24no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389C2314yes—
141#386—24no—
142#386—24no—
143#385—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#382C2312yes—
155#382C2312yes—
156#381—24no—
157#380—22no—
158#380—22no—
159#380—22no—
160#380—22no—
161#380C2314yes—
162C4×C4⋊C4—22no—
163C4×C4⋊C4D412yes—
164C4×C4⋊C4C2311yescharacteristic
165C4×C22⋊C4D412yes—
166C4×C22⋊C4D412yes—
167C4×C22⋊C4C2312yes—
168C2×#95—22no—
169C43C2311yescharacteristic
170#340C2318yes—
171#340C2318yes—
172#340C2318yes—
173#336C2312yes—
174#336C2314yes—
175C2×#41—48no—
176C2×#41—48no—
177C2×#41—84no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8C2412yes—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4C2×D411yescharacteristic
184C22×D4C2412yes—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2412yes—
191C2×SD16—28no—
192C2×SD16—28no—
193C2×SD16—44no—
194C22×C8C2414yes—
195C22×C8C2412yes—
196C22×C8C2412yes—
197C4⋊Q8—24no—
198C4⋊Q8—22no—
199C4⋊Q8—24no—
200C4⋊Q8—28no—
201C4⋊Q8—22no—
202C4⋊Q8—84no—
203#127—22no—
204#127#4114yes—
205#125—48no—
206#124—48no—
207C22⋊Q8—416no—
208C22⋊Q8—48no—
209C22⋊Q8—84no—
210#121—28no—
211#121—44no—
212#121—48no—
213C4×Q8—28no—
214C4×D4—28no—
215C4×D4—28no—
216C4×D4—24no—
217C4×D4—28no—
218C4×D4—44no—
219C4×D4—48no—
220C4×D4—48no—
221C4×D4—84no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—84no—
238C2×C4⋊C4—84no—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2414yes—
243C2×C4⋊C4C2414yes—
244C2×C4⋊C4C2412yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4C2×D412yes—
252C2×C22⋊C4C2×D412yes—
253C2×C22⋊C4C2×D412yes—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2412yes—
262C2×C42C2411yescharacteristic
263C2×C42C2411yescharacteristic
264C2×C42C2411yescharacteristic
265#107—44no—
266#107—48no—
267C4⋊C8—24no—
268C4⋊C8—44no—
269#103—44no—
270#103—48no—
271#102—44no—
272#102—48no—
273C22⋊C8—28no—
274C22⋊C8—44no—
275#97—24no—
276#97—24no—
277C4×C8—22no—
278C4×C8—22no—
279C4×C8—24no—
280#95—24no—
281#95—22no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—22no—
288#95C2×D411yescharacteristic
289#95C2×D411yescharacteristic
290#95C2×D412yes—
291#95C2×D412yes—
292C24—22no—
293C24C22×D412yes—
294#41—832no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—84no—
304C2×D4—24no—
305C2×D4—22no—
306C2×D4—28no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—24no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—84no—
315C2×D4C2×#4114yes—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—44no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4#13718yes—
339C22×C4C22×D412yes—
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D412yes—
345C22×C4C22×D412yes—
346C22×C4C2511yescharacteristic
347SD16—416no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C82−1+414yes—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—48no—
384C22⋊C4—44no—
385C22⋊C4—84no—
386C22⋊C4—84no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—44no—
395C42C2×#4112yes—
396C42H(5,𝔽2)14yes—
397C42H(5,𝔽2)12yes—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—42no—
406C23—42no—
407C23—44no—
408C23—44no—
409C23C2×#13714yes—
410C23C23×D411yescharacteristic
411Q8—44no—
412Q8—44no—
413Q8—88no—
414Q8—164no—
415D4—28no—
416D4—28no—
417D4—44no—
418D4—416no—
419D4—816no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—216no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—816no—
451C2×C4—164no—
452C2×C4—168no—
453C2×C4#53512yes—
454C2×C4#53512yes—
455C2×C4#54614yes—
456C2×C4C2×#13714yes—
457C2×C4C2×#13714yes—
458C2×C4#57614yes—
459C2×C4C22×#4111yescharacteristic
460C2×C4C2×H(5,𝔽2)12yes—
461C2×C4C2×H(5,𝔽2)11yescharacteristic
462C2×C4C2×2−1+412yes—
463C8—24no—
464C8—42no—
465C8—44no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—44no—
474C22—48no—
475C22—84no—
476C22—82no—
477C22—82no—
478C22C2×#53511yescharacteristic
479C22C2×#54611yescharacteristic
480C22#349911yescharacteristic
481C22C22×#13712yes—
482C22C2×#57612yes—
483C4—22no—
484C4—22no—
485C4—24no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—84no—
493C4—84no—
494C4#331418yes—
495C4#352012yes—
496C4#354812yes—
497C2—44no—
498C2—84no—
499C2#3895111yescharacteristic
500C2#6080011yescharacteristic
501C2#6081711yescharacteristic
502C2C2×#331412yes—
503C2#6248012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36149C2
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
1. #36180C2semidirect
3. #33293C2semidirect
4. #31954C2semidirect
5. #29844C2semidirect
6. #28719C2semidirect
7. #28508C2semidirect
8. #27950C2semidirect
9. #25597C2semidirect
10. #24833C2semidirect
11. #24130C2semidirect
12. #21859C2semidirect
13. #21692C2semidirect
14. #21265C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#361492
#332932
#319542
#298442
#287192
#285082
#279502
#255972
#248332
#241302
#218592
#216922
#212652