Finite Group Databasebeta

← #7231137

Subgroups — #7231137

Order 512.

1405 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic
2#36508C211yesmaximal, characteristic
3#33073C212yesmaximal, complemented
4#32820C212yesmaximal, complemented
5#32683C212yesmaximal, complemented
6#32037C212yesmaximal
7#31512C214yesmaximal, complemented
8#30633C214yesmaximal, complemented
9#27045C214yesmaximal, complemented
10#26355C214yesmaximal, complemented
11#25651C212yesmaximal, complemented
12#24108C212yesmaximal
13#23347C211yesmaximal, characteristic
14C2×#557C2211yescharacteristic
15C2×#556C2211yescharacteristic
16C2×#487C2212yes—
17C2×#483C2212yes—
18C2×#477C2212yes—
19C2×#428C2212yes—
20#2862C2211yescharacteristic
21#2861—21no—
22#2859C2212yes—
23#2844—21no—
24#2809—21no—
25#2720—24no—
26#2683—21no—
27#2567C2212yes—
28#2565—21no—
29#2546C2211yescharacteristic
30#2545—21no—
31#2542C2211yescharacteristic
32#2519—24no—
33#2497C2214yes—
34#2487C2212yes—
35#2440C2214yes—
36#2429C2214yes—
37#2379—21no—
38#2362C2214yes—
39C4×C4⋊Q8—21no—
40C4×#124C2211yescharacteristic
41C42×Q8C2211yescharacteristic
42#2096C2214yes—
43#2088C2214yes—
44#2072C2214yes—
45#2066C2214yes—
46#2066C2214yes—
47#2058C2214yes—
48#2058C2214yes—
49#2043C2214yes—
50#2043C2214yes—
51#2035C2214yes—
52#2035C2214yes—
53#2035C2214yes—
54#1984C2212yes—
55#1981C2212yes—
56#1948C2212yes—
57#1947C2212yes—
58#1941C2214yes—
59#1923C2214yes—
60#1920C2214yes—
61#1919C2214yes—
62#1918C2214yes—
63#1894C2212yes—
64#1894C2214yes—
65#1872C2214yes—
66#1870C2212yes—
67#1865C2214yes—
68#1863C2214yes—
69#1863C2214yes—
70#1854C2214yes—
71#1851C2214yes—
72#1831C2212yes—
73#1825C2212yes—
74#1803C2212yes—
75#1803C2212yes—
76C4×#103C2212yes—
77C22×M16C2312yes—
78#557—42no—
79#557—42no—
80#556—42no—
81#556—42no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#126—22no—
88C2×#125—21no—
89C2×#125—22no—
90C2×#125—22no—
91C2×#124—21no—
92C2×#124D411yescharacteristic
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8C2312yes—
95C2×C22⋊Q8C2314yes—
96C2×C4×Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8C2311yescharacteristic
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2311yescharacteristic
102C2×C4×D4C2311yescharacteristic
103C2×#117C2312yes—
104C22×C4⋊C4C2312yes—
105#487—44no—
106#483—44no—
107#477—44no—
108#428—44no—
109C2×#107C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112C2×#106C2314yes—
113C2×C4⋊C8C2312yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×C22⋊C8C2312yes—
123C2×C22⋊C8C2314yes—
124C2×C4×C8C2312yes—
125#400—22no—
126#400—24no—
127#399—22no—
128#399—22no—
129#398—22no—
130#398—24no—
131#398—24no—
132#396—24no—
133#396—24no—
134#395—22no—
135#395—22no—
136#395—24no—
137#391—22no—
138#391—21no—
139#391—21no—
140#391—21no—
141#391D411yescharacteristic
142#391C2314yes—
143#390—21no—
144#390D411yescharacteristic
145#389—22no—
146#389—22no—
147#389—24no—
148#389C2314yes—
149#389C2314yes—
150#389C2314yes—
151#389C2314yes—
152#388—24no—
153#388—24no—
154#388—24no—
155#388—24no—
156#388D414yes—
157#387—24no—
158#387—24no—
159#386—22no—
160#385—24no—
161#385—22no—
162#384—21no—
163#384—22no—
164#384C2312yes—
165#384C2312yes—
166#382—22no—
167#382—22no—
168#382D411yescharacteristic
169#380C2312yes—
170#380C2314yes—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—24no—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2311yescharacteristic
177C4×C22⋊C4C2314yes—
178C43—21no—
179C43C2311yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C22×Q8—21no—
190C22×Q8—22no—
191C22×Q8C2×D411yescharacteristic
192C22×Q8C2412yes—
193C22×Q8C2411yescharacteristic
194C22×D4C2×D411yescharacteristic
195C23×C4—22no—
196C23×C4C2412yes—
197C2×M16—44no—
198C2×M16—42no—
199C22×C8C2414yes—
200C22×C8C2414yes—
201C4⋊Q8—24no—
202C4⋊Q8—41no—
203C4⋊Q8—41no—
204C4⋊Q8—42no—
205C4⋊Q8—42no—
206C4⋊Q8—42no—
207C4⋊Q8—41no—
208C4⋊Q8—41no—
209C4⋊Q8—44no—
210#126—48no—
211#125—44no—
212#125—48no—
213#125—84no—
214#124—24no—
215#124—44no—
216C22⋊Q8—24no—
217C22⋊Q8—28no—
218C22⋊Q8—44no—
219C22⋊Q8—42no—
220C22⋊Q8—88no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—24no—
224C4×Q8—24no—
225C4×Q8—41no—
226C4×Q8—41no—
227C4×Q8—42no—
228C4×Q8—42no—
229C4×Q8—44no—
230C4×Q8—44no—
231C4×D4—41no—
232C4×D4—41no—
233#117—24no—
234#117—42no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2411yescharacteristic
264C2×C4⋊C4C2412yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4C2×D414yes—
271C2×C22⋊C4C2×D414yes—
272C2×C22⋊C4C2×D412yes—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D412yes—
282C2×C42C2×D411yescharacteristic
283C2×C42C2411yescharacteristic
284C2×C42C2412yes—
285C2×C42C2412yes—
286#107—28no—
287#106—28no—
288#106—44no—
289#106—44no—
290C4⋊C8—42no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293#103—28no—
294#103—44no—
295#103—44no—
296#103—44no—
297#103—44no—
298#103—44no—
299C22⋊C8—44no—
300C22⋊C8—42no—
301C4×C8—42no—
302#95—22no—
303#95—21no—
304#95—24no—
305#95—22no—
306#95—24no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95C2×D414yes—
314#95C2×D414yes—
315C24C22×D412yes—
316C2×Q8—21no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—84no—
329C2×Q8—82no—
330C2×Q8#13714yes—
331C2×Q8H(5,𝔽2)12yes—
332C2×D4—22no—
333C2×D4—41no—
334C2×D4—41no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—82no—
347C22×C4—84no—
348C22×C4C22×D412yes—
349C22×C4C22×D412yes—
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C2511yescharacteristic
356M16—48no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C4⋊C4—24no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—48no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—48no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—84no—
390C4⋊C4—82no—
391C4⋊C4—84no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4H(5,𝔽2)12yes—
395C4⋊C42−1+414yes—
396C22⋊C4—28no—
397C22⋊C4—28no—
398C22⋊C4—42no—
399C22⋊C4—44no—
400C22⋊C4—48no—
401C22⋊C4—44no—
402C22⋊C4—82no—
403C22⋊C4—84no—
404C42—24no—
405C42—22no—
406C42—24no—
407C42—22no—
408C42—22no—
409C42—21no—
410C42—21no—
411C42—22no—
412C42—22no—
413C42—41no—
414C42—44no—
415C42—44no—
416C42—44no—
417C42—42no—
418C42—44no—
419C42—82no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—22no—
424C23—24no—
425C23C23×D411yescharacteristic
426Q8—24no—
427Q8—44no—
428Q8—41no—
429Q8—41no—
430Q8—42no—
431Q8—42no—
432Q8—44no—
433Q8—84no—
434Q8—164no—
435D4—82no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—164no—
473C2×C4C2×#13712yes—
474C2×C4C2×#13714yes—
475C2×C4#57714yes—
476C2×C4C2×H(5,𝔽2)11yescharacteristic
477C2×C4C2×H(5,𝔽2)11yescharacteristic
478C2×C4C2×2−1+412yes—
479C2×C4#58512yes—
480C8—44no—
481C8—44no—
482C22—42no—
483C22—42no—
484C22—44no—
485C22—44no—
486C22—42no—
487C22#348411yescharacteristic
488C22#348912yes—
489C22C22×#13711yescharacteristic
490C22C2×#57712yes—
491C22#358611yescharacteristic
492C4—22no—
493C4—24no—
494C4—24no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—44no—
499C4—41no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—82no—
504C4—84no—
505C4—84no—
506C4#352911yescharacteristic
507C4#355011yescharacteristic
508C4#355912yes—
509C2—82no—
510C2#4622811yescharacteristic
511C2#6157311yescharacteristic
512C2#6166411yescharacteristic
513C2#6214712yes—
514C2#6216712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33073C2semidirect
4. #32820C2semidirect
5. #32683C2semidirect
7. #31512C2semidirect
8. #30633C2semidirect
9. #27045C2semidirect
10. #26355C2semidirect
11. #25651C2semidirect

Maximal subgroups

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

subgroupindex
#461522
#365082
#330732
#328202
#326832
#320372
#315122
#306332
#270452
#263552
#256512
#241082
#233472