Finite Group Databasebeta

← #6497304

Subgroups — #6497304

Order 512.

1541 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#33157C214yesmaximal, complemented
4#29949C214yesmaximal, complemented
5#27962C212yesmaximal, complemented
6#27948C212yesmaximal, complemented
7#27876C212yesmaximal, complemented
8#27875C212yesmaximal, complemented
9#27621C214yesmaximal, complemented
10#27609C214yesmaximal
11#21981C212yesmaximal, complemented
12#21835C212yesmaximal, complemented
13#20718C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15C2×C4×#41C2211yescharacteristic
16C2×#431C2212yes—
17#2857C2212yes—
18#2857C2212yes—
19#2629C2212yes—
20#2629C2212yes—
21#2582—22no—
22#2538—22no—
23#2417—22no—
24#2417—22no—
25#2355—22no—
26#2354—22no—
27#2352—22no—
28#2352—22no—
29#2316C2212yes—
30#2314C2212yes—
31#2314C2212yes—
32#2311C2212yes—
33#2311C2212yes—
34#2285C2212yes—
35#2285C2212yes—
36#2285C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#384C2211yescharacteristic
39#2274C2212yes—
40#2101C2218yes—
41#2084C2218yes—
42#2077C2218yes—
43#2077C2218yes—
44#1990C2212yes—
45#1990C2212yes—
46#1990C2212yes—
47#1990C2212yes—
48#1989C2218yes—
49#1948C2218yes—
50#1920C2218yes—
51#1919C2218yes—
52#1853C2218yes—
53#1853C2218yes—
54#1844C2212yes—
55#1844C2212yes—
56#1834C2212yes—
57#1834C2212yes—
58#1825C2218yes—
59#1825C2218yes—
60#1818C2212yes—
61#1798C2212yes—
62#1798C2212yes—
63#1780C2212yes—
64C4×C22⋊C8C2212yes—
65C4×C22⋊C8C2211yescharacteristic
66C4×C22⋊C8C2211yescharacteristic
67C22×#41C2311yescharacteristic
68#533—28no—
69C2×C4⋊Q8C2312yes—
70C2×#125C2312yes—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C4×#41—28no—
75C2×C4×Q8—22no—
76C2×C4×Q8D412yes—
77C2×C4×D4D412yes—
78C2×C4×D4D412yes—
79C2×C4×D4D412yes—
80C2×#117—22no—
81C2×#117—22no—
82C2×#117C2312yes—
83C2×#117C2312yes—
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#431—28no—
95C2×#106C2318yes—
96C2×#106C2318yes—
97C2×C4⋊C8C2314yes—
98C2×C22⋊C8C2314yes—
99C2×C22⋊C8C2314yes—
100C2×C22⋊C8C2312yes—
101C2×C22⋊C8C2312yes—
102C2×C4×C8C2312yes—
103C2×C4×C8C2312yes—
104#398—24no—
105#398—24no—
106#396—24no—
107#396—24no—
108#391—22no—
109#391—22no—
110#391—22no—
111#391—22no—
112#390—22no—
113#390—22no—
114#389—24no—
115#389—22no—
116#389—22no—
117#389C2314yes—
118#389C2314yes—
119#388—24no—
120#388—24no—
121#387—24no—
122#387—24no—
123#386—22no—
124#386—22no—
125#386—22no—
126#386—22no—
127#384—22no—
128#384—22no—
129#384C2312yes—
130#384C2312yes—
131#384C2312yes—
132#384C2312yes—
133#384C2312yes—
134#382—22no—
135#382—22no—
136#382C2312yes—
137#380—22no—
138#380—22no—
139#380—22no—
140#380—22no—
141#380—22no—
142#380—24no—
143#380C2314yes—
144#380C2314yes—
145#380C2314yes—
146#380C2314yes—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4C2314yes—
152C4×C4⋊C4C2314yes—
153C4×C22⋊C4—24no—
154#340C2318yes—
155#340C2318yes—
156#340C2318yes—
157#340C2318yes—
158#340C2318yes—
159#340C2318yes—
160#336C2314yes—
161#336C2312yes—
162#336C2312yes—
163C2×#41—24no—
164C2×#41—24no—
165C2×#41—42no—
166C22×Q8—22no—
167C22×Q8C2×D411yescharacteristic
168C22×D4C2×D411yescharacteristic
169C22×D4C2×D411yescharacteristic
170C22×D4C2×D411yescharacteristic
171C23×C4—22no—
172C23×C4—22no—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C22×C8C2414yes—
180C22×C8C2414yes—
181C4⋊Q8—28no—
182#125—28no—
183C22⋊Q8—48no—
184C22⋊Q8—84no—
185C22⋊Q8—84no—
186C4×Q8—28no—
187C4×Q8—48no—
188C4×D4—28no—
189C4×D4—28no—
190C4×D4—28no—
191#117—24no—
192#117—24no—
193#117—24no—
194#117—28no—
195#117—28no—
196#117—84no—
197#117—84no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—84no—
215C2×C4⋊C4—84no—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2414yes—
220C2×C4⋊C4C2414yes—
221C2×C4⋊C4C2414yes—
222C2×C4⋊C4C2414yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4C2×D412yes—
230C2×C22⋊C4C2×D412yes—
231C2×C22⋊C4C2×D412yes—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42#4118yes—
242C2×C42C2411yescharacteristic
243C2×C42C2411yescharacteristic
244C2×C42C2411yescharacteristic
245C2×C42C2411yescharacteristic
246#106—48no—
247#106—48no—
248C4⋊C8—28no—
249C22⋊C8—28no—
250C22⋊C8—28no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C4×C8—24no—
254C4×C8—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264C24C22×D411yescharacteristic
265C24C22×D411yescharacteristic
266C24C22×D411yescharacteristic
267#41—416no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4#12318yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D412yes—
309C22×C4C22×D412yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D412yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D412yes—
314C22×C4C2×#4114yes—
315C22×C42−1+414yes—
316C22×C4C2511yescharacteristic
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—28no—
329C4⋊C4—28no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—48no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—84no—
350C22⋊C4—84no—
351C22⋊C4—84no—
352C22⋊C4—84no—
353C42—24no—
354C42—24no—
355C42—22no—
356C42—22no—
357C42—44no—
358C42—44no—
359C42—84no—
360C42—84no—
361C42#12318yes—
362C42C2×#4114yes—
363C42C2×#4114yes—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—22no—
376C23—22no—
377C23C2×#12312yes—
378C23#53612yes—
379C23C23×D411yescharacteristic
380Q8—44no—
381Q8—88no—
382D4—44no—
383D4—44no—
384D4—44no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—216no—
392C2×C4—28no—
393C2×C4—28no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—164no—
419C2×C4—164no—
420C2×C4C2×#12312yes—
421C2×C4C2×#12314yes—
422C2×C4C2×#12314yes—
423C2×C4#53612yes—
424C2×C4#57714yes—
425C2×C4#57714yes—
426C2×C4C22×#4112yes—
427C2×C4C2×2−1+412yes—
428C2×C4#58512yes—
429C8—44no—
430C8—44no—
431C22—24no—
432C22—24no—
433C22—22no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—42no—
440C22—42no—
441C22—41no—
442C22—41no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22C22×#12311yescharacteristic
447C22C2×#53611yescharacteristic
448C22#344911yescharacteristic
449C22C2×#57712yes—
450C22C2×#57712yes—
451C4—24no—
452C4—22no—
453C4—22no—
454C4—42no—
455C4—42no—
456C4—41no—
457C4—84no—
458C4—84no—
459C4—84no—
460C4—84no—
461C4#309918yes—
462C4#349514yes—
463C2—42no—
464C2—42no—
465C2—81no—
466C2#3531111yescharacteristic
467C2C2×#309912yes—
468C2#6088112yes—
469C2#6101211yescharacteristic
470C2#6101211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #33157C2semidirect
4. #29949C2semidirect
5. #27962C2semidirect
6. #27948C2semidirect
7. #27876C2semidirect
8. #27875C2semidirect
9. #27621C2semidirect
11. #21981C2semidirect
12. #21835C2semidirect

Maximal subgroups

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

subgroupindex
#352802
#341882
#331572
#299492
#279622
#279482
#278762
#278752
#276212
#276092
#219812
#218352
#207182