Finite Group Databasebeta

← #164144

Subgroups — #164144

Order 512.

849 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36302C211yesmaximal, characteristic
2#22639C211yesmaximal, characteristic, complemented
3#20368C211yesmaximal, characteristic, complemented
4#7684C211yesmaximal, characteristic
5#7676C212yesmaximal, complemented
6#7664C211yesmaximal, characteristic
7#7656C212yesmaximal, complemented
8#7617C211yesmaximal, characteristic, complemented
9#7608C212yesmaximal, complemented
10#7566C211yesmaximal, characteristic, complemented
11#7560C212yesmaximal, complemented
12C2×#405—21no—
13#2861C2212yes—
14#2856C2211yescharacteristic
15#2552C411yescharacteristic
16#2548C411yescharacteristic
17#2531—22no—
18#2513—21no—
19#2500—21no—
20#2490C412yes—
21#2476—22no—
22#2427C412yes—
23#2366C411yescharacteristic
24#2363—21no—
25#2355C411yescharacteristic
26#2354—21no—
27C4×C22⋊Q8C2211yescharacteristic
28#2293C2212yes—
29C4×#117C2211yescharacteristic
30#1939—21no—
31#1923—22no—
32#1839C2212yes—
33#1797C2211yescharacteristic
34C8×C4⋊C4—21no—
35C4×C4⋊C8C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37#1746—22no—
38#1745C2212yes—
39#1744C2212yes—
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×M16—21no—
54C23×C8—21no—
55C2×#127C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×#124C2×C412yes—
58C2×C22⋊Q8C2×C411yescharacteristic
59C2×#121—21no—
60C2×#121—21no—
61C2×#121C2×C412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×#117—22no—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73#405—28no—
74C2×#97—22no—
75C2×C4×C8—22no—
76C2×C4×C8—21no—
77C2×C4×C8—22no—
78#398—22no—
79#397—22no—
80#397—22no—
81#394—22no—
82#394—22no—
83#394—22no—
84#391C2×C412yes—
85#391C2311yescharacteristic
86#390—21no—
87#390—21no—
88#390—21no—
89#390—22no—
90#390—21no—
91#390—21no—
92#390C2×C412yes—
93#390C2×C412yes—
94#390C2312yes—
95#389—21no—
96#389C2×C412yes—
97#389C2311yescharacteristic
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#387—21no—
106#387—22no—
107#386—22no—
108#386—22no—
109#385—21no—
110#384—21no—
111#384C2×C411yescharacteristic
112#382—21no—
113#380—22no—
114#380C2×C411yescharacteristic
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—21no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4C2×C412yes—
126C4×C22⋊C4D412yes—
127#376D412yes—
128C43C2312yes—
129#336—22no—
130#336—22no—
131#336—21no—
132#336—21no—
133#336—22no—
134#336—22no—
135#336C2312yes—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141C22×Q8C22×C411yescharacteristic
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C22×C412yes—
146C23×C4—22no—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—42no—
153C22×C8—24no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—42no—
161C22×C8—42no—
162#127—42no—
163#125—82no—
164#124—82no—
165C22⋊Q8—22no—
166C22⋊Q8—41no—
167#121—42no—
168#121—42no—
169#121—81no—
170#121—81no—
171#121—82no—
172C4×Q8—42no—
173C4×D4—24no—
174C4×D4—42no—
175C4×D4—44no—
176#117—22no—
177#117—22no—
178#117—24no—
179#117—82no—
180#117—82no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—42no—
215C2×C42C22×C412yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2411yescharacteristic
225C4⋊C8—42no—
226C4⋊C8—44no—
227C22⋊C8—42no—
228C22⋊C8—81no—
229#97—28no—
230C4×C8—28no—
231C4×C8—42no—
232C4×C8—82no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—22no—
237#95—21no—
238#95—22no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246C24—22no—
247C24C2×C22⋊C411yescharacteristic
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—44no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—81no—
262C2×D4—81no—
263C2×D4—82no—
264C2×D4—82no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—82no—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4#13512yes—
291C22×C4#13512yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295M16—48no—
296C2×C8—22no—
297C2×C8—24no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—48no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—41no—
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—82no—
328C4⋊C4—81no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4#10414yes—
332C4⋊C4#13112yes—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—44no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—81no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—24no—
358C42—42no—
359C42—41no—
360C42—41no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—82no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23#41612yes—
379C23C22×C22⋊C411yescharacteristic
380Q8—81no—
381Q8—81no—
382D4—82no—
383D4—82no—
384D4—84no—
385D4—162no—
386D4—164no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—88no—
432C2×C4#41012yes—
433C2×C4#41612yes—
434C2×C4C2×#10414yes—
435C2×C4C2×#10412yes—
436C2×C4C2×#13111yescharacteristic
437C2×C4#56811yescharacteristic
438C2×C4C2×#13511yescharacteristic
439C2×C4C2×#13511yescharacteristic
440C2×C4#57611yescharacteristic
441C2×C4#57711yescharacteristic
442C8—42no—
443C8—42no—
444C8—42no—
445C8—44no—
446C8—82no—
447C8—81no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22#288011yescharacteristic
463C22C2×#41011yescharacteristic
464C22#288811yescharacteristic
465C22C2×#41611yescharacteristic
466C22#289711yescharacteristic
467C22C22×#10411yescharacteristic
468C22#290111yescharacteristic
469C4—21no—
470C4—21no—
471C4—24no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—41no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—41no—
481C4—42no—
482C4—81no—
483C4—82no—
484C4—81no—
485C4—81no—
486C4#150014yes—
487C2—42no—
488C2—162no—
489C2#1741811yescharacteristic
490C2#1751011yescharacteristic
491C2#1751711yescharacteristic
492C2C2×#150011yescharacteristic
493C2#1760711yescharacteristic
494C2#1765511yescharacteristic
495C2#1772411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22639C2semidirect
3. #20368C2semidirect
5. #7676C2semidirect
7. #7656C2semidirect
8. #7617C2semidirect
9. #7608C2semidirect
10. #7566C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#363022
#226392
#203682
#76842
#76762
#76642
#76562
#76172
#76082
#75662
#75602