Finite Group Databasebeta

← #315886

Subgroups — #315886

Order 512.

658 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33212C211yesmaximal, characteristic
2#27972C211yesmaximal, characteristic
3#27960C211yesmaximal, characteristic
4#27892C211yesmaximal, characteristic
5#27878C211yesmaximal, characteristic
6#20764C211yesmaximal, characteristic
7#10682C211yesmaximal, characteristic, complemented
8#10682C211yesmaximal, characteristic, complemented
9#10679C211yesmaximal, characteristic, complemented
10#10676C211yesmaximal, characteristic, complemented
11#9834C211yesmaximal, characteristic, complemented
12#9831C211yesmaximal, characteristic, complemented
13#8821C211yesmaximal, characteristic, complemented
14#8821C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#2285C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2099—22no—
23#2076—22no—
24#2072—22no—
25#2036—22no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1956C2211yescharacteristic
35#1956C2211yescharacteristic
36#1944—22no—
37#1928—22no—
38#1921—22no—
39#1872—22no—
40#1845C2211yescharacteristic
41#1845C2211yescharacteristic
42#1844C2211yescharacteristic
43#1836—22no—
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1823—22no—
55#1809—22no—
56C4×C22⋊C8C2211yescharacteristic
57C4×C22⋊C8C2211yescharacteristic
58#1736—22no—
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×M16D412yes—
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C42C2311yescharacteristic
73#500—22no—
74#500—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#106—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×C22⋊C8D412yes—
84C2×C22⋊C8D412yes—
85C4×M16—22no—
86C4×M16—22no—
87C2×#97D412yes—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#391—21no—
95#391—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#382—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#336D412yes—
135#336D412yes—
136#322—22no—
137#322—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—24no—
145C2×M16—24no—
146C2×M16—42no—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C22⋊Q8—44no—
162C22⋊Q8—82no—
163#117—44no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—82no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—41no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2411yescharacteristic
206#107—44no—
207#107—44no—
208#107—82no—
209#106—44no—
210#103—44no—
211#103—82no—
212#103—82no—
213#103—82no—
214C22⋊C8—42no—
215C22⋊C8—42no—
216#97—24no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8#4112yes—
228C4×C8#4112yes—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235C24#12711yescharacteristic
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4#12711yescharacteristic
267C22×C4#12711yescharacteristic
268C22×C4#12711yescharacteristic
269C22×C4#13714yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273M16—48no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8#12314yes—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—82no—
333C42C2×#4111yescharacteristic
334C42C2×#4111yescharacteristic
335C422−1+411yescharacteristic
336C422−1+411yescharacteristic
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23#49712yes—
344C23C2×#12711yescharacteristic
345Q8—82no—
346Q8—82no—
347Q8—82no—
348Q8—82no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—164no—
388C2×C4#49712yes—
389C2×C4C2×#12311yescharacteristic
390C2×C4#53611yescharacteristic
391C2×C4#54711yescharacteristic
392C2×C4#54711yescharacteristic
393C2×C4#54711yescharacteristic
394C2×C4#54711yescharacteristic
395C2×C4C2×#13712yes—
396C2×C4#57512yes—
397C2×C4#57512yes—
398C2×C4#57512yes—
399C2×C4#57612yes—
400C2×C4#57812yes—
401C8—24no—
402C8—42no—
403C8—42no—
404C8—42no—
405C8—42no—
406C8—42no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22#250111yescharacteristic
415C22C2×#49711yescharacteristic
416C22#315311yescharacteristic
417C22#315311yescharacteristic
418C22#315311yescharacteristic
419C22#315511yescharacteristic
420C22#315611yescharacteristic
421C4—22no—
422C4—41no—
423C4—41no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4#309512yes—
433C4#323612yes—
434C4#334712yes—
435C4#334912yes—
436C4#335212yes—
437C4#335512yes—
438C2—42no—
439C2#1968111yescharacteristic
440C2#1968211yescharacteristic
441C2#1973011yescharacteristic
442C2#1975411yescharacteristic
443C2#2791911yescharacteristic
444C2#2794211yescharacteristic
445C2#2794311yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
7. #10682C2semidirect
8. #10682C2semidirect
9. #10679C2semidirect
10. #10676C2semidirect
11. #9834C2semidirect
12. #9831C2semidirect
13. #8821C2semidirect
14. #8821C2semidirect

Maximal subgroups

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

subgroupindex
#332122
#279722
#279602
#278922
#278782
#207642
#106822
#106792
#106762
#98342
#98312
#88212
#81472