Finite Group Databasebeta

← #6484958

Subgroups — #6484958

Order 512.

1442 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35336C211yesmaximal, characteristic, complemented
2#35332C211yesmaximal, characteristic
3#33050C212yesmaximal, complemented
4#33044C212yesmaximal, complemented
5#30191C214yesmaximal, complemented
6#29361C212yesmaximal, complemented
7#29360C212yesmaximal, complemented
8#29221C212yesmaximal, complemented
9#29140C212yesmaximal, complemented
10#29134C214yesmaximal, complemented
11#28752C212yesmaximal, complemented
12#28720C212yesmaximal, complemented
13#21898C212yesmaximal, complemented
14#21897C212yesmaximal
15C4×#424C211yesmaximal, characteristic, complemented
16C2×#541C2211yescharacteristic
17C2×#540C2211yescharacteristic
18C2×#488C2212yes—
19C2×#488C2212yes—
20C2×#424C2212yes—
21C2×C4×M16C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2859C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2858C2211yescharacteristic
27#2851C2212yes—
28#2850C2212yes—
29#2647C2212yes—
30#2642C2211yescharacteristic
31#2641C2211yescharacteristic
32#2629C2212yes—
33#2620C2212yes—
34#2590—24no—
35#2587—24no—
36#2559—24no—
37#2558—24no—
38#2330C2212yes—
39#2326C2212yes—
40#2314C2212yes—
41C4×#125C2212yes—
42C4×#124C2212yes—
43C4×#117C2211yescharacteristic
44#2100C2214yes—
45#2099C2214yes—
46#2072C2214yes—
47#2071C2214yes—
48#2042C2214yes—
49#2042C2214yes—
50#2026C2214yes—
51#2025C2214yes—
52#1991C2212yes—
53#1991C2212yes—
54#1989C2214yes—
55#1989C2214yes—
56#1982C2214yes—
57#1981C2214yes—
58#1958C2212yes—
59#1958C2212yes—
60#1936C2214yes—
61#1935C2214yes—
62#1934C2214yes—
63#1933C2214yes—
64#1895C2214yes—
65#1894C2214yes—
66#1893C2214yes—
67#1892C2214yes—
68#1835C2212yes—
69#1835C2212yes—
70#1805C2214yes—
71#1804C2214yes—
72#1803C2214yes—
73#1802C2214yes—
74#1768C2212yes—
75#1768C2212yes—
76C4×C4⋊C8C2212yes—
77C4×C4⋊C8C2212yes—
78#1729C2212yes—
79C22×M16C2312yes—
80C23×C8C2312yes—
81#541—28no—
82#540—28no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×#127C2311yescharacteristic
85C2×#126—22no—
86C2×#125—22no—
87C2×#125C2312yes—
88C2×#125C2311yescharacteristic
89C2×#125C2311yescharacteristic
90C2×#125C2312yes—
91C2×#124C2312yes—
92C2×#124C2312yes—
93C2×#123—22no—
94C2×C22⋊Q8C2312yes—
95C2×#121—22no—
96C2×#121C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×D4C2312yes—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C42C2311yescharacteristic
102#488—44no—
103#488—44no—
104#424—28no—
105C2×C4⋊C8C2314yes—
106C2×C4⋊C8C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#102C2314yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C4×M16—42no—
116C2×#97C2312yes—
117C2×C4×C8—42no—
118C2×C4×C8C2312yes—
119C2×C4×C8C2312yes—
120C2×C4×C8C2312yes—
121#400—24no—
122#400—24no—
123#398—24no—
124#398—24no—
125#396—24no—
126#396—24no—
127#394—24no—
128#394—24no—
129#391C2312yes—
130#391C2312yes—
131#390—24no—
132#390C2312yes—
133#390C2312yes—
134#389—24no—
135#389—24no—
136#389—24no—
137#389—24no—
138#389C2314yes—
139#389C2314yes—
140#388—24no—
141#388—24no—
142#388—24no—
143#388—24no—
144#387—24no—
145#387—24no—
146#387—24no—
147#387—24no—
148#386—24no—
149#385—24no—
150#385—24no—
151#384C2312yes—
152#384C2312yes—
153#384C2312yes—
154#382C2312yes—
155#382C2312yes—
156#382C2312yes—
157#380C2314yes—
158C4×C4⋊C4—24no—
159C4×C4⋊C4D414yes—
160C4×C4⋊C4C2314yes—
161C4×C22⋊C4—24no—
162C4×C22⋊C4D414yes—
163#376C2311yescharacteristic
164#376C2311yescharacteristic
165C43C2311yescharacteristic
166C43C2311yescharacteristic
167#340C2314yes—
168#340C2314yes—
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#336C2314yes—
176#336C2314yes—
177C22×Q8C2412yes—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4C2412yes—
181C23×C4—22no—
182C23×C4C2×D412yes—
183C23×C4C2411yescharacteristic
184C2×M16—24no—
185C2×M16—42no—
186C2×M16—42no—
187C22×C8—24no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8C2414yes—
191C22×C8C2414yes—
192C4⋊Q8—24no—
193#127—24no—
194#126—48no—
195#125—24no—
196#125—24no—
197#125—24no—
198#125—24no—
199#125—44no—
200#125—48no—
201#124—24no—
202#124—24no—
203#124—44no—
204#123—48no—
205C22⋊Q8—28no—
206#121—28no—
207#121—48no—
208C4×Q8—28no—
209C4×D4—28no—
210#117—42no—
211#117#4118yes—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2412yes—
227C2×C4⋊C4C2412yes—
228C2×C4⋊C4C2414yes—
229C2×C4⋊C4C2414yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4C2×D412yes—
239C2×C22⋊C4C2×D412yes—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42C2×D412yes—
245C2×C42C2×D414yes—
246C2×C42C2×D414yes—
247C2×C42C2411yescharacteristic
248C2×C42C2411yescharacteristic
249C2×C42C2411yescharacteristic
250C2×C42C2411yescharacteristic
251C2×C42C2411yescharacteristic
252C2×C42C2411yescharacteristic
253C4⋊C8—28no—
254C4⋊C8—28no—
255#103—44no—
256#103—44no—
257#103—44no—
258#103—44no—
259#102—44no—
260#102—44no—
261#102—44no—
262#102—44no—
263#97—42no—
264C4×C8—24no—
265C4×C8—24no—
266C4×C8—42no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95C2×D414yes—
277#95C2×D414yes—
278C24—22no—
279C24C22×D411yescharacteristic
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×D4—24no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—84no—
291C2×D4—84no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—84no—
304C22×C4C22×D412yes—
305C22×C4C22×D412yes—
306C22×C4C22×D412yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D412yes—
309C22×C4C22×D412yes—
310C22×C4C22×D412yes—
311C22×C4C2×#4114yes—
312C22×C4C2511yescharacteristic
313M16—48no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—48no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—42no—
328C4⋊C4—48no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—48no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C4⋊C4—84no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C4⋊C4#12418yes—
341C22⋊C4—42no—
342C22⋊C4—48no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—48no—
347C22⋊C4—84no—
348C22⋊C4—84no—
349C22⋊C4—84no—
350C22⋊C4#12418yes—
351C42—24no—
352C42—21no—
353C42—21no—
354C42—22no—
355C42—22no—
356C42—44no—
357C42—44no—
358C42—44no—
359C42—84no—
360C42C2×#4114yes—
361C42C2×#4114yes—
362C42H(5,𝔽2)12yes—
363C422−1+412yes—
364C23—24no—
365C23—21no—
366C23—21no—
367C23—44no—
368C23—44no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23C2×#12412yes—
373C23C23×D411yescharacteristic
374Q8—44no—
375Q8—44no—
376D4—44no—
377D4—44no—
378D4—164no—
379D4—164no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—48no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—164no—
408C2×C4C2×#12414yes—
409C2×C4C2×#12414yes—
410C2×C4C2×#12412yes—
411C2×C4#53714yes—
412C2×C4C22×#4112yes—
413C2×C4C2×H(5,𝔽2)11yescharacteristic
414C2×C4C2×2−1+411yescharacteristic
415C2×C4#58512yes—
416C8—44no—
417C8—44no—
418C22—22no—
419C22—24no—
420C22—42no—
421C22—42no—
422C22—84no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—84no—
427C22C22×#12411yescharacteristic
428C22C2×#53711yescharacteristic
429C22#344911yescharacteristic
430C22#358414yes—
431C4—22no—
432C4—24no—
433C4—24no—
434C4—24no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—84no—
439C4—82no—
440C4—82no—
441C4—82no—
442C4—84no—
443C4#350612yes—
444C4#350712yes—
445C2—42no—
446C2—162no—
447C2#3518011yescharacteristic
448C2#6062412yes—
449C2#6091512yes—
450C2#6100912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
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
1. #35336C2semidirect
3. #33050C2semidirect
4. #33044C2semidirect
5. #30191C2semidirect
6. #29361C2semidirect
7. #29360C2semidirect
8. #29221C2semidirect
9. #29140C2semidirect
10. #29134C2semidirect
11. #28752C2semidirect
12. #28720C2semidirect
13. #21898C2semidirect
15. C4×#424C2semidirect

Maximal subgroups

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

subgroupindex
#353362
#353322
#330502
#330442
#301912
#293612
#293602
#292212
#291402
#291342
#287522
#287202
#218982
#218972
C4×#4242