Finite Group Databasebeta

← #298660

Subgroups — #298660

Order 512.

1129 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic, complemented
2#33042C211yesmaximal, characteristic
3#25903C211yesmaximal, characteristic
4#25903C211yesmaximal, characteristic
5#25902C211yesmaximal, characteristic, complemented
6#25856C211yesmaximal, characteristic, complemented
7C2×#1711C212yesmaximal, complemented
8#10319C212yesmaximal, complemented
9#10311C212yesmaximal, complemented
10#9821C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#487—28no—
15C2×#487—22no—
16C2×#487C2212yes—
17C2×#487C2212yes—
18C2×#487C2212yes—
19C2×#487C2212yes—
20C2×#486C2211yescharacteristic
21C2×#486C2212yes—
22C2×#486C2212yes—
23C22×#103—22no—
24C22×C4×C8C2211yescharacteristic
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2043—22no—
28#2024—22no—
29#2023—22no—
30#2023—22no—
31#1990C2211yescharacteristic
32#1990C2212yes—
33#1990C2212yes—
34#1990C2212yes—
35#1957C2211yescharacteristic
36#1956C2212yes—
37#1934—22no—
38#1893—22no—
39#1874—22no—
40#1874—22no—
41#1872—22no—
42#1834C2211yescharacteristic
43#1787—22no—
44C4×C22⋊C8C2211yescharacteristic
45C4×C22⋊C8C2211yescharacteristic
46#1711—28no—
47C2×C82C2212yes—
48#1273C2212yes—
49C23×Q8—21no—
50C23×Q8—21no—
51C23×C8D412yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8—42no—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×#127C2311yescharacteristic
66C2×C22⋊Q8—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#487—216no—
76#487—24no—
77#487—24no—
78#487—24no—
79#487—24no—
80#487—24no—
81#487—42no—
82#487—42no—
83#487D418yescomplemented
84#486—24no—
85#486—24no—
86#486—24no—
87#486—42no—
88#486—42no—
89C2×#103—216no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—22no—
98C2×#102—22no—
99C2×C22⋊C8D412yes—
100C2×C22⋊C8D412yes—
101C2×C4×C8—24no—
102C2×C4×C8D412yes—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2312yes—
105C2×C4×C8C2312yes—
106C2×C4×C8C2311yescharacteristic
107#391—21no—
108#390—21no—
109#390—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#380—21no—
116#380—21no—
117#340—22no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—22no—
122#336D412yes—
123#336D412yes—
124C82—24no—
125C22×Q8—24no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—44no—
130C22×Q8—42no—
131C22×Q8—44no—
132C22×Q8—41no—
133C22×Q8—41no—
134C22×Q8—44no—
135C22×Q8—44no—
136C22×Q8—82no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—41no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C22×C8—24no—
147C22×C8—24no—
148C22×C8—42no—
149C22×C8C2×D412yes—
150C22×C8C2×D414yes—
151C22×C8C2×D414yes—
152C22×C8C2×D412yes—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—44no—
164C4⋊Q8—44no—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8C2×D414yes—
170C4⋊Q8C2×D414yes—
171#127—22no—
172#127—22no—
173C22⋊Q8—44no—
174#121—44no—
175#121—44no—
176#121—44no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2411yescharacteristic
209#103—216no—
210#103—216no—
211#103—48no—
212#103—48no—
213#103—84no—
214#102—48no—
215#102—48no—
216#102—44no—
217#102—44no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C4×C8—24no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8C2×D414yes—
227#95—41no—
228#95—41no—
229C24—41no—
230C24#12711yescharacteristic
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—416no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—42no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—42no—
245C2×Q8—816no—
246C2×Q8—82no—
247C2×Q8—88no—
248C2×Q8—82no—
249C2×Q8—82no—
250C2×Q8—82no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—82no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4#12711yescharacteristic
283C22×C4#12711yescharacteristic
284C22×C4#12711yescharacteristic
285C22×C4#13514yes—
286C22×C4#13514yes—
287C22×C4#13514yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—48no—
303C2×C8#12014yes—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C42—21no—
331C42—21no—
332C42—22no—
333C42—22no—
334C42C22×D412yes—
335C42H(5,𝔽2)12yes—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23#49512yes—
348C23#49512yes—
349C23#49512yes—
350C23C2×#12711yescharacteristic
351Q8—48no—
352Q8—48no—
353Q8—88no—
354Q8—88no—
355Q8—84no—
356Q8—84no—
357Q8—84no—
358D4—82no—
359D4—82no—
360D4—84no—
361D4—84no—
362D4—164no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—28no—
372C2×C4—28no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—88no—
376C2×C4—88no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4#49512yes—
391C2×C4#49512yes—
392C2×C4#49512yes—
393C2×C4C2×#12011yescharacteristic
394C2×C4#53511yescharacteristic
395C2×C4D4212yes—
396C2×C4D4212yes—
397C2×C4C2×#13512yes—
398C2×C4C2×#13512yes—
399C2×C4C2×#13512yes—
400C2×C4#57512yes—
401C2×C4#57712yes—
402C2×C4#57812yes—
403C8—24no—
404C8—42no—
405C8—44no—
406C8—44no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22#171114yescomplemented
421C22#243311yescharacteristic
422C22C2×#49511yescharacteristic
423C22C2×#49511yescharacteristic
424C22C2×#49511yescharacteristic
425C22#315311yescharacteristic
426C22#315411yescharacteristic
427C22#315611yescharacteristic
428C4—22no—
429C4—22no—
430C4—22no—
431C4—84no—
432C4—84no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4#300212yes—
437C4#321112yes—
438C4#328214yes—
439C4#328514yes—
440C2—24no—
441C2—321no—
442C2C2×#171111yescharacteristic
443C2#1962811yescharacteristic
444C2#1964011yescharacteristic
445C2#1965911yescharacteristic
446C2#2572411yescharacteristic
447C2#2577611yescharacteristic
448C2#2578211yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#487C2semidirect
5. #25902C2semidirect
6. #25856C2semidirect
7. C2×#1711C2semidirect
8. #10319C2semidirect
9. #10311C2semidirect
10. #9821C2semidirect
11. C8×C22⋊C8C2semidirect
83. #487D4semidirect
420. C22#1711semidirect

Maximal subgroups

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

subgroupindex
C22×#4872
#330422
#259032
#259022
#258562
C2×#17112
#103192
#103112
#98212
C8×C22⋊C82