Finite Group Databasebeta

← #164945

Subgroups — #164945

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36415C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic, complemented
3#22696C211yesmaximal, characteristic, complemented
4#7730C212yesmaximal, complemented
5#7690C212yesmaximal, complemented
6#7662C211yesmaximal, characteristic, complemented
7#7662C211yesmaximal, characteristic, complemented
8#7651C211yesmaximal, characteristic, complemented
9#7612C212yesmaximal, complemented
10#7578C212yesmaximal, complemented
11#7569C211yesmaximal, characteristic, complemented
12C2×#555C411yescharacteristic
13C2×#547C411yescharacteristic
14C2×C8×D4—21no—
15#2862C2211yescharacteristic
16#2850C2212yes—
17#2563C411yescharacteristic
18#2557—22no—
19#2557—22no—
20#2547C411yescharacteristic
21#2535—21no—
22#2535—21no—
23#2499C412yes—
24#2431C412yes—
25#2352—21no—
26#2352—21no—
27C4×#127C2211yescharacteristic
28#2297C2212yes—
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#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1270C2211yescharacteristic
48#1270C2212yes—
49#1270C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×#41C2×C412yes—
54C23×C8—22no—
55#555—44no—
56#547—44no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#127C2×C411yescharacteristic
60C2×#124—22no—
61C2×#124C2×C412yes—
62C2×#123—22no—
63C2×C22⋊Q8C2×C412yes—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121C2×C412yes—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×#117—21no—
75C2×#117C2×C412yes—
76C22×C4⋊C4—21no—
77C22×C42C2312yes—
78C8×D4—44no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×#97—22no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90#397—22no—
91#397—22no—
92#397—22no—
93#393—22no—
94#392—22no—
95#392—22no—
96#391—21no—
97#391—21no—
98#391C2×C412yes—
99#391C2×C412yes—
100#391C2312yes—
101#390—21no—
102#390—21no—
103#390C2×C412yes—
104#390C2311yescharacteristic
105#390C2311yescharacteristic
106#388—22no—
107#388—22no—
108#387—22no—
109#387—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#384C2×C411yescharacteristic
117#382—21no—
118#382—21no—
119#382C2×C412yes—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4C2×C412yes—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4D412yes—
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—21no—
136#336—22no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C2×#41—24no—
144C2×#41—42no—
145C2×#41—44no—
146C2×#41—82no—
147C22×Q8—21no—
148C22×Q8C22×C412yes—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C22⋊C411yescharacteristic
154C22×D4C22×C411yescharacteristic
155C22×D4C22×C411yescharacteristic
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4C22×C412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—44no—
169C22×C8—44no—
170C22×C8—82no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173#127—22no—
174#127—41no—
175#124—48no—
176#124—82no—
177#123—48no—
178C22⋊Q8—44no—
179#121—44no—
180#121—44no—
181#121—44no—
182C4×D4—24no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—42no—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—42no—
193#117—44no—
194#117—44no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4C22⋊C412yes—
222C2×C22⋊C4C22⋊C412yes—
223C2×C22⋊C4C22⋊C412yes—
224C2×C42—22no—
225C2×C42—82no—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2411yescharacteristic
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8—82no—
241C22⋊C8—44no—
242C22⋊C8—44no—
243C22⋊C8—82no—
244#97—82no—
245C4×C8—42no—
246C4×C8—42no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258C24—21no—
259C24—21no—
260C24C2×C22⋊C412yes—
261#41—88no—
262#41—88no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—81no—
269C2×Q8—81no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—81no—
287C2×D4—81no—
288C2×D4—82no—
289C2×D4—82no—
290C2×D4—82no—
291C2×D4#10014yes—
292C2×D4#13112yes—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—82no—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—88no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—44no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C42—21no—
380C42—21no—
381C42—22no—
382C42—42no—
383C42—41no—
384C42—41no—
385C42—42no—
386C42—41no—
387C42—42no—
388C42—44no—
389C42#13512yes—
390C42#13512yes—
391C42AGL(1,ℤ/8)12yes—
392C42#13712yes—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23C2×#10014yes—
414C23C22×C22⋊C411yescharacteristic
415Q8—82no—
416Q8—82no—
417Q8—162no—
418D4—28no—
419D4—42no—
420D4—42no—
421D4—84no—
422D4—84no—
423D4—81no—
424D4—81no—
425D4—88no—
426D4—164no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—88no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4C2×#10012yes—
471C2×C4#41612yes—
472C2×C4#41612yes—
473C2×C4#41712yes—
474C2×C4#41712yes—
475C2×C4C2×#13111yescharacteristic
476C2×C4#56811yescharacteristic
477C2×C4C2×#13511yescharacteristic
478C2×C4C2×#13511yescharacteristic
479C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
480C2×C4C2×#13711yescharacteristic
481C8—42no—
482C8—42no—
483C8—81no—
484C8—82no—
485C8—81no—
486C8—82no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22#288011yescharacteristic
509C22#288311yescharacteristic
510C22C22×#10011yescharacteristic
511C22C2×#41611yescharacteristic
512C22C2×#41711yescharacteristic
513C22#290111yescharacteristic
514C22#290111yescharacteristic
515C4—21no—
516C4—21no—
517C4—22no—
518C4—22no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—42no—
524C4—41no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—82no—
529C4—81no—
530C4—81no—
531C4#155114yes—
532C2—44no—
533C2—82no—
534C2—161no—
535C2#1742111yescharacteristic
536C2#1750811yescharacteristic
537C2#1751211yescharacteristic
538C2#1761711yescharacteristic
539C2#1763011yescharacteristic
540C2#1768511yescharacteristic
541C2C2×#155111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36415C2
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. #22839C2semidirect
3. #22696C2semidirect
4. #7730C2semidirect
5. #7690C2semidirect
6. #7662C2semidirect
7. #7662C2semidirect
8. #7651C2semidirect
9. #7612C2semidirect
10. #7578C2semidirect
11. #7569C2semidirect

Maximal subgroups

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

subgroupindex
#364152
#228392
#226962
#77302
#76902
#76622
#76512
#76122
#75782
#75692