Finite Group Databasebeta

← #315967

Subgroups — #315967

Order 512.

704 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29876C211yesmaximal, characteristic
2#27950C211yesmaximal, characteristic
3#27908C211yesmaximal, characteristic
4#27905C211yesmaximal, characteristic
5#27877C211yesmaximal, characteristic
6#21982C211yesmaximal, characteristic
7#10675C211yesmaximal, characteristic, complemented
8#10675C211yesmaximal, characteristic, complemented
9#10675C211yesmaximal, characteristic, complemented
10#10674C211yesmaximal, characteristic, complemented
11#9831C211yesmaximal, characteristic, complemented
12#9823C211yesmaximal, characteristic, complemented
13#8820C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#485—22no—
18C2×#483—22no—
19C2×#431C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2097—22no—
25#2096—22no—
26#2086—22no—
27#2083—22no—
28#2075—22no—
29#2075—22no—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1956C2211yescharacteristic
37#1956C2211yescharacteristic
38#1956C2211yescharacteristic
39#1956C2211yescharacteristic
40#1947—22no—
41#1918—22no—
42#1852—22no—
43#1852—22no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1845C2211yescharacteristic
49#1845C2211yescharacteristic
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1844C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1834C2211yescharacteristic
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C42C2311yescharacteristic
72#500—22no—
73#500—22no—
74#485—48no—
75#483—48no—
76#431—22no—
77#431—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×#106—22no—
88C2×C4⋊C8D412yes—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×C22⋊C8D412yes—
94C2×C22⋊C8D412yes—
95C2×C22⋊C8D412yes—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#391—21no—
103#391—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#384—21no—
109#384—21no—
110#384—21no—
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#382—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
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
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C22⋊Q8—44no—
161C22⋊Q8—44no—
162#117—44no—
163C2×C4⋊C4—21no—
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—44no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—41no—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2411yescharacteristic
205#107—44no—
206#107—44no—
207#107—44no—
208#107—82no—
209#107—82no—
210#107—82no—
211#107—82no—
212#106—44no—
213#106—44no—
214#106—44no—
215C4⋊C8—24no—
216#103—44no—
217#103—44no—
218#103—44no—
219#103—44no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222C22⋊C8D818yes—
223C4×C8—21no—
224C4×C8—21no—
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#4112yes—
234C4×C8#4112yes—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241C24#12711yescharacteristic
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4#12711yescharacteristic
273C22×C4#12711yescharacteristic
274C22×C4#12711yescharacteristic
275C22×C4C2×D814yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
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—22no—
292C2×C8—22no—
293C2×C8#12314yes—
294C2×C8C2×D814yes—
295C2×C8C2×D814yes—
296C2×C8#13714yes—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
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—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—82no—
338C42C2×#4111yescharacteristic
339C42C2×#4111yescharacteristic
340C422−1+411yescharacteristic
341C422−1+411yescharacteristic
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23#49312yes—
349C23C2×#12711yescharacteristic
350Q8—82no—
351Q8—82no—
352Q8—82no—
353Q8—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
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—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—164no—
393C2×C4#49312yes—
394C2×C4C2×#12311yescharacteristic
395C2×C4#53611yescharacteristic
396C2×C4#54711yescharacteristic
397C2×C4#54711yescharacteristic
398C2×C4#54711yescharacteristic
399C2×C4#54711yescharacteristic
400C2×C4C22×D812yes—
401C2×C4C2×#13712yes—
402C2×C4#57712yes—
403C2×C4#57712yes—
404C2×C4#57812yes—
405C2×C4#57812yes—
406C8—24no—
407C8—24no—
408C8—24no—
409C8—24no—
410C8—42no—
411C8—42no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22#250111yescharacteristic
420C22C2×#49311yescharacteristic
421C22#315211yescharacteristic
422C22#315411yescharacteristic
423C22#315411yescharacteristic
424C22#315611yescharacteristic
425C22#315611yescharacteristic
426C4—22no—
427C4—41no—
428C4—41no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4#309712yes—
438C4#324212yes—
439C4#333412yes—
440C4#333412yes—
441C4#334412yes—
442C4#335612yes—
443C2—42no—
444C2#1963011yescharacteristic
445C2#1966011yescharacteristic
446C2#1972011yescharacteristic
447C2#1973611yescharacteristic
448C2#2792211yescharacteristic
449C2#2792911yescharacteristic
450C2#2794011yescharacteristic
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. #10675C2semidirect
8. #10675C2semidirect
9. #10675C2semidirect
10. #10674C2semidirect
11. #9831C2semidirect
12. #9823C2semidirect
13. #8820C2semidirect
14. #8814C2semidirect

Maximal subgroups

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

subgroupindex
#298762
#279502
#279082
#279052
#278772
#219822
#106752
#106742
#98312
#98232
#88202
#88142
#81482