Finite Group Databasebeta

← #6479898

Subgroups — #6479898

Order 512.

1456 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35287C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#33156C212yesmaximal, complemented
4#33116C212yesmaximal
5#29906C212yesmaximal, complemented
6#29644C212yesmaximal, complemented
7#29518C212yesmaximal, complemented
8#29514C212yesmaximal, complemented
9#29404C214yesmaximal, complemented
10#29360C212yesmaximal, complemented
11#29326C212yesmaximal, complemented
12#29280C212yesmaximal, complemented
13#29254C212yesmaximal, complemented
14#22088C214yesmaximal, complemented
15#20780C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#482C2214yes—
18#2860C2212yes—
19#2854C2212yes—
20#2647C2211yescharacteristic
21#2641C2211yescharacteristic
22#2629C2211yescharacteristic
23#2619C2211yescharacteristic
24#2593—22no—
25#2554—22no—
26#2468—22no—
27#2457—22no—
28#2360—22no—
29#2360—22no—
30#2355—22no—
31Q8×C4⋊C4—22no—
32#2330C2212yes—
33#2330C2212yes—
34#2324C2212yes—
35#2319C2212yes—
36#2317C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×#127C2211yescharacteristic
40C4×#125C2211yescharacteristic
41C2×#384C2211yescharacteristic
42#2274C2212yes—
43C4×#117C2211yescharacteristic
44#2099C2212yes—
45#2099C2212yes—
46#2093C2212yes—
47#2093C2212yes—
48#2083C2214yes—
49#2071C2214yes—
50#2059C2214yes—
51#2058C2212yes—
52#2057C2212yes—
53#2025C2214yes—
54#2023C2212yes—
55#2023C2212yes—
56#1989C2214yes—
57#1982C2214yes—
58#1949C2214yes—
59#1947C2214yes—
60#1944C2214yes—
61#1942C2214yes—
62#1936C2214yes—
63#1934C2214yes—
64#1926C2214yes—
65#1925C2214yes—
66#1921C2214yes—
67#1921C2214yes—
68#1920C2214yes—
69#1918C2214yes—
70#1874C2214yes—
71#1862C2214yes—
72#1862C2214yes—
73#1852C2214yes—
74#1840C2212yes—
75#1831C2212yes—
76#1825C2214yes—
77#1817C2212yes—
78#1798C2212yes—
79#1768C2212yes—
80#1750C2212yes—
81#1745C2211yescharacteristic
82#1744C2211yescharacteristic
83#533—28no—
84C2×C4⋊Q8C2312yes—
85C2×#127C2311yescharacteristic
86C2×#126—22no—
87C2×#125C2311yescharacteristic
88C2×#125C2312yes—
89C2×#123—22no—
90C2×C22⋊Q8—22no—
91C2×#121C2312yes—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×D4—22no—
96C2×C4×D4C2312yes—
97C2×#117C2312yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C22×C4⋊C4—22no—
101C22×C4⋊C4C2312yes—
102C22×C42C2311yescharacteristic
103#482—216no—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#107C2314yes—
107C2×#106C2314yes—
108C2×C4⋊C8C2314yes—
109C2×#102C2314yes—
110C2×#102C2314yes—
111C2×#102C2314yes—
112C2×#102C2314yes—
113C2×C22⋊C8C2314yes—
114C2×#97C2312yes—
115C2×C4×C8C2312yes—
116#400—22no—
117#398—22no—
118#398—22no—
119#396—22no—
120#394—22no—
121#394—22no—
122#394—22no—
123#392—22no—
124#391—22no—
125#391—22no—
126#390—22no—
127#390—22no—
128#390C2312yes—
129#390C2312yes—
130#390C2312yes—
131#390C2312yes—
132#389—22no—
133#389—22no—
134#389—24no—
135#389—24no—
136#389C2312yes—
137#389C2314yes—
138#389C2314yes—
139#389C2312yes—
140#389C2312yes—
141#389C2312yes—
142#388—22no—
143#388—22no—
144#388—22no—
145#388—22no—
146#388—22no—
147#388—22no—
148#388—22no—
149#388—22no—
150#387—22no—
151#387—24no—
152#387—24no—
153#386—22no—
154#386—22no—
155#385—22no—
156#385—22no—
157#385—22no—
158#384—22no—
159#384—22no—
160#384C2311yescharacteristic
161#384C2312yes—
162#384C2312yes—
163#382—22no—
164#382C2311yescharacteristic
165#381—22no—
166#380—22no—
167#380C2312yes—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2314yes—
177C4×C22⋊C4—24no—
178C4×C22⋊C4D412yes—
179C4×C22⋊C4D412yes—
180#376C2311yescharacteristic
181#376C2311yescharacteristic
182C43C2311yescharacteristic
183C43C2311yescharacteristic
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2312yes—
193#336C2312yes—
194#336C2312yes—
195#336C2312yes—
196#336C2314yes—
197#336C2314yes—
198C22×Q8—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4C2412yes—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4C2×D412yes—
205C23×C4C2411yescharacteristic
206C22×C8C2414yes—
207C22×C8C2414yes—
208C4⋊Q8—28no—
209#127—24no—
210#126—84no—
211#125—28no—
212#125—24no—
213#123—84no—
214C22⋊Q8—216no—
215#121—28no—
216C4×Q8—216no—
217C4×Q8—84no—
218C4×Q8—84no—
219C4×D4—44no—
220C4×D4—84no—
221#117—24no—
222#117—24no—
223#117—42no—
224#117#4118yes—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—84no—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2414yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2414yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4C2×D412yes—
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2×D411yescharacteristic
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280#107—28no—
281#107—44no—
282#107—44no—
283#106—28no—
284C4⋊C8—28no—
285#102—28no—
286#102—28no—
287#102—44no—
288#102—44no—
289C22⋊C8—28no—
290#97—42no—
291C4×C8—42no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95C2×D412yes—
304#95C2×D412yes—
305#95C2×D412yes—
306#95C2×D412yes—
307C24—22no—
308C24C22×D411yescharacteristic
309C2×Q8—28no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4—82no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—84no—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D412yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D412yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D412yes—
344C22×C4C22×D412yes—
345C22×C4C2×#4114yes—
346C22×C4C2511yescharacteristic
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C4⋊C4—216no—
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—84no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4C2×#4118yes—
377C22⋊C4—28no—
378C22⋊C4—24no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C42—24no—
393C42—28no—
394C42—24no—
395C42—24no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—42no—
402C42—42no—
403C42—82no—
404C42—84no—
405C42—84no—
406C422−1+414yes—
407C23—22no—
408C23—21no—
409C23—21no—
410C23—22no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23#52512yes—
419C23C23×D411yescharacteristic
420Q8—416no—
421D4—48no—
422D4—88no—
423D4—164no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—28no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—28no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—164no—
462C2×C4—164no—
463C2×C4#52512yes—
464C2×C4#53614yes—
465C2×C4#57614yes—
466C2×C4C22×#4112yes—
467C2×C4C2×2−1+412yes—
468C2×C4#58512yes—
469C8—44no—
470C8—44no—
471C22—22no—
472C22—24no—
473C22—41no—
474C22—41no—
475C22—42no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22C2×#52511yescharacteristic
484C22C2×#53611yescharacteristic
485C22#344911yescharacteristic
486C22C2×#57612yes—
487C22#358612yes—
488C4—22no—
489C4—24no—
490C4—24no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—42no—
496C4—48no—
497C4—42no—
498C4—82no—
499C4—84no—
500C4—82no—
501C4—82no—
502C4#350714yes—
503C2—42no—
504C2—162no—
505C2#3511111yescharacteristic
506C2#6058412yes—
507C2#6088812yes—
508C2#6101111yescharacteristic
509C2#6102011yescharacteristic
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. #34214C2semidirect
3. #33156C2semidirect
5. #29906C2semidirect
6. #29644C2semidirect
7. #29518C2semidirect
8. #29514C2semidirect
9. #29404C2semidirect
10. #29360C2semidirect
11. #29326C2semidirect
12. #29280C2semidirect
13. #29254C2semidirect
14. #22088C2semidirect
15. #20780C2semidirect

Maximal subgroups

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

subgroupindex
#352872
#342142
#331562
#331162
#299062
#296442
#295182
#295142
#294042
#293602
#293262
#292802
#292542
#220882
#207802