Finite Group Databasebeta

← #6475451

Subgroups — #6475451

Order 512.

1984 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3#33043C214yesmaximal, complemented
4#30190C214yesmaximal, complemented
5#29332C214yesmaximal, complemented
6#29106C214yesmaximal, complemented
7#27975C212yesmaximal, complemented
8#27960C212yesmaximal, complemented
9#25904C212yesmaximal, complemented
10#25856C212yesmaximal, complemented
11#21762C214yesmaximal
12C2×C4×C22⋊C8C211yesmaximal, characteristic
13C2×#533C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C23×C42C2211yescharacteristic
17C2×#487C2214yescomplemented
18C22×C22⋊C8C2212yes—
19C22×C4×C8C2212yes—
20#2857—22no—
21#2857—22no—
22#2857C2212yes—
23#2857C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2855—22no—
29#2851C2212yes—
30#2851C2212yes—
31#2848—22no—
32C4×C22⋊Q8—22no—
33C4×C22⋊Q8—22no—
34#2296C2212yes—
35C2×#391C2212yes—
36C2×#389—22no—
37#2285C2211yescharacteristic
38C2×#384C2211yescharacteristic
39#2280C2212yes—
40C2×#380—22no—
41#2099C2218yescomplemented
42#2074C2218yescomplemented
43#2043C2218yescomplemented
44#2026C2218yescomplemented
45#1990C2214yescomplemented
46#1956C2214yescomplemented
47#1933C22116yescomplemented
48#1892C22116yes—
49#1873C2218yes—
50#1873C2218yes—
51#1834C2214yescomplemented
52#1823C2218yes—
53#1787C2218yes—
54C4×C22⋊C8C2212yes—
55C4×C22⋊C8C2212yes—
56C4×C22⋊C8C2212yes—
57C4×C22⋊C8C2212yes—
58C2×#336C2212yes—
59C23×Q8C2312yes—
60C24×C4D412yes—
61C24×C4C2311yescharacteristic
62C23×C8C2314yes—
63#533—44no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—44no—
66C2×C4⋊Q8C2312yes—
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2312yes—
69C2×#125C2312yes—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—216no—
76C2×C22⋊Q8C2314yes—
77C2×C22⋊Q8C2314yes—
78C2×C4×Q8—22no—
79C2×#117C2312yes—
80C2×#117C2312yes—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—22no—
86C22×C42—44no—
87C22×C42D412yes—
88C22×C42D412yes—
89C22×C42D412yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#487—48no—
97C2×#103C23116yes—
98C2×#103C23116yes—
99C2×C22⋊C8—28no—
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C2×C4×C8—44no—
103C2×C4×C8C2314yes—
104C2×C4×C8C2314yes—
105#391—22no—
106#391—22no—
107#391—24no—
108#391C2314yes—
109#391C2314yes—
110#389—22no—
111#389—22no—
112#389—24no—
113#389—24no—
114#389—24no—
115#389—24no—
116#389C2314yes—
117#389C2314yes—
118#387—24no—
119#387—24no—
120#385—24no—
121#385—24no—
122#385—24no—
123#384—22no—
124#384C2312yes—
125#384C2312yes—
126#384C2312yes—
127#382C2312yes—
128#381—22no—
129#380—22no—
130#380—22no—
131#380—22no—
132#380—24no—
133#380—24no—
134#380—24no—
135#380—24no—
136#380C2314yes—
137#380C2314yes—
138#379—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—24no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C2×#95—24no—
144#340C23116yes—
145#340C23116yes—
146#336C2314yes—
147#336C2314yes—
148C25C2×D411yescharacteristic
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—28no—
152C22×Q8—44no—
153C22×Q8—44no—
154C22×Q8C2414yes—
155C23×C4—28no—
156C23×C4—28no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—44no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4#4118yes—
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2411yescharacteristic
175C22×C8—28no—
176C22×C8—44no—
177C22×C8—44no—
178C22×C8C2418yes—
179C4⋊Q8—28no—
180C4⋊Q8—28no—
181C4⋊Q8—44no—
182C4⋊Q8—48no—
183#125—44no—
184C22⋊Q8—216no—
185C22⋊Q8—216no—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C22⋊Q8—48no—
191C4×Q8—48no—
192#117—44no—
193#117—44no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—48no—
211C2×C4⋊C4C2414yes—
212C2×C4⋊C4C2414yes—
213C2×C4⋊C4C2414yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—48no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—44no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2411yescharacteristic
239C2×C42C2411yescharacteristic
240C2×C42C2411yescharacteristic
241C2×C42C2411yescharacteristic
242#103—416no—
243#103—416no—
244C22⋊C8—28no—
245C22⋊C8—28no—
246C4×C8—28no—
247C4×C8—44no—
248#95—28no—
249#95—28no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255C24—24no—
256C24—24no—
257C24—41no—
258C24—41no—
259C24#12414yes—
260C24C22×D411yescharacteristic
261C24C22×D411yescharacteristic
262C24C22×D411yescharacteristic
263C2×Q8—28no—
264C2×Q8—28no—
265C2×Q8—28no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—816no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—416no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—432no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—48no—
312C22×C4#12414yes—
313C22×C4#12414yes—
314C22×C4#12414yes—
315C22×C4#13518yes—
316C22×C4#13718yes—
317C22×C4C22×D412yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C22×D412yes—
324C22×C4C2×#4114yes—
325C22×C4C2×#4114yes—
326C22×C4C2×#4114yes—
327C22×C4C2511yescharacteristic
328C2×C8—28no—
329C2×C8—28no—
330C2×C8—28no—
331C2×C8—416no—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—48no—
336C4⋊C4—48no—
337C4⋊C4—44no—
338C4⋊C4—48no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—44no—
342C4⋊C4—48no—
343C4⋊C4—48no—
344C4⋊C4—44no—
345C4⋊C4—48no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C22⋊C4—48no—
354C42—24no—
355C42—24no—
356C42—24no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—44no—
364C42—44no—
365C42H(5,𝔽2)14yes—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—22no—
372C23—28no—
373C23—28no—
374C23—21no—
375C23—21no—
376C23—22no—
377C23—22no—
378C23—44no—
379C23—48no—
380C23—48no—
381C23C2×#12412yes—
382C23C2×#12412yes—
383C23C2×#12412yes—
384C23C2×#13514yes—
385C23C2×#13714yes—
386C23C23×D411yescharacteristic
387Q8—416no—
388Q8—88no—
389Q8—88no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—216no—
401C2×C4—216no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—48no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—48no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—88no—
427C2×C4—88no—
428C2×C4C2×#12412yes—
429C2×C4C2×#12412yes—
430C2×C4C2×#12412yes—
431C2×C4#53414yes—
432C2×C4C2×#13514yes—
433C2×C4C2×#13514yes—
434C2×C4C2×#13714yes—
435C2×C4C2×#13714yes—
436C2×C4C22×#4112yes—
437C2×C4C2×H(5,𝔽2)12yes—
438C2×C4#58512yes—
439C8—48no—
440C22—24no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—416no—
452C22—42no—
453C22—42no—
454C22#313114yes—
455C22#313914yes—
456C22C22×#12411yescharacteristic
457C22C2×#53411yescharacteristic
458C22#344911yescharacteristic
459C22C22×#13512yes—
460C22C22×#13712yes—
461C4—24no—
462C4—24no—
463C4—22no—
464C4—22no—
465C4—22no—
466C4—42no—
467C4—84no—
468C4—84no—
469C4—84no—
470C4—84no—
471C4#319018yes—
472C4#350514yes—
473C2—24no—
474C2—24no—
475C2—42no—
476C2#3501911yescharacteristic
477C2C2×#313111yescharacteristic
478C2C2×#313911yescharacteristic
479C2C2×#319012yes—
480C2#6094712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33043C2semidirect
4. #30190C2semidirect
5. #29332C2semidirect
6. #29106C2semidirect
7. #27975C2semidirect
8. #27960C2semidirect
9. #25904C2semidirect
10. #25856C2semidirect
17. C2×#487C22semidirect
41. #2099C22semidirect
42. #2074C22semidirect
43. #2043C22semidirect
44. #2026C22semidirect
45. #1990C22semidirect
46. #1956C22semidirect
47. #1933C22semidirect
51. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#28572
#330432
#301902
#293322
#291062
#279752
#279602
#259042
#258562
#217622
C2×C4×C22⋊C82