Finite Group Databasebeta

← #164613

Subgroups — #164613

Order 512.

961 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36505C211yesmaximal, characteristic
2#22473C211yesmaximal, characteristic
3#20630C211yesmaximal, characteristic
4#7731C212yesmaximal, complemented
5#7711C212yesmaximal, complemented
6#7685C211yesmaximal, characteristic, complemented
7#7668C211yesmaximal, characteristic, complemented
8#7617C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic, complemented
10#7600C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic, complemented
12C2×#557C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#436—21no—
15C2×#424—21no—
16C2×#408—21no—
17#2857C2211yescharacteristic
18#2851C2212yes—
19#2562C412yes—
20#2547C411yescharacteristic
21#2538C411yescharacteristic
22#2536C412yes—
23#2475—22no—
24#2460—22no—
25#2446—21no—
26#2432—21no—
27#2367—21no—
28#2362—21no—
29#2355C411yescharacteristic
30C4×C22⋊Q8C2211yescharacteristic
31C4×C22⋊Q8C2212yes—
32#1984—21no—
33#1870—22no—
34#1817C2212yes—
35#1798—22no—
36#1797C2212yes—
37#1797C2212yes—
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40#1728C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2212yes—
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C22×M16—22no—
56C23×C8—21no—
57#557—44no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#125—22no—
61C2×#125C2×C412yes—
62C2×#124—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×C22⋊Q8C2×C412yes—
66C4×#41—44no—
67C2×C4×Q8—21no—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×D4C2×C412yes—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D412yes—
76C2×#117D411yescharacteristic
77C2×#117D412yes—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#436—44no—
81#424—44no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85#408—44no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C4×C8—21no—
92#400—22no—
93#398—22no—
94#397—22no—
95#396—22no—
96#394—22no—
97#394—22no—
98#391—21no—
99#391—21no—
100#391—21no—
101#391C2311yescharacteristic
102#390—21no—
103#389—21no—
104#389—21no—
105#389C2311yescharacteristic
106#389C2312yes—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#387—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#382C2×C412yes—
122#380—21no—
123#380C2×C411yescharacteristic
124#380C2×C412yes—
125#380C2×C412yes—
126#380C2×C412yes—
127#380C2×C412yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135#336—21no—
136#336—21no—
137#336—22no—
138#336—22no—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2312yes—
143#336C2311yescharacteristic
144#336C2311yescharacteristic
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—42no—
148C2×#41—42no—
149C22×Q8—21no—
150C22×Q8C22⋊C411yescharacteristic
151C22×Q8C22×C411yescharacteristic
152C22×D4C22⋊C411yescharacteristic
153C22×D4C22⋊C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C2×M16—44no—
161C2×M16—42no—
162C2×M16—42no—
163C2×M16—44no—
164C2×M16—42no—
165C2×M16—82no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8—42no—
175C4⋊Q8—42no—
176C4⋊Q8—44no—
177#125—44no—
178#125—48no—
179#124—44no—
180C22⋊Q8—22no—
181C22⋊Q8—24no—
182C22⋊Q8—41no—
183C22⋊Q8—42no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C4×Q8—24no—
187C4×Q8—42no—
188C4×Q8—44no—
189C4×Q8—42no—
190C4×Q8—44no—
191C4×D4—24no—
192C4×D4—44no—
193C4×D4—42no—
194C4×D4—44no—
195#117—24no—
196#117—44no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—42no—
232C2×C42—44no—
233C2×C42C22×C412yes—
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D412yes—
239C2×C42C2411yescharacteristic
240C4⋊C8—44no—
241C4⋊C8—44no—
242C4⋊C8—42no—
243C22⋊C8—42no—
244C22⋊C8—44no—
245C22⋊C8—44no—
246C22⋊C8—44no—
247C22⋊C8—81no—
248C4×C8—42no—
249#95—21no—
250#95—22no—
251#95—22no—
252#95—21no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—21no—
259#95—21no—
260#95—22no—
261#95—21no—
262C24C2×C22⋊C412yes—
263C24C2×C22⋊C411yescharacteristic
264#41—48no—
265#41—48no—
266C2×Q8—21no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8#13112yes—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—42no—
281C2×D4—42no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4AGL(1,ℤ/8)12yes—
317C22×C4#13712yes—
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321M16—88no—
322M16—84no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—84no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—81no—
365C4⋊C4#9914yes—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—81no—
375C22⋊C4—82no—
376C42—22no—
377C42—24no—
378C42—24no—
379C42—22no—
380C42—41no—
381C42—41no—
382C42—42no—
383C42—44no—
384C42—42no—
385C42—42no—
386C42—82no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23#41712yes—
401C23C22×C22⋊C411yescharacteristic
402Q8—42no—
403Q8—42no—
404Q8—44no—
405Q8—84no—
406D4—48no—
407D4—44no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
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—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4C2×#9914yes—
459C2×C4C2×#9912yes—
460C2×C4#41712yes—
461C2×C4#42112yes—
462C2×C4C2×#13111yescharacteristic
463C2×C4#56811yescharacteristic
464C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
465C2×C4C2×#13711yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57811yescharacteristic
468C8—42no—
469C8—81no—
470C8—81no—
471C8—81no—
472C8—82no—
473C8—82no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22#288011yescharacteristic
489C22C22×#9911yescharacteristic
490C22#288911yescharacteristic
491C22C2×#41711yescharacteristic
492C22#289811yescharacteristic
493C22C2×#42111yescharacteristic
494C22#290111yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—24no—
502C4—41no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—41no—
507C4—81no—
508C4—82no—
509C4—81no—
510C4—82no—
511C4—81no—
512C4#151314yes—
513C2—42no—
514C2—82no—
515C2#1741411yescharacteristic
516C2#1751511yescharacteristic
517C2#1752511yescharacteristic
518C2C2×#151311yescharacteristic
519C2#1760811yescharacteristic
520C2#1768211yescharacteristic
521C2#1776311yescharacteristic
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. #7731C2semidirect
5. #7711C2semidirect
6. #7685C2semidirect
7. #7668C2semidirect
8. #7617C2semidirect
9. #7617C2semidirect
10. #7600C2semidirect
11. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#365052
#224732
#206302
#77312
#77112
#76852
#76682
#76172
#76002
#75662