Finite Group Databasebeta

← #6490608

Subgroups — #6490608

Order 512.

1536 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34621C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#33157C212yesmaximal, complemented
4#33116C212yesmaximal
5#30157C214yesmaximal, complemented
6#29358C212yesmaximal, complemented
7#29323C212yesmaximal, complemented
8#29289C212yesmaximal, complemented
9#28280C212yesmaximal, complemented
10#27646C212yesmaximal, complemented
11#24617C214yesmaximal, complemented
12#24299C212yesmaximal, complemented
13#22117C212yesmaximal, complemented
14#22111C212yesmaximal, complemented
15#20812C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#482C2214yes—
18C2×#476C2212yes—
19C2×#476C2212yes—
20C2×#433C2212yes—
21#2651—22no—
22#2648—22no—
23#2647C2211yescharacteristic
24#2644C2212yes—
25#2641C2211yescharacteristic
26#2629C2211yescharacteristic
27#2624C2212yes—
28#2619C2211yescharacteristic
29#2395—22no—
30#2392—22no—
31#2369—22no—
32#2365—22no—
33#2345—22no—
34Q8×C22⋊C4—22no—
35#2331C2212yes—
36#2331C2212yes—
37#2316C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×#127C2211yescharacteristic
41C4×#125C2211yescharacteristic
42C4×#121C2212yes—
43#2293C2212yes—
44#2285C2211yescharacteristic
45#2274C2212yes—
46C4×#117C2211yescharacteristic
47#2088C2214yes—
48#2086C2214yes—
49#2085C2214yes—
50#2073C2214yes—
51#2065C2214yes—
52#2065C2214yes—
53#2058C2214yes—
54#2023C2214yes—
55#1989C2214yes—
56#1987C2212yes—
57#1987C2212yes—
58#1982C2214yes—
59#1949C2214yes—
60#1948C2212yes—
61#1948C2214yes—
62#1947C2212yes—
63#1944C2214yes—
64#1941C2214yes—
65#1936C2214yes—
66#1934C2214yes—
67#1926C2214yes—
68#1925C2214yes—
69#1920C2214yes—
70#1919C2214yes—
71#1874C2214yes—
72#1853C2214yes—
73#1842C2212yes—
74#1832C2212yes—
75#1825C2214yes—
76#1802C2212yes—
77#1802C2212yes—
78#1798C2212yes—
79#1797C2212yes—
80#1768C2212yes—
81#1767C2211yescharacteristic
82C4×C4⋊C8C2211yescharacteristic
83C23×Q8—21no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#127C2311yescharacteristic
90C2×#125C2311yescharacteristic
91C2×#121C2312yes—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4C2312yes—
97C2×#117—22no—
98C2×#117C2312yes—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2312yes—
103C22×C42C2311yescharacteristic
104#482—48no—
105#476—28no—
106#476—28no—
107#433—44no—
108C2×#107C2314yes—
109C2×#106C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112C2×C4⋊C8C2312yes—
113C2×C4⋊C8C2312yes—
114C2×C4⋊C8C2314yes—
115C2×C4⋊C8C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2312yes—
122C2×C4×C8C2312yes—
123#400—22no—
124#399—22no—
125#398—22no—
126#398—22no—
127#397—22no—
128#397—22no—
129#397—22no—
130#397—22no—
131#396—22no—
132#396—22no—
133#394—22no—
134#394—22no—
135#394—22no—
136#394—22no—
137#394—22no—
138#392—22no—
139#391—22no—
140#391—22no—
141#391—24no—
142#391—24no—
143#390—22no—
144#390C2312yes—
145#390C2312yes—
146#390C2312yes—
147#390C2312yes—
148#389—24no—
149#389—22no—
150#389—22no—
151#389C2312yes—
152#389C2314yes—
153#389C2314yes—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#387—22no—
161#387—24no—
162#387—24no—
163#386—22no—
164#384C2311yescharacteristic
165#384C2312yes—
166#382C2311yescharacteristic
167#382C2312yes—
168#380—21no—
169#380—21no—
170#380—24no—
171#380C2312yes—
172C4×C4⋊C4—24no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4D412yes—
179C4×C4⋊C4D412yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C4×C22⋊C4—24no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4D412yes—
188#376C2311yescharacteristic
189#376C2311yescharacteristic
190C43C2311yescharacteristic
191C43C2311yescharacteristic
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2312yes—
202#336C2312yes—
203C22×Q8—28no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—84no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4C2412yes—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2411yescharacteristic
215C22×C8C2414yes—
216C22×C8C2414yes—
217C4⋊Q8—28no—
218C4⋊Q8—28no—
219C4⋊Q8—28no—
220C4⋊Q8—24no—
221C4⋊Q8#4118yes—
222#127—24no—
223#125—24no—
224#121—24no—
225#121—24no—
226C4×Q8—84no—
227C4×Q8—84no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—84no—
231C4×D4—84no—
232#117—22no—
233#117—22no—
234#117—24no—
235#117—24no—
236#117—24no—
237#117—84no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2414yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2414yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
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⋊C4C2×D412yes—
273C2×C22⋊C4C2×D412yes—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291#107—44no—
292#106—28no—
293#106—28no—
294#106—44no—
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—28no—
298C4⋊C8—44no—
299#102—28no—
300#102—28no—
301#102—44no—
302#102—44no—
303C22⋊C8—28no—
304#97—42no—
305C4×C8—42no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95C2×D412yes—
320#95C2×D412yes—
321#95C2×D412yes—
322#95C2×D412yes—
323C24—22no—
324C24C22×D411yescharacteristic
325C2×Q8—28no—
326C2×Q8—28no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—84no—
330C2×Q8—816no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—82no—
340C2×D4—82no—
341C2×D4—82no—
342C2×D4—82no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—84no—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D412yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4H(5,𝔽2)12yes—
369C22×C42−1+412yes—
370C22×C4C2511yescharacteristic
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C4⋊C4—28no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4C2×#4118yes—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C42—22no—
416C42—22no—
417C42—24no—
418C42—24no—
419C42—22no—
420C42—21no—
421C42—21no—
422C42—22no—
423C42—21no—
424C42—21no—
425C42—42no—
426C42—42no—
427C42—44no—
428C42—44no—
429C42—84no—
430C42—84no—
431C42C2×#4114yes—
432C422−1+414yes—
433C23—22no—
434C23—21no—
435C23—21no—
436C23—22no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23#53712yes—
445C23C23×D411yescharacteristic
446Q8—416no—
447Q8—88no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—88no—
453D4—88no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—28no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—88no—
492C2×C4—88no—
493C2×C4#52512yes—
494C2×C4#52512yes—
495C2×C4#53614yes—
496C2×C4#53712yes—
497C2×C4#57714yes—
498C2×C4C22×#4112yes—
499C2×C4C2×H(5,𝔽2)11yescharacteristic
500C2×C4C2×2−1+412yes—
501C2×C4C2×2−1+411yescharacteristic
502C8—44no—
503C8—44no—
504C22—24no—
505C22—22no—
506C22—41no—
507C22—41no—
508C22—42no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22C2×#52511yescharacteristic
517C22C2×#53611yescharacteristic
518C22C2×#53711yescharacteristic
519C22C2×#57712yes—
520C22#358512yes—
521C4—22no—
522C4—22no—
523C4—24no—
524C4—42no—
525C4—42no—
526C4—41no—
527C4—41no—
528C4—48no—
529C4—42no—
530C4—84no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4#312614yes—
535C4#352212yes—
536C4#352312yes—
537C2—42no—
538C2—162no—
539C2#3524711yescharacteristic
540C2C2×#312611yescharacteristic
541C2#6056411yescharacteristic
542C2#6089212yes—
543C2#6092912yes—
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. #33157C2semidirect
5. #30157C2semidirect
6. #29358C2semidirect
7. #29323C2semidirect
8. #29289C2semidirect
9. #28280C2semidirect
10. #27646C2semidirect
11. #24617C2semidirect
12. #24299C2semidirect
13. #22117C2semidirect
14. #22111C2semidirect
15. #20812C2semidirect

Maximal subgroups

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

subgroupindex
#346212
#342162
#331572
#331162
#301572
#293582
#293232
#292892
#282802
#276462
#246172
#242992
#221172
#221112
#208122