Finite Group Databasebeta

← #164288

Subgroups — #164288

Order 512.

1115 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#36505C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic
3#20336C211yesmaximal, characteristic
4#7711C212yesmaximal, complemented
5#7691C212yesmaximal, complemented
6#7685C211yesmaximal, characteristic, complemented
7#7668C211yesmaximal, characteristic, complemented
8#7634C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic, complemented
10#7589C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic, complemented
12C22×#131—21no—
13C2×#557C411yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×#446—21no—
16#2857C2211yescharacteristic
17#2851C2212yes—
18#2562C412yes—
19#2547C411yescharacteristic
20#2538C411yescharacteristic
21#2536C412yes—
22#2475—22no—
23#2460—22no—
24#2446—21no—
25#2432—21no—
26#2367—21no—
27#2362—21no—
28#2355C411yescharacteristic
29C4×C22⋊Q8C2211yescharacteristic
30C4×C22⋊Q8C2212yes—
31#1955—22no—
32#1954—21no—
33#1797C2212yes—
34#1797C2211yescharacteristic
35#1749C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37#1729—21no—
38#1729—22no—
39#1728C2211yescharacteristic
40#1728C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2212yes—
53C22×#41D411yescharacteristic
54C2×#131—28no—
55C2×#131—44no—
56C2×#131—44no—
57C22×M16—22no—
58C22×M16—21no—
59C23×C8—22no—
60C23×C8—21no—
61#557—44no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#125—22no—
65C2×#125C2×C412yes—
66C2×#124—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C22⋊Q8C2×C412yes—
70C4×#41—44no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2×C412yes—
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×D4C2×C412yes—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D412yes—
80C2×#117D411yescharacteristic
81C2×#117D412yes—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#446—44no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—21no—
91C2×#97—22no—
92C2×C4×C8—21no—
93#400—22no—
94#398—22no—
95#397—22no—
96#396—22no—
97#394—22no—
98#394—22no—
99#391—21no—
100#391—21no—
101#391—21no—
102#391C2311yescharacteristic
103#390—21no—
104#389—21no—
105#389—21no—
106#389C2311yescharacteristic
107#389C2312yes—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#387—21no—
117#385—22no—
118#385—22no—
119#385—22no—
120#385—21no—
121#384C2×C411yescharacteristic
122#382C2×C412yes—
123#380—21no—
124#380C2×C411yescharacteristic
125#380C2×C412yes—
126#380C2×C412yes—
127#380C2×C412yes—
128#380C2×C412yes—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2×C412yes—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136#336—22no—
137#336—21no—
138#336—22no—
139#336—21no—
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2311yescharacteristic
143#336C2312yes—
144#336C2311yescharacteristic
145#336C2311yescharacteristic
146C2×#41—24no—
147C2×#41—24no—
148C2×#41—22no—
149C2×#41—22no—
150C22×Q8—21no—
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C22×C411yescharacteristic
153C22×D4C22⋊C412yes—
154C22×D4C22⋊C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161#131—432no—
162C2×M16—28no—
163C2×M16—24no—
164C2×M16—44no—
165C2×M16—42no—
166C2×M16—44no—
167C2×M16—42no—
168C22×C8—28no—
169C22×C8—24no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—44no—
177C22×C8—42no—
178C22×C8—44no—
179C22×C8—42no—
180C4⋊Q8—42no—
181C4⋊Q8—44no—
182#125—44no—
183#125—48no—
184#124—44no—
185C22⋊Q8—22no—
186C22⋊Q8—24no—
187C22⋊Q8—41no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—44no—
194C4×Q8—42no—
195C4×Q8—44no—
196C4×D4—44no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—44no—
200#117—42no—
201#117—44no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C22⋊C412yes—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—42no—
237C2×C42—44no—
238C2×C42C22×C412yes—
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2411yescharacteristic
245C4⋊C8—42no—
246C4⋊C8—44no—
247C22⋊C8—44no—
248C22⋊C8—42no—
249C22⋊C8—82no—
250C22⋊C8—81no—
251#97—44no—
252C4×C8—42no—
253#95—21no—
254#95—22no—
255#95—22no—
256#95—21no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—21no—
266C24C2×C22⋊C411yescharacteristic
267C24C2×C22⋊C412yes—
268#41—216no—
269#41—216no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8#9914yes—
279C2×Q8#13112yescomplemented
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4#10414yes—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4AGL(1,ℤ/8)12yes—
321C22×C4#13712yes—
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325M16—416no—
326M16—48no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—24no—
330C2×C8—22no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—416no—
336C2×C8—48no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—42no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—81no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—81no—
380C22⋊C4—82no—
381C42—22no—
382C42—24no—
383C42—24no—
384C42—22no—
385C42—41no—
386C42—42no—
387C42—41no—
388C42—44no—
389C42—42no—
390C42—42no—
391C42—82no—
392C23—21no—
393C23—22no—
394C23—21no—
395C23—22no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—21no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23—22no—
404C23#41712yes—
405C23C2×#10414yes—
406C23C22×C22⋊C411yescharacteristic
407Q8—28no—
408Q8—42no—
409Q8—42no—
410Q8—84no—
411D4—216no—
412D4—28no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4C2×#9914yes—
464C2×C4C2×#9912yes—
465C2×C4#41712yes—
466C2×C4C2×#10412yes—
467C2×C4C2×#13111yescharacteristic
468C2×C4#56811yescharacteristic
469C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
470C2×C4C2×#13711yescharacteristic
471C2×C4#57711yescharacteristic
472C2×C4#57811yescharacteristic
473C8—42no—
474C8—42no—
475C8—44no—
476C8—42no—
477C8—81no—
478C8—82no—
479C22—22no—
480C22—22no—
481C22—24no—
482C22—22no—
483C22—24no—
484C22—24no—
485C22—41no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22—42no—
492C22—41no—
493C22#288011yescharacteristic
494C22C22×#9911yescharacteristic
495C22#288911yescharacteristic
496C22C2×#41711yescharacteristic
497C22#289811yescharacteristic
498C22C22×#10411yescharacteristic
499C22#290111yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—24no—
505C4—21no—
506C4—21no—
507C4—41no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—41no—
512C4—82no—
513C4—81no—
514C4—82no—
515C4—81no—
516C4—81no—
517C4#150414yes—
518C2—42no—
519C2—44no—
520C2#1741411yescharacteristic
521C2#1751511yescharacteristic
522C2#1751911yescharacteristic
523C2C2×#150411yescharacteristic
524C2#1760611yescharacteristic
525C2#1765311yescharacteristic
526C2#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. #7711C2semidirect
5. #7691C2semidirect
6. #7685C2semidirect
7. #7668C2semidirect
8. #7634C2semidirect
9. #7617C2semidirect
10. #7589C2semidirect
11. #7566C2semidirect
279. C2×Q8#131semidirect

Maximal subgroups

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

subgroupindex
#365052
#225322
#203362
#77112
#76912
#76852
#76682
#76342
#76172
#75892
#75662