Finite Group Databasebeta

← #6355384

Subgroups — #6355384

Order 512.

1640 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic, complemented
2#60562C212yesmaximal
3C2×#2990C211yesmaximal, characteristic
4#19490C218yesmaximal, complemented
5#19485C218yesmaximal, complemented
6#19290C214yesmaximal, complemented
7#19259C214yesmaximal, complemented
8#17819C211yesmaximal, characteristic, complemented
9#17751C212yesmaximal, complemented
10C2×#533C2212yes—
11C2×C4×#41C2211yescharacteristic
12#3249—21no—
13#3249—21no—
14#3230C2214yescomplemented
15#3209C2214yescomplemented
16#3189—22no—
17#3170C2214yes—
18#3116—24no—
19#3113C2214yes—
20#3110C2214yes—
21#3107C2218yes—
22#3104—24no—
23#3104—22no—
24#3101—22no—
25#3095—22no—
26#3079C2214yes—
27#3039C2214yes—
28#3007—22no—
29#2990—24no—
30#2990C2218yes—
31C2×#446C2212yes—
32C4×#137—24no—
33C2×#435—22no—
34C2×C8×D4C2212yes—
35#2920C2212yes—
36C2×#432—21no—
37C2×#431C2211yescharacteristic
38C2×#423C2211yescharacteristic
39C22×C4⋊C8—21no—
40#1700C22116yescomplemented
41#1694C2218yescomplemented
42#1690C2218yescomplemented
43#1679C22116yescomplemented
44#1673C2218yescomplemented
45#1669C2218yescomplemented
46#1574C2218yescomplemented
47#1568C2214yes—
48#1561C2218yescomplemented
49#1552C2218yes—
50#1549C2214yescomplemented
51#1546C2214yescomplemented
52C22×#41D411yescharacteristic
53C2×#137—24no—
54C22×M16—21no—
55C23×C8—22no—
56#533—22no—
57#533—22no—
58#533—22no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533C2314yes—
63#533C2314yes—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×#125—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C4×#41—22no—
70C4×#41—22no—
71C4×#41D414yes—
72C4×#41C2314yes—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4D411yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C2×#117—21no—
82C2×#117—22no—
83C2×#117D411yescharacteristic
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#500C2314yes—
87#494C2314yes—
88#487C2318yes—
89#485—28no—
90#484—24no—
91#484—28no—
92#483—24no—
93#477—24no—
94#477—24no—
95#477—24no—
96#476—24no—
97#475C2318yes—
98#474C2318yes—
99#467—24no—
100#467—24no—
101#462—24no—
102#462—24no—
103#462—24no—
104#462C2318yes—
105#461—24no—
106#461C2318yes—
107#452—28no—
108#451—24no—
109#451—28no—
110#450—24no—
111#446—24no—
112#446C2318yes—
113#441—28no—
114#440—28no—
115C4×Q16—28no—
116C4×Q16—24no—
117C4×Q16—24no—
118C4×SD16—28no—
119#435—216no—
120C8×D4—24no—
121C8×D4C2318yes—
122#432—22no—
123#432—28no—
124#432C2314yes—
125#431—22no—
126#431D414yes—
127#431C2314yes—
128#431C2314yes—
129#428—24no—
130#423—22no—
131#423C2314yes—
132#423C2314yes—
133C2×C4⋊C8—22no—
134C2×C4⋊C8—21no—
135C2×C4⋊C8—28no—
136C2×C4⋊C8D411yescharacteristic
137C2×C4⋊C8D411yescharacteristic
138C2×C4⋊C8C2312yes—
139#419—24no—
140#417—24no—
141#407—22no—
142C2×C22⋊C8—22no—
143C2×C22⋊C8D412yes—
144C4×M16—22no—
145C2×#97—22no—
146C2×C4×C8C2312yes—
147#335C2318yes—
148#334C2318yes—
149#332C23116yes—
150#332C23116yes—
151C2×#41—22no—
152C2×#41—44no—
153C2×#41C2×D414yes—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8C2×D411yescharacteristic
157C22×D4—21no—
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4#4111yescharacteristic
165#137—416no—
166C2×Q16—28no—
167C2×Q16—24no—
168C2×Q16—24no—
169C2×SD16—28no—
170C2×M16—24no—
171C2×M16—24no—
172C2×M16—82no—
173C22×C8—22no—
174C22×C8—28no—
175C22×C8—84no—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8C2414yes—
185C4⋊Q8C2414yes—
186#125—24no—
187#125—24no—
188#125—42no—
189#125—42no—
190C22⋊Q8—24no—
191C22⋊Q8—24no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C22⋊Q8—42no—
195C22⋊Q8—42no—
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C4×Q8—22no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—24no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×Q8C2×D414yes—
210C4×Q8C2×D412yes—
211C4×Q8C2412yes—
212C4×D4—24no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—22no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4C2×D414yes—
220C4×D4C2×D412yes—
221C4×D4C2412yes—
222#117—22no—
223#117—24no—
224#117—24no—
225#117—24no—
226#117—24no—
227#117—42no—
228#117—42no—
229#117C2×D414yes—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—42no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—42no—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42#4111yescharacteristic
255C2×C42#4112yes—
256C2×C42C2411yescharacteristic
257C2×C42C2412yes—
258#107—24no—
259#107—24no—
260#107—28no—
261#106—28no—
262C4⋊C8—28no—
263C4⋊C8—24no—
264C4⋊C8—22no—
265C4⋊C8—24no—
266C4⋊C8C2×D414yes—
267C4⋊C8C2×D414yes—
268C4⋊C8C2414yes—
269C4⋊C8C2418yes—
270#103—28no—
271#103—28no—
272#103—28no—
273#103—28no—
274#103—28no—
275#102—28no—
276C22⋊C8—24no—
277C22⋊C8—24no—
278C22⋊C8—28no—
279C22⋊C8C2×D418yes—
280#97—28no—
281#97—24no—
282C4×C8—24no—
283C4×C8C2×D414yes—
284C4×C8C2414yes—
285C24—21no—
286C24#12111yescharacteristic
287#41—28no—
288#41—44no—
289#41—48no—
290C2×Q8—21no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—42no—
294C2×Q8—48no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—82no—
298C2×Q8—82no—
299C2×Q8C22×D412yes—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—21no—
303C2×D4—44no—
304C2×D4C22×D412yes—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4#12112yes—
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12112yes—
326C22×C4#12112yes—
327C22×C4#12111yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4112yes—
334Q16—416no—
335Q16—84no—
336Q16—84no—
337SD16—416no—
338M16—48no—
339M16—84no—
340C2×C8—28no—
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—44no—
345C2×C8—88no—
346C2×C8C22×D414yes—
347C2×C8C22×D414yes—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4C22×D412yes—
366C4⋊C4C22×D412yes—
367C22⋊C4—24no—
368C22⋊C4—22no—
369C22⋊C4—24no—
370C22⋊C4—22no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C42—24no—
374C42—22no—
375C42—24no—
376C42—22no—
377C42—44no—
378C42—44no—
379C42#12112yes—
380C42#12112yes—
381C42#12112yes—
382C42C22×D412yes—
383C42C22×D411yescharacteristic
384C42C22×D411yescharacteristic
385C42C22×D412yes—
386C42C2×#4112yes—
387C42C2×#4111yescharacteristic
388C42C2511yescharacteristic
389C23—24no—
390C23—22no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23C2×#12112yes—
397C23C2×#12111yescharacteristic
398Q8—24no—
399Q8—42no—
400Q8—44no—
401Q8—44no—
402Q8—48no—
403Q8—84no—
404Q8—84no—
405D4—24no—
406D4—22no—
407D4—48no—
408D4#57514yes—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—48no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4#47314yes—
438C2×C4C2×#12111yescharacteristic
439C2×C4C2×#12112yes—
440C2×C4C2×#12111yescharacteristic
441C2×C4C2×#12112yes—
442C2×C4C2×#12111yescharacteristic
443C2×C4C2×#12112yes—
444C2×C4C23×D411yescharacteristic
445C2×C4C23×D411yescharacteristic
446C2×C4C22×#4111yescharacteristic
447C8—24no—
448C8—24no—
449C8—44no—
450C8—48no—
451C22—21no—
452C22—21no—
453C22—22no—
454C22—22no—
455C22—24no—
456C22—22no—
457C22—48no—
458C22—42no—
459C22—42no—
460C22—82no—
461C22C2×#47312yes—
462C22C22×#12111yescharacteristic
463C4—21no—
464C4—22no—
465C4—22no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—84no—
473C4—84no—
474C4C2×#47314yes—
475C4C2×#57512yes—
476C4#358511yescharacteristic
477C4#358611yescharacteristic
478C2—22no—
479C2—42no—
480C2—44no—
481C2#6042511yescharacteristic
482C2#6044411yescharacteristic
483C2C22×#47311yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60943C2semidirect
4. #19490C2semidirect
5. #19485C2semidirect
6. #19290C2semidirect
7. #19259C2semidirect
8. #17819C2semidirect
9. #17751C2semidirect
14. #3230C22semidirect
15. #3209C22semidirect
40. #1700C22semidirect
41. #1694C22semidirect
42. #1690C22semidirect
43. #1679C22semidirect
44. #1673C22semidirect
45. #1669C22semidirect
46. #1574C22semidirect
48. #1561C22semidirect
50. #1549C22semidirect
51. #1546C22semidirect

Maximal subgroups

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

subgroupindex
#609432
#605622
C2×#29902
#194902
#194852
#192902
#192592
#178192
#177512