Finite Group Databasebeta

← #160897

Subgroups — #160897

Order 512.

1120 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34313C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic, complemented
3#20570C211yesmaximal, characteristic, complemented
4#7637C212yesmaximal, complemented
5#7628C212yesmaximal, complemented
6#7621C212yesmaximal, complemented
7#7608C212yesmaximal, complemented
8#7587C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic, complemented
12C2×#550C411yescharacteristic
13C2×#520—21no—
14C2×#446—21no—
15#2366C412yes—
16#2366C411yescharacteristic
17#2361—21no—
18#2361—22no—
19Q8×C4⋊C4C411yescharacteristic
20D4×C4⋊C4C412yes—
21D4×C4⋊C4C411yescharacteristic
22#2341—21no—
23#2324—22no—
24#2324—21no—
25C4×#127C2211yescharacteristic
26C4×#127C2212yes—
27#2293C2211yescharacteristic
28#2293C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1917—22no—
32#1839C2212yes—
33#1831C2211yescharacteristic
34#1771—22no—
35#1770—21no—
36#1770—21no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1745C2212yes—
41#1745C2212yes—
42#1744C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2211yescharacteristic
45#1271C2212yes—
46#1270C2211yescharacteristic
47#1270C2211yescharacteristic
48#1270C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×#41—21no—
54#550—24no—
55#550—24no—
56C2×#127C2×C412yes—
57C2×#127C2×C411yescharacteristic
58C2×#121—21no—
59C2×#121—22no—
60#520—44no—
61#520—44no—
62C2×C4×Q8—21no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4—22no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—22no—
72C2×C4×D4C2×C412yes—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C412yes—
75C2×C4×D4C2×C412yes—
76C2×C4×D4C2×C412yes—
77C2×#117—22no—
78C2×#117—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81#446—28no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×#97—22no—
87C2×#97—22no—
88C2×#97—22no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92#390—22no—
93#390—21no—
94#390—22no—
95#390—21no—
96#390—21no—
97#390—22no—
98#390C2×C411yescharacteristic
99#390C2×C412yes—
100#390C2312yes—
101#390C2311yescharacteristic
102#389—21no—
103#389—22no—
104#389C2×C411yescharacteristic
105#389C2×C412yes—
106#389C2311yescharacteristic
107#388—22no—
108#388—21no—
109#388—21no—
110#388—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#384C2×C412yes—
118#384C2×C411yescharacteristic
119#380—22no—
120#380—21no—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D412yes—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—22no—
131C43C2311yescharacteristic
132C43C2312yes—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—21no—
138#336—22no—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144#336C2312yes—
145C2×#41—28no—
146C2×#41—44no—
147C2×#41—44no—
148C22×Q8C22⋊C411yescharacteristic
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4C22×C412yes—
154C22×D4C22×C411yescharacteristic
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166#127—24no—
167#127—24no—
168#127—22no—
169#127M1614yes—
170#121—28no—
171#121—216no—
172C4×Q8—24no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—44no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—22no—
181C4×D4—44no—
182C4×D4—44no—
183C4×D4—48no—
184C4×D4—44no—
185C4×D4—42no—
186C4×D4—42no—
187C4×D4—81no—
188C4×D4—82no—
189C4×D4—81no—
190C4×D4—82no—
191C4×D4—82no—
192C4×D4—82no—
193C4×D4—82no—
194C4×D4M1614yes—
195#117—44no—
196#117—44no—
197#117—42no—
198#117—42no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42C22×C412yes—
234C2×C42C22×C412yes—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2411yescharacteristic
246C4⋊C8—24no—
247C4⋊C8—28no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250#97—28no—
251#97—44no—
252#97—82no—
253C4×C8—24no—
254C4×C8—42no—
255C4×C8—81no—
256#95—21no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—21no—
261#95C22⋊C412yes—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264C24—22no—
265C24—21no—
266#41—432no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8#10414yes—
270C2×D4—28no—
271C2×D4—28no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—44no—
283C2×D4—42no—
284C2×D4—81no—
285C2×D4—82no—
286C2×D4—81no—
287C2×D4—82no—
288C2×D4C2×M1614yes—
289C2×D4C2×M1612yes—
290C22×C4—28no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—42no—
338C4⋊C4—21no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4C2×M1612yes—
357C22⋊C4—28no—
358C22⋊C4—24no—
359C22⋊C4—44no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—42no—
365C22⋊C4—41no—
366C22⋊C4—42no—
367C22⋊C4—41no—
368C22⋊C4—42no—
369C22⋊C4—82no—
370C22⋊C4—81no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—21no—
379C42—22no—
380C42—21no—
381C42—22no—
382C42—22no—
383C42—24no—
384C42—42no—
385C42—41no—
386C42—41no—
387C42—42no—
388C42—41no—
389C42—44no—
390C42—42no—
391C42C2×M1612yes—
392C42C2×M1612yes—
393C42AGL(1,ℤ/8)12yes—
394C42AGL(1,ℤ/8)12yes—
395C23—24no—
396C23—22no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23—42no—
402C23—41no—
403C23—42no—
404C23—41no—
405C23—42no—
406C23—41no—
407C23—42no—
408C23—41no—
409C23C22×C22⋊C411yescharacteristic
410Q8—28no—
411D4—44no—
412D4—42no—
413D4—42no—
414D4—44no—
415D4—416no—
416D4—48no—
417D4—84no—
418D4—88no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—416no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—48no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—164no—
458C2×C4—164no—
459C2×C4—164no—
460C2×C4#40712yes—
461C2×C4#40712yes—
462C2×C4#41812yes—
463C2×C4#41812yes—
464C2×C4C2×#10414yes—
465C2×C4C2×#10412yes—
466C2×C4C22×M1611yescharacteristic
467C2×C4C22×M1611yescharacteristic
468C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
469C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
470C2×C4#57611yescharacteristic
471C2×C4#57711yescharacteristic
472C8—42no—
473C8—44no—
474C8—42no—
475C8—42no—
476C8—81no—
477C8—82no—
478C22—44no—
479C22—42no—
480C22—44no—
481C22—42no—
482C22—44no—
483C22—42no—
484C22—82no—
485C22—81no—
486C22—82no—
487C22—81no—
488C22—82no—
489C22—81no—
490C22—82no—
491C22—81no—
492C22C2×#40711yescharacteristic
493C22#288411yescharacteristic
494C22#288811yescharacteristic
495C22C2×#41811yescharacteristic
496C22#289711yescharacteristic
497C22C22×#10411yescharacteristic
498C22#290111yescharacteristic
499C4—24no—
500C4—22no—
501C4—22no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—41no—
512C4—42no—
513C4—82no—
514C4—81no—
515C4—81no—
516C4#149014yes—
517C2—84no—
518C2—82no—
519C2#1737211yescharacteristic
520C2C2×#149011yescharacteristic
521C2#1745311yescharacteristic
522C2#1755011yescharacteristic
523C2#1763111yescharacteristic
524C2#1765511yescharacteristic
525C2#1775811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#550C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22832C2semidirect
3. #20570C2semidirect
4. #7637C2semidirect
5. #7628C2semidirect
6. #7621C2semidirect
7. #7608C2semidirect
8. #7587C2semidirect
9. #7574C2semidirect
10. #7569C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#343132
#228322
#205702
#76372
#76282
#76212
#76082
#75872
#75742
#75692
#75602