Finite Group Databasebeta

← #164711

Subgroups — #164711

Order 512.

961 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36405C211yesmaximal, characteristic
2#22469C211yesmaximal, characteristic
3#20684C211yesmaximal, characteristic
4#7721C212yesmaximal, complemented
5#7685C211yesmaximal, characteristic, complemented
6#7668C211yesmaximal, characteristic, complemented
7#7655C212yesmaximal, complemented
8#7646C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic, complemented
10#7579C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic, complemented
12C2×#556C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C8×D4—21no—
15C2×#408—21no—
16#2857C2211yescharacteristic
17#2855C2212yes—
18#2582C411yescharacteristic
19#2546C411yescharacteristic
20#2485C412yes—
21#2478—22no—
22#2468—22no—
23#2447—21no—
24#2432—21no—
25#2415C412yes—
26#2365—21no—
27#2362—21no—
28#2354C411yescharacteristic
29C4×C22⋊Q8C2211yescharacteristic
30#2296C2212yes—
31#1984—21no—
32#1869—22no—
33#1818—22no—
34#1817C2212yes—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2211yescharacteristic
38C4×C22⋊C8C2211yescharacteristic
39#1729—21no—
40#1728C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2212yes—
52#1271C2212yes—
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56C23×C8—22no—
57#556—44no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#126—22no—
61C2×#124—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×C22⋊Q8C2×C412yes—
65C4×#41—44no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C412yes—
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D412yes—
74C2×#117C2×C412yes—
75C2×#117C2×C412yes—
76C2×#117D412yes—
77C2×#117D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2312yes—
80C8×D4—44no—
81C2×C4⋊C8—21no—
82#408—44no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C4×C8—21no—
89#398—22no—
90#398—22no—
91#397—22no—
92#397—22no—
93#394—22no—
94#394—22no—
95#391—21no—
96#391—21no—
97#391—21no—
98#391C2×C412yes—
99#391C2311yescharacteristic
100#390—21no—
101#389—21no—
102#389—22no—
103#389—21no—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2312yes—
108#389C2311yescharacteristic
109#388—22no—
110#388—22no—
111#388—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—21no—
116#385—22no—
117#385—22no—
118#385—22no—
119#385—21no—
120#384C2×C411yescharacteristic
121#381—22no—
122#380—21no—
123#380C2×C412yes—
124#380C2×C412yes—
125#380C2×C411yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C22⋊C4—21no—
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×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C2×M16—42no—
160C2×M16—42no—
161C2×M16—42no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—44no—
169C22×C8—42no—
170C22×C8—42no—
171C22×C8—44no—
172C22×C8—42no—
173C22×C8—82no—
174C4⋊Q8—42no—
175C4⋊Q8—44no—
176#126—48no—
177#124—44no—
178C22⋊Q8—22no—
179C22⋊Q8—41no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C22⋊Q8—44no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—44no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—44no—
190C4×D4—44no—
191#117—24no—
192#117—44no—
193#117—42no—
194#117—42no—
195#117—44no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4C22⋊C412yes—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—42no—
231C2×C42—44no—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2411yescharacteristic
239C4⋊C8—42no—
240C22⋊C8—42no—
241C22⋊C8—44no—
242C22⋊C8—44no—
243C22⋊C8—44no—
244C22⋊C8—81no—
245C4×C8—42no—
246#95—22no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95—21no—
257#95—22no—
258#95—21no—
259C24C2×C22⋊C412yes—
260C24C2×C22⋊C411yescharacteristic
261#41—48no—
262#41—48no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—42no—
278C2×D4#9914yes—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4#13512yes—
314C22×C4#13512yes—
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318M16—84no—
319C2×C8—22no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—88no—
338C2×C8—84no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—82no—
361C4⋊C4—81no—
362C4⋊C4#13112yes—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—42no—
366C22⋊C4—44no—
367C22⋊C4—42no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—81no—
372C22⋊C4—82no—
373C42—22no—
374C42—22no—
375C42—41no—
376C42—42no—
377C42—41no—
378C42—42no—
379C42—42no—
380C42—44no—
381C42—44no—
382C42—42no—
383C42—42no—
384C23—22no—
385C23—21no—
386C23—21no—
387C23—22no—
388C23—21no—
389C23—22no—
390C23—21no—
391C23—22no—
392C23—21no—
393C23—22no—
394C23—21no—
395C23—22no—
396C23C2×#9914yes—
397C23#41612yes—
398C23C22×C22⋊C411yescharacteristic
399Q8—44no—
400Q8—81no—
401Q8—81no—
402Q8—84no—
403D4—28no—
404D4—48no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4C2×#9912yes—
457C2×C4#41612yes—
458C2×C4#42112yes—
459C2×C4C2×#13111yescharacteristic
460C2×C4#56811yescharacteristic
461C2×C4C2×#13511yescharacteristic
462C2×C4C2×#13511yescharacteristic
463C2×C4#57711yescharacteristic
464C2×C4#57811yescharacteristic
465C8—42no—
466C8—81no—
467C8—81no—
468C8—81no—
469C8—82no—
470C8—82no—
471C22—24no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—24no—
476C22—24no—
477C22—41no—
478C22—41no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—41no—
483C22—42no—
484C22—41no—
485C22#288011yescharacteristic
486C22C22×#9911yescharacteristic
487C22#288911yescharacteristic
488C22C2×#41611yescharacteristic
489C22#289811yescharacteristic
490C22C2×#42111yescharacteristic
491C22#290111yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—42no—
499C4—41no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—41no—
504C4—81no—
505C4—81no—
506C4—82no—
507C4—82no—
508C4—81no—
509C4#151614yes—
510C2—44no—
511C2—42no—
512C2#1741611yescharacteristic
513C2#1751011yescharacteristic
514C2#1752311yescharacteristic
515C2C2×#151611yescharacteristic
516C2#1760811yescharacteristic
517C2#1768311yescharacteristic
518C2#1772511yescharacteristic
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. #7685C2semidirect
6. #7668C2semidirect
7. #7655C2semidirect
8. #7646C2semidirect
9. #7617C2semidirect
10. #7579C2semidirect
11. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#364052
#224692
#206842
#77212
#76852
#76682
#76552
#76462
#76172
#75792
#75662