Finite Group Databasebeta

← #480272

Subgroups — #480272

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34172C211yesmaximal, characteristic
2#17369C212yesmaximal, complemented
3#11502C214yesmaximal, complemented
4#11500C214yesmaximal, complemented
5#11428C212yesmaximal, complemented
6#11425C212yesmaximal, complemented
7C23×#41C411yescharacteristic
8C2×#518C411yescharacteristic
9C2×#518C2211yescharacteristic
10#2975—22no—
11#2339—24no—
12#2339—22no—
13#2339—22no—
14#2310C411yescharacteristic
15#2310C2211yescharacteristic
16#2308C412yes—
17#2308C2212yes—
18#2280C411yescharacteristic
19#2280C2211yescharacteristic
20C2×#381C412yes—
21C2×#381C2212yes—
22#1465—24no—
23#1462C2218yes—
24#1460—22no—
25#1458C2214yes—
26#1317C2214yes—
27#1317C2218yes—
28#1316C2214yes—
29C22×#41—24no—
30C22×#41—22no—
31C22×#41—21no—
32C22×#41—21no—
33C22×#41—24no—
34C22×#41—44no—
35C22×#41—44no—
36C23×Q8C2×C411yescharacteristic
37C23×D4C2×C411yescharacteristic
38C23×D4C2×C412yes—
39C24×C4C2×C412yes—
40C24×C4C2×C411yescharacteristic
41#518—44no—
42#518—44no—
43C2×C4×D4—22no—
44C2×C4×D4—21no—
45C2×C4×D4—21no—
46C2×C4×D4D414yes—
47C2×C4×D4D412yes—
48C2×C4×D4D412yes—
49C2×#117C2×C411yescharacteristic
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2311yescharacteristic
53C22×C22⋊C4C2×C412yes—
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2312yes—
57#436—48no—
58#431—24no—
59#407—28no—
60#391—22no—
61#391—22no—
62#391—22no—
63#391—22no—
64#386—24no—
65#386—24no—
66#386—24no—
67#386—24no—
68#381—24no—
69#381—24no—
70#381C2×C412yes—
71#381C2×C412yes—
72#381C2×C412yes—
73#381C2×C411yescharacteristic
74#381C2×C412yes—
75#381C2312yes—
76#381C2311yescharacteristic
77#380—22no—
78#380—22no—
79#379C2×C411yescharacteristic
80#379C2×C412yes—
81#379C2311yescharacteristic
82C4×C22⋊C4—22no—
83C4×C22⋊C4—22no—
84C2×#95—24no—
85C2×#95—24no—
86#324—24no—
87#323—28no—
88#323C2318yes—
89C25C22×C412yes—
90C25C22×C411yescharacteristic
91C2×#41—216no—
92C2×#41—28no—
93C2×#41—28no—
94C2×#41—432no—
95C2×#41—48no—
96C2×#41—48no—
97C2×#41—432no—
98C2×#41—44no—
99C2×#41—44no—
100C2×#41—44no—
101C2×#41—44no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—22no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8M1614yescomplemented
108C22×D4—24no—
109C22×D4—24no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—24no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—24no—
116C22×D4—44no—
117C22×D4—44no—
118C22×D4C22⋊C412yes—
119C22×D4C22⋊C411yescharacteristic
120C22×D4C22⋊C411yescharacteristic
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—44no—
133C23×C4—44no—
134C23×C4C22⋊C414yes—
135C23×C4M1614yescomplemented
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C412yes—
139C23×C4C2×D414yes—
140C2×M16—48no—
141C4×D4—28no—
142C4×D4—44no—
143C4×D4—48no—
144C4×D4—48no—
145C4×D4—44no—
146C4×D4—44no—
147#117—42no—
148#117—42no—
149#117—42no—
150#117—42no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C2×D412yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4C22⋊C414yes—
165C2×C22⋊C4C22×C411yescharacteristic
166C2×C22⋊C4C22×C412yes—
167C2×C22⋊C4C22×C411yescharacteristic
168C2×C22⋊C4C22×C411yescharacteristic
169C2×C22⋊C4C22×C412yes—
170C2×C22⋊C4C2×D414yes—
171C2×C22⋊C4C2411yescharacteristic
172C2×C42—21no—
173C2×C42—21no—
174C2×C42C22⋊C412yes—
175C2×C42C2×D412yes—
176C4⋊C8—44no—
177C22⋊C8—28no—
178C22⋊C8—48no—
179C4×C8—44no—
180#95—24no—
181#95—22no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—22no—
186#95—24no—
187C24—22no—
188C24—22no—
189C24—24no—
190C24—24no—
191C24—21no—
192C24—21no—
193C24—41no—
194C24—41no—
195C24—84no—
196C24—82no—
197C24C2×C22⋊C412yes—
198C24C2×C22⋊C412yes—
199C24C2×M1612yes—
200C24C23×C411yescharacteristic
201#41—464no—
202#41—432no—
203#41—432no—
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—48no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—48no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×D4—216no—
213C2×D4—216no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—28no—
217C2×D4—28no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—416no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—416no—
227C2×D4—44no—
228C2×D4—44no—
229C2×D4#9914yes—
230C2×D4#10014yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—28no—
234C22×C4—28no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—28no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—416no—
251C22×C4—416no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—84no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C414yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×M1612yes—
273C22×C4C2×M1612yes—
274C22×C4C2×M1612yes—
275C22×C4C2×M1614yes—
276C22×C4C22×D412yes—
277M16—168no—
278C2×C8—48no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C22⋊C4—28no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—41no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—84no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C42—24no—
297C42—42no—
298C42—42no—
299C42—42no—
300C23—28no—
301C23—28no—
302C23—22no—
303C23—22no—
304C23—21no—
305C23—22no—
306C23—21no—
307C23—22no—
308C23—22no—
309C23—44no—
310C23—44no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—41no—
317C23—41no—
318C23—88no—
319C23—82no—
320C23—88no—
321C23—84no—
322C23—84no—
323C23—82no—
324C23—84no—
325C23—84no—
326C23C2×#9914yes—
327C23C2×#10014yes—
328C23C22×C22⋊C411yescharacteristic
329C23C22×M1611yescharacteristic
330C23C22×M1611yescharacteristic
331Q8—48no—
332Q8—48no—
333D4—216no—
334D4—28no—
335D4—28no—
336D4—416no—
337D4—416no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—416no—
351C2×C4—416no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—48no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—88no—
369C2×C4—164no—
370C2×C4—164no—
371C2×C4—164no—
372C2×C4#40712yes—
373C2×C4#40712yes—
374C2×C4C2×#9912yes—
375C2×C4C2×#10012yes—
376C8—88no—
377C22—22no—
378C22—28no—
379C22—21no—
380C22—21no—
381C22—24no—
382C22—24no—
383C22—44no—
384C22—41no—
385C22—41no—
386C22—44no—
387C22—44no—
388C22—44no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—44no—
393C22—42no—
394C22—42no—
395C22—82no—
396C22—88no—
397C22—88no—
398C22—84no—
399C22—84no—
400C22—82no—
401C22C2×#40711yescharacteristic
402C22C22×#9911yescharacteristic
403C22C22×#10011yescharacteristic
404C4—22no—
405C4—42no—
406C4—44no—
407C4—42no—
408C4—42no—
409C4—44no—
410C4—82no—
411C4—81no—
412C4—81no—
413C4#146214yes—
414C2—21no—
415C2—21no—
416C2—44no—
417C2—44no—
418C2—41no—
419C2—41no—
420C2—82no—
421C2C2×#146211yescharacteristic
422C2#3335012yes—
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×#41C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17369C2semidirect
3. #11502C2semidirect
4. #11500C2semidirect
5. #11428C2semidirect
6. #11425C2semidirect
107. C22×Q8M16semidirect
135. C23×C4M16semidirect

Maximal subgroups

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

subgroupindex
#341722
#173692
#115022
#115002
#114282
#114252