Finite Group Databasebeta

← #375685

Subgroups — #375685

Order 512.

1657 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic, complemented
3C42×C22⋊C4C211yesmaximal, characteristic, complemented
4#11201C212yesmaximal, complemented
5#10971C212yesmaximal
6#10933C212yesmaximal, complemented
7#10923C212yesmaximal, complemented
8#10886C212yesmaximal, complemented
9#10878C212yesmaximal, 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⋊C4—22no—
29C2×C4×C22⋊C4C2212yes—
30C2×C43C2212yes—
31#1749—22no—
32C4×C22⋊C8—22no—
33C2×#339C2214yes—
34C2×#339C2214yes—
35C2×#339C2214yes—
36#1728C2212yes—
37C4×#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—24no—
56C22×C42—44no—
57C22×C42C2×C412yes—
58C22×C42C2×C412yes—
59C22×C42D412yes—
60C22×C42D412yes—
61C22×C42D412yes—
62C22×C42C2314yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#104—28no—
66C2×#104—28no—
67C2×#104—44no—
68#407—28no—
69C2×C22⋊C8—24no—
70C4×M16—44no—
71C2×#97—22no—
72C2×C4×C8—22no—
73#391C2×C411yescharacteristic
74#391C2×C411yescharacteristic
75#391C2×C412yes—
76#390C2×C411yescharacteristic
77#390C2×C411yescharacteristic
78#390C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2×C412yes—
82#389C2×C412yes—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#386—22no—
92#386—22no—
93#386—22no—
94#386—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#381—22no—
100#381—22no—
101#381—21no—
102#380C2×C412yes—
103#380C2×C412yes—
104#379—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C22⋊C4—216no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—28no—
114C2×#95—22no—
115C2×#95—24no—
116C43—44no—
117#339—28no—
118#339—28no—
119#339—44no—
120#336—24no—
121C25C2×D411yescharacteristic
122C2×#41—24no—
123C2×#41—24no—
124C2×#41—24no—
125C22×Q8C22×C411yescharacteristic
126C22×D4—22no—
127C22×D4—21no—
128C22×D4C22×C411yescharacteristic
129C23×C4—24no—
130C23×C4—28no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—21no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2411yescharacteristic
147C2×M16—24no—
148C2×M16—24no—
149C2×M16—24no—
150C2×M16—24no—
151C2×M16—24no—
152C2×M16#4118yes—
153C22×C8—24no—
154#123—48no—
155#123—44no—
156#123—48no—
157C22⋊Q8—44no—
158#121—44no—
159#120—44no—
160C4×D4—48no—
161#117—24no—
162#117—24no—
163#117—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C22⋊C4—216no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—216no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—24no—
197C2×C42—44no—
198C2×C42—44no—
199C2×C42—44no—
200C2×C42—44no—
201C2×C42—44no—
202C2×C42—48no—
203C2×C42—48no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22⋊C412yes—
207C2×C42C22⋊C412yes—
208C2×C42C22×C412yes—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4118yes—
215#104—416no—
216C22⋊C8—216no—
217#97—44no—
218C4×C8—44no—
219#95—216no—
220#95—28no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231C24—22no—
232C24—24no—
233C24—22no—
234C24—22no—
235C24—21no—
236C24—42no—
237C24C2×C22⋊C412yes—
238C24C22×D411yescharacteristic
239#41—44no—
240#41—44no—
241#41—44no—
242#41—44no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—44no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—216no—
259C22×C4—22no—
260C22×C4—28no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—432no—
289C22×C4—48no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4#10418yescomplemented
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C2×#4114yes—
308C22×C4C2×#4114yes—
309C22×C4H(5,𝔽2)14yes—
310M16—28no—
311M16—28no—
312M16—44no—
313M16—44no—
314C2×C8—28no—
315C2×C8—44no—
316C2×C8—44no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C22⋊C4—232no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C42—22no—
339C42—22no—
340C42—24no—
341C42—22no—
342C42—22no—
343C42—24no—
344C42—28no—
345C42—28no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—432no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—44no—
355C42—44no—
356C23—22no—
357C23—22no—
358C23—28no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—48no—
366C23—48no—
367C23—44no—
368C23—44no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23C2×#10414yes—
375C23C22×C22⋊C411yescharacteristic
376C23#52512yes—
377C23#53212yes—
378C23#53412yes—
379Q8—44no—
380D4—44no—
381D4—88no—
382D4—84no—
383C2×C4—216no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—216no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—48no—
401C2×C4—48no—
402C2×C4—416no—
403C2×C4—416no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—44no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—84no—
423C2×C4C2×#10414yes—
424C2×C4C2×#10414yes—
425C2×C4#42114yes—
426C2×C4C4×#4112yes—
427C2×C4C4×#4112yes—
428C2×C4#51812yes—
429C2×C4#52512yes—
430C2×C4#53212yes—
431C2×C4#53412yes—
432C2×C4#55114yes—
433C2×C4#55314yes—
434C2×C4#55314yes—
435C2×C4#55914yes—
436C8—48no—
437C22—24no—
438C22—24no—
439C22—22no—
440C22—22no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—416no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—82no—
450C22—81no—
451C22—82no—
452C22—81no—
453C22—81no—
454C22#229711yescharacteristic
455C22#229711yescharacteristic
456C22#230811yescharacteristic
457C22#260214yes—
458C22C22×#10412yes—
459C22C2×#42112yes—
460C4—24no—
461C4—24no—
462C4—24no—
463C4—416no—
464C4—42no—
465C4—42no—
466C4—82no—
467C4—81no—
468C4—82no—
469C4—81no—
470C4—81no—
471C4#237414yes—
472C4#260214yes—
473C2—24no—
474C2—44no—
475C2—161no—
476C2#1416711yescharacteristic
477C2#2138212yes—
478C2#2141511yescharacteristic
479C2#2141811yescharacteristic
480C2#2177312yes—
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. C42×C22⋊C4C2semidirect
4. #11201C2semidirect
6. #10933C2semidirect
7. #10923C2semidirect
8. #10886C2semidirect
9. #10878C2semidirect
15. #2855C4semidirect
16. #2855C4semidirect
17. #2854C4semidirect
18. #2854C4semidirect
299. C22×C4#104semidirect

Maximal subgroups

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

subgroupindex
#366262
C4×#4072
C42×C22⋊C42
#112012
#109712
#109332
#109232
#108862
#108782