Finite Group Databasebeta

← #6506507

Subgroups — #6506507

Order 512.

2014 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35011C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic
3#29329C212yesmaximal, complemented
4#29329C212yesmaximal, complemented
5#27972C212yesmaximal, complemented
6#27697C218yesmaximal, complemented
7#27258C218yesmaximal, complemented
8#25902C212yesmaximal, complemented
9#21759C212yesmaximal
10C2×#1872C212yesmaximal
11#20523C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×C22⋊Q8C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C2×#487C2214yescomplemented
17C22×#103C2214yes—
18C22×C22⋊C8C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2659—21no—
22#2659—21no—
23#2583—24no—
24#2502C2214yes—
25#2440—24no—
26#2440C2218yes—
27#2434C2214yes—
28C2×#398—21no—
29C2×#395—21no—
30#2315—21no—
31#2315—21no—
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#385—21no—
35C2×#384C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#2100C2214yescomplemented
39#2100C2214yescomplemented
40#2072C2218yescomplemented
41#2043C2214yescomplemented
42#2043C2214yescomplemented
43#2024C2218yescomplemented
44#1990C2214yescomplemented
45#1956C2214yescomplemented
46#1933C2218yescomplemented
47#1933C2218yescomplemented
48#1892C2218yes—
49#1872C2214yes—
50#1872C2214yes—
51#1872C2218yes—
52#1872C2214yes—
53#1872C2214yes—
54#1834C2214yescomplemented
55#1824C2214yes—
56#1787C2214yes—
57C8×C22⋊C4C2218yes—
58C4×C22⋊C8C2212yes—
59C4×C22⋊C8C2212yes—
60C2×#340C2214yes—
61C23×Q8—21no—
62C23×Q8C2312yes—
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C23×C8C2314yes—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—44no—
68C2×C4⋊Q8C2312yes—
69C2×C4⋊Q8C2312yes—
70C2×C4⋊Q8C2314yes—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—48no—
77C2×C22⋊Q8C2314yes—
78C2×C22⋊Q8C2314yes—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4—22no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4D411yescharacteristic
97C22×C22⋊C4D411yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#487—48no—
101C2×#103—48no—
102C2×#103C2318yes—
103C2×#103C2318yes—
104C2×#103C2318yes—
105C2×#103C2318yes—
106C2×C22⋊C8—28no—
107C2×C22⋊C8C2314yes—
108C2×C22⋊C8C2314yes—
109C2×C4×C8C2314yes—
110C2×C4×C8C2314yes—
111#398—24no—
112#398—24no—
113#398—28no—
114#395—24no—
115#395—24no—
116#395—28no—
117#391C2314yes—
118#391C2314yes—
119#389—22no—
120#389—22no—
121#389—22no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389C2314yes—
126#389C2314yes—
127#385—24no—
128#385—24no—
129#385—24no—
130#385—24no—
131#385—28no—
132#384—22no—
133#384C2314yes—
134#384C2312yes—
135#384C2312yes—
136#380—22no—
137#380—22no—
138#380—22no—
139#380—22no—
140#380—22no—
141#380C2314yes—
142C4×C22⋊C4D414yes—
143C4×C22⋊C4C2314yes—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—21no—
147C2×#95D411yescharacteristic
148C2×#95D411yescharacteristic
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#336C2314yes—
154#336C2314yes—
155C25C2×D411yescharacteristic
156C22×Q8—24no—
157C22×Q8—22no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—44no—
161C22×Q8—44no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×Q8—82no—
165C22×Q8C2414yes—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—82no—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4#4118yes—
185C23×C4C2411yescharacteristic
186C23×C4C2412yes—
187C22×C8—28no—
188C22×C8—44no—
189C22×C8—44no—
190C22×C8C2418yes—
191C4⋊Q8—28no—
192C4⋊Q8—44no—
193C4⋊Q8—44no—
194C4⋊Q8—44no—
195C4⋊Q8—48no—
196C22⋊Q8—28no—
197C22⋊Q8—28no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C22⋊Q8—48no—
201C22⋊Q8—48no—
202C22⋊Q8—432no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2414yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4SD1618yes—
251C2×C22⋊C4Q1618yes—
252C2×C22⋊C4C2×D414yes—
253C2×C22⋊C4C2×D414yes—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2412yes—
259C2×C42C2412yes—
260#103—216no—
261#103—48no—
262#103—48no—
263#103—48no—
264C22⋊C8—28no—
265C22⋊C8—28no—
266C4×C8—44no—
267C4×C8—44no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95C2×D414yes—
276#95C2×D414yes—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24#12414yes—
282C24C2×SD1614yes—
283C24C2×Q1614yes—
284C24C22×D412yes—
285C24C22×D411yescharacteristic
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—28no—
289C2×Q8—28no—
290C2×Q8—48no—
291C2×Q8—416no—
292C2×Q8—44no—
293C2×Q8—48no—
294C2×Q8—82no—
295C2×Q8—82no—
296C2×Q8—84no—
297C2×Q8—88no—
298C2×Q8—816no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—416no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—48no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—48no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—84no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—816no—
344C22×C4#12418yes—
345C22×C4#12414yes—
346C22×C4C2×SD1614yes—
347C22×C4C2×SD1614yes—
348C22×C4C2×Q1614yes—
349C22×C4C2×Q1614yes—
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D412yes—
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C2×#4114yes—
357C22×C4C2×#4114yes—
358C22×C4C2×#4114yes—
359C22×C4C2511yescharacteristic
360C2×C8—28no—
361C2×C8—28no—
362C2×C8—28no—
363C2×C8—88no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—48no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—84no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4H(5,𝔽2)14yes—
392C22⋊C4—48no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4C2×SD16116yes—
396C22⋊C4C2×Q16116yes—
397C42—24no—
398C42—24no—
399C42—42no—
400C42—42no—
401C23—21no—
402C23—22no—
403C23—21no—
404C23—22no—
405C23—21no—
406C23—22no—
407C23—21no—
408C23—22no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23#487116yescomplemented
416C23C2×#12412yes—
417C23C2×#12414yes—
418C23C22×SD1614yes—
419C23C22×SD1612yes—
420C23C22×Q1614yes—
421C23C22×Q1612yes—
422C23C23×D411yescharacteristic
423Q8—48no—
424Q8—48no—
425Q8—816no—
426Q8—88no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—48no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—48no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—816no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—88no—
472C2×C4—88no—
473C2×C4C2×#12412yes—
474C2×C4C2×#12414yes—
475C2×C4C22×SD1614yes—
476C2×C4C22×SD1614yes—
477C2×C4C22×Q1614yes—
478C2×C4C22×Q1614yes—
479C2×C4C22×#4112yes—
480C2×C4C2×H(5,𝔽2)12yes—
481C2×C4#58512yes—
482C8—48no—
483C22—22no—
484C22—24no—
485C22—24no—
486C22—22no—
487C22—22no—
488C22—24no—
489C22—22no—
490C22—24no—
491C22—24no—
492C22—22no—
493C22—22no—
494C22—24no—
495C22—416no—
496C22C2×#48714yes—
497C22C2×#48718yes—
498C22C22×#12411yescharacteristic
499C22#348411yescharacteristic
500C22#348511yescharacteristic
501C22C23×SD1611yescharacteristic
502C22C23×Q1611yescharacteristic
503C22#358511yescharacteristic
504C22#358611yescharacteristic
505C4—24no—
506C4—22no—
507C4—24no—
508C4—24no—
509C4—24no—
510C4—24no—
511C4—44no—
512C4—84no—
513C4—82no—
514C4—82no—
515C4—84no—
516C4—82no—
517C4#350514yes—
518C2—24no—
519C2—28no—
520C2#3537711yescharacteristic
521C2C22×#48711yescharacteristic
522C2#6062111yescharacteristic
523C2#6156011yescharacteristic
524C2#6156311yescharacteristic
525C2#6169511yescharacteristic
526C2#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. #29329C2semidirect
4. #29329C2semidirect
5. #27972C2semidirect
6. #27697C2semidirect
7. #27258C2semidirect
8. #25902C2semidirect
16. C2×#487C22semidirect
38. #2100C22semidirect
39. #2100C22semidirect
40. #2072C22semidirect
41. #2043C22semidirect
42. #2043C22semidirect
43. #2024C22semidirect
44. #1990C22semidirect
45. #1956C22semidirect
46. #1933C22semidirect
47. #1933C22semidirect
54. #1834C22semidirect
415. C23#487semidirect

Maximal subgroups

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

subgroupindex
#350112
C2×#24402
#293292
#279722
#276972
#272582
#259022
#217592
C2×#18722
#205232