Finite Group Databasebeta

← #6468465

Subgroups — #6468465

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34281C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#29294C212yesmaximal, complemented
4#29278C212yesmaximal, complemented
5#29259C212yesmaximal, complemented
6#29253C214yesmaximal
7#28294C212yesmaximal, complemented
8#27681C212yesmaximal
9#24306C212yesmaximal
10#23782C212yesmaximal, complemented
11#23777C212yesmaximal, complemented
12#21621C214yesmaximal, complemented
13#21398C212yesmaximal
14#21368C212yesmaximal
15#20813C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×#479C2214yes—
19C2×#433C2212yes—
20#2373—22no—
21#2373—22no—
22#2371—22no—
23#2370—22no—
24#2360—22no—
25#2360—22no—
26#2355—22no—
27Q8×C4⋊C4—22no—
28#2330C2211yescharacteristic
29#2330C2212yes—
30#2330C2212yes—
31#2327C2211yescharacteristic
32#2323C2212yes—
33#2317C2212yes—
34#2314C2211yescharacteristic
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2211yescharacteristic
38C4×#125C2211yescharacteristic
39C4×C22⋊Q8C2212yes—
40#2295C2212yes—
41C2×#384C2211yescharacteristic
42#2276C2212yes—
43#2274C2212yes—
44C4×#117C2211yescharacteristic
45#2100C2212yes—
46#2100C2212yes—
47#2095C2212yes—
48#2095C2212yes—
49#2087C2214yes—
50#2086C2214yes—
51#2068C2214yes—
52#2066C2214yes—
53#2058C2212yes—
54#2057C2212yes—
55#2024C2212yes—
56#2024C2212yes—
57#1989C2214yes—
58#1989C2214yes—
59#1981C2214yes—
60#1949C2214yes—
61#1947C2214yes—
62#1947C2214yes—
63#1919C2214yes—
64#1918C2214yes—
65#1873C2214yes—
66#1872C2214yes—
67#1863C2214yes—
68#1863C2214yes—
69#1854C2214yes—
70#1852C2214yes—
71#1843C2212yes—
72#1831C2212yes—
73#1803C2214yes—
74#1798C2212yes—
75#1797C2212yes—
76#1776C2214yes—
77C4×#107C2214yes—
78#1768C2212yes—
79#1767C2211yescharacteristic
80C4×C4⋊C8C2211yescharacteristic
81#1763C2214yes—
82C4×#103C2214yes—
83#541—24no—
84#541Q818yescomplemented
85#533—28no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×#126—22no—
89C2×#125—22no—
90C2×#125C2311yescharacteristic
91C2×#125C2312yes—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8C2312yes—
94C2×C4×Q8—22no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2312yes—
100C2×#117C2312yes—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C22×C4⋊C4—22no—
104C22×C4⋊C4C2312yes—
105C22×C42C2311yescharacteristic
106#479—216no—
107#433—28no—
108C2×#107C2314yes—
109C2×#107C2314yes—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×C4⋊C8C2314yes—
113C2×C4⋊C8C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2312yes—
122C2×C4×C8C2312yes—
123#391—22no—
124#391—22no—
125#391C2312yes—
126#391C2312yes—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—24no—
132#389—24no—
133#389—22no—
134#389C2312yes—
135#389C2314yes—
136#389C2314yes—
137#389C2312yes—
138#388—22no—
139#388—22no—
140#388—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—24no—
150#387—24no—
151#387—22no—
152#385—22no—
153#385—22no—
154#385—22no—
155#384—22no—
156#384—22no—
157#384C2312yes—
158#384C2311yescharacteristic
159#384C2312yes—
160#382—22no—
161#382C2311yescharacteristic
162#380C2312yes—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2314yes—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—24no—
182C4×C22⋊C4D412yes—
183C4×C22⋊C4D412yes—
184#376—22no—
185#376—22no—
186#376C2311yescharacteristic
187#376C2311yescharacteristic
188C43C2311yescharacteristic
189C43C2311yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2312yes—
199#336C2312yes—
200#336C2314yes—
201C22×Q8—22no—
202C22×Q8C2412yes—
203C23×C4—22no—
204C23×C4C2×D412yes—
205C23×C4C2411yescharacteristic
206C22×C8C2414yes—
207C22×C8C2414yes—
208C4⋊Q8—28no—
209C4⋊Q8—22no—
210C4⋊Q8C2×Q814yes—
211#126—216no—
212#125—216no—
213#125—24no—
214#125—22no—
215#125C2×Q814yes—
216#125#4118yes—
217C22⋊Q8—216no—
218C22⋊Q8—24no—
219C22⋊Q8C2×Q818yes—
220C4×Q8—216no—
221C4×Q8—24no—
222C4×Q8—28no—
223C4×Q8—44no—
224C4×Q8—84no—
225C4×Q8—84no—
226C4×Q8C2×Q818yes—
227#117—28no—
228#117—42no—
229#117C2×Q814yes—
230#117#4114yes—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—84no—
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2414yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2×D412yes—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2411yescharacteristic
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282C2×C42C2411yescharacteristic
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285#107—28no—
286#107—44no—
287#107—44no—
288#106—28no—
289C4⋊C8—28no—
290C4⋊C8—28no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293#103—28no—
294#103—28no—
295#103—44no—
296#103—44no—
297C22⋊C8—28no—
298#97—24no—
299C4×C8—24no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—22no—
315#95C2×D412yes—
316#95C2×D412yes—
317#95C2×D412yes—
318#95C2×D412yes—
319C24C22×D411yescharacteristic
320C2×Q8—28no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8C22×Q814yes—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—84no—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D412yes—
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×Q812yes—
350C22×C4C2×#4112yes—
351C22×C4C2511yescharacteristic
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C4⋊C4—28no—
359C4⋊C4—28no—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—28no—
363C4⋊C4—216no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4—84no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4C22⋊Q818yes—
384C4⋊C4C22×Q814yes—
385C4⋊C4C2×#4118yes—
386C22⋊C4—28no—
387C22⋊C4—28no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—82no—
391C22⋊C4C22⋊Q818yes—
392C42—28no—
393C42—28no—
394C42—24no—
395C42—24no—
396C42—24no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—42no—
401C42—42no—
402C42—44no—
403C42—44no—
404C42—82no—
405C42—82no—
406C42—82no—
407C42—84no—
408C42—84no—
409C42C22×Q812yes—
410C42C22×Q812yes—
411C42C2×#4112yes—
412C42C2×#4112yes—
413C42C2×#4114yes—
414C23—22no—
415C23—21no—
416C23—21no—
417C23—22no—
418C23C2×C22⋊Q812yes—
419C23C23×D411yescharacteristic
420Q8—28no—
421Q8—416no—
422Q8—44no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—164no—
463C2×C4C2×C22⋊Q814yes—
464C2×C4C2×C22⋊Q814yes—
465C2×C4C2×C22⋊Q812yes—
466C2×C4#52514yes—
467C2×C4#57814yes—
468C2×C4C23×Q811yescharacteristic
469C2×C4C22×#4112yes—
470C2×C4C22×#4111yescharacteristic
471C2×C4#58512yes—
472C8—44no—
473C8—44no—
474C22—24no—
475C22—22no—
476C22—41no—
477C22—41no—
478C22—42no—
479C22C22×C22⋊Q811yescharacteristic
480C22C2×#52511yescharacteristic
481C22#344911yescharacteristic
482C22C2×#57812yes—
483C22#358512yes—
484C4—24no—
485C4—24no—
486C4—22no—
487C4—24no—
488C4—24no—
489C4—44no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—44no—
495C4—48no—
496C4—82no—
497C4—84no—
498C4—82no—
499C4Q8×#4112yes—
500C4#348012yes—
501C2—42no—
502C2#3495211yescharacteristic
503C2#6049812yes—
504C2#6058912yes—
505C2#6101311yescharacteristic
506C2#6101911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29294C2semidirect
4. #29278C2semidirect
5. #29259C2semidirect
7. #28294C2semidirect
10. #23782C2semidirect
11. #23777C2semidirect
12. #21621C2semidirect
84. #541Q8semidirect

Maximal subgroups

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

subgroupindex
#342812
#342142
#292942
#292782
#292592
#292532
#282942
#276812
#243062
#237822
#237772
#216212
#213982
#213682
#208132