Finite Group Databasebeta

← #165053

Subgroups — #165053

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36416C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic
3#22696C211yesmaximal, characteristic
4#7731C212yesmaximal, complemented
5#7691C212yesmaximal, complemented
6#7663C211yesmaximal, characteristic, complemented
7#7660C211yesmaximal, characteristic, complemented
8#7652C211yesmaximal, characteristic, complemented
9#7613C212yesmaximal, complemented
10#7579C212yesmaximal, complemented
11#7570C211yesmaximal, characteristic, complemented
12C2×C8×D4—21no—
13#2859C2211yescharacteristic
14#2851C2212yes—
15#2589—22no—
16#2587—22no—
17#2580—21no—
18#2571C411yescharacteristic
19#2566—21no—
20#2552C411yescharacteristic
21#2441C412yes—
22#2439C412yes—
23#2370—21no—
24#2355—21no—
25#2354C411yescharacteristic
26#2354C411yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28#2296C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1924—22no—
33#1850—22no—
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1817C2212yes—
37#1797C2212yes—
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1749C2212yes—
42C2×#336C2212yes—
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C23×C8—22no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125—21no—
56C2×#125C2×C412yes—
57C2×C4×Q8—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×#117C2×C412yes—
62C2×#117C2×C412yes—
63C22×C42C2312yes—
64C8×D4—44no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—22no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76#400—22no—
77#399—22no—
78#398—22no—
79#396—22no—
80#396—22no—
81#393—22no—
82#391—21no—
83#391—22no—
84#390—21no—
85#390—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2311yescharacteristic
96#389C2311yescharacteristic
97#389C2312yes—
98#388—22no—
99#388—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#384—21no—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#384C2×C411yescharacteristic
112#382—21no—
113#382—21no—
114#382C2×C412yes—
115#380C2×C412yes—
116#380C2×C412yes—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130C43C2311yescharacteristic
131#336—22no—
132#336—21no—
133#336—21no—
134#336—22no—
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141C22×Q8—21no—
142C22×D4C22⋊C411yescharacteristic
143C23×C4C22×C412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—44no—
153C22×C8—44no—
154C22×C8—82no—
155C4⋊Q8—22no—
156C4⋊Q8—41no—
157#125—42no—
158#125—42no—
159#125—82no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×D4—24no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167#117—24no—
168#117—42no—
169#117—82no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—82no—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2411yescharacteristic
216C4⋊C8—42no—
217C4⋊C8—42no—
218C4⋊C8—42no—
219C4⋊C8—42no—
220C4⋊C8—82no—
221C22⋊C8—44no—
222C22⋊C8—44no—
223C22⋊C8—82no—
224#97—82no—
225C4×C8—42no—
226C4×C8—42no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242C24C2×C22⋊C412yes—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—81no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4#10114yes—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—88no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—81no—
319C4⋊C4—81no—
320C4⋊C4—81no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4#13112yes—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—44no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C42—21no—
334C42—21no—
335C42—22no—
336C42—42no—
337C42—41no—
338C42—41no—
339C42—42no—
340C42—41no—
341C42—42no—
342C42—44no—
343C42—42no—
344C42—42no—
345C42—82no—
346C42#13512yes—
347C42#13512yes—
348C42AGL(1,ℤ/8)12yes—
349C42#13712yes—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23C2×#10114yes—
357C23C22×C22⋊C411yescharacteristic
358Q8—84no—
359D4—28no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4C2×#10112yes—
407C2×C4#41612yes—
408C2×C4#41612yes—
409C2×C4#41712yes—
410C2×C4#41712yes—
411C2×C4C2×#13111yescharacteristic
412C2×C4#56811yescharacteristic
413C2×C4C2×#13511yescharacteristic
414C2×C4C2×#13511yescharacteristic
415C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
416C2×C4C2×#13711yescharacteristic
417C8—42no—
418C8—42no—
419C8—81no—
420C8—82no—
421C8—81no—
422C8—82no—
423C22—24no—
424C22—24no—
425C22—24no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22#288011yescharacteristic
431C22#288311yescharacteristic
432C22C22×#10111yescharacteristic
433C22C2×#41611yescharacteristic
434C22C2×#41711yescharacteristic
435C22#290111yescharacteristic
436C22#290111yescharacteristic
437C4—21no—
438C4—21no—
439C4—22no—
440C4—22no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—42no—
446C4—41no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—81no—
452C4—82no—
453C4—81no—
454C4—81no—
455C4#155214yes—
456C2—44no—
457C2#1742611yescharacteristic
458C2#1751011yescharacteristic
459C2#1751211yescharacteristic
460C2#1761711yescharacteristic
461C2#1763011yescharacteristic
462C2#1768911yescharacteristic
463C2C2×#155211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
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
4. #7731C2semidirect
5. #7691C2semidirect
6. #7663C2semidirect
7. #7660C2semidirect
8. #7652C2semidirect
9. #7613C2semidirect
10. #7579C2semidirect
11. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#364162
#228392
#226962
#77312
#76912
#76632
#76602
#76522
#76132
#75792
#75702