Finite Group Databasebeta

← #481847

Subgroups — #481847

Order 512.

1033 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36020C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic, complemented
3#17370C211yesmaximal, characteristic, complemented
4#11805C214yesmaximal, complemented
5#11803C214yesmaximal, complemented
6#11700C212yesmaximal, complemented
7#11697C212yesmaximal, complemented
8C2×#547—21no—
9C2×#547C411yescharacteristic
10C2×#520C2211yescharacteristic
11#2975—21no—
12#2879—21no—
13#2621C2211yescharacteristic
14#2618C2212yes—
15#2546C411yescharacteristic
16#2543C412yes—
17#2536C411yescharacteristic
18#2486C412yes—
19#2429—21no—
20#2425—22no—
21#2417—21no—
22#2413—22no—
23#2392C2211yescharacteristic
24C2×#398C2212yes—
25#2349C411yescharacteristic
26#2339—21no—
27#1822—22no—
28#1821C2214yes—
29#1742C2212yes—
30C2×#343—21no—
31#1465—22no—
32#1463C2214yes—
33#1461—21no—
34#1458C2212yes—
35#1346C2214yes—
36#1346C2218yes—
37#1345C2214yes—
38C22×#41—21no—
39C22×#41D411yescharacteristic
40C22×M16—22no—
41#547—44no—
42#547—48no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#123—22no—
46C2×#123—22no—
47C2×#123—22no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×#121—22no—
52C2×#121C2×C412yes—
53#520—44no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C2×C4×D4D411yescharacteristic
63C2×#117D412yes—
64C2×#117C2311yescharacteristic
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69#436—44no—
70#431—22no—
71#408—44no—
72#407—24no—
73#407—24no—
74#398—21no—
75#398—22no—
76#398—22no—
77#398C2×C412yes—
78#398C2×C412yes—
79#398C2×C412yes—
80#398C2312yes—
81#398C2311yescharacteristic
82#397—21no—
83#397C2×C412yes—
84#397C2×C411yescharacteristic
85#396—22no—
86#396—22no—
87#396—21no—
88#394—21no—
89#394C2×C412yes—
90#394C2311yescharacteristic
91#392C2×C411yescharacteristic
92#391—21no—
93#391—21no—
94#389—21no—
95#389—22no—
96#389—22no—
97#388—22no—
98#388—22no—
99#386—22no—
100#386—22no—
101#386—22no—
102#385—22no—
103#382C2×C411yescharacteristic
104#381—22no—
105#380—22no—
106#380—21no—
107#380C2×C412yes—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—21no—
110C2×#95—22no—
111C2×#95C2×C412yes—
112#343—24no—
113#343—22no—
114#343C2314yes—
115#324—24no—
116#324C2314yes—
117#323—22no—
118C2×#41—24no—
119C2×#41—24no—
120C2×#41—42no—
121C2×#41—44no—
122C2×#41—44no—
123C2×#41—44no—
124C22×Q8—21no—
125C22×Q8C22⋊C411yescharacteristic
126C22×D4—22no—
127C22×D4—21no—
128C22×D4C22⋊C412yes—
129C22×D4C22⋊C411yescharacteristic
130C22×D4C22×C411yescharacteristic
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4C22⋊C412yes—
135C23×C4C22×C412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C2×M16—24no—
140C2×M16—42no—
141C2×M16—44no—
142C2×M16—44no—
143C2×M16—44no—
144C22×C8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—44no—
147#123—48no—
148#123—48no—
149#123—48no—
150C22⋊Q8—44no—
151C22⋊Q8—44no—
152C22⋊Q8—44no—
153#121—44no—
154#121—48no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—42no—
160C4×D4—44no—
161C4×D4—44no—
162C4×D4—44no—
163C4×D4—44no—
164#117—42no—
165#117—44no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2411yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22×C411yescharacteristic
202C2×C22⋊C4C2×D411yescharacteristic
203C2×C42—21no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C4⋊C8—42no—
209C22⋊C8—24no—
210C22⋊C8—48no—
211C22⋊C8—44no—
212C4×C8—42no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—21no—
220#95—21no—
221#95—22no—
222#95C22×C411yescharacteristic
223#95C22×C412yes—
224#95C22×C411yescharacteristic
225C24—21no—
226C24C2×C22⋊C412yes—
227C24C2×C22⋊C411yescharacteristic
228#41—48no—
229#41—48no—
230#41—816no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—42no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—42no—
248C2×D4—81no—
249C2×D4—81no—
250C2×D4#9914yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4AGL(1,ℤ/8)14yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286M16—44no—
287M16—44no—
288M16—84no—
289M16—164no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—84no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—82no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—42no—
322C22⋊C4—44no—
323C22⋊C4—42no—
324C22⋊C4—82no—
325C22⋊C4—81no—
326C42—22no—
327C42—22no—
328C42—42no—
329C42—42no—
330C42—42no—
331C23—21no—
332C23—22no—
333C23—22no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23C2×#9914yes—
347C23#41812yes—
348C23C22×C22⋊C411yescharacteristic
349Q8—44no—
350Q8—84no—
351D4—28no—
352D4—48no—
353D4—84no—
354D4—82no—
355D4—88no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—48no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—88no—
394C2×C4—88no—
395C2×C4—84no—
396C2×C4—162no—
397C2×C4C2×#9912yes—
398C2×C4#41812yes—
399C2×C4#42112yes—
400C2×C4C2×AGL(1,ℤ/8)12yes—
401C8—84no—
402C8—84no—
403C22—21no—
404C22—21no—
405C22—24no—
406C22—24no—
407C22—22no—
408C22—24no—
409C22—42no—
410C22—44no—
411C22—44no—
412C22—41no—
413C22—41no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22C22×#9911yescharacteristic
423C22C2×#41811yescharacteristic
424C22C2×#42111yescharacteristic
425C4—22no—
426C4—42no—
427C4—41no—
428C4—41no—
429C4—42no—
430C4—42no—
431C4—41no—
432C4—41no—
433C4—42no—
434C4—81no—
435C4—82no—
436C4—81no—
437C4—82no—
438C4#151014yes—
439C2—21no—
440C2—21no—
441C2—44no—
442C2—42no—
443C2—161no—
444C2C2×#151011yescharacteristic
445C2#3335011yescharacteristic
446C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#36020C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20695C2semidirect
3. #17370C2semidirect
4. #11805C2semidirect
5. #11803C2semidirect
6. #11700C2semidirect
7. #11697C2semidirect

Maximal subgroups

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

subgroupindex
#360202
#206952
#173702
#118052
#118032
#117002
#116972