Finite Group Databasebeta

← #164564

Subgroups — #164564

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36275C211yesmaximal, characteristic
2#22473C211yesmaximal, characteristic
3#20684C211yesmaximal, characteristic
4#7721C212yesmaximal, complemented
5#7695C212yesmaximal, complemented
6#7688C211yesmaximal, characteristic, complemented
7#7665C211yesmaximal, characteristic, complemented
8#7617C211yesmaximal, characteristic, complemented
9#7610C212yesmaximal, complemented
10#7593C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic, complemented
12C2×#554C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#436—21no—
15C2×#408—21no—
16#2857C2211yescharacteristic
17#2855C2212yes—
18#2582C411yescharacteristic
19#2567C411yescharacteristic
20#2487C412yes—
21#2453—21no—
22#2449—22no—
23#2441—21no—
24#2438—22no—
25#2417C412yes—
26D4×C4⋊C4—21no—
27D4×C4⋊C4C411yescharacteristic
28#2339—21no—
29C4×C22⋊Q8C2211yescharacteristic
30C2×#389C2212yes—
31#1984—21no—
32#1870—22no—
33#1818—22no—
34#1817C2212yes—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2211yescharacteristic
38C4×C22⋊C8C2211yescharacteristic
39#1729—21no—
40#1728C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C22×M16—22no—
56C23×C8—21no—
57#554—44no—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8C2×C412yes—
64C4×#41—44no—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C412yes—
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D412yes—
74C2×#117D412yes—
75C2×#117D411yescharacteristic
76C22×C4⋊C4—22no—
77C22×C4⋊C4C2×C412yes—
78C22×C4⋊C4C2×C412yes—
79C22×C42C2311yescharacteristic
80C22×C42C2312yes—
81#436—44no—
82C2×C4⋊C8—21no—
83#408—44no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90#400—22no—
91#399—22no—
92#398—22no—
93#398—22no—
94#396—22no—
95#395—22no—
96#391—21no—
97#391—21no—
98#391—21no—
99#391C2×C412yes—
100#391C2311yescharacteristic
101#389—21no—
102#389—21no—
103#389—21no—
104#389—22no—
105#389—22no—
106#389C2×C412yes—
107#389C2×C412yes—
108#389C2×C412yes—
109#389C2312yes—
110#389C2311yescharacteristic
111#387—22no—
112#387—22no—
113#387—21no—
114#386—22no—
115#386—22no—
116#385—22no—
117#385—21no—
118#385—22no—
119#385—22no—
120#384—21no—
121#384C2×C411yescharacteristic
122#381—22no—
123#380—21no—
124#380—22no—
125#380C2×C412yes—
126#380C2×C412yes—
127#380C2×C411yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C22⋊C4—21no—
131C2×#95—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—21no—
136#336—21no—
137#336—22no—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—42no—
148C22×Q8—21no—
149C22×Q8C22⋊C411yescharacteristic
150C22×Q8C22×C411yescharacteristic
151C22×D4C22⋊C412yes—
152C22×D4C22⋊C411yescharacteristic
153C23×C4—22no—
154C23×C4—22no—
155C23×C4C22×C412yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×M16—44no—
162C2×M16—42no—
163C2×M16—42no—
164C2×M16—44no—
165C2×M16—42no—
166C2×M16—82no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—42no—
176C4⋊Q8—42no—
177#125—44no—
178C22⋊Q8—22no—
179C22⋊Q8—48no—
180C22⋊Q8—41no—
181C22⋊Q8—44no—
182C22⋊Q8—81no—
183C22⋊Q8—81no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—44no—
190C4×D4—44no—
191C4×D4—44no—
192C4×D4—44no—
193#117—44no—
194#117—42no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4C22⋊C412yes—
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—42no—
233C2×C42—44no—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C412yes—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2411yescharacteristic
241C4⋊C8—42no—
242C22⋊C8—42no—
243C22⋊C8—44no—
244C22⋊C8—44no—
245C22⋊C8—44no—
246C22⋊C8—81no—
247C4×C8—42no—
248#95—22no—
249#95—22no—
250#95—21no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—21no—
258#95—22no—
259#95—21no—
260#95—21no—
261C24C2×C22⋊C412yes—
262C24C2×C22⋊C411yescharacteristic
263#41—48no—
264#41—48no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—81no—
274C2×Q8—81no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—42no—
280C2×D4#9914yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4#13712yes—
322C22×C4#13712yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326M16—88no—
327M16—84no—
328C2×C8—22no—
329C2×C8—41no—
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—42no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—84no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—81no—
372C4⋊C4#13112yes—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—81no—
380C22⋊C4—82no—
381C42—22no—
382C42—22no—
383C42—41no—
384C42—42no—
385C42—41no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C23—21no—
391C23—22no—
392C23—21no—
393C23—22no—
394C23—21no—
395C23—22no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—21no—
400C23—21no—
401C23—22no—
402C23C2×#9914yes—
403C23#41912yes—
404C23C22×C22⋊C411yescharacteristic
405Q8—44no—
406Q8—81no—
407Q8—81no—
408Q8—162no—
409D4—28no—
410D4—48no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
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—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
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—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—88no—
463C2×C4—164no—
464C2×C4C2×#9912yes—
465C2×C4#41912yes—
466C2×C4#42112yes—
467C2×C4C2×#13111yescharacteristic
468C2×C4#56811yescharacteristic
469C2×C4C2×#13711yescharacteristic
470C2×C4C2×#13711yescharacteristic
471C2×C4#57611yescharacteristic
472C2×C4#57711yescharacteristic
473C8—42no—
474C8—81no—
475C8—81no—
476C8—81no—
477C8—82no—
478C8—82no—
479C22—24no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—24no—
484C22—24no—
485C22—41no—
486C22—42no—
487C22—41no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—41no—
492C22—41no—
493C22#288011yescharacteristic
494C22C22×#9911yescharacteristic
495C22#288911yescharacteristic
496C22C2×#41911yescharacteristic
497C22#289711yescharacteristic
498C22C2×#42111yescharacteristic
499C22#290111yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—42no—
507C4—41no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—41no—
512C4—81no—
513C4—81no—
514C4—82no—
515C4—81no—
516C4—82no—
517C4#151114yes—
518C2—44no—
519C2—42no—
520C2#1741611yescharacteristic
521C2#1751411yescharacteristic
522C2#1752211yescharacteristic
523C2C2×#151111yescharacteristic
524C2#1760311yescharacteristic
525C2#1768211yescharacteristic
526C2#1776111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7721C2semidirect
5. #7695C2semidirect
6. #7688C2semidirect
7. #7665C2semidirect
8. #7617C2semidirect
9. #7610C2semidirect
10. #7593C2semidirect
11. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#362752
#224732
#206842
#77212
#76952
#76882
#76652
#76172
#76102
#75932
#75662