Finite Group Databasebeta

← #6506562

Subgroups — #6506562

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35011C211yesmaximal, characteristic
2C2×#2441C211yesmaximal, characteristic
3#29335C212yesmaximal, complemented
4#29329C212yesmaximal, complemented
5#27972C212yesmaximal, complemented
6#27707C218yesmaximal, complemented
7#27283C218yesmaximal, complemented
8#25912C212yesmaximal, complemented
9#21765C212yesmaximal
10#20911C212yesmaximal
11#20524C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#117C2211yescharacteristic
15C2×#491C2214yescomplemented
16C2×#419C2214yes—
17C2×#407C2212yes—
18#2857C2212yes—
19#2857C2212yes—
20#2659—21no—
21#2659—21no—
22#2583—24no—
23#2502C2214yes—
24#2441—24no—
25#2441C2218yes—
26#2434C2214yes—
27C2×#399—21no—
28C2×#396—21no—
29#2315—21no—
30#2315—21no—
31C2×#389—21no—
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#387—21no—
35C2×#384C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#2100C2214yescomplemented
39#2099C2214yescomplemented
40#2072C2218yescomplemented
41#2043C2214yescomplemented
42#2043C2214yescomplemented
43#2026C2218yescomplemented
44#1992C2214yescomplemented
45#1956C2214yescomplemented
46#1935C2218yescomplemented
47#1933C2218yescomplemented
48#1894C2218yes—
49#1873C2218yes—
50#1872C2214yes—
51#1872C2214yes—
52#1872C2214yes—
53#1872C2214yes—
54#1834C2214yescomplemented
55#1823C2214yes—
56#1787C2214yes—
57#1753C2218yes—
58#1749C2212yes—
59C4×C22⋊C8C2212yes—
60#1736C2214yes—
61C23×Q8C2312yes—
62C24×C4C2311yescharacteristic
63C22×M16C2314yes—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×#125—22no—
67C2×#125—44no—
68C2×#125C2314yes—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8C2314yes—
74C2×C22⋊Q8C2314yes—
75C2×#117—22no—
76C2×#117—44no—
77C2×#117C2314yes—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—22no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4D411yescharacteristic
94C22×C22⋊C4D411yescharacteristic
95C22×C42—21no—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#491—48no—
99#419—48no—
100C2×#103C2318yes—
101C2×#103C2318yes—
102C2×#103C2318yes—
103C2×#103C2318yes—
104#407—28no—
105C2×C22⋊C8C2314yes—
106C2×C22⋊C8C2314yes—
107C2×#97C2314yes—
108C2×C4×C8C2314yes—
109#399—28no—
110#398—24no—
111#398—24no—
112#396—28no—
113#395—24no—
114#395—24no—
115#391C2314yes—
116#391C2314yes—
117#389—22no—
118#389—22no—
119#389—22no—
120#389—22no—
121#389—22no—
122#389—22no—
123#389—28no—
124#389C2314yes—
125#389C2314yes—
126#387—28no—
127#385—24no—
128#385—24no—
129#385—24no—
130#385—24no—
131#384—22no—
132#384C2314yes—
133#384C2312yes—
134#384C2312yes—
135#380—22no—
136#380—22no—
137#380—22no—
138#380—22no—
139#380—22no—
140#380C2314yes—
141C4×C22⋊C4D414yes—
142C4×C22⋊C4C2314yes—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95D411yescharacteristic
147C2×#95D411yescharacteristic
148#340C2318yes—
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#336C2314yes—
153#336C2314yes—
154C25C2×D411yescharacteristic
155C22×Q8—22no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8C2414yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4#4118yes—
176C23×C4C2411yescharacteristic
177C23×C4C2412yes—
178C2×M16—28no—
179C2×M16—44no—
180C2×M16—44no—
181C22×C8C2418yes—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184#125—28no—
185#125—44no—
186#125—48no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—28no—
190C22⋊Q8—28no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193#117—28no—
194#117—44no—
195#117—48no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C2412yes—
231C2×C4⋊C4C2414yes—
232C2×C4⋊C4C2412yes—
233C2×C4⋊C4C2412yes—
234C2×C4⋊C4C2412yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4C2×D414yes—
245C2×C22⋊C4C2×D414yes—
246C2×C42—24no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—44no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2412yes—
253C2×C42C2412yes—
254#103—216no—
255#103—48no—
256#103—48no—
257#103—48no—
258C22⋊C8—28no—
259C22⋊C8—28no—
260#97—44no—
261C4×C8—44no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95C2×D414yes—
270#95C2×D414yes—
271C24—22no—
272C24—22no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24#12414yes—
278C24C22×D412yes—
279C24C22×D411yescharacteristic
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—28no—
283C2×Q8—28no—
284C2×Q8—416no—
285C2×Q8—44no—
286C2×Q8—82no—
287C2×Q8—82no—
288C2×Q8—88no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—416no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4#12418yes—
333C22×C4#12414yes—
334C22×C4#13714yes—
335C22×C4#13714yes—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D412yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C2×#4114yes—
343C22×C4C2×#4114yes—
344C22×C4C2×#4114yes—
345C22×C4C2511yescharacteristic
346M16—88no—
347C2×C8—28no—
348C2×C8—28no—
349C2×C8—28no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—48no—
358C4⋊C4—48no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4H(5,𝔽2)14yes—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—84no—
384C42—24no—
385C42—24no—
386C42—42no—
387C42—42no—
388C42—48no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—28no—
400C23—44no—
401C23—44no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23C2×#12414yes—
407C23C2×#12412yes—
408C23C2×#13712yes—
409C23C2×#13712yes—
410C23C23×D411yescharacteristic
411Q8—48no—
412Q8—48no—
413Q8—88no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—88no—
460C2×C4—88no—
461C2×C4C2×#12412yes—
462C2×C4C2×#12414yes—
463C2×C4C2×#13714yes—
464C2×C4C2×#13714yes—
465C2×C4C22×#4112yes—
466C2×C4C2×H(5,𝔽2)12yes—
467C2×C4#58512yes—
468C8—48no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—24no—
476C22—24no—
477C22—24no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—88no—
483C22#313914yes—
484C22C22×#12411yescharacteristic
485C22#348411yescharacteristic
486C22#348511yescharacteristic
487C22C22×#13711yescharacteristic
488C22C22×#13711yescharacteristic
489C22#358511yescharacteristic
490C22#358611yescharacteristic
491C4—24no—
492C4—24no—
493C4—22no—
494C4—24no—
495C4—42no—
496C4—44no—
497C4—42no—
498C4—82no—
499C4—84no—
500C4—82no—
501C4—82no—
502C4—84no—
503C4#350514yes—
504C2—24no—
505C2—44no—
506C2#3537711yescharacteristic
507C2C2×#313911yescharacteristic
508C2#6062111yescharacteristic
509C2#6158611yescharacteristic
510C2#6158811yescharacteristic
511C2#6170511yescharacteristic
512C2#6170811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29335C2semidirect
4. #29329C2semidirect
5. #27972C2semidirect
6. #27707C2semidirect
7. #27283C2semidirect
8. #25912C2semidirect
15. C2×#491C22semidirect
38. #2100C22semidirect
39. #2099C22semidirect
40. #2072C22semidirect
41. #2043C22semidirect
42. #2043C22semidirect
43. #2026C22semidirect
44. #1992C22semidirect
45. #1956C22semidirect
46. #1935C22semidirect
47. #1933C22semidirect
54. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#350112
C2×#24412
#293352
#293292
#279722
#277072
#272832
#259122
#217652
#209112
#205242