Finite Group Databasebeta

← #315709

Subgroups — #315709

Order 512.

680 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic
2#27975C211yesmaximal, characteristic
3#27908C211yesmaximal, characteristic
4#27877C211yesmaximal, characteristic
5#27876C211yesmaximal, characteristic
6#20816C211yesmaximal, characteristic
7#10685C211yesmaximal, characteristic, complemented
8#10684C211yesmaximal, characteristic, complemented
9#10681C211yesmaximal, characteristic, complemented
10#10680C211yesmaximal, characteristic, complemented
11#9834C211yesmaximal, characteristic, complemented
12#9834C211yesmaximal, characteristic, complemented
13#8821C211yesmaximal, characteristic, complemented
14#8820C211yesmaximal, characteristic, complemented
15#8145C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#485—22no—
18C22×C4×C8C2211yescharacteristic
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#2097—22no—
24#2077—22no—
25#2075—22no—
26#2074—22no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1956C2211yescharacteristic
34#1956C2211yescharacteristic
35#1956C2211yescharacteristic
36#1956C2211yescharacteristic
37#1942—22no—
38#1919—22no—
39#1918—22no—
40#1873—22no—
41#1845C2211yescharacteristic
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1844C2211yescharacteristic
46#1844C2211yescharacteristic
47#1844C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1808—22no—
55#1807—22no—
56C4×#106—22no—
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C23×C8D412yes—
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×#117—21no—
69C2×#117—21no—
70C22×C42C2311yescharacteristic
71#500—22no—
72#500—22no—
73#485—48no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#106—22no—
79C2×#106—22no—
80C2×#106—22no—
81C2×#106—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×C22⋊C8D412yes—
87C2×C22⋊C8D412yes—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8D412yes—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#391—21no—
98#391—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#382—21no—
110#382—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
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#340—22no—
135#336D412yes—
136#336D412yes—
137#322—22no—
138#322—22no—
139C22×Q8—21no—
140C22×Q8—21no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C22×C8—24no—
145C22×C8—24no—
146C22×C8—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—44no—
163#117—44no—
164#117—82no—
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—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2411yescharacteristic
203#107—44no—
204#107—44no—
205#107—44no—
206#107—44no—
207#106—44no—
208#106—44no—
209#106—44no—
210#106—44no—
211#103—44no—
212#103—44no—
213#103—82no—
214#103—82no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C4×C8—24no—
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—21no—
228C4×C8#4112yes—
229C4×C8#4112yes—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238C24#12711yescharacteristic
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4#12711yescharacteristic
267C22×C4#12711yescharacteristic
268C22×C4#12711yescharacteristic
269C22×C4AGL(1,ℤ/8)14yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C2×C8—22no—
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—48no—
289C2×C8#12314yes—
290C2×C8#13514yes—
291C2×C8#13714yes—
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—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—82no—
331C42—82no—
332C42C2×#4111yescharacteristic
333C42C2×#4111yescharacteristic
334C422−1+411yescharacteristic
335C422−1+411yescharacteristic
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23#49612yes—
343C23C2×#12711yescharacteristic
344Q8—82no—
345Q8—82no—
346Q8—82no—
347Q8—82no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—81no—
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#49612yes—
387C2×C4C2×#12311yescharacteristic
388C2×C4#53611yescharacteristic
389C2×C4#54711yescharacteristic
390C2×C4#54711yescharacteristic
391C2×C4#54711yescharacteristic
392C2×C4#54711yescharacteristic
393C2×C4C2×#13512yes—
394C2×C4C2×AGL(1,ℤ/8)12yes—
395C2×C4C2×#13712yes—
396C2×C4#57512yes—
397C2×C4#57712yes—
398C2×C4#57812yes—
399C8—24no—
400C8—24no—
401C8—24no—
402C8—42no—
403C8—42no—
404C8—42no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22#250111yescharacteristic
413C22#314711yescharacteristic
414C22C2×#49611yescharacteristic
415C22#315011yescharacteristic
416C22#315211yescharacteristic
417C22#315411yescharacteristic
418C22#315611yescharacteristic
419C4—22no—
420C4—41no—
421C4—41no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4#309512yes—
431C4#323712yes—
432C4#333612yes—
433C4#334212yes—
434C4#334512yes—
435C4#334512yes—
436C2—42no—
437C2#1961911yescharacteristic
438C2#1964211yescharacteristic
439C2#1967511yescharacteristic
440C2#1970711yescharacteristic
441C2#2791611yescharacteristic
442C2#2792711yescharacteristic
443C2#2793411yescharacteristic
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. #10685C2semidirect
8. #10684C2semidirect
9. #10681C2semidirect
10. #10680C2semidirect
11. #9834C2semidirect
12. #9834C2semidirect
13. #8821C2semidirect
14. #8820C2semidirect

Maximal subgroups

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

subgroupindex
#332092
#279752
#279082
#278772
#278762
#208162
#106852
#106842
#106812
#106802
#98342
#88212
#88202
#81452