Finite Group Databasebeta

← #6526844

Subgroups — #6526844

Order 512.

2200 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3236C212yesmaximal, complemented
2C2×#3110C212yesmaximal
3#35173C211yesmaximal, characteristic, complemented
4C2×#2566C211yesmaximal, characteristic
5#28543C218yesmaximal, complemented
6#28433C218yesmaximal, complemented
7#28311C212yesmaximal
8#28302C212yesmaximal
9#27919C212yesmaximal, complemented
10#27910C212yesmaximal, complemented
11#20534C211yesmaximal, characteristic, complemented
12C2×#533C2212yes—
13C22×#125C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15#3236—216no—
16C2×#501C2214yescomplemented
17C2×#491C2214yescomplemented
18#3110—216no—
19C2×#484C2214yes—
20C2×#484C2214yes—
21C2×#483C2214yes—
22C2×#479C2218yescomplemented
23C2×#474C2214yes—
24C2×#461C2214yes—
25C2×#450C2214yes—
26C2×#440C2218yes—
27C2×#432C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2659—22no—
31#2584—24no—
32#2567C2214yescomplemented
33#2566—24no—
34#2566C2218yes—
35#2546C2214yescomplemented
36C2×#396—22no—
37#2316—22no—
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#385—22no—
41C2×#384C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44#2097C2218yescomplemented
45#2095C2218yescomplemented
46#2094C2214yes—
47#2093C2214yes—
48#2013C2214yes—
49#2013C2214yes—
50#1973C2218yes—
51#1968C2218yes—
52#1960C2214yescomplemented
53#1956C2214yescomplemented
54#1847C2214yescomplemented
55#1845C2214yescomplemented
56#1771C2218yescomplemented
57#1749C2212yes—
58C4×C22⋊C8C2212yes—
59C23×D4C2311yescharacteristic
60C24×C4—21no—
61C22×SD16C2318yes—
62#533—28no—
63#533—28no—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×C4⋊Q8C2312yes—
68C2×#125—22no—
69C2×#125—22no—
70C2×#125—28no—
71C2×#125C2314yes—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8C2314yes—
74C2×C4×Q8—22no—
75C2×C4×Q8C2314yes—
76C2×C4×D4—22no—
77C2×C4×D4—28no—
78C2×C4×D4C2314yes—
79C2×#117—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4D412yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4—22no—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#501—216no—
94#491—216no—
95#484—216no—
96#484—216no—
97#483—216no—
98#479—232no—
99#474—216no—
100#461—216no—
101#450—216no—
102#440—232no—
103#432D4116yes—
104C2×#107C2318yes—
105C2×#107C2318yes—
106C2×#106C2318yes—
107C2×C4⋊C8C2314yes—
108C2×#103C2318yes—
109C2×#103C2318yes—
110C2×#103C2318yes—
111C2×#102C2318yes—
112C2×C22⋊C8C2314yes—
113C2×#97C2314yes—
114C2×#97C2314yes—
115C2×C4×C8C2314yes—
116#398—28no—
117#396—216no—
118#395—28no—
119#391C2314yes—
120#391C2314yes—
121#390—22no—
122#390—22no—
123#390C2314yes—
124#390C2314yes—
125#389—28no—
126#389—28no—
127#389—24no—
128#389—24no—
129#385—216no—
130#384—22no—
131#384C2314yes—
132#384C2312yes—
133#384C2312yes—
134#382—22no—
135#382C2314yes—
136#380—24no—
137#380—24no—
138C4×C4⋊C4D414yes—
139C4×C4⋊C4C2314yes—
140C2×#95—22no—
141C2×#95D412yes—
142#336C2314yes—
143#336C2314yes—
144C25—21no—
145C22×Q8—22no—
146C22×Q8C2414yes—
147C22×D4—24no—
148C22×D4—21no—
149C22×D4—24no—
150C22×D4—42no—
151C22×D4C2412yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—44no—
157C23×C4—84no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2411yescharacteristic
163C2×SD16—216no—
164C2×SD16—216no—
165C2×SD16—48no—
166C22×C8C2418yes—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8—44no—
172C4⋊Q8—44no—
173#125—28no—
174#125—28no—
175#125—48no—
176#125—48no—
177C22⋊Q8—28no—
178C22⋊Q8—28no—
179C22⋊Q8—48no—
180C4×Q8—28no—
181C4×Q8—28no—
182C4×Q8—48no—
183C4×D4—28no—
184C4×D4—28no—
185C4×D4—48no—
186#117—48no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—84no—
209C2×C4⋊C4C2×D418yes—
210C2×C4⋊C4C2412yes—
211C2×C4⋊C4C2412yes—
212C2×C4⋊C4C2412yes—
213C2×C4⋊C4C2414yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—48no—
220C2×C22⋊C4—84no—
221C2×C42—24no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D418yes—
228C2×C42C2412yes—
229C2×C42C2412yes—
230C2×C42C2412yes—
231#107—216no—
232#107—216no—
233#106—216no—
234C4⋊C8C2×D4116yes—
235#103—216no—
236#103—216no—
237#103—216no—
238#102—216no—
239C22⋊C8C2×D4116yes—
240#97—44no—
241#97C2×D4116yes—
242C4×C8—44no—
243#95—28no—
244#95—28no—
245#95—28no—
246#95C2×D418yes—
247C24—24no—
248C24—22no—
249C24—44no—
250C24—42no—
251C24—42no—
252C24—44no—
253C24C22×D411yescharacteristic
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—28no—
257C2×Q8—28no—
258C2×Q8—44no—
259C2×Q8—84no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—416no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D42−1+414yes—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—48no—
289C22×C4—88no—
290C22×C4—816no—
291C22×C4#12714yes—
292C22×C4#12714yes—
293C22×C4#13718yes—
294C22×C4C22×D414yes—
295C22×C4C22×D412yes—
296C22×C4C22×D412yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D414yes—
299C22×C4C22×D414yes—
300C22×C4C2511yescharacteristic
301SD16—432no—
302C2×C8—28no—
303C2×C8—28no—
304C2×C8C22×D4116yes—
305C4⋊C4—216no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—48no—
315C4⋊C4—48no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—44no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—88no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C22⋊C4—48no—
327C22⋊C4—48no—
328C22⋊C4—416no—
329C22⋊C4—88no—
330C42—24no—
331C42—24no—
332C42—42no—
333C42—42no—
334C42—48no—
335C42#12718yes—
336C42C22×D414yes—
337C42C22×D414yes—
338C23—22no—
339C23—21no—
340C23—21no—
341C23—24no—
342C23—48no—
343C23—416no—
344C23—44no—
345C23—42no—
346C23—42no—
347C23—44no—
348C23—84no—
349C23—88no—
350C23—84no—
351C23C2×#12712yes—
352C23C2×#13714yes—
353C23C23×D411yescharacteristic
354Q8—48no—
355Q8—48no—
356Q8—88no—
357D4—28no—
358D4—48no—
359D4—44no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—28no—
363C2×C4—24no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—28no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—48no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—816no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4C2×#12714yes—
390C2×C4C2×#12712yes—
391C2×C4C2×#12714yes—
392C2×C4C2×#13718yes—
393C2×C4C23×D412yes—
394C2×C4C2×2−1+412yes—
395C2×C4#58512yes—
396C8—216no—
397C22—24no—
398C22—22no—
399C22—24no—
400C22—22no—
401C22—44no—
402C22—48no—
403C22—816no—
404C22—84no—
405C22—82no—
406C22—82no—
407C22#315214yes—
408C22C22×#12711yescharacteristic
409C22#348511yescharacteristic
410C22#348611yescharacteristic
411C22C22×#13712yes—
412C22#358512yes—
413C4—28no—
414C4—22no—
415C4—24no—
416C4—24no—
417C4—44no—
418C4—84no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4—84no—
423C4#348914yes—
424C2—24no—
425C2—84no—
426C2#3563411yescharacteristic
427C2C2×#315211yescharacteristic
428C2#6064411yescharacteristic
429C2#6170812yes—
430C2#6182612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3236C2semidirect
3. #35173C2semidirect
5. #28543C2semidirect
6. #28433C2semidirect
9. #27919C2semidirect
10. #27910C2semidirect
11. #20534C2semidirect
16. C2×#501C22semidirect
17. C2×#491C22semidirect
22. C2×#479C22semidirect
32. #2567C22semidirect
35. #2546C22semidirect
44. #2097C22semidirect
45. #2095C22semidirect
52. #1960C22semidirect
53. #1956C22semidirect
54. #1847C22semidirect
55. #1845C22semidirect
56. #1771C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32362
C2×#31102
#351732
C2×#25662
#285432
#284332
#283112
#283022
#279192
#279102
#205342