Finite Group Databasebeta

← #497501

Subgroups — #497501

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38955C211yesmaximal, characteristic
2#22406C212yesmaximal
3#13299C212yesmaximal, complemented
4#13091C214yesmaximal, complemented
5#12859C212yesmaximal, complemented
6#12798C211yesmaximal, characteristic, complemented
7C4×#354C211yesmaximal, characteristic, complemented
8#12583C212yesmaximal, complemented
9C2×#554C411yescharacteristic, complemented
10C2×#547C2211yescharacteristic
11C2×#410C2214yes—
12#2862C411yescharacteristic, complemented
13#2687—24no—
14#2621C412yescomplemented
15#2563C2211yescharacteristic
16#2546C411yescharacteristic, complemented
17#2503C412yescomplemented
18#2442—22no—
19#2432C2212yes—
20#2420—22no—
21#2344—22no—
22#2311C2212yes—
23C4×#124C2211yescharacteristic
24#2293C411yescharacteristic, 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#554—42no—
39#554—42no—
40#547—42no—
41#547—42no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×#124C2311yescharacteristic
47C2×#123D414yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8C2×C412yes—
50C2×C22⋊Q8C2×C412yes—
51C2×#121D412yes—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×D4—21no—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×#117—24no—
57C2×#117C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4D412yes—
60#410—28no—
61#410—44no—
62C2×#99—24no—
63C2×#99—24no—
64C2×#99C2314yes—
65C2×#99C2312yes—
66C2×#99C2312yes—
67#400—24no—
68#398—24no—
69#398—22no—
70#397—24no—
71#394—24no—
72#393—22no—
73#391—22no—
74#391C2×C412yes—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2×C411yescharacteristic
78#390—21no—
79#390C2312yes—
80#390C2311yescharacteristic
81#389—22no—
82#389—21no—
83#389—42no—
84#388—22no—
85#388—22no—
86#387—44no—
87#386—24no—
88#385—24no—
89#382—22no—
90#382C2×C412yes—
91#382C2×C411yescharacteristic
92#381—24no—
93#381—24no—
94#380—22no—
95#380—24no—
96#380—22no—
97C4×C4⋊C4—42no—
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4C2×C412yes—
102C43C2×C411yescharacteristic
103#354—24no—
104#354—42no—
105#342—22no—
106#342—22no—
107C2×#41—24no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—44no—
111C2×#41—82no—
112C22×Q8—21no—
113C22×Q8C22×C411yescharacteristic
114C22×Q8C22×C412yes—
115C22×D4C22⋊C411yescharacteristic
116C22×D4C2×D414yes—
117C22×D4C2411yescharacteristic
118C23×C4—22no—
119C23×C4C22×C412yes—
120C23×C4C2×D414yes—
121C23×C4C2×D412yes—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8—41no—
126C4⋊Q8—82no—
127#125—42no—
128#125—42no—
129#124—22no—
130#124—21no—
131#124—21no—
132#123—44no—
133#123—44no—
134C22⋊Q8—48no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137#121—42no—
138#121—42no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—42no—
146C4×D4—42no—
147#117—48no—
148#117—82no—
149#117—84no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C414yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C22⋊C414yes—
181C2×C22⋊C4C2×D412yes—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—44no—
186C2×C42—42no—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C412yes—
191#99—22no—
192#99—22no—
193#99—24no—
194#99—24no—
195#99—22no—
196#99—48no—
197#99—84no—
198#99#4114yes—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—21no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—42no—
208#95—44no—
209#95—42no—
210#95—44no—
211C24C2×C22⋊C412yes—
212C24C22×D412yes—
213#41—84no—
214#41—84no—
215#41—84no—
216#41—84no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—21no—
220C2×Q8—21no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—82no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—42no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4C2×#4112yes—
237C2×D4H(5,𝔽2)11yescharacteristic
238C2×D42−1+411yescharacteristic
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—84no—
262C22×C4C2×C22⋊C414yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C23×C411yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—44no—
282C4⋊C4—82no—
283C4⋊C4—84no—
284C4⋊C4—88no—
285C4⋊C4—84no—
286C4⋊C4#9912yes—
287C4⋊C4#9912yes—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—44no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—44no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—42no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—84no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—41no—
309C42—42no—
310C42—42no—
311C42—41no—
312C42—41no—
313C42—81no—
314C42—88no—
315C42—84no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23C22×C22⋊C411yescharacteristic
326C23#54612yes—
327C23#54712yes—
328Q8—82no—
329Q8—82no—
330Q8—81no—
331Q8—81no—
332Q8—162no—
333D4—42no—
334D4—41no—
335D4—41no—
336D4—42no—
337D4—42no—
338D4—84no—
339D4—84no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—84no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—84no—
383C2×C4—82no—
384C2×C4—84no—
385C2×C4—164no—
386C2×C4C2×#9911yescharacteristic
387C2×C4C2×#9914yes—
388C2×C4C2×#9911yescharacteristic
389C2×C4C4×#4111yescharacteristic
390C2×C4#52011yescharacteristic
391C22—21no—
392C22—21no—
393C22—22no—
394C22—22no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22#233911yescharacteristic
406C22C22×#9911yescharacteristic
407C22#288311yescharacteristic
408C4—21no—
409C4—24no—
410C4—21no—
411C4—21no—
412C4—42no—
413C4—41no—
414C4—44no—
415C4—81no—
416C4—82no—
417C4—81no—
418C4—84no—
419C4—81no—
420C4—82no—
421C4—81no—
422C4—164no—
423C4—162no—
424C4#212912yes—
425C2—21no—
426C2—21no—
427C2—42no—
428C2—42no—
429C2—82no—
430C2#2247611yescharacteristic
431C2C2×#212911yescharacteristic
432C2#3336211yescharacteristic
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. #13299C2semidirect
4. #13091C2semidirect
5. #12859C2semidirect
6. #12798C2semidirect
7. C4×#354C2semidirect
8. #12583C2semidirect
9. C2×#554C4semidirect
12. #2862C4semidirect
14. #2621C4semidirect
16. #2546C4semidirect
17. #2503C4semidirect
24. #2293C4semidirect

Maximal subgroups

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

subgroupindex
#389552
#224062
#132992
#130912
#128592
#127982
C4×#3542
#125832