Finite Group Databasebeta

← #161960

Subgroups — #161960

Order 512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic, complemented
3#20752C211yesmaximal, characteristic, complemented
4#7710C214yesmaximal, complemented
5#7662C212yesmaximal, complemented
6#7659C212yesmaximal, complemented
7#7601C214yesmaximal, complemented
8C2×#533C411yescharacteristic
9C2×#532C411yescharacteristic
10C2×#431—21no—
11C2×#423—21no—
12#2862C411yescharacteristic
13#2862C2211yescharacteristic
14#2861—21no—
15#2861—21no—
16#2858C411yescharacteristic
17#2858C2211yescharacteristic
18#2850—21no—
19#2850—21no—
20#2850C412yes—
21#2850C2212yes—
22C4×#124—21no—
23C4×#124—21no—
24C4×#124C412yes—
25C4×#124C2212yes—
26#2297—21no—
27#2297—21no—
28C2×C43C2211yescharacteristic
29#1842—21no—
30#1831—21no—
31#1831—21no—
32#1831C2212yes—
33#1831C2212yes—
34#1831C2212yes—
35#1797—21no—
36#1797C2212yes—
37C4×C4⋊C8—21no—
38C4×C4⋊C8C2212yes—
39#1728—21no—
40#1728C2212yes—
41#1272C2218yes—
42#1270C2218yes—
43C22×#41—21no—
44C22×M16—22no—
45#533—44no—
46#532—44no—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#127C2×C411yescharacteristic
51C2×#125—21no—
52C2×#124—21no—
53C2×#124—21no—
54C2×#124—21no—
55C2×#124C2×C412yes—
56C2×#124C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8C2×C412yes—
59C2×#121—21no—
60C2×#121C2×C412yes—
61C2×C4×Q8—21no—
62C2×C4×Q8C2×C412yes—
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×#117—21no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73#431—44no—
74#423—44no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C4×C8—22no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391—21no—
86#391—21no—
87#391—21no—
88#391C2×C412yes—
89#391C2312yes—
90#390—21no—
91#390—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390C2×C412yes—
97#390C2312yes—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#382—21no—
111#382—21no—
112#382C2×C412yes—
113#382C2×C412yes—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43C2311yescharacteristic
125C43C2311yescharacteristic
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336C2314yes—
131#336C2314yes—
132C2×#41—44no—
133C2×#41—44no—
134C2×#41—44no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8C22×C412yes—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4C22×C412yes—
142C23×C4—22no—
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—44no—
151C2×M16—44no—
152C2×M16—44no—
153C22×C8—24no—
154C22×C8—24no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158#127—42no—
159#125—44no—
160#124—28no—
161#124—44no—
162#124—44no—
163#124—42no—
164#124—42no—
165#124—42no—
166#124—42no—
167C22⋊Q8—44no—
168C22⋊Q8—44no—
169#121—44no—
170#121—44no—
171C4×Q8—44no—
172C4×Q8—44no—
173C4×D4—44no—
174C4×D4—44no—
175#117—44no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
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×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
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×C42C22⋊C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2411yescharacteristic
233C4⋊C8—44no—
234C4⋊C8—44no—
235C4⋊C8—44no—
236C4⋊C8—44no—
237C4⋊C8—44no—
238C22⋊C8—44no—
239C4×C8—44no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248C24—22no—
249C24C2×C22⋊C411yescharacteristic
250#41—816no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×D4—24no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315M16—88no—
316C2×C8—44no—
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—44no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C42—22no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—44no—
370C42#10014yes—
371C42#13114yes—
372C42AGL(1,ℤ/8)12yes—
373C42AGL(1,ℤ/8)12yes—
374C42#13712yes—
375C42#13712yes—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23#42112yes—
390C23C22×C22⋊C411yescharacteristic
391Q8—84no—
392Q8—84no—
393Q8—84no—
394D4—84no—
395D4—84no—
396D4—88no—
397D4—84no—
398C2×C4—24no—
399C2×C4—28no—
400C2×C4—24no—
401C2×C4—28no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—88no—
445C2×C4#40812yes—
446C2×C4#40812yes—
447C2×C4#41012yes—
448C2×C4C2×#10012yes—
449C2×C4C2×#10012yes—
450C2×C4C2×#10012yes—
451C2×C4#41712yes—
452C2×C4#41712yes—
453C2×C4#41712yes—
454C2×C4#41712yes—
455C2×C4#42112yes—
456C2×C4C2×#13112yes—
457C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
458C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4C2×#13711yescharacteristic
461C8—84no—
462C8—84no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22C2×#40811yescharacteristic
478C22C2×#41011yescharacteristic
479C22C22×#10011yescharacteristic
480C22C2×#41711yescharacteristic
481C22C2×#41711yescharacteristic
482C22C2×#42111yescharacteristic
483C22C2×#42111yescharacteristic
484C4—22no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—22no—
489C4—22no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—41no—
498C4—41no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4#156014yes—
503C2—42no—
504C2—162no—
505C2#1738411yescharacteristic
506C2#1746611yescharacteristic
507C2#1748111yescharacteristic
508C2#1756311yescharacteristic
509C2#1758811yescharacteristic
510C2#1766611yescharacteristic
511C2C2×#156011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C4
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. #20808C2semidirect
3. #20752C2semidirect
4. #7710C2semidirect
5. #7662C2semidirect
6. #7659C2semidirect
7. #7601C2semidirect

Maximal subgroups

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

subgroupindex
#366392
#208082
#207522
#77102
#76622
#76592
#76012