Finite Group Databasebeta

← #497495

Subgroups — #497495

Order 512.

1043 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36406C211yesmaximal, characteristic
2#22406C212yesmaximal
3#13298C212yesmaximal, complemented
4#13091C214yesmaximal, complemented
5#12857C212yesmaximal, complemented
6#12798C211yesmaximal, characteristic, complemented
7C4×#354C211yesmaximal, characteristic, complemented
8#12581C212yesmaximal, complemented
9C2×#556C411yescharacteristic, complemented
10C2×#547C2211yescharacteristic
11C2×#410C2214yes—
12#2862C411yescharacteristic, complemented
13#2857C412yescomplemented
14#2553—24no—
15#2546C411yescharacteristic, complemented
16#2542C2211yescharacteristic
17#2499C412yescomplemented
18#2432C2212yes—
19#2427—22no—
20#2360—22no—
21#2344—22no—
22C4×#124C2211yescharacteristic
23#2297C2212yes—
24C42×D4C411yescharacteristic, complemented
25C2×#354C2212yes—
26C2×#354C2212yes—
27#1964—22no—
28#1899—24no—
29#1799C2212yes—
30#1782C2214yes—
31C4×#99C2212yes—
32#1393C2214yes—
33#1393C2214yes—
34#1393C2211yescharacteristic
35#1393C2212yes—
36#1393C2211yescharacteristic
37C22×#41C2312yes—
38#556—42no—
39#556—41no—
40#556—41no—
41#547—41no—
42#547—42no—
43#547—41no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#126—22no—
48C2×#124—21no—
49C2×#124C2311yescharacteristic
50C2×#123D414yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8C2×C412yes—
53C2×C22⋊Q8C2×C412yes—
54C2×#121D412yes—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2311yescharacteristic
60C2×#117—24no—
61C2×#117C2×C412yes—
62C22×C4⋊C4D412yes—
63C22×C42C2×C412yes—
64#410—28no—
65#410—44no—
66C2×#99—24no—
67C2×#99—24no—
68C2×#99C2314yes—
69C2×#99C2312yes—
70C2×#99C2312yes—
71#399—22no—
72#398—22no—
73#397—24no—
74#394—24no—
75#391—22no—
76#391C2×C411yescharacteristic
77#391C2×C412yes—
78#391C2×C412yes—
79#391C2×C412yes—
80#390C2312yes—
81#390C2311yescharacteristic
82#389—24no—
83#389—22no—
84#389—42no—
85#388—22no—
86#387—24no—
87#387—44no—
88#386—24no—
89#385—24no—
90#384C2×C412yes—
91#382—22no—
92#382C2×C411yescharacteristic
93#381—24no—
94#380—24no—
95#380—22no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—42no—
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C22⋊C4—22no—
100C4×C22⋊C4—24no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4C2×C412yes—
104C43C2×C411yescharacteristic
105#354—24no—
106#354—42no—
107#342—22no—
108#342—22no—
109C2×#41—24no—
110C2×#41—22no—
111C2×#41—22no—
112C2×#41—42no—
113C2×#41—42no—
114C2×#41—82no—
115C22×Q8—21no—
116C22×Q8C22×C411yescharacteristic
117C22×Q8C22×C412yes—
118C22×D4C22⋊C411yescharacteristic
119C22×D4C2×D414yes—
120C22×D4C2411yescharacteristic
121C23×C4—22no—
122C23×C4C22×C412yes—
123C23×C4C2×D414yes—
124C23×C4C2×D412yes—
125C4⋊Q8—41no—
126C4⋊Q8—41no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129C4⋊Q8—82no—
130#126—44no—
131#126—44no—
132#124—22no—
133#124—21no—
134#124—21no—
135#124—42no—
136#124—42no—
137#123—44no—
138#123—44no—
139C22⋊Q8—42no—
140C22⋊Q8—42no—
141C22⋊Q8—48no—
142C22⋊Q8—42no—
143C22⋊Q8—42no—
144#121—42no—
145#121—42no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—42no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×D4—22no—
152C4×D4—21no—
153C4×D4—21no—
154C4×D4—41no—
155C4×D4—41no—
156C4×D4—44no—
157#117—42no—
158#117—42no—
159#117—48no—
160#117—84no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C414yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C414yes—
190C2×C22⋊C4C2×D412yes—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—44no—
197C2×C42—42no—
198C2×C42—82no—
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C412yes—
202C2×C42C22×C411yescharacteristic
203#99—22no—
204#99—22no—
205#99—24no—
206#99—24no—
207#99—22no—
208#99—48no—
209#99—84no—
210#99#4114yes—
211#95—21no—
212#95—22no—
213#95—22no—
214#95—24no—
215#95—22no—
216#95—24no—
217#95—24no—
218#95—42no—
219#95—44no—
220#95—42no—
221#95—44no—
222C24C2×C22⋊C412yes—
223C24C22×D412yes—
224#41—82no—
225#41—84no—
226#41—82no—
227#41—84no—
228#41—82no—
229#41—82no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—44no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4C2×#4112yes—
251C2×D4H(5,𝔽2)11yescharacteristic
252C2×D42−1+411yescharacteristic
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—82no—
278C22×C4C2×C22⋊C414yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C23×C411yescharacteristic
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—48no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—84no—
299C4⋊C4—88no—
300C4⋊C4—84no—
301C4⋊C4#9912yes—
302C4⋊C4#9911yescharacteristic
303C4⋊C4#9911yescharacteristic
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—24no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—48no—
312C22⋊C4—44no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—42no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—41no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—41no—
332C42—41no—
333C42—82no—
334C42—88no—
335C42—84no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23C22×C22⋊C411yescharacteristic
346C23#54612yes—
347C23#54712yes—
348Q8—82no—
349Q8—82no—
350Q8—81no—
351Q8—81no—
352Q8—82no—
353Q8—82no—
354D4—42no—
355D4—41no—
356D4—41no—
357D4—42no—
358D4—42no—
359D4—84no—
360D4—84no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—82no—
400C2×C4—88no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—84no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—84no—
407C2×C4—82no—
408C2×C4—84no—
409C2×C4C2×#9911yescharacteristic
410C2×C4C2×#9912yes—
411C2×C4C2×#9912yes—
412C2×C4C2×#9911yescharacteristic
413C2×C4C4×#4111yescharacteristic
414C2×C4#52011yescharacteristic
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22#233911yescharacteristic
430C22C22×#9911yescharacteristic
431C22#288311yescharacteristic
432C4—21no—
433C4—22no—
434C4—22no—
435C4—21no—
436C4—21no—
437C4—44no—
438C4—42no—
439C4—42no—
440C4—41no—
441C4—44no—
442C4—81no—
443C4—84no—
444C4—82no—
445C4—81no—
446C4—81no—
447C4—164no—
448C4—162no—
449C4#212612yes—
450C2—21no—
451C2—21no—
452C2—42no—
453C2—42no—
454C2—82no—
455C2#2247611yescharacteristic
456C2C2×#212611yescharacteristic
457C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#547 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13298C2semidirect
4. #13091C2semidirect
5. #12857C2semidirect
6. #12798C2semidirect
7. C4×#354C2semidirect
8. #12581C2semidirect
9. C2×#556C4semidirect
12. #2862C4semidirect
13. #2857C4semidirect
15. #2546C4semidirect
17. #2499C4semidirect
24. C42×D4C4semidirect

Maximal subgroups

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

subgroupindex
#364062
#224062
#132982
#130912
#128572
#127982
C4×#3542
#125812