Finite Group Databasebeta

← #6506759

Subgroups — #6506759

Order 512.

1982 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic
3#29296C212yesmaximal, complemented
4#29296C212yesmaximal, complemented
5#28769C218yesmaximal, complemented
6#27950C212yesmaximal, complemented
7#26065C218yesmaximal, complemented
8#25813C212yesmaximal, complemented
9#21940C212yesmaximal
10C2×#1852C212yesmaximal
11#20523C211yesmaximal, characteristic
12C2×#543—21no—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—21no—
15C23×C4⋊C4C2211yescharacteristic
16C2×#500C2214yescomplemented
17C22×#107C2214yes—
18C22×C22⋊C8C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2659—21no—
22#2659—21no—
23#2549—24no—
24#2440—24no—
25#2440C2214yes—
26#2440C2218yes—
27#2440C2214yes—
28C2×#398—21no—
29C2×#395—21no—
30#2315—21no—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#385—21no—
34C2×#384C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#2083C2218yescomplemented
39#2083C2214yescomplemented
40#2083C2214yescomplemented
41#2057C2214yescomplemented
42#2057C2218yescomplemented
43#2057C2214yescomplemented
44#1990C2214yescomplemented
45#1990C2214yescomplemented
46#1942C2218yescomplemented
47#1942C2218yescomplemented
48#1852C2214yes—
49#1852C2214yes—
50#1852C2214yes—
51#1852C2218yes—
52#1852C2214yes—
53#1852C2214yes—
54#1852C2214yes—
55#1845C2214yescomplemented
56#1808C2218yes—
57C8×C22⋊C4C2218yes—
58C4×C22⋊C8C2212yes—
59C4×C22⋊C8C2212yes—
60C2×#340C2214yes—
61C23×Q8—21no—
62C24×C4—21no—
63C24×C4C2311yescharacteristic
64C23×C8C2314yes—
65#543—48no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—44no—
68C2×C4⋊Q8C2314yes—
69C2×C4⋊Q8C2312yes—
70C2×C4⋊Q8C2312yes—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—48no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—44no—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4—22no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4D411yescharacteristic
95C22×C22⋊C4D411yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#500—48no—
99C2×#107—48no—
100C2×#107C2318yes—
101C2×#107C2318yes—
102C2×#107C2318yes—
103C2×#107C2318yes—
104C2×C22⋊C8Q8116yescomplemented
105C2×C22⋊C8C2314yes—
106C2×C22⋊C8C2314yes—
107C2×C4×C8C2314yes—
108C2×C4×C8C2314yes—
109#398—28no—
110#398—24no—
111#398—24no—
112#395—28no—
113#395—24no—
114#395—24no—
115#391—22no—
116#391—22no—
117#389—22no—
118#389—22no—
119#389—22no—
120#389—22no—
121#389C2314yes—
122#389C2314yes—
123#389C2314yes—
124#389C2314yes—
125#385—24no—
126#385—24no—
127#385—28no—
128#384—22no—
129#384C2312yes—
130#384C2312yes—
131#384C2314yes—
132#380—22no—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380C2314yes—
138#380C2314yes—
139#380C2314yes—
140C4×C22⋊C4D414yes—
141C4×C22⋊C4C2314yes—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95D411yescharacteristic
147C2×#95D411yescharacteristic
148#340C2318yes—
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#336C2314yes—
153#336C2314yes—
154C25C2×D411yescharacteristic
155C22×Q8—24no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8—82no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—82no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×Q818yes—
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182C22×C8—44no—
183C22×C8—44no—
184C22×C8C2×Q8116yescomplemented
185C22×C8C2418yes—
186C4⋊Q8—28no—
187C4⋊Q8—44no—
188C4⋊Q8—44no—
189C4⋊Q8—44no—
190C4⋊Q8—48no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193C22⋊Q8—432no—
194C22⋊Q8—416no—
195C22⋊Q8—416no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
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—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4C2412yes—
232C2×C4⋊C4C2412yes—
233C2×C4⋊C4C2414yes—
234C2×C4⋊C4C2414yes—
235C2×C4⋊C4C2412yes—
236C2×C4⋊C4C2412yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—48no—
242C2×C22⋊C4D818yes—
243C2×C22⋊C4Q1618yes—
244C2×C22⋊C4C2×D414yes—
245C2×C22⋊C4C2×D414yes—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2412yes—
251C2×C42C2412yes—
252#107—216no—
253#107—48no—
254#107—48no—
255#107—48no—
256C22⋊C8C2×Q8116yes—
257C22⋊C8C2×Q8116yescomplemented
258C4×C8—44no—
259C4×C8—44no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95C2×D414yes—
268#95C2×D414yes—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24C4⋊Q814yes—
274C24C2×D814yes—
275C24C2×Q1614yes—
276C24C22×D412yes—
277C24C22×D411yescharacteristic
278C2×Q8—48no—
279C2×Q8—48no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×Q8—84no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—82no—
286C2×Q8—82no—
287C2×Q8—816no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—416no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—48no—
313C22×C4—42no—
314C22×C4—42no—
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—42no—
324C22×C4—48no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—84no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—816no—
333C22×C4C4⋊Q814yes—
334C22×C4C4⋊Q818yes—
335C22×C4C2×D814yes—
336C22×C4C2×D814yes—
337C22×C4C2×Q1614yes—
338C22×C4C2×Q1614yes—
339C22×C4C22×D412yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C22×D412yes—
345C22×C4C22×Q814yes—
346C22×C4C22×Q814yes—
347C22×C4C22×Q814yes—
348C22×C4C2511yescharacteristic
349C2×C8—28no—
350C2×C8—28no—
351C2×C8—416no—
352C2×C8C22×Q8116yes—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—28no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—28no—
360C4⋊C4—48no—
361C4⋊C4—48no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—44no—
369C4⋊C4—48no—
370C4⋊C4—44no—
371C4⋊C4—48no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—84no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4H(5,𝔽2)14yes—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4C2×D8116yes—
384C22⋊C4C2×Q16116yes—
385C42—24no—
386C42—24no—
387C42—42no—
388C42—42no—
389C23—21no—
390C23—22no—
391C23—21no—
392C23—22no—
393C23—21no—
394C23—22no—
395C23—21no—
396C23—22no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23#500116yescomplemented
404C23C2×C4⋊Q812yes—
405C23C2×C4⋊Q814yes—
406C23C22×D814yes—
407C23C22×D812yes—
408C23C22×Q1614yes—
409C23C22×Q1612yes—
410C23C23×D411yescharacteristic
411Q8—816no—
412Q8—164no—
413Q8—164no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—48no—
431C2×C4—44no—
432C2×C4—44no—
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—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—816no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—164no—
459C2×C4—164no—
460C2×C4C2×C4⋊Q814yes—
461C2×C4C2×C4⋊Q812yes—
462C2×C4C22×D814yes—
463C2×C4C22×D814yes—
464C2×C4C22×Q1614yes—
465C2×C4C22×Q1614yes—
466C2×C4C23×Q812yes—
467C2×C4C2×H(5,𝔽2)12yes—
468C2×C4#58512yes—
469C8—216no—
470C22—22no—
471C22—24no—
472C22—24no—
473C22—22no—
474C22—22no—
475C22—24no—
476C22—22no—
477C22—24no—
478C22—22no—
479C22—24no—
480C22—22no—
481C22—24no—
482C22—416no—
483C22C2×#50018yes—
484C22C2×#50014yes—
485C22C22×C4⋊Q811yescharacteristic
486C22#348411yescharacteristic
487C22#348511yescharacteristic
488C22C23×D811yescharacteristic
489C22C23×Q1611yescharacteristic
490C22#358511yescharacteristic
491C22#358611yescharacteristic
492C4—24no—
493C4—22no—
494C4—24no—
495C4—24no—
496C4—24no—
497C4—24no—
498C4—44no—
499C4—82no—
500C4—82no—
501C4—84no—
502C4—82no—
503C4—84no—
504C4#350114yes—
505C2—24no—
506C2—28no—
507C2#3539111yescharacteristic
508C2C22×#50011yescharacteristic
509C2#6066411yescharacteristic
510C2#6155911yescharacteristic
511C2#6156311yescharacteristic
512C2#6169411yescharacteristic
513C2#6169811yescharacteristic
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. #29296C2semidirect
4. #29296C2semidirect
5. #28769C2semidirect
6. #27950C2semidirect
7. #26065C2semidirect
8. #25813C2semidirect
16. C2×#500C22semidirect
38. #2083C22semidirect
39. #2083C22semidirect
40. #2083C22semidirect
41. #2057C22semidirect
42. #2057C22semidirect
43. #2057C22semidirect
44. #1990C22semidirect
45. #1990C22semidirect
46. #1942C22semidirect
47. #1942C22semidirect
55. #1845C22semidirect
104. C2×C22⋊C8Q8semidirect
184. C22×C8C2×Q8semidirect
257. C22⋊C8C2×Q8semidirect
403. C23#500semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#24402
#292962
#287692
#279502
#260652
#258132
#219402
C2×#18522
#205232