Finite Group Databasebeta

← #375686

Subgroups — #375686

Order 512.

1481 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic, complemented
3#13753C211yesmaximal, characteristic, complemented
4#11201C212yesmaximal, complemented
5#10971C212yesmaximal
6#10933C212yesmaximal, complemented
7#10924C212yesmaximal, complemented
8#10886C212yesmaximal, complemented
9#10879C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C22×#104C2214yes—
13C2×#407C2212yes—
14C2×C4×M16C2212yes—
15#2855C412yescomplemented
16#2855C412yescomplemented
17#2854C412yescomplemented
18#2854C412yescomplemented
19#2853—22no—
20#2849—22no—
21C4×#123—22no—
22C4×#123—22no—
23#2297C2211yescharacteristic
24#2297C2211yescharacteristic
25#2296C2211yescharacteristic
26#2296C2211yescharacteristic
27#2280C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29C2×#376C2212yes—
30#1749—22no—
31C4×C22⋊C8—22no—
32C2×#339C2214yes—
33C2×#339C2214yes—
34C2×#339C2214yes—
35#1728C2212yes—
36#1434—22no—
37C22⋊#95—22no—
38C4×#95C2212yes—
39C22×#41C2311yescharacteristic
40C24×C4D412yes—
41C24×C4C2311yescharacteristic
42C22×M16C2314yes—
43#525—44no—
44C2×#123—22no—
45C2×#123—22no—
46C2×#123—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×#121C2×C412yes—
49C2×#120—21no—
50C2×C4×D4—22no—
51C2×#117C2311yescharacteristic
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C22×C22⋊C4—22no—
55C22×C42—44no—
56C22×C42C2×C412yes—
57C22×C42C2×C412yes—
58C22×C42D412yes—
59C22×C42D412yes—
60C22×C42D412yes—
61C22×C42C2314yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#104—28no—
65C2×#104—28no—
66C2×#104—44no—
67#407—28no—
68C2×C22⋊C8—24no—
69C4×M16—44no—
70C2×#97—22no—
71C2×C4×C8—22no—
72#391C2×C411yescharacteristic
73#391C2×C411yescharacteristic
74#391C2×C412yes—
75#390C2×C411yescharacteristic
76#390C2×C411yescharacteristic
77#390C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2×C412yes—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#386—22no—
91#386—22no—
92#386—22no—
93#386—22no—
94#385—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#381—22no—
99#381—22no—
100#381—21no—
101#380C2×C412yes—
102#380C2×C412yes—
103#379—21no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—28no—
112#376—44no—
113C2×#95—22no—
114C2×#95—24no—
115C2×#95—24no—
116#339—28no—
117#339—28no—
118#339—44no—
119#336—24no—
120C25C2×D411yescharacteristic
121C2×#41—24no—
122C2×#41—24no—
123C2×#41—24no—
124C22×Q8C22×C411yescharacteristic
125C22×D4—22no—
126C22×D4—21no—
127C22×D4C22×C411yescharacteristic
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—21no—
133C23×C4—44no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C22×C412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2411yescharacteristic
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—24no—
149C2×M16—24no—
150C2×M16—24no—
151C2×M16#4118yes—
152C22×C8—24no—
153#123—48no—
154#123—48no—
155#123—44no—
156C22⋊Q8—44no—
157#121—44no—
158#120—44no—
159C4×D4—48no—
160#117—24no—
161#117—24no—
162#117—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C22⋊C4—28no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—48no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—24no—
195C2×C42—44no—
196C2×C42—44no—
197C2×C42—44no—
198C2×C42—44no—
199C2×C42—44no—
200C2×C42C22⋊C412yes—
201C2×C42C22⋊C412yes—
202C2×C42C22⋊C412yes—
203C2×C42C22⋊C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4118yes—
211#104—416no—
212C22⋊C8—216no—
213#97—44no—
214C4×C8—44no—
215#95—28no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—48no—
227#95—48no—
228C24—22no—
229C24—24no—
230C24—22no—
231C24—22no—
232C24—21no—
233C24—42no—
234C24C2×C22⋊C412yes—
235C24C22×D411yescharacteristic
236#41—44no—
237#41—44no—
238#41—44no—
239#41—44no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—44no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—28no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—432no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4114yes—
305C22×C4C2×#4114yes—
306C22×C4H(5,𝔽2)14yes—
307M16—28no—
308M16—28no—
309M16—44no—
310M16—44no—
311C2×C8—28no—
312C2×C8—44no—
313C2×C8—44no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—416no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C42—24no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—24no—
341C42—28no—
342C42—28no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—44no—
351C42—44no—
352C23—22no—
353C23—22no—
354C23—28no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—48no—
362C23—48no—
363C23—44no—
364C23—44no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—41no—
369C23—41no—
370C23#42114yes—
371C23C22×C22⋊C411yescharacteristic
372C23#52512yes—
373C23#53212yes—
374C23#53412yes—
375Q8—44no—
376D4—44no—
377D4—88no—
378D4—84no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—48no—
401C2×C4—48no—
402C2×C4—48no—
403C2×C4—48no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—48no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—88no—
419C2×C4—88no—
420C2×C4—84no—
421C2×C4#42114yes—
422C2×C4C4×#4112yes—
423C2×C4C4×#4112yes—
424C2×C4#51812yes—
425C2×C4#52512yes—
426C2×C4#53212yes—
427C2×C4#53412yes—
428C2×C4#55114yes—
429C2×C4#55314yes—
430C2×C4#55314yes—
431C2×C4#55914yes—
432C8—48no—
433C22—24no—
434C22—24no—
435C22—22no—
436C22—22no—
437C22—24no—
438C22—24no—
439C22—24no—
440C22—416no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—82no—
446C22—81no—
447C22—82no—
448C22—81no—
449C22—81no—
450C22#229711yescharacteristic
451C22#229711yescharacteristic
452C22#230811yescharacteristic
453C22#260214yes—
454C22C2×#42112yes—
455C22C2×#42112yes—
456C4—24no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—88no—
462C4—82no—
463C4—81no—
464C4—82no—
465C4—81no—
466C4—81no—
467C4#237414yes—
468C4#260214yes—
469C2—24no—
470C2—44no—
471C2—161no—
472C2#1416711yescharacteristic
473C2#2138312yes—
474C2#2141611yescharacteristic
475C2#2141711yescharacteristic
476C2#2177412yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#407C2semidirect
3. #13753C2semidirect
4. #11201C2semidirect
6. #10933C2semidirect
7. #10924C2semidirect
8. #10886C2semidirect
9. #10879C2semidirect
15. #2855C4semidirect
16. #2855C4semidirect
17. #2854C4semidirect
18. #2854C4semidirect

Maximal subgroups

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

subgroupindex
#366262
C4×#4072
#137532
#112012
#109712
#109332
#109242
#108862
#108792