Finite Group Databasebeta

← #165670

Subgroups — #165670

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36539C211yesmaximal, characteristic
2#22698C212yesmaximal, complemented
3#7714C212yesmaximal, complemented
4#7697C212yesmaximal, complemented
5#7678C212yesmaximal, complemented
6#7635C211yesmaximal, characteristic, complemented
7#7617C212yesmaximal, complemented
8#7601C211yesmaximal, characteristic, complemented
9#7589C212yesmaximal, complemented
10C2×#435—22no—
11#2864C2211yescharacteristic
12#2854C2212yes—
13#2602—22no—
14#2594—22no—
15#2576—22no—
16#2536C411yescharacteristic
17#2535C411yescharacteristic
18#2494C412yes—
19#2426C412yes—
20#2374—22no—
21#2357C411yescharacteristic
22#2351C411yescharacteristic
23C4×#124C2211yescharacteristic
24C4×C22⋊Q8C2212yes—
25C42×D4C2211yescharacteristic
26#1940—22no—
27#1851—22no—
28#1850—22no—
29#1831C2212yes—
30#1797C2212yes—
31#1797C2212yes—
32#1767C2212yes—
33#1728C2212yes—
34C2×#336C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1271C2212yes—
39#1271C2212yes—
40#1271C2212yes—
41#1271C2212yes—
42#1270C2211yescharacteristic
43#1270C2212yes—
44#1270C2211yescharacteristic
45C22×M16—22no—
46C23×C8—22no—
47C2×#124C2×C411yescharacteristic
48C2×#123—22no—
49C2×C22⋊Q8C2×C412yes—
50C2×#121C2×C412yes—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C412yes—
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C22×C42C2312yes—
57#435—48no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×#97—22no—
65#401—22no—
66#398—21no—
67#397—21no—
68#397—22no—
69#396—22no—
70#394—21no—
71#394—22no—
72#392—21no—
73#391C2311yescharacteristic
74#390—22no—
75#390C2×C412yes—
76#390C2311yescharacteristic
77#389—22no—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2312yes—
81#388—22no—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#386—22no—
93#386—22no—
94#385—22no—
95#382—21no—
96#382C2×C411yescharacteristic
97#382C2×C411yescharacteristic
98#382C2×C411yescharacteristic
99#381—21no—
100#380C2×C412yes—
101#380C2×C412yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4D412yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4C2×C412yes—
111C4×C22⋊C4D412yes—
112C4×C22⋊C4D412yes—
113C43C2311yescharacteristic
114#336—22no—
115#336—22no—
116#336—22no—
117#336C2312yes—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121C22×Q8C22×C411yescharacteristic
122C22×D4—22no—
123C22×D4C22⋊C411yescharacteristic
124C22×D4C22×C411yescharacteristic
125C23×C4—21no—
126C23×C4C22×C412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C2×M16—44no—
130C2×M16—44no—
131C2×M16—82no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—44no—
137C22×C8—44no—
138C22×C8—82no—
139#124—22no—
140#124—41no—
141#123—44no—
142#123—44no—
143C22⋊Q8—24no—
144C22⋊Q8—42no—
145#121—82no—
146C4×Q8—42no—
147C4×Q8—82no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—42no—
151C4×D4—42no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4C22⋊C412yes—
174C2×C22⋊C4C22⋊C412yes—
175C2×C22⋊C4C22⋊C412yes—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—82no—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2411yescharacteristic
188C4⋊C8—44no—
189C4⋊C8—44no—
190C4⋊C8—82no—
191C4⋊C8—82no—
192C22⋊C8—44no—
193C22⋊C8—44no—
194#97—44no—
195#95—21no—
196#95—22no—
197#95—22no—
198#95—21no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—21no—
203#95—21no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95C22⋊C411yescharacteristic
208#95C22⋊C412yes—
209#95C22⋊C412yes—
210#95C22⋊C411yescharacteristic
211C24—21no—
212C24C2×C22⋊C412yes—
213C2×Q8—21no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8#13112yes—
217C2×D4—24no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—44no—
224C2×D4—82no—
225C2×D4—82no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—82no—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4AGL(1,ℤ/8)14yes—
250C22×C4#13714yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D412yes—
253M16—88no—
254C2×C8—24no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—88no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—44no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—82no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C42—21no—
295C42—21no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—41no—
303C42—42no—
304C42—42no—
305C42—82no—
306C42—82no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23#41714yes—
319C23C22×C22⋊C411yescharacteristic
320Q8—42no—
321Q8—42no—
322D4—44no—
323D4—81no—
324D4—81no—
325D4—164no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—162no—
364C2×C4#41312yes—
365C2×C4#41714yes—
366C2×C4C2×#13111yescharacteristic
367C2×C4#56811yescharacteristic
368C2×C4C2×AGL(1,ℤ/8)12yes—
369C2×C4C2×#13712yes—
370C8—44no—
371C8—82no—
372C8—82no—
373C8—82no—
374C22—24no—
375C22—24no—
376C22—24no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—82no—
385C22—82no—
386C22#288011yescharacteristic
387C22#288311yescharacteristic
388C22C2×#41311yescharacteristic
389C22C2×#41712yes—
390C22#290112yes—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—24no—
397C4—42no—
398C4—42no—
399C4—41no—
400C4—42no—
401C4—42no—
402C4—81no—
403C4—81no—
404C4—82no—
405C4—82no—
406C2—44no—
407C2—161no—
408C2#1742411yescharacteristic
409C2#1751512yes—
410C2#1762212yes—
411C2#1768611yescharacteristic
412C2#1775211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2535C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22698C2semidirect
3. #7714C2semidirect
4. #7697C2semidirect
5. #7678C2semidirect
6. #7635C2semidirect
7. #7617C2semidirect
8. #7601C2semidirect
9. #7589C2semidirect

Maximal subgroups

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

subgroupindex
#365392
#226982
#77142
#76972
#76782
#76352
#76172
#76012
#75892