Finite Group Databasebeta

← #161842

Subgroups — #161842

Order 512.

942 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3#20777C211yesmaximal, characteristic, complemented
4#7718C214yesmaximal, complemented
5#7706C214yesmaximal, complemented
6#7588C212yesmaximal, complemented
7#7587C212yesmaximal, complemented
8C2×#529—21no—
9#2861—21no—
10#2860—21no—
11#2860C2212yes—
12#2859C411yescharacteristic
13#2858C411yescharacteristic
14#2857C411yescharacteristic
15#2856C411yescharacteristic
16#2854—21no—
17#2854C2212yes—
18#2850—21no—
19C4×C4⋊Q8C2211yescharacteristic
20C4×#127C2211yescharacteristic
21C4×#124—21no—
22C4×#121—21no—
23C4×#121C412yes—
24#2293—21no—
25#2293C412yes—
26C2×C43C2211yescharacteristic
27#1839—21no—
28#1839—21no—
29#1831C2212yes—
30#1831C2212yes—
31#1817C2212yes—
32#1817C2212yes—
33#1797—21no—
34#1797—21no—
35#1767—21no—
36C4×C4⋊C8—21no—
37#1749—21no—
38C4×C22⋊C8—21no—
39#1745C2212yes—
40#1744C2212yes—
41#1271C2218yes—
42#1270C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45#529—48no—
46C2×#126—22no—
47C2×#125—21no—
48C2×#124—21no—
49C2×#124—21no—
50C2×#123—22no—
51C2×C22⋊Q8—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121C2×C412yes—
55C2×C4×Q8—21no—
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C412yes—
60C2×#117—22no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×#97—22no—
73C2×C4×C8—22no—
74#391—21no—
75#391—21no—
76#390—21no—
77#390—21no—
78#390—21no—
79#390—21no—
80#390—21no—
81#390—21no—
82#390C2×C412yes—
83#390C2×C412yes—
84#390C2×C412yes—
85#390C2312yes—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2312yes—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#386—22no—
103#386—22no—
104#384C2×C411yescharacteristic
105#384C2×C411yescharacteristic
106#382—21no—
107#380—21no—
108#380C2×C412yes—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C412yes—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C43C2×C411yescharacteristic
114C43C2×C411yescharacteristic
115C43D411yescharacteristic
116C43D411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43C2311yescharacteristic
120C43C2311yescharacteristic
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336C2314yes—
128#336C2314yes—
129C22×Q8—21no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4C22×C412yes—
134C23×C4—22no—
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C22×C8—24no—
143C22×C8—24no—
144C4⋊Q8—24no—
145#127—24no—
146#126—48no—
147#125—44no—
148#124—44no—
149#124—44no—
150#123—48no—
151C22⋊Q8—44no—
152#121—44no—
153#121—44no—
154#121—44no—
155C4×Q8—44no—
156C4×D4—44no—
157C4×D4—44no—
158C4×D4—48no—
159C4×D4—44no—
160#117—48no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—22no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2411yescharacteristic
220C4⋊C8—44no—
221C4⋊C8—44no—
222C22⋊C8—44no—
223C22⋊C8—44no—
224#97—44no—
225C4×C8—44no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232C24—22no—
233C24C2×C22⋊C411yescharacteristic
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×D4—24no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—44no—
295C2×C8—44no—
296C4⋊C4—24no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—44no—
346C42#13114yes—
347C42AGL(1,ℤ/8)14yes—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23#41012yes—
362C23C22×C22⋊C411yescharacteristic
363Q8—84no—
364D4—84no—
365D4—84no—
366D4—88no—
367D4—84no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
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—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—88no—
417C2×C4#40812yes—
418C2×C4#40812yes—
419C2×C4#41312yes—
420C2×C4#41812yes—
421C2×C4#41812yes—
422C2×C4#42112yes—
423C2×C4#42112yes—
424C2×C4C2×#13112yes—
425C2×C4C2×AGL(1,ℤ/8)12yes—
426C2×C4#57512yes—
427C8—84no—
428C8—84no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—82no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22C2×#40811yescharacteristic
444C22C2×#41011yescharacteristic
445C22C2×#41311yescharacteristic
446C22C2×#41811yescharacteristic
447C22#289611yescharacteristic
448C22C2×#42111yescharacteristic
449C22C2×#42111yescharacteristic
450C4—22no—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—82no—
464C4—82no—
465C4—82no—
466C4#149614yes—
467C2—42no—
468C2—162no—
469C2#1738311yescharacteristic
470C2#1746511yescharacteristic
471C2C2×#149611yescharacteristic
472C2#1758911yescharacteristic
473C2#1759111yescharacteristic
474C2#1767411yescharacteristic
475C2#1774311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. #20777C2semidirect
4. #7718C2semidirect
5. #7706C2semidirect
6. #7588C2semidirect
7. #7587C2semidirect

Maximal subgroups

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

subgroupindex
#366312
#207782
#207772
#77182
#77062
#75882
#75872