Finite Group Databasebeta

← #493973

Subgroups — #493973

Order 512.

1255 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#535C211yesmaximal, characteristic
2#20723C212yesmaximal, complemented
3#12859C212yesmaximal, complemented
4#12797C214yesmaximal, complemented
5#12697C212yesmaximal
6#12628C214yesmaximal, complemented
7C2×#535C2211yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C22×C4×D4C411yescharacteristic, complemented
10#2328C2212yes—
11#2326C412yescomplemented
12#2324—22no—
13#2321—22no—
14#2317C411yescharacteristic, complemented
15#2311C2211yescharacteristic
16#2309C411yescharacteristic, complemented
17C4×#127C2212yes—
18C4×#124C412yescomplemented
19C4×#121—22no—
20C4×#120—22no—
21#2283—22no—
22C4×#117C411yescharacteristic, complemented
23C2×#353C2218yes—
24#1820—22no—
25#1799—22no—
26#1799C2214yescomplemented
27#1782C2214yes—
28C4×#99—22no—
29C4×#99C2214yescomplemented
30#1393C2218yescomplemented
31C22×#41C2311yescharacteristic
32C23×D4C2×C411yescharacteristic
33C24×C4—21no—
34#535—28no—
35C2×#127C2312yes—
36C2×#124C2×C412yes—
37C2×#121—22no—
38C2×#120—22no—
39C4×#41—42no—
40C4×#41—42no—
41C2×C4×Q8C2×C411yescharacteristic
42C2×C4×Q8C2×C411yescharacteristic
43C2×C4×D4—24no—
44C2×C4×D4—21no—
45C2×C4×D4—21no—
46C2×C4×D4—42no—
47C2×C4×D4—42no—
48C2×C4×D4C2×C412yes—
49C2×C4×D4C2×C412yes—
50C2×C4×D4D412yes—
51C2×C4×D4D412yes—
52C2×C4×D4C2311yescharacteristic
53C2×C4×D4C2311yescharacteristic
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C2×#117D412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C22⋊C4—21no—
60C22×C42C2×C411yescharacteristic
61C22×C42D412yes—
62C2×#99—24no—
63C2×#99C2318yes—
64#391C2×C412yes—
65#390—24no—
66#390—22no—
67#390C2×C412yes—
68#390C2×C412yes—
69#390C2312yes—
70#389—22no—
71#388—24no—
72#388—24no—
73#386—22no—
74#386—24no—
75#386—24no—
76#385—24no—
77#385—22no—
78#384C2×C412yes—
79#383—42no—
80#382—42no—
81#382C2×C412yes—
82#381—24no—
83#380—22no—
84#380—24no—
85#379—22no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—42no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4—24no—
93#376—42no—
94#376C2×C412yes—
95C43C2×C412yes—
96#353—216no—
97#342—24no—
98#342—24no—
99C25—21no—
100C2×#41—24no—
101C2×#41—42no—
102C2×#41—42no—
103C22×Q8C22×C411yescharacteristic
104C22×D4—24no—
105C22×D4—24no—
106C22×D4—21no—
107C22×D4—42no—
108C22×D4—42no—
109C22×D4C22×C412yes—
110C22×D4C2×D412yes—
111C22×D4C2411yescharacteristic
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—84no—
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124#127—24no—
125#127#4118yes—
126#124—24no—
127#124—24no—
128#121—216no—
129#120—216no—
130C4×Q8—42no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×D4—42no—
134C4×D4—42no—
135C4×D4—42no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—44no—
141C4×D4—42no—
142C4×D4—42no—
143C4×D4—44no—
144C4×D4—42no—
145C4×D4—42no—
146C4×D4—88no—
147#117—24no—
148#117—42no—
149#117—42no—
150#117—42no—
151#117—42no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C22⋊C4—21no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4C22⋊C412yes—
174C2×C22⋊C4C2×D412yes—
175C2×C22⋊C4C2×D412yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—44no—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C412yes—
193C2×C42C22×C412yes—
194#99—216no—
195#99—48no—
196#95—22no—
197#95—22no—
198#95—24no—
199#95—24no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204C24—24no—
205C24—22no—
206C24—42no—
207C24—42no—
208C24—82no—
209C24—82no—
210C24C2×C22⋊C411yescharacteristic
211C24C22×D412yes—
212#41—416no—
213C2×Q8—22no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—216no—
220C2×D4—22no—
221C2×D4—42no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—416no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—84no—
230C2×D4—84no—
231C2×D4#11714yes—
232C2×D4C2×#4114yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—48no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—48no—
258C22×C4—82no—
259C22×C4—84no—
260C22×C4—82no—
261C22×C4—81no—
262C22×C4—88no—
263C22×C4—88no—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C23×C411yescharacteristic
273C4⋊C4—24no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—84no—
282C4⋊C4—88no—
283C22⋊C4—24no—
284C22⋊C4—216no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—44no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—48no—
292C22⋊C4—48no—
293C22⋊C4—84no—
294C22⋊C4—84no—
295C22⋊C4—84no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—21no—
305C42—21no—
306C42—22no—
307C42—42no—
308C42—44no—
309C42—82no—
310C42—88no—
311C42#9912yes—
312C42#9912yes—
313C42#11714yes—
314C23—22no—
315C23—21no—
316C23—21no—
317C23—22no—
318C23—24no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23—44no—
324C23—44no—
325C23—48no—
326C23—42no—
327C23—42no—
328C23—88no—
329C23—84no—
330C23—88no—
331C23—84no—
332C23#41012yes—
333C23C22×C22⋊C411yescharacteristic
334C23#52514yes—
335Q8—44no—
336D4—44no—
337D4—44no—
338D4—48no—
339D4—44no—
340D4—48no—
341D4—816no—
342C2×C4—28no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4—88no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—88no—
382C2×C4—88no—
383C2×C4—88no—
384C2×C4C2×#9912yes—
385C2×C4C2×#9912yes—
386C2×C4C2×#9911yescharacteristic
387C2×C4C2×#9911yescharacteristic
388C2×C4C2×#11712yes—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—24no—
393C22—44no—
394C22—42no—
395C22—42no—
396C22—48no—
397C22—41no—
398C22—42no—
399C22—42no—
400C22—41no—
401C22—44no—
402C22—82no—
403C22—84no—
404C22—816no—
405C22—82no—
406C22—82no—
407C22#228011yescharacteristic
408C22C22×#9911yescharacteristic
409C22C2×#41011yescharacteristic
410C4—21no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—44no—
415C4—41no—
416C4—42no—
417C4—42no—
418C4—41no—
419C4—42no—
420C4—82no—
421C4—82no—
422C4—168no—
423C4#212812yes—
424C2—21no—
425C2—21no—
426C2—44no—
427C2—42no—
428C2—84no—
429C2#2072211yescharacteristic
430C2C2×#212811yescharacteristic
431C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20723C2semidirect
3. #12859C2semidirect
4. #12797C2semidirect
6. #12628C2semidirect
9. C22×C4×D4C4semidirect
11. #2326C4semidirect
14. #2317C4semidirect
16. #2309C4semidirect
18. C4×#124C4semidirect
22. C4×#117C4semidirect
26. #1799C22semidirect
29. C4×#99C22semidirect
30. #1393C22semidirect

Maximal subgroups

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

subgroupindex
C4×#5352
#207232
#128592
#127972
#126972
#126282