Finite Group Databasebeta

← #315769

Subgroups — #315769

Order 512.

744 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33176C211yesmaximal, characteristic
2#27960C211yesmaximal, characteristic
3#27950C211yesmaximal, characteristic
4#27877C211yesmaximal, characteristic
5#27877C211yesmaximal, characteristic
6#20800C211yesmaximal, characteristic
7#10676C211yesmaximal, characteristic, complemented
8#10675C211yesmaximal, characteristic, complemented
9#10675C211yesmaximal, characteristic, complemented
10#10674C211yesmaximal, characteristic, complemented
11#9839C211yesmaximal, characteristic, complemented
12#9825C211yesmaximal, characteristic, complemented
13#8820C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic
16C2×#494C2211yescharacteristic
17C2×#470—22no—
18C2×#428—22no—
19C2×C4×M16C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2099—22no—
25#2083—22no—
26#2075—22no—
27#2075—22no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1971—22no—
37#1956C2211yescharacteristic
38#1956C2211yescharacteristic
39#1918—22no—
40#1918—22no—
41#1852—22no—
42#1847—22no—
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1845C2211yescharacteristic
49#1844C2211yescharacteristic
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1823—22no—
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×Q16—22no—
64C22×M16D412yes—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#494—22no—
75#494—22no—
76#470—48no—
77#428—48no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×#103—22no—
88C2×#103—22no—
89C2×C22⋊C8D412yes—
90C2×C22⋊C8D412yes—
91C4×M16—22no—
92C4×M16—22no—
93C2×#97D412yes—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#391—21no—
101#391—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#384—21no—
107#384—21no—
108#384—21no—
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#382—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
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#340—22no—
135#340—22no—
136#336D412yes—
137#336D412yes—
138#322—22no—
139C82—22no—
140C22×Q8—21no—
141C22×Q8—21no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×Q16—44no—
147C2×Q16—44no—
148C2×Q16—44no—
149C2×M16—24no—
150C2×M16—24no—
151C2×M16D818yes—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C22⋊Q8—44no—
167C22⋊Q8—44no—
168#117—44no—
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—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—82no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—41no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2411yescharacteristic
211#107—44no—
212#107—44no—
213#107—44no—
214#107—44no—
215#107—82no—
216#107—82no—
217#106—44no—
218#106—44no—
219#106—44no—
220#103—44no—
221#103—44no—
222C22⋊C8—24no—
223C22⋊C8—42no—
224#97—24no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8#4112yes—
236C4×C8#4112yes—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243C24#12711yescharacteristic
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4#12711yescharacteristic
275C22×C4#12711yescharacteristic
276C22×C4#12711yescharacteristic
277C22×C4C2×D814yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281Q16—88no—
282M16—216no—
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—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8#12314yes—
297C2×C8C2×D814yes—
298C2×C8C2×D814yes—
299C2×C8#13714yes—
300C2×C8#13714yes—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
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—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—82no—
342C42C2×#4111yescharacteristic
343C42C2×#4111yescharacteristic
344C422−1+411yescharacteristic
345C422−1+411yescharacteristic
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23#49312yes—
353C23C2×#12711yescharacteristic
354Q8—82no—
355Q8—82no—
356Q8—82no—
357Q8—82no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
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—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—164no—
397C2×C4#49312yes—
398C2×C4C2×#12311yescharacteristic
399C2×C4#53611yescharacteristic
400C2×C4#54711yescharacteristic
401C2×C4#54711yescharacteristic
402C2×C4#54711yescharacteristic
403C2×C4#54711yescharacteristic
404C2×C4C22×D812yes—
405C2×C4C2×#13712yes—
406C2×C4C2×#13712yes—
407C2×C4#57512yes—
408C2×C4#57712yes—
409C2×C4#57812yes—
410C8—24no—
411C8—24no—
412C8—42no—
413C8—42no—
414C8—42no—
415C8#48018yes—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22#250111yescharacteristic
424C22C2×#49311yescharacteristic
425C22#315211yescharacteristic
426C22#315211yescharacteristic
427C22#315311yescharacteristic
428C22#315411yescharacteristic
429C22#315611yescharacteristic
430C4—22no—
431C4—41no—
432C4—41no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4—161no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4C2×#48012yes—
442C4#322612yes—
443C4#334412yes—
444C4#334912yes—
445C4#335212yes—
446C4#335612yes—
447C2—42no—
448C2#1963011yescharacteristic
449C2#1963111yescharacteristic
450C2#1972011yescharacteristic
451C2#1975111yescharacteristic
452C2#2790911yescharacteristic
453C2#2794011yescharacteristic
454C2#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. #10676C2semidirect
8. #10675C2semidirect
9. #10675C2semidirect
10. #10674C2semidirect
11. #9839C2semidirect
12. #9825C2semidirect
13. #8820C2semidirect
14. #8814C2semidirect

Maximal subgroups

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

subgroupindex
#331762
#279602
#279502
#278772
#208002
#106762
#106752
#106742
#98392
#98252
#88202
#88142
#81462