Finite Group Databasebeta

← #160912

Subgroups — #160912

Order 512.

1062 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34330C211yesmaximal, characteristic
2#22700C211yesmaximal, characteristic, complemented
3#20568C211yesmaximal, characteristic, complemented
4#7633C212yesmaximal, complemented
5#7631C212yesmaximal, complemented
6#7616C212yesmaximal, complemented
7#7612C212yesmaximal, complemented
8#7587C211yesmaximal, characteristic, complemented
9#7577C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic, complemented
11#7562C211yesmaximal, characteristic, complemented
12C2×#550C411yescharacteristic
13C2×#436—21no—
14C2×#436—21no—
15#2366—21no—
16#2366C411yescharacteristic
17#2364—22no—
18#2363C412yes—
19#2357—21no—
20#2355C411yescharacteristic
21#2354C411yescharacteristic
22#2339C412yes—
23#2333—22no—
24#2328—21no—
25C4×#127C2211yescharacteristic
26C4×#121C2212yes—
27#2297C2212yes—
28#2295C2211yescharacteristic
29#2275—21no—
30C42×D4C2211yescharacteristic
31#1939—21no—
32#1851—22no—
33#1831C2211yescharacteristic
34#1797C2212yes—
35#1770—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1753—22no—
39#1749C2212yes—
40#1744C2211yescharacteristic
41#1728C2212yes—
42#1728C2212yes—
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×#41C2×C412yes—
54C22×M16—22no—
55C22×M16—22no—
56#550—24no—
57#550—24no—
58C2×#127—21no—
59C2×#127C2×C411yescharacteristic
60C2×#121—22no—
61C2×#121C2×C412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4—22no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2×C412yes—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×#117C2×C412yes—
74C22×C42C2312yes—
75#436—28no—
76#436—44no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×#97—22no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#391C2×C411yescharacteristic
88#391C2×C412yes—
89#391C2311yescharacteristic
90#390—21no—
91#390—21no—
92#390—22no—
93#390—21no—
94#390—21no—
95#390C2×C412yes—
96#390C2312yes—
97#389C2×C411yescharacteristic
98#389C2311yescharacteristic
99#388—22no—
100#388—22no—
101#388—22no—
102#387—21no—
103#387—21no—
104#387—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#384C2×C411yescharacteristic
113#382—21no—
114#381—22no—
115#380C2×C412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
131#376—21no—
132C2×#95—22no—
133C43C2311yescharacteristic
134#336—21no—
135#336—22no—
136#336—22no—
137#336—21no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—24no—
146C2×#41—82no—
147C2×#41—82no—
148C22×Q8C22×C411yescharacteristic
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C22⋊C411yescharacteristic
154C22×D4C22×C412yes—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4C22×C412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C2×M16—28no—
162C2×M16—44no—
163C2×M16—44no—
164C2×M16—82no—
165C2×M16—82no—
166C2×M16—82no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173#127—28no—
174#127—22no—
175#127M1614yescomplemented
176#121—216no—
177#121—24no—
178#121—24no—
179C4×Q8—22no—
180C4×Q8M1614yescomplemented
181C4×D4—28no—
182C4×D4—24no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—44no—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—81no—
191C4×D4—81no—
192C4×D4—82no—
193C4×D4—82no—
194C4×D4—82no—
195C4×D4—82no—
196#117—82no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—82no—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C412yes—
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2411yescharacteristic
236C4⋊C8—24no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8—82no—
240C22⋊C8—28no—
241C22⋊C8—44no—
242#97—82no—
243C4×C8—24no—
244C4×C8—42no—
245C4×C8—81no—
246#95—21no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—22no—
251#95—21no—
252#95—22no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260C24—22no—
261C24C2×C22⋊C412yes—
262#41—88no—
263#41—88no—
264C2×Q8—21no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8C2×M1612yes—
268C2×D4—28no—
269C2×D4—28no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—81no—
281C2×D4—81no—
282C2×D4—82no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4—82no—
286C2×D4C2×M1614yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—28no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323M16—88no—
324M16—88no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—24no—
328C2×C8—22no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—42no—
343C4⋊C4—21no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4#10414yes—
356C4⋊C4C2×M1612yes—
357C22⋊C4—28no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—44no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C42—21no—
372C42—21no—
373C42—24no—
374C42—22no—
375C42—42no—
376C42—41no—
377C42—42no—
378C42—41no—
379C42—42no—
380C42—41no—
381C42—42no—
382C42—82no—
383C42—82no—
384C42C2×M1612yes—
385C42C2×M1612yes—
386C42AGL(1,ℤ/8)12yes—
387C42#13712yes—
388C23—22no—
389C23—24no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23C22×C22⋊C411yescharacteristic
403Q8—42no—
404Q8—42no—
405D4—44no—
406D4—44no—
407D4—44no—
408D4—48no—
409D4—84no—
410D4—88no—
411D4—88no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—88no—
455C2×C4—164no—
456C2×C4—164no—
457C2×C4#40712yes—
458C2×C4#40712yes—
459C2×C4#41712yes—
460C2×C4#41712yes—
461C2×C4C2×#10414yes—
462C2×C4C2×#10412yes—
463C2×C4C22×M1611yescharacteristic
464C2×C4C22×M1611yescharacteristic
465C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
466C2×C4C2×#13711yescharacteristic
467C2×C4#57611yescharacteristic
468C2×C4#57711yescharacteristic
469C8—42no—
470C8—44no—
471C8—42no—
472C8—42no—
473C8—81no—
474C8—82no—
475C22—42no—
476C22—44no—
477C22—42no—
478C22—44no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—44no—
483C22—42no—
484C22—42no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22C2×#40711yescharacteristic
490C22#288311yescharacteristic
491C22#288811yescharacteristic
492C22C2×#41711yescharacteristic
493C22#289711yescharacteristic
494C22C22×#10411yescharacteristic
495C22#290111yescharacteristic
496C4—22no—
497C4—22no—
498C4—21no—
499C4—21no—
500C4—24no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—41no—
510C4—82no—
511C4—81no—
512C4—81no—
513C4#148914yes—
514C2—82no—
515C2—84no—
516C2#1736911yescharacteristic
517C2C2×#148911yescharacteristic
518C2#1745311yescharacteristic
519C2#1754711yescharacteristic
520C2#1763011yescharacteristic
521C2#1765511yescharacteristic
522C2#1776211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#550C4
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
2. #22700C2semidirect
3. #20568C2semidirect
4. #7633C2semidirect
5. #7631C2semidirect
6. #7616C2semidirect
7. #7612C2semidirect
8. #7587C2semidirect
9. #7577C2semidirect
10. #7569C2semidirect
11. #7562C2semidirect
175. #127M16semidirect
180. C4×Q8M16semidirect

Maximal subgroups

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

subgroupindex
#343302
#227002
#205682
#76332
#76312
#76162
#76122
#75872
#75772
#75692
#75622