Finite Group Databasebeta

← #493716

Subgroups — #493716

Order 512.

1299 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34275C211yesmaximal, characteristic
2C2×#2128C212yesmaximal
3#20634C212yesmaximal, complemented
4#12799C214yesmaximal, complemented
5#12797C214yesmaximal, complemented
6#12581C212yesmaximal, complemented
7C2×#535C2211yescharacteristic
8C2×#520C2211yescharacteristic
9C2×#518C411yescharacteristic, complemented
10C2×#410C2214yes—
11#2363—22no—
12#2361—22no—
13#2358—24no—
14#2328C2212yes—
15#2326C412yescomplemented
16#2317C411yescharacteristic, complemented
17#2316—22no—
18C4×#127C2212yes—
19C4×#124C412yescomplemented
20#2297C2211yescharacteristic
21C2×#390C411yescharacteristic, complemented
22C4×#117C411yescharacteristic, complemented
23#2128—28no—
24C2×#353C2218yes—
25#1799—24no—
26#1799C2218yescomplemented
27#1740—22no—
28#1393C2214yescomplemented
29#1393C2214yescomplemented
30C22×#41C2311yescharacteristic
31C23×D4C2×C411yescharacteristic
32#535—24no—
33#535D414yes—
34#535Q814yes—
35C2×#127C2312yes—
36C2×#124C2×C412yes—
37C2×#121—22no—
38C2×#120—22no—
39#520—44no—
40#518—44no—
41C2×C4×Q8C2×C412yes—
42C2×C4×D4—24no—
43C2×C4×D4—21no—
44C2×C4×D4—21no—
45C2×C4×D4C2×C412yes—
46C2×C4×D4C2×C412yes—
47C2×C4×D4D414yes—
48C2×C4×D4C2312yes—
49C2×#117—24no—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117D412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4D412yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C42C2×C411yescharacteristic
59#410—28no—
60#410—28no—
61C2×#99—24no—
62C2×#99C2318yes—
63#391C2×C411yescharacteristic
64#391C2×C411yescharacteristic
65#390—22no—
66#390—24no—
67#390C2×C412yes—
68#390C2×C412yes—
69#390C2311yescharacteristic
70#390C2311yescharacteristic
71#389—22no—
72#389—48no—
73#388—24no—
74#388—24no—
75#386—22no—
76#386—24no—
77#386—24no—
78#385—22no—
79#385—24no—
80#384C2×C412yes—
81#382C2×C412yes—
82#380—28no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—24no—
88#376C2×C412yes—
89C43C2×C412yes—
90#353—216no—
91#342—28no—
92C25—21no—
93C2×#41—42no—
94C2×#41—42no—
95C2×#41C2×D412yes—
96C2×#41C2×Q812yes—
97C2×#41#4114yes—
98C22×Q8C22×C411yescharacteristic
99C22×D4—24no—
100C22×D4—21no—
101C22×D4—22no—
102C22×D4—42no—
103C22×D4—42no—
104C22×D4C4⋊C414yescomplemented
105C22×D4C22×C412yes—
106C22×D4C2×D412yes—
107C22×D4C2411yescharacteristic
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4C22⋊C412yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D414yes—
115#127—24no—
116#127C2×D414yes—
117#127C2×Q814yes—
118#124—24no—
119#124C4⋊C418yescomplemented
120#121—28no—
121#121—28no—
122#120—28no—
123#120—28no—
124C4×Q8—44no—
125C4×D4—44no—
126C4×D4—44no—
127C4×D4—44no—
128C4×D4—48no—
129C4×D4—44no—
130C4×D4—44no—
131C4×D4—88no—
132#117—22no—
133#117—24no—
134#117—44no—
135#117—42no—
136#117—48no—
137#117—48no—
138#117C4⋊C414yescomplemented
139C2×C4⋊C4—21no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—48no—
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4C22⋊C414yes—
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22⋊C411yescharacteristic
149C2×C4⋊C4C22×C411yescharacteristic
150C2×C4⋊C4C22×C411yescharacteristic
151C2×C4⋊C4C22×C412yes—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—28no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—48no—
163C2×C22⋊C4—84no—
164C2×C22⋊C4C22⋊C412yes—
165C2×C22⋊C4C2×D414yes—
166C2×C42—21no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—42no—
171C2×C42—44no—
172C2×C42C22⋊C414yes—
173C2×C42C22×C412yes—
174C2×C42C22×C412yes—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177#99—28no—
178#99—28no—
179#99—48no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—24no—
184#95—22no—
185#95—48no—
186C24—22no—
187C24—24no—
188C24—42no—
189C24—42no—
190C24—82no—
191C24—82no—
192C24C2×C22⋊C411yescharacteristic
193C24C22×D412yes—
194#41—44no—
195#41—44no—
196#41—48no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8C2×C4⋊C412yes—
200C2×Q8#11712yes—
201C2×D4—28no—
202C2×D4—28no—
203C2×D4—22no—
204C2×D4—24no—
205C2×D4—42no—
206C2×D4—41no—
207C2×D4—42no—
208C2×D4—41no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—48no—
212C2×D4—48no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—84no—
216C2×D4—84no—
217C2×D4C2×C4⋊C414yes—
218C2×D4#12114yes—
219C2×D4C22⋊Q814yes—
220C2×D4C22×D411yescharacteristic
221C2×D4C22×Q811yescharacteristic
222C2×D4C2×#4112yes—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—28no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—81no—
240C22×C4—84no—
241C22×C4—84no—
242C22×C4—84no—
243C22×C4—82no—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C414yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C4⋊C412yes—
250C22×C4#11712yes—
251C22×C4#12114yes—
252C22×C4C22⋊Q814yes—
253C22×C4C23×C411yescharacteristic
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—28no—
257C4⋊C4—22no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—88no—
262C4⋊C4—84no—
263C4⋊C4—168no—
264C4⋊C4#9914yes—
265C22⋊C4—24no—
266C22⋊C4—216no—
267C22⋊C4—24no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—48no—
271C22⋊C4—44no—
272C22⋊C4—42no—
273C22⋊C4—44no—
274C22⋊C4—48no—
275C22⋊C4—48no—
276C22⋊C4—82no—
277C22⋊C4—84no—
278C42—22no—
279C42—24no—
280C42—44no—
281C42—41no—
282C42—41no—
283C42—44no—
284C42—42no—
285C42—42no—
286C42—88no—
287C42—82no—
288C42C2×C4⋊C414yes—
289C23—21no—
290C23—22no—
291C23—21no—
292C23—22no—
293C23—24no—
294C23—41no—
295C23—41no—
296C23—42no—
297C23—42no—
298C23—44no—
299C23—44no—
300C23—48no—
301C23—42no—
302C23—42no—
303C23—88no—
304C23—164no—
305C23—162no—
306C23—162no—
307C23#38012yes—
308C23C22×C22⋊C411yescharacteristic
309C23C2×#12112yes—
310C23C2×C22⋊Q812yes—
311Q8—42no—
312Q8—42no—
313D4—44no—
314D4—44no—
315D4—41no—
316D4—41no—
317D4—42no—
318D4—44no—
319D4—44no—
320D4—48no—
321D4—88no—
322D4—88no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—28no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—28no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—88no—
350C2×C4—84no—
351C2×C4—88no—
352C2×C4—88no—
353C2×C4—88no—
354C2×C4—164no—
355C2×C4—162no—
356C2×C4—164no—
357C2×C4#38014yes—
358C2×C4#38012yes—
359C2×C4C2×#9914yes—
360C2×C4C2×#9912yes—
361C2×C4#41012yes—
362C2×C4C22×C4⋊C411yescharacteristic
363C2×C4C2×#11711yescharacteristic
364C22—21no—
365C22—21no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—48no—
374C22—42no—
375C22—41no—
376C22—42no—
377C22—41no—
378C22—44no—
379C22—82no—
380C22—84no—
381C22—82no—
382C22—82no—
383C22—168no—
384C22C2×#38011yescharacteristic
385C22C22×#9911yescharacteristic
386C22C2×#41011yescharacteristic
387C4—24no—
388C4—21no—
389C4—21no—
390C4—24no—
391C4—42no—
392C4—44no—
393C4—44no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—168no—
399C4#179914yes—
400C2—21no—
401C2—21no—
402C2—42no—
403C2—42no—
404C2—42no—
405C2—84no—
406C2C2×#179911yescharacteristic
407C2#3335911yescharacteristic
408C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20634C2semidirect
4. #12799C2semidirect
5. #12797C2semidirect
6. #12581C2semidirect
9. C2×#518C4semidirect
15. #2326C4semidirect
16. #2317C4semidirect
19. C4×#124C4semidirect
21. C2×#390C4semidirect
22. C4×#117C4semidirect
26. #1799C22semidirect
28. #1393C22semidirect
29. #1393C22semidirect
104. C22×D4C4⋊C4semidirect
119. #124C4⋊C4semidirect
138. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342752
C2×#21282
#206342
#127992
#127972
#125812