Finite Group Databasebeta

← #208163

Subgroups — #208163

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic, complemented
3C4×#432C211yesmaximal, characteristic, complemented
4#9897C212yesmaximal, complemented
5#9696C212yesmaximal, complemented
6#9142C212yesmaximal
7#9140C212yesmaximal, complemented
8#8236C212yesmaximal, complemented
9#8234C212yesmaximal, complemented
10C2×#529—21no—
11C2×#432C2212yes—
12#2861—21no—
13#2860—21no—
14#2860C412yes—
15#2859C2211yescharacteristic
16#2858C2211yescharacteristic
17#2857C2211yescharacteristic
18#2856C2211yescharacteristic
19#2854—21no—
20#2854C412yes—
21#2850—21no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×#127C2211yescharacteristic
24C4×#124—21no—
25C4×#121—21no—
26C4×#121C412yes—
27#2293—21no—
28#2293C412yes—
29C2×C43C2211yescharacteristic
30#1839—21no—
31#1831C2212yes—
32#1817—21no—
33#1797C2212yes—
34#1767C2212yes—
35C4×C4⋊C8C2212yes—
36C4×C4⋊C8C2212yes—
37#1749—21no—
38C4×C22⋊C8—21no—
39C4×C22⋊C8—21no—
40#1745—21no—
41#1744—21no—
42C4×#97—21no—
43C2×#332C2214yes—
44C2×#331C2214yes—
45#1296C2214yes—
46#1296C2214yes—
47C2×C4⋊Q8C2311yescharacteristic
48C2×#127C2311yescharacteristic
49#529—48no—
50C2×#126—22no—
51C2×#125—21no—
52C2×#124—21no—
53C2×#124—21no—
54C2×#123—22no—
55C2×C22⋊Q8—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121C2×C412yes—
59C2×C4×Q8—21no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C412yes—
64C2×#117—22no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72#432—28no—
73C2×C4⋊C8C2314yes—
74C2×C4⋊C8C2314yes—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×#97—22no—
78C2×#97—22no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81#391—21no—
82#391—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390—21no—
88#390—21no—
89#390C2×C412yes—
90#390C2×C412yes—
91#390C2×C412yes—
92#390C2×C412yes—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#386—22no—
110#386—22no—
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#382—21no—
114#380—21no—
115#380C2×C412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C43C2×C411yescharacteristic
121C43C2×C411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43C2311yescharacteristic
127C43C2311yescharacteristic
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#332—28no—
135#331—28no—
136C22×Q8—21no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C22×C412yes—
141C23×C4—22no—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C22×C8—24no—
150C22×C8—24no—
151C4⋊Q8—24no—
152#127—24no—
153#126—48no—
154#125—44no—
155#124—44no—
156#124—44no—
157#123—48no—
158C22⋊Q8—44no—
159#121—44no—
160#121—44no—
161#121—44no—
162C4×Q8—44no—
163C4×D4—44no—
164C4×D4—44no—
165C4×D4—48no—
166C4×D4—44no—
167#117—48no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—22no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2411yescharacteristic
227C4⋊C8—24no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230C4⋊C8#4118yes—
231C22⋊C8—28no—
232C22⋊C8—44no—
233#97—28no—
234#97—44no—
235C4×C8—44no—
236C4×C8—44no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243C24—22no—
244C24C2×C22⋊C411yescharacteristic
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×D4—24no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C2×C8—28no—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—44no—
307C4⋊C4—24no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—41no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—44no—
356C42#13514yes—
357C42AGL(1,ℤ/8)14yes—
358C42C2×#4114yes—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23#41312yes—
373C23C22×C22⋊C411yescharacteristic
374Q8—84no—
375D4—84no—
376D4—88no—
377D4—84no—
378D4—84no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—88no—
426C2×C4#41312yes—
427C2×C4#41612yes—
428C2×C4#41612yes—
429C2×C4#41812yes—
430C2×C4#41812yes—
431C2×C4C4×#4112yes—
432C2×C4#52512yes—
433C2×C4#53212yes—
434C2×C4C2×#13512yes—
435C2×C4C2×AGL(1,ℤ/8)12yes—
436C2×C4#57512yes—
437C2×C4#57512yes—
438C8—48no—
439C8—84no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22#229711yescharacteristic
455C22C2×#41311yescharacteristic
456C22C2×#41311yescharacteristic
457C22C2×#41611yescharacteristic
458C22C2×#41811yescharacteristic
459C22#289611yescharacteristic
460C22#289611yescharacteristic
461C4—22no—
462C4—22no—
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4#310614yes—
477C4#318314yes—
478C2—42no—
479C2—162no—
480C2#1771311yescharacteristic
481C2#1771311yescharacteristic
482C2#1774311yescharacteristic
483C2#1774411yescharacteristic
484C2#2139011yescharacteristic
485C2#2141111yescharacteristic
486C2#2142611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20777C2semidirect
3. C4×#432C2semidirect
4. #9897C2semidirect
5. #9696C2semidirect
7. #9140C2semidirect
8. #8236C2semidirect
9. #8234C2semidirect

Maximal subgroups

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

subgroupindex
#366312
#207772
C4×#4322
#98972
#96962
#91422
#91402
#82362
#82342