Finite Group Databasebeta

← #6436854

Subgroups — #6436854

Order 512.

1666 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#36645C211yesmaximal, characteristic, complemented
2#34026C211yesmaximal, characteristic
3#29999C212yesmaximal, complemented
4#29998C212yesmaximal, complemented
5#29983C212yesmaximal, complemented
6#29980C212yesmaximal, complemented
7#29971C214yesmaximal, complemented
8#29955C214yesmaximal
9#23783C214yesmaximal, complemented
10#23781C214yesmaximal, complemented
11#21364C212yesmaximal, complemented
12#20810C211yesmaximal, characteristic, complemented
13#20743C212yesmaximal
14C2×#533—21no—
15C2×C4×#41C2211yescharacteristic
16C2×#432C2212yes—
17#2868C2212yes—
18#2862—21no—
19#2859—21no—
20#2859C2212yes—
21#2857—22no—
22#2857C2214yes—
23C4×C4⋊Q8—22no—
24C4×C4⋊Q8C2214yes—
25#2298—22no—
26#2297C2212yes—
27#2296C2212yes—
28#2295C2211yescharacteristic
29#2295C2211yescharacteristic
30#2295C2212yes—
31#2293C2211yescharacteristic
32#2293C2211yescharacteristic
33#2293C2212yes—
34#2290—21no—
35#2290—21no—
36#2290—22no—
37#2285—21no—
38#2285C2212yes—
39C4×#117—21no—
40C4×#117—21no—
41C4×#117C2211yescharacteristic
42C2×C43C2211yescharacteristic
43#2099C2214yes—
44#2099C2214yes—
45#2096C2214yes—
46#2093C2214yes—
47#2088C2218yes—
48#2087C2218yes—
49#2068C2218yes—
50#2065C2218yes—
51#2058C2214yes—
52#2057C2214yes—
53#2024C2214yes—
54#2023C2214yes—
55#1948C2214yes—
56#1947C2214yes—
57#1854C2218yes—
58#1842C2212yes—
59#1831C2212yes—
60#1825C2214yes—
61#1803C2214yes—
62#1802C2214yes—
63#1797C2214yes—
64#1788C2214yes—
65#1776C2218yes—
66#1767C2214yes—
67#1763C2214yes—
68#1762C2214yes—
69C22×#41C2311yescharacteristic
70#533—48no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8C2314yes—
74C2×#125—22no—
75C2×C22⋊Q8—24no—
76C4×#41—24no—
77C4×#41Q818yescomplemented
78C2×C4×Q8—24no—
79C2×C4×Q8C2311yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4C2311yescharacteristic
84C2×C4×D4C2311yescharacteristic
85C2×#117—22no—
86C2×#117—21no—
87C2×#117—22no—
88C2×#117—21no—
89C2×#117—21no—
90C2×#117C2311yescharacteristic
91C2×#117C2312yes—
92C22×C42—21no—
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D412yes—
96C22×C42C2312yes—
97C22×C42C2311yescharacteristic
98#432—28no—
99C2×#107C2318yes—
100C2×#106C2318yes—
101C2×C4⋊C8C2314yes—
102C2×C4⋊C8C2318yes—
103C2×#103C2318yes—
104C2×#102C2318yes—
105C2×C22⋊C8C2314yes—
106C2×#97C2314yes—
107#391—24no—
108#391—24no—
109#391C2312yes—
110#391C2312yes—
111#390C2312yes—
112#390C2312yes—
113#389—24no—
114#389—24no—
115#389—24no—
116#389C2318yes—
117#389C2312yes—
118#389C2312yes—
119#389C2312yes—
120#389C2312yes—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#387—22no—
126#387—22no—
127#387—22no—
128#387—22no—
129#386—22no—
130#386—22no—
131#385—22no—
132#385—22no—
133#384—22no—
134#384—22no—
135#384C2314yes—
136#384C2314yes—
137#382—22no—
138#382—22no—
139#380—22no—
140#380—24no—
141#380C2314yes—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4C2314yes—
150C4×C4⋊C4C2311yescharacteristic
151C4×C4⋊C4C2311yescharacteristic
152C4×C4⋊C4C2312yes—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—22no—
158C4×C22⋊C4—22no—
159C4×C22⋊C4—22no—
160#376—22no—
161#376—21no—
162#376—21no—
163#376—21no—
164C43—21no—
165C43D412yes—
166C43D411yescharacteristic
167C43D411yescharacteristic
168C43C2312yes—
169C43C2311yescharacteristic
170C43C2311yescharacteristic
171#340C2318yes—
172#340C2318yes—
173#340C2318yes—
174#340C2318yes—
175#336C2314yes—
176#336C2314yes—
177C2×#41—22no—
178C2×#41—44no—
179C2×#41C2×Q814yes—
180C22×Q8—24no—
181C22×Q8C2411yescharacteristic
182C22×D4—21no—
183C22×D4C2411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2411yescharacteristic
192C22×C8C2418yes—
193C4⋊Q8—48no—
194C4⋊Q8—48no—
195C4⋊Q8—48no—
196#125—48no—
197C22⋊Q8—416no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—416no—
201C4×Q8C2×Q814yes—
202C4×Q8C2×Q814yes—
203C4×D4—28no—
204C4×D4—28no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4C2×Q814yes—
208C4×D4C2×Q814yes—
209#117—22no—
210#117—28no—
211#117—48no—
212#117—44no—
213#117—48no—
214#117—44no—
215#117—44no—
216#117C2×Q814yes—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4C2412yes—
230C2×C4⋊C4C2412yes—
231C2×C4⋊C4C2418yes—
232C2×C4⋊C4C2411yescharacteristic
233C2×C4⋊C4C2411yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—28no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D412yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×Q814yes—
271C2×C42#4114yes—
272C2×C42C2412yes—
273C2×C42C2412yes—
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276#107—48no—
277#106—48no—
278C4⋊C8—28no—
279C4⋊C8—48no—
280#103—48no—
281#102—48no—
282C22⋊C8—28no—
283#97—28no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—21no—
294#95—21no—
295#95—28no—
296#95—24no—
297#95—24no—
298C24—21no—
299C24C22×D411yescharacteristic
300#41—48no—
301#41—48no—
302C2×Q8—21no—
303C2×Q8—24no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—48no—
307C2×Q8C22×Q812yes—
308C2×D4—24no—
309C2×D4—21no—
310C2×D4—24no—
311C2×D4—44no—
312C2×D4C22×Q812yes—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—28no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4C22⋊Q814yes—
337C22×C4C22⋊Q814yes—
338C22×C4C22×D412yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D412yes—
344C22×C4C22×D412yes—
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×Q812yes—
348C22×C4C2×#4112yes—
349C22×C4C2511yescharacteristic
350C2×C8—28no—
351C2×C8—28no—
352C2×C8—28no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—28no—
358C4⋊C4—28no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—44no—
362C4⋊C4—416no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—44no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4C22×Q812yes—
372C4⋊C4C22×Q812yes—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—48no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—22no—
387C42—24no—
388C42—22no—
389C42—24no—
390C42—24no—
391C42—24no—
392C42—24no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—44no—
398C42—44no—
399C42—48no—
400C42—42no—
401C42—42no—
402C42C22⋊Q814yes—
403C42C22⋊Q814yes—
404C42AGL(1,ℤ/8)14yes—
405C42#13714yes—
406C42C22×Q812yes—
407C42C22×Q812yes—
408C42C2×#4112yes—
409C42C2×#4112yes—
410C23—22no—
411C23—21no—
412C23—21no—
413C23—22no—
414C23—22no—
415C23—42no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23C2×C22⋊Q812yes—
420C23C23×D411yescharacteristic
421Q8—24no—
422Q8—42no—
423Q8—816no—
424D4—24no—
425D4—48no—
426D4—42no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—48no—
460C2×C4—48no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—48no—
466C2×C4C2×C22⋊Q812yes—
467C2×C4C2×C22⋊Q812yes—
468C2×C4C2×C22⋊Q812yes—
469C2×C4C2×C22⋊Q812yes—
470C2×C4C2×C22⋊Q812yes—
471C2×C4C2×AGL(1,ℤ/8)14yes—
472C2×C4C2×AGL(1,ℤ/8)12yes—
473C2×C4C2×#13714yes—
474C2×C4C2×#13712yes—
475C2×C4#57514yes—
476C2×C4C23×Q811yescharacteristic
477C2×C4C22×#4111yescharacteristic
478C2×C4#58514yes—
479C8—48no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22—82no—
490C22C22×C22⋊Q811yescharacteristic
491C22#344912yes—
492C22C22×AGL(1,ℤ/8)11yescharacteristic
493C22C22×#13711yescharacteristic
494C22C2×#57512yes—
495C4—24no—
496C4—24no—
497C4—24no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—82no—
507C4—82no—
508C4—88no—
509C4#307914yes—
510C4#345512yes—
511C4#345812yes—
512C2—42no—
513C2—82no—
514C2#3463411yescharacteristic
515C2C2×#307911yescharacteristic
516C2#6048411yescharacteristic
517C2#6096312yes—
518C2#6096512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #29999C2semidirect
4. #29998C2semidirect
5. #29983C2semidirect
6. #29980C2semidirect
7. #29971C2semidirect
9. #23783C2semidirect
10. #23781C2semidirect
11. #21364C2semidirect
12. #20810C2semidirect
77. C4×#41Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
#340262
#299992
#299982
#299832
#299802
#299712
#299552
#237832
#237812
#213642
#208102
#207432