Finite Group Databasebeta

← #6484843

Subgroups — #6484843

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic
3#33155C214yesmaximal
4#29979C214yesmaximal
5#29293C214yesmaximal, complemented
6#29282C214yesmaximal, complemented
7#29226C211yesmaximal, characteristic, complemented
8#29225C211yesmaximal, characteristic, complemented
9#29214C211yesmaximal, characteristic, complemented
10#29214C211yesmaximal, characteristic, complemented
11#29184C212yesmaximal, complemented
12#29183C212yesmaximal, complemented
13#21986C212yesmaximal
14#21839C212yesmaximal
15#20811C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2212yes—
18#2868C2211yescharacteristic
19#2859C2212yes—
20#2859C2211yescharacteristic
21#2629C2212yes—
22#2628C2212yes—
23#2570—22no—
24#2550—22no—
25#2441—22no—
26#2439—22no—
27#2368—22no—
28#2368—22no—
29#2355—22no—
30#2355—22no—
31#2331C2212yes—
32#2331C2212yes—
33#2331C2212yes—
34#2330C2212yes—
35#2317C2212yes—
36C4×#125C2211yescharacteristic
37C4×#125C2211yescharacteristic
38C4×#125C2211yescharacteristic
39C4×#125C2211yescharacteristic
40#2285C2212yes—
41C2×#382C2211yescharacteristic
42#2274C2212yes—
43C4×#117C2211yescharacteristic
44#2100C2214yes—
45#2099C2214yes—
46#2088C2214yes—
47#2088C2214yes—
48#2087C2214yes—
49#2086C2214yes—
50#2059C2214yes—
51#2059C2214yes—
52#1992C2212yes—
53#1992C2212yes—
54#1992C2212yes—
55#1992C2212yes—
56#1988C2214yes—
57#1988C2214yes—
58#1987C2214yes—
59#1987C2214yes—
60#1949C2214yes—
61#1949C2214yes—
62#1949C2214yes—
63#1949C2214yes—
64#1921C2214yes—
65#1920C2214yes—
66#1919C2214yes—
67#1918C2214yes—
68#1853C2214yes—
69#1852C2214yes—
70#1847C2212yes—
71#1847C2212yes—
72#1843C2212yes—
73#1836C2212yes—
74#1836C2212yes—
75#1832C2212yes—
76#1825C2214yes—
77#1825C2214yes—
78#1798C2212yes—
79#1798C2212yes—
80#1767C2212yes—
81#1767C2211yescharacteristic
82#1767C2211yescharacteristic
83#533—24no—
84#533Q818yescomplemented
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#125—22no—
89C2×#125C2311yescharacteristic
90C2×#125C2311yescharacteristic
91C2×#125C2312yes—
92C2×C22⋊Q8—22no—
93C2×C4×Q8—22no—
94C2×#117—22no—
95C2×#117C2312yes—
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C42C2311yescharacteristic
100#432—28no—
101C2×#107C2314yes—
102C2×#107C2314yes—
103C2×#106C2314yes—
104C2×#106C2314yes—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2314yes—
108C2×C4⋊C8C2314yes—
109C2×C22⋊C8C2314yes—
110C2×#97C2312yes—
111C2×#97C2312yes—
112#400—22no—
113#399—22no—
114#398—24no—
115#396—24no—
116#395—22no—
117#393—22no—
118#391—22no—
119#391—22no—
120#391—22no—
121#391—22no—
122#391—24no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—24no—
128#389C2312yes—
129#389C2312yes—
130#389C2312yes—
131#389C2312yes—
132#389C2314yes—
133#389C2314yes—
134#387—22no—
135#387—22no—
136#387—22no—
137#387—22no—
138#387—24no—
139#387—24no—
140#387—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#385—22no—
145#385—22no—
146#384—22no—
147#384—22no—
148#384C2312yes—
149#384C2312yes—
150#384C2312yes—
151#382—22no—
152#382—22no—
153#382—22no—
154#382—22no—
155#382C2311yescharacteristic
156#382C2311yescharacteristic
157#382C2312yes—
158#382C2312yes—
159#380—22no—
160#380C2314yes—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4D412yes—
169C4×C4⋊C4D412yes—
170C4×C4⋊C4C2312yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4—22no—
176C4×C22⋊C4—22no—
177C4×C22⋊C4D412yes—
178C4×C22⋊C4D412yes—
179#376C2311yescharacteristic
180#376C2311yescharacteristic
181C2×#95—22no—
182C43C2311yescharacteristic
183C43C2311yescharacteristic
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2312yes—
197#336C2312yes—
198#336C2314yes—
199C22×Q8—22no—
200C22×Q8—22no—
201C23×C4—22no—
202C23×C4C2×D412yes—
203C23×C4C2411yescharacteristic
204C22×C8C2414yes—
205C22×C8C2414yes—
206C4⋊Q8—24no—
207C4⋊Q8—28no—
208C4⋊Q8—84no—
209C4⋊Q8C2×Q818yes—
210#125—22no—
211#125—22no—
212#125—42no—
213#125—42no—
214#125—84no—
215#125C2×Q814yes—
216#125C2×Q814yes—
217C22⋊Q8—216no—
218C4×Q8—216no—
219#117—24no—
220#117—24no—
221#117—42no—
222#117—84no—
223#117C2×Q818yes—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4C2×D411yescharacteristic
254C2×C22⋊C4C2×D411yescharacteristic
255C2×C22⋊C4C2×D412yes—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×Q814yes—
268C2×C42#4114yes—
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2411yescharacteristic
275#107—44no—
276#107—44no—
277#106—44no—
278#106—44no—
279C4⋊C8—28no—
280C4⋊C8—24no—
281C4⋊C8—24no—
282C4⋊C8—44no—
283C22⋊C8—28no—
284#97—24no—
285#97—24no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—24no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—22no—
303#95—22no—
304#95C2×D412yes—
305#95C2×D412yes—
306#95C2×D412yes—
307#95C2×D412yes—
308C24C22×D411yescharacteristic
309C2×Q8—28no—
310C2×Q8—44no—
311C2×Q8—82no—
312C2×Q8—82no—
313C2×Q8—84no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4C22⋊Q818yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D412yes—
332C22×C4C22×D412yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D412yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×Q812yes—
338C22×C4C2×#4112yes—
339C22×C4C2511yescharacteristic
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C4⋊C4—28no—
347C4⋊C4—28no—
348C4⋊C4—28no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—28no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4C22×Q814yes—
374C4⋊C4C22×Q814yes—
375C22⋊C4—28no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—42no—
379C22⋊C4—84no—
380C42—28no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—82no—
389C42—82no—
390C42—82no—
391C42—82no—
392C42—84no—
393C42C22⋊Q818yes—
394C42C22×Q812yes—
395C42C22×Q812yes—
396C42C2×#4112yes—
397C42C2×#4112yes—
398C422−1+414yes—
399C23—22no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23C2×C22⋊Q812yes—
404C23C23×D411yescharacteristic
405Q8—88no—
406Q8—164no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—48no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—164no—
447C2×C4C2×C22⋊Q812yes—
448C2×C4C2×C22⋊Q814yes—
449C2×C4C2×C22⋊Q814yes—
450C2×C4#53612yes—
451C2×C4#53612yes—
452C2×C4C23×Q811yescharacteristic
453C2×C4C22×#4111yescharacteristic
454C2×C4C2×2−1+412yes—
455C2×C4#58512yes—
456C8—44no—
457C8—44no—
458C22—24no—
459C22—22no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22C22×C22⋊Q811yescharacteristic
464C22C2×#53611yescharacteristic
465C22#344911yescharacteristic
466C22#358512yes—
467C22#358612yes—
468C4—24no—
469C4—24no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—44no—
474C4—44no—
475C4—42no—
476C4—42no—
477C4—41no—
478C4—41no—
479C4—84no—
480C4—82no—
481C4—82no—
482C4—84no—
483C4#350212yes—
484C4#350712yes—
485C2—42no—
486C2#3517911yescharacteristic
487C2#6049312yes—
488C2#6088311yescharacteristic
489C2#6088411yescharacteristic
490C2#6101612yes—
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
5. #29293C2semidirect
6. #29282C2semidirect
7. #29226C2semidirect
8. #29225C2semidirect
9. #29214C2semidirect
10. #29214C2semidirect
11. #29184C2semidirect
12. #29183C2semidirect
84. #533Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#342152
#331552
#299792
#292932
#292822
#292262
#292252
#292142
#291842
#291832
#219862
#218392
#208112