Finite Group Databasebeta

← #6806366

Subgroups — #6806366

Order 512.

1324 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36584C211yesmaximal, characteristic, complemented
2#36573C211yesmaximal, characteristic
3#33295C211yesmaximal, characteristic, complemented
4#31467C214yesmaximal, complemented
5#31463C214yesmaximal, complemented
6#30235C212yesmaximal, complemented
7#29214C211yesmaximal, characteristic, complemented
8#29184C212yesmaximal, complemented
9#28724C211yesmaximal, characteristic
10#28722C212yesmaximal
11#26125C214yesmaximal, complemented
12#25002C214yesmaximal, complemented
13#22226C212yesmaximal
14#21656C211yesmaximal, characteristic
15#21346C211yesmaximal, characteristic, complemented
16C2×#490C2211yescharacteristic
17C2×#490C2212yes—
18#2859C2212yes—
19#2859C2211yescharacteristic
20#2859C2211yescharacteristic
21#2859C2212yes—
22#2580—22no—
23#2580C2212yes—
24#2571C2212yes—
25#2571C2212yes—
26#2571C2212yes—
27#2509—22no—
28#2509—22no—
29#2509—22no—
30#2454—22no—
31#2454—22no—
32#2454C2214yes—
33#2351—22no—
34Q8×C4⋊C4C2212yes—
35#2349—22no—
36#2349C2212yes—
37C4×#127C2212yes—
38C4×#127C2211yescharacteristic
39C4×#125C2212yes—
40C4×#125C2211yescharacteristic
41C42×Q8C2211yescharacteristic
42#2101C2214yes—
43#2101C2214yes—
44#2100C2214yes—
45#2099C2214yes—
46#2088C2214yes—
47#2086C2214yes—
48#2073C2212yes—
49#2073C2212yes—
50#2071C2214yes—
51#2071C2214yes—
52#2059C2214yes—
53#1992C2212yes—
54#1992C2212yes—
55#1992C2212yes—
56#1992C2211yescharacteristic
57#1992C2212yes—
58#1991C2212yes—
59#1991C2212yes—
60#1985C2212yes—
61#1961C2211yescharacteristic
62#1949C2214yes—
63#1940C2212yes—
64#1934C2214yes—
65#1926C2214yes—
66#1925C2214yes—
67#1921C2214yes—
68#1920C2214yes—
69#1916C2214yes—
70#1864C2214yes—
71#1862C2214yes—
72#1847C2212yes—
73#1836C2211yescharacteristic
74#1836C2214yes—
75#1804C2214yes—
76#1804C2214yes—
77#1802C2214yes—
78#1767C2212yes—
79#1767C2211yescharacteristic
80#1762C2214yes—
81#1745C2212yes—
82C4×#97C2211yescharacteristic
83C2×#127C2311yescharacteristic
84C2×#125—22no—
85C2×#125C2312yes—
86C2×#125C2311yescharacteristic
87C2×#125C2312yes—
88C2×C4×Q8—22no—
89C2×C4×Q8D412yes—
90C2×C4×Q8C2311yescharacteristic
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×C4×D4—22no—
94C2×C4×D4C2312yes—
95C2×C4×D4C2314yes—
96#490—24no—
97#490—28no—
98C2×#107C2314yes—
99C2×#106C2314yes—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2314yes—
102C2×#102C2314yes—
103C2×#102C2314yes—
104C2×#102C2314yes—
105C2×#102C2314yes—
106C2×#97C2312yes—
107C2×#97C2311yescharacteristic
108C2×#97C2312yes—
109C2×#97C2311yescharacteristic
110C2×C4×C8C2312yes—
111#400—22no—
112#400—22no—
113#400—22no—
114#400—22no—
115#400—22no—
116#400—22no—
117#399—24no—
118#399—22no—
119#396—24no—
120#393—22no—
121#390—22no—
122#390—22no—
123#390—22no—
124#390C2314yes—
125#390C2312yes—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389C2312yes—
143#389C2312yes—
144#389C2312yes—
145#389C2314yes—
146#389C2314yes—
147#387—22no—
148#387—22no—
149#387—22no—
150#385—24no—
151#385—24no—
152#385—24no—
153#384C2311yescharacteristic
154#384C2312yes—
155#384C2312yes—
156#384C2312yes—
157#384C2312yes—
158#382—22no—
159#382—22no—
160#382—22no—
161#382—22no—
162#382C2311yescharacteristic
163#382C2312yes—
164#382C2312yes—
165#382C2311yescharacteristic
166#382C2311yescharacteristic
167#382C2312yes—
168#380—24no—
169#380—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2311yescharacteristic
180C4×C4⋊C4C2312yes—
181C2×#95—22no—
182C43C2311yescharacteristic
183C43C2312yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2312yes—
195#336C2314yes—
196#336C2314yes—
197#336C2314yes—
198C22×Q8C2×D411yescharacteristic
199C22×D4—22no—
200C22×D4C2412yes—
201C23×C4—24no—
202C23×C4—22no—
203C22×C8C2412yes—
204C22×C8C2414yes—
205C22×C8C2412yes—
206#127—22no—
207#127#4114yes—
208#125—24no—
209#125—22no—
210#125—24no—
211#125—22no—
212#125—42no—
213#125—42no—
214#125—84no—
215C4×Q8—24no—
216C4×Q8—44no—
217C4×Q8—48no—
218C4×D4—28no—
219C4×D4—48no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4—84no—
223C4×D4—84no—
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—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
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⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2412yes—
268C2×C42C2412yes—
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272#107—44no—
273#106—44no—
274C4⋊C8—24no—
275C4⋊C8—44no—
276#102—28no—
277#102—28no—
278#102—28no—
279#102—44no—
280#97—22no—
281#97—24no—
282#97—24no—
283#97—22no—
284C4×C8—42no—
285#95—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—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—22no—
302#95C2×D411yescharacteristic
303#95C2×D412yes—
304#95C2×D411yescharacteristic
305#95C2×D412yes—
306C24—22no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×D4—28no—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4C2×#4114yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—48no—
323C22×C4—44no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4—84no—
327C22×C4C22×D412yes—
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D412yes—
334C22×C4C22×D412yes—
335C22×C4C2511yescharacteristic
336C2×C8—24no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—24no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C22⋊C4—44no—
379C22⋊C4—48no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C42—24no—
384C42—24no—
385C42—22no—
386C42—24no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—44no—
393C42—44no—
394C42—44no—
395C42C2×#4112yes—
396C422−1+414yes—
397C422−1+412yes—
398C23—24no—
399C23—44no—
400C23—42no—
401C23—42no—
402C23—44no—
403C23C23×D411yescharacteristic
404Q8—28no—
405D4—28no—
406D4—416no—
407D4—44no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—164no—
444C2×C4—168no—
445C2×C4#53612yes—
446C2×C4#53612yes—
447C2×C4#54814yes—
448C2×C4#57514yes—
449C2×C4C22×#4111yescharacteristic
450C2×C4C2×2−1+412yes—
451C2×C4C2×2−1+411yescharacteristic
452C2×C4#58512yes—
453C8—42no—
454C8—44no—
455C8—42no—
456C22—44no—
457C22—48no—
458C22—84no—
459C22—82no—
460C22—82no—
461C22C2×#53611yescharacteristic
462C22C2×#54811yescharacteristic
463C22#350011yescharacteristic
464C22C2×#57512yes—
465C22#358512yes—
466C4—24no—
467C4—22no—
468C4—24no—
469C4—22no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—82no—
475C4—84no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4#351512yes—
481C4#355512yes—
482C2—84no—
483C2#3917311yescharacteristic
484C2#6086111yescharacteristic
485C2#6088311yescharacteristic
486C2#6131812yes—
487C2#6258012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36584C2semidirect
3. #33295C2semidirect
4. #31467C2semidirect
5. #31463C2semidirect
6. #30235C2semidirect
7. #29214C2semidirect
8. #29184C2semidirect
11. #26125C2semidirect
12. #25002C2semidirect
15. #21346C2semidirect

Maximal subgroups

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

subgroupindex
#365842
#365732
#332952
#314672
#314632
#302352
#292142
#291842
#287242
#287222
#261252
#250022
#222262
#216562
#213462