Finite Group Databasebeta

← #298650

Subgroups — #298650

Order 512.

1181 conjugacy classes of subgroups.

The subgroup lattice has 454 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#25902C211yesmaximal, characteristic, complemented
5#25811C211yesmaximal, characteristic, complemented
6#25718C211yesmaximal, characteristic
7C2×#1708C212yesmaximal, complemented
8#10317C212yesmaximal, complemented
9#10310C212yesmaximal, complemented
10#9821C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#498C2212yes—
15C2×#492C2212yes—
16C2×#487—28no—
17C2×#487—22no—
18C2×#487C2212yes—
19C2×#487C2212yes—
20C2×#487C2212yes—
21C2×#486C2211yescharacteristic
22C2×#486C2212yes—
23C2×#448—22no—
24C22×#103—22no—
25C22×C4×C8C2211yescharacteristic
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2058—22no—
29#2024—22no—
30#2023—22no—
31#2013—22no—
32#1990C2211yescharacteristic
33#1990C2212yes—
34#1990C2212yes—
35#1990C2212yes—
36#1957C2211yescharacteristic
37#1956C2212yes—
38#1934—22no—
39#1893—22no—
40#1874—22no—
41#1872—22no—
42#1853—22no—
43#1834C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45C4×C22⋊C8C2211yescharacteristic
46#1708—28no—
47C2×C82C2212yes—
48#1273C2212yes—
49C23×Q8—21no—
50C23×Q8—21no—
51C22×SD16—24no—
52C23×C8D412yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—42no—
62C2×C4⋊Q8—42no—
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×#127C2311yescharacteristic
67C2×C22⋊Q8—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#498—24no—
77#498—42no—
78#492—24no—
79#492—42no—
80#487—216no—
81#487—24no—
82#487—24no—
83#487—24no—
84#487—24no—
85#487—42no—
86#487D418yescomplemented
87#486—24no—
88#486—24no—
89#486—42no—
90#448—48no—
91C2×#106—24no—
92C2×#103—216no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#102—24no—
98C2×#102—22no—
99C2×#102—22no—
100C2×C22⋊C8D412yes—
101C2×C22⋊C8D412yes—
102C2×C4×C8—24no—
103C2×C4×C8D412yes—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2312yes—
106C2×C4×C8C2312yes—
107C2×C4×C8C2311yescharacteristic
108#391—21no—
109#390—21no—
110#390—21no—
111#390—21no—
112#389—21no—
113#389—21no—
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#380—21no—
117#380—21no—
118#340—24no—
119#340—22no—
120#340—24no—
121#340—24no—
122#340—22no—
123#336D412yes—
124#336D412yes—
125C82—24no—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—44no—
131C22×Q8—42no—
132C22×Q8—44no—
133C22×Q8—41no—
134C22×Q8—41no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8—82no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—41no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×SD16—48no—
148C2×SD16—48no—
149C2×SD16—84no—
150C22×C8—24no—
151C22×C8—24no—
152C22×C8—42no—
153C22×C8C2×D412yes—
154C22×C8C2×D414yes—
155C22×C8C2×D412yes—
156C22×C8C2×D414yes—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170C4⋊Q8—44no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8C2×D414yes—
174C4⋊Q8C2×D414yes—
175#127—22no—
176#127—22no—
177C22⋊Q8—44no—
178#121—44no—
179#121—44no—
180#121—44no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213#106—48no—
214#103—216no—
215#103—216no—
216#103—48no—
217#103—48no—
218#102—48no—
219#102—44no—
220#102—44no—
221C22⋊C8—24no—
222C22⋊C8SD1618yes—
223C4×C8—24no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—22no—
228C4×C8—22no—
229C4×C8C2×D414yes—
230#95—41no—
231#95—41no—
232C24—41no—
233C24#12711yescharacteristic
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—416no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—42no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—42no—
248C2×Q8—816no—
249C2×Q8—82no—
250C2×Q8—88no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×Q8—82no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—82no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4#12711yescharacteristic
286C22×C4#12711yescharacteristic
287C22×C4#12711yescharacteristic
288C22×C4C2×SD1614yes—
289C22×C4#13514yes—
290C22×C4#13514yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294SD16—816no—
295C2×C8—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—24no—
299C2×C8—22no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—48no—
306C2×C8#12014yes—
307C2×C8C2×SD1618yes—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C42—21no—
335C42—21no—
336C42—22no—
337C42—22no—
338C42C22×D412yes—
339C42H(5,𝔽2)12yes—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23#49212yes—
352C23#49512yes—
353C23#49512yes—
354C23C2×#12711yescharacteristic
355Q8—48no—
356Q8—48no—
357Q8—88no—
358Q8—88no—
359Q8—84no—
360Q8—84no—
361Q8—84no—
362D4—82no—
363D4—82no—
364D4—84no—
365D4—84no—
366D4—164no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—28no—
376C2×C4—28no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—88no—
380C2×C4—88no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4#49212yes—
395C2×C4#49512yes—
396C2×C4#49512yes—
397C2×C4C2×#12011yescharacteristic
398C2×C4#53511yescharacteristic
399C2×C4D4212yes—
400C2×C4D4212yes—
401C2×C4C22×SD1612yes—
402C2×C4C2×#13512yes—
403C2×C4C2×#13512yes—
404C2×C4#57512yes—
405C2×C4#57712yes—
406C2×C4#57812yes—
407C8—24no—
408C8—28no—
409C8—42no—
410C8—44no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22#170814yescomplemented
425C22#243311yescharacteristic
426C22C2×#49211yescharacteristic
427C22C2×#49511yescharacteristic
428C22C2×#49511yescharacteristic
429C22#315311yescharacteristic
430C22#315411yescharacteristic
431C22#315611yescharacteristic
432C4—22no—
433C4—22no—
434C4—22no—
435C4—84no—
436C4—84no—
437C4—161no—
438C4—161no—
439C4—161no—
440C4#300212yes—
441C4#321112yes—
442C4D4×SD1614yes—
443C4#328514yes—
444C2—24no—
445C2—321no—
446C2C2×#170811yescharacteristic
447C2#1962411yescharacteristic
448C2#1964011yescharacteristic
449C2#1965711yescharacteristic
450C2#2572411yescharacteristic
451C2#2577411yescharacteristic
452C2#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
4. #25902C2semidirect
5. #25811C2semidirect
7. C2×#1708C2semidirect
8. #10317C2semidirect
9. #10310C2semidirect
10. #9821C2semidirect
11. C8×C22⋊C8C2semidirect
86. #487D4semidirect
424. C22#1708semidirect

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
#258112
#257182
C2×#17082
#103172
#103102
#98212
C8×C22⋊C82