Finite Group Databasebeta

← #6490792

Subgroups — #6490792

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35331C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#33117C212yesmaximal, complemented
4#33116C212yesmaximal
5#30068C214yesmaximal, complemented
6#29264C212yesmaximal, complemented
7#29263C212yesmaximal, complemented
8#29211C212yesmaximal, complemented
9#28754C212yesmaximal, complemented
10#27915C212yesmaximal, complemented
11#25908C212yesmaximal, complemented
12#24075C214yesmaximal, complemented
13#22224C212yesmaximal, complemented
14#22223C212yesmaximal, complemented
15#20783C211yesmaximal, characteristic, complemented
16C2×#542C2212yes—
17C22×C4⋊Q8C2211yescharacteristic
18C2×#489C2212yes—
19C2×#489C2212yes—
20C2×#485C2214yes—
21C2×#482C2214yes—
22#2862C2211yescharacteristic
23#2858C2211yescharacteristic
24#2850C2212yes—
25#2640C2212yes—
26#2621C2211yescharacteristic
27#2619C2211yescharacteristic
28#2560—22no—
29#2560—22no—
30#2557—22no—
31#2557—22no—
32#2369—22no—
33#2365—22no—
34#2345—22no—
35Q8×C22⋊C4—22no—
36#2331C2212yes—
37#2331C2212yes—
38#2328C2211yescharacteristic
39#2327C2211yescharacteristic
40#2316C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×C4⋊Q8C2212yes—
43C4×#124C2212yes—
44#2285C2211yescharacteristic
45#2274C2212yes—
46C4×#117C2211yescharacteristic
47#2097C2214yes—
48#2092C2214yes—
49#2058C2214yes—
50#2057C2214yes—
51#2042C2214yes—
52#2042C2214yes—
53#2024C2214yes—
54#2023C2214yes—
55#1993C2212yes—
56#1993C2212yes—
57#1982C2214yes—
58#1981C2214yes—
59#1959C2212yes—
60#1959C2212yes—
61#1949C2214yes—
62#1949C2214yes—
63#1928C2214yes—
64#1927C2214yes—
65#1926C2214yes—
66#1925C2214yes—
67#1874C2214yes—
68#1872C2214yes—
69#1865C2214yes—
70#1864C2214yes—
71#1863C2214yes—
72#1862C2214yes—
73#1847C2212yes—
74#1847C2212yes—
75#1840C2212yes—
76#1832C2212yes—
77#1818C2212yes—
78#1798C2212yes—
79#1767C2212yes—
80#1749C2212yes—
81#1745C2211yescharacteristic
82#1745C2211yescharacteristic
83C23×Q8—21no—
84#542—48no—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×#127C2311yescharacteristic
93C2×#126—22no—
94C2×#124—22no—
95C2×#124C2312yes—
96C2×#124C2312yes—
97C2×#123—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8C2312yes—
100C2×#121C2312yes—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8C2312yes—
104C2×C4×D4C2312yes—
105C2×#117C2312yes—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C22×C4⋊C4—21no—
109C22×C4⋊C4C2312yes—
110C22×C42C2311yescharacteristic
111#489—28no—
112#489—28no—
113#485—48no—
114#482—48no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2312yes—
130C2×#97C2312yes—
131#398—22no—
132#397—22no—
133#397—22no—
134#397—22no—
135#394—22no—
136#393—22no—
137#393—22no—
138#392—22no—
139#391—22no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—24no—
144#391—24no—
145#391C2312yes—
146#391C2312yes—
147#390—22no—
148#390—22no—
149#390C2312yes—
150#390C2312yes—
151#389—24no—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#387—22no—
161#387—22no—
162#387—24no—
163#387—24no—
164#386—22no—
165#386—22no—
166#386—22no—
167#386—22no—
168#384C2312yes—
169#382C2311yescharacteristic
170#382C2311yescharacteristic
171#382C2312yes—
172#382C2312yes—
173#380—21no—
174#380—21no—
175#380—24no—
176C4×C4⋊C4—24no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4D412yes—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4C2314yes—
184C4×C4⋊C4C2314yes—
185C4×C22⋊C4—24no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4D412yes—
189C4×C22⋊C4D412yes—
190#376C2311yescharacteristic
191#376C2311yescharacteristic
192C43C2311yescharacteristic
193C43C2311yescharacteristic
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2312yes—
199#336C2312yes—
200#336C2314yes—
201#336C2312yes—
202#336C2312yes—
203#336C2314yes—
204C22×Q8—28no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—84no—
209C22×Q8C2412yes—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4C2412yes—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2411yescharacteristic
218C22×C8C2414yes—
219C22×C8C2414yes—
220C4⋊Q8—28no—
221C4⋊Q8—28no—
222C4⋊Q8—28no—
223C4⋊Q8—24no—
224C4⋊Q8—24no—
225C4⋊Q8—44no—
226C4⋊Q8#4118yes—
227#127—24no—
228#126—48no—
229#124—24no—
230#124—24no—
231#124—44no—
232#124—48no—
233#123—48no—
234C22⋊Q8—28no—
235C22⋊Q8—48no—
236#121—28no—
237C4×Q8—44no—
238C4×Q8—84no—
239C4×Q8—84no—
240C4×D4—44no—
241#117—24no—
242#117—24no—
243#117—24no—
244#117—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2414yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D412yes—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295#107—44no—
296#107—44no—
297#106—44no—
298#106—44no—
299C4⋊C8—28no—
300#103—28no—
301#103—28no—
302#103—44no—
303#103—44no—
304#102—28no—
305#102—28no—
306#102—44no—
307#102—44no—
308C22⋊C8—28no—
309#97—24no—
310#97—24no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95C2×D412yes—
326#95C2×D412yes—
327#95C2×D412yes—
328#95C2×D412yes—
329C24—22no—
330C24C22×D411yescharacteristic
331C2×Q8—28no—
332C2×Q8—28no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—84no—
341C2×Q8—82no—
342C2×Q8—82no—
343C2×Q8—816no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4—82no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—84no—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D412yes—
379C22×C4H(5,𝔽2)14yes—
380C22×C4C2511yescharacteristic
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C4⋊C4—28no—
388C4⋊C4—28no—
389C4⋊C4—28no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C4⋊C4C2×#4118yes—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—44no—
431C42—44no—
432C42—84no—
433C42—84no—
434C42C2×#4114yes—
435C422−1+414yes—
436C23—22no—
437C23—22no—
438C23—21no—
439C23—21no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23#53512yes—
448C23C23×D411yescharacteristic
449Q8—416no—
450Q8—44no—
451Q8—44no—
452Q8—88no—
453Q8—164no—
454D4—44no—
455D4—44no—
456D4—164no—
457D4—164no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—28no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—28no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—88no—
496C2×C4—164no—
497C2×C4#52512yes—
498C2×C4#52512yes—
499C2×C4#53512yes—
500C2×C4#53614yes—
501C2×C4C22×#4112yes—
502C2×C4C2×H(5,𝔽2)12yes—
503C2×C4C2×2−1+412yes—
504C8—44no—
505C8—44no—
506C22—24no—
507C22—22no—
508C22—41no—
509C22—41no—
510C22—42no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22C2×#52511yescharacteristic
519C22C2×#53511yescharacteristic
520C22C2×#53611yescharacteristic
521C22#358512yes—
522C22#358612yes—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—24no—
527C4—41no—
528C4—48no—
529C4—41no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—84no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4#352214yes—
538C2—42no—
539C2—162no—
540C2#3524911yescharacteristic
541C2#6056411yescharacteristic
542C2#6056611yescharacteristic
543C2#6082912yes—
544C2#6088512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
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
2. #34216C2semidirect
3. #33117C2semidirect
5. #30068C2semidirect
6. #29264C2semidirect
7. #29263C2semidirect
8. #29211C2semidirect
9. #28754C2semidirect
10. #27915C2semidirect
11. #25908C2semidirect
12. #24075C2semidirect
13. #22224C2semidirect
14. #22223C2semidirect
15. #20783C2semidirect

Maximal subgroups

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

subgroupindex
#353312
#342162
#331172
#331162
#300682
#292642
#292632
#292112
#287542
#279152
#259082
#240752
#222242
#222232
#207832