Finite Group Databasebeta

← #193261

Subgroups — #193261

Order 512.

1122 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#34209C211yesmaximal, characteristic
2#21964C211yesmaximal, characteristic
3#21362C211yesmaximal, characteristic
4#21324C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8925C212yesmaximal, complemented
8#8923C212yesmaximal, complemented
9#8922C212yesmaximal, complemented
10#7903C211yesmaximal, characteristic
11#7896C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#519—21no—
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2365—21no—
18#2365—21no—
19#2365C2212yes—
20#2365C2212yes—
21#2344—22no—
22#2344—22no—
23#2327—22no—
24#2310—21no—
25C4×C4⋊Q8—21no—
26C4×C4⋊Q8—21no—
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1993—22no—
31#1990C2212yes—
32#1990C2212yes—
33#1948—21no—
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1948C2212yes—
37#1947—21no—
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1947C2212yes—
41#1854—21no—
42#1854—21no—
43#1809—22no—
44C4×#107—21no—
45C4×#106—21no—
46#1736—22no—
47#1549C418yescomplemented
48C2×#332C2212yes—
49C2×#332C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C23×Q8—21no—
55C22×M16—21no—
56C22×M16—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yes—
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8Q814yes—
64C2×C4⋊Q8C2312yes—
65#519—44no—
66#519—44no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×#117—24no—
72C2×#117—21no—
73C2×#117—21no—
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C22×C4⋊C4—21no—
77C22×C4⋊C4D412yes—
78C22×C42C2311yescharacteristic
79#431—22no—
80#431C2×C414yes—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—22no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—22no—
87#423—22no—
88#423C2×C414yes—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C22⋊C8—21no—
94C4×M16—22no—
95C4×M16—22no—
96C2×#97—21no—
97C2×C4×C8—21no—
98C2×C4×C8C2312yes—
99#391—22no—
100#391—22no—
101#391—22no—
102#391—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#387—24no—
108#387—24no—
109#387—24no—
110#387—24no—
111#384D411yescharacteristic
112#384D411yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#382—21no—
116#382—21no—
117#381—24no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380D412yes—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2312yes—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136#376—21no—
137C43—21no—
138#340—22no—
139#340—22no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—24no—
145#340—24no—
146#332C2×C418yes—
147#332C2×C418yes—
148C22×Q8—24no—
149C22×Q8—22no—
150C22×Q8—24no—
151C22×Q8—22no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8—42no—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×M16—24no—
161C2×M16—24no—
162C2×M16—42no—
163C2×M16—42no—
164C2×M16—81no—
165C2×M16—81no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8C4⋊C418yes—
178C4⋊Q8C22×C414yes—
179C4⋊Q8C2×Q814yes—
180C4×Q8—48no—
181C4×Q8—48no—
182C4×Q8—44no—
183C4×Q8—44no—
184C4×Q8—44no—
185C4×Q8—44no—
186#117—22no—
187#117—22no—
188#117—416no—
189#117—41no—
190#117—41no—
191#117—42no—
192#117—42no—
193#117—44no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C4⋊C418yes—
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—41no—
224C2×C42—21no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—41no—
234C2×C42C22×C412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×Q812yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42C2411yescharacteristic
242#107—42no—
243#107—42no—
244#107—82no—
245#106—42no—
246#106—42no—
247#106—82no—
248C4⋊C8—24no—
249C4⋊C8—22no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8C22×C414yes—
253C4⋊C8C22×C412yes—
254C4⋊C8C22×C412yes—
255C22⋊C8—24no—
256#97—42no—
257C4×C8—24no—
258C4×C8—22no—
259C4×C8—22no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—21no—
265#95—21no—
266#95—22no—
267#95—21no—
268#95—21no—
269#95—22no—
270#95—41no—
271#95—41no—
272C24#12011yescharacteristic
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—28no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—48no—
279C2×Q8—48no—
280C2×Q8—48no—
281C2×Q8—48no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—28no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4C2×C4⋊C412yes—
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4#12011yescharacteristic
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4C22⋊Q812yes—
315C22×C4C22⋊Q812yes—
316C22×C4#12312yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320M16—48no—
321M16—84no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—24no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—28no—
340C4⋊C4—28no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4C2×C4⋊C418yes—
366C4⋊C4AGL(1,ℤ/8)12yes—
367C4⋊C4AGL(1,ℤ/8)12yes—
368C22⋊C4—24no—
369C22⋊C4—42no—
370C22⋊C4—44no—
371C22⋊C4—82no—
372C42—24no—
373C42—42no—
374C42—48no—
375C42—48no—
376C42—42no—
377C42—42no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—42no—
383C42—42no—
384C42—82no—
385C42C2×C4⋊C412yes—
386C42C4×D412yes—
387C42C4×D412yes—
388C42C22⋊Q812yes—
389C42C22⋊Q812yes—
390C42#12312yes—
391C42C23×C411yescharacteristic
392C42C22×Q811yescharacteristic
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23#38512yes—
402C23C2×#12011yescharacteristic
403Q8—416no—
404Q8—44no—
405Q8—44no—
406Q8—44no—
407Q8—44no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—28no—
427C2×C4—416no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4#38512yes—
453C2×C4#38512yes—
454C2×C4#38512yes—
455C2×C4#45412yes—
456C2×C4#45612yes—
457C2×C4C22×C4⋊C411yescharacteristic
458C2×C4C2×C4×D411yescharacteristic
459C2×C4C2×C4×D411yescharacteristic
460C2×C4C2×C22⋊Q811yescharacteristic
461C2×C4C2×C22⋊Q811yescharacteristic
462C2×C4C2×#12311yescharacteristic
463C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
464C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
465C2×C4#57511yescharacteristic
466C2×C4#57511yescharacteristic
467C2×C4#57711yescharacteristic
468C2×C4#57711yescharacteristic
469C2×C4#57811yescharacteristic
470C2×C4#57811yescharacteristic
471C8—44no—
472C8—42no—
473C8—42no—
474C8—44no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22C2×#38511yescharacteristic
483C22#300311yescharacteristic
484C22#300511yescharacteristic
485C22#301311yescharacteristic
486C22#301311yescharacteristic
487C22C2×#45411yescharacteristic
488C22C2×#45611yescharacteristic
489C4—21no—
490C4—21no—
491C4—22no—
492C4—22no—
493C4—48no—
494C4—42no—
495C4—42no—
496C4—82no—
497C4—81no—
498C4—82no—
499C4—81no—
500C4—161no—
501C4—161no—
502C4#185714yes—
503C4C4×AGL(1,ℤ/8)11yescharacteristic
504C4C4×AGL(1,ℤ/8)11yescharacteristic
505C4#294911yescharacteristic
506C4#295011yescharacteristic
507C4#307711yescharacteristic
508C4#307711yescharacteristic
509C4#308211yescharacteristic
510C4#308211yescharacteristic
511C2—42no—
512C2#1858211yescharacteristic
513C2#1858411yescharacteristic
514C2#1859011yescharacteristic
515C2#1859211yescharacteristic
516C2C2×#185711yescharacteristic
517C2#2087311yescharacteristic
518C2#2088011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8926C2semidirect
7. #8925C2semidirect
8. #8923C2semidirect
9. #8922C2semidirect
47. #1549C4semidirect

Maximal subgroups

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

subgroupindex
#342092
#219642
#213622
#213242
C2×#15492
#89262
#89252
#89232
#89222
#79032
#78962