Finite Group Databasebeta

← #215363

Subgroups — #215363

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic
3C4×#382C211yesmaximal, characteristic
4#9711C214yesmaximal
5#9679C214yesmaximal
6#8333C214yesmaximal
7C2×#533C2212yes—
8#2868C411yescharacteristic, complemented
9#2868C411yescharacteristic, complemented
10#2862C412yescomplemented
11#2859—21no—
12#2859—21no—
13#2857—21no—
14#2857—21no—
15#2857C2212yes—
16#2857C2212yes—
17C4×C4⋊Q8—21no—
18C4×C4⋊Q8—21no—
19C4×C4⋊Q8C412yescomplemented
20C4×C4⋊Q8C412yescomplemented
21#2285—21no—
22#2285—21no—
23C2×#382—21no—
24C2×C4×C4⋊C4—21no—
25C2×C43C2211yescharacteristic
26#1839—21no—
27#1831—21no—
28#1817C2212yes—
29#1797C2212yes—
30C4×C4⋊C8—21no—
31C4×C22⋊C8C2212yes—
32#1744—21no—
33C2×#328C22116yes—
34C4⋊#95C2212yes—
35#1434—21no—
36#1434C2212yes—
37C4×#95—21no—
38C4×#95C2212yes—
39#533—48no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×#125—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8C2314yes—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C414yes—
51C2×#117—22no—
52C22×C4⋊C4—22no—
53C22×C42—22no—
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×C4⋊C8—22no—
62C2×C22⋊C8C2314yes—
63C2×C4×C8—22no—
64#391—22no—
65#391—22no—
66#391C2×C414yes—
67#391C2×C414yes—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#384—21no—
77#384—21no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384C2×C412yes—
83#384C2×C412yes—
84#382—28no—
85#382—22no—
86#382—22no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—44no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95C2314yes—
101C43C2×C411yescharacteristic
102C43C2×C411yescharacteristic
103C43C2×C411yescharacteristic
104C43C2×C411yescharacteristic
105C43D411yescharacteristic
106C43D411yescharacteristic
107C43D411yescharacteristic
108C43D411yescharacteristic
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#328—416no—
114C22×Q8—22no—
115C22×Q8C22×C414yes—
116C23×C4—24no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—24no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131#125—48no—
132C22⋊Q8—216no—
133C22⋊Q8—48no—
134C4×Q8—48no—
135C4×Q8—48no—
136#117—48no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4C22×C414yes—
154C2×C22⋊C4—24no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42—44no—
163C2×C42—44no—
164C2×C42—44no—
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C411yescharacteristic
180C2×C42C22⋊C411yescharacteristic
181C2×C42SD1614yes—
182C2×C42Q1614yes—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C4⋊C8—44no—
196C22⋊C8—28no—
197C4×C8—44no—
198#95—28no—
199#95—28no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—28no—
205#95—44no—
206#95—44no—
207C24C22×D411yescharacteristic
208C2×Q8—28no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—44no—
213C2×Q8—44no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—28no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×SD1612yes—
252C22×C4C2×SD1612yes—
253C22×C4C2×SD1612yes—
254C22×C4C2×Q1612yes—
255C22×C4C2×Q1612yes—
256C22×C4C2×Q1612yes—
257C22×C4#13714yes—
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4H(5,𝔽2)14yes—
261C2×C8—44no—
262C2×C8—44no—
263C2×C8—44no—
264C4⋊C4—28no—
265C4⋊C4—416no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—48no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—48no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C22⋊C4—48no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—41no—
298C42—41no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42—41no—
303C42—48no—
304C42—88no—
305C42#10314yes—
306C42#10314yes—
307C42#10314yes—
308C42#10314yes—
309C42#10414yes—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23C22×C22⋊C411yescharacteristic
315C23#53512yes—
316C23C22×SD1611yescharacteristic
317C23C22×Q1611yescharacteristic
318C23C2×#13712yes—
319Q8—88no—
320Q8—88no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4#41012yes—
371C2×C4#41012yes—
372C2×C4C2×#10312yes—
373C2×C4C2×#10312yes—
374C2×C4C2×#10312yes—
375C2×C4C2×#10312yes—
376C2×C4C2×#10312yes—
377C2×C4C2×#10312yes—
378C2×C4#41912yes—
379C2×C4#41912yes—
380C2×C4C2×#10412yes—
381C2×C4C2×#10412yes—
382C2×C4C2×#10412yes—
383C2×C4#42112yes—
384C2×C4#51812yes—
385C2×C4#53512yes—
386C8—84no—
387C22—22no—
388C22—22no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22#230911yescharacteristic
396C22C2×#41011yescharacteristic
397C22C2×#41011yescharacteristic
398C22C22×#10311yescharacteristic
399C22C2×#41911yescharacteristic
400C22C22×#10411yescharacteristic
401C22C2×#42111yescharacteristic
402C22#320612yes—
403C22#320912yes—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—41no—
415C4—41no—
416C4—88no—
417C4—84no—
418C4—84no—
419C4#150314yes—
420C4#151814yes—
421C2—22no—
422C2—22no—
423C2C2×#150311yescharacteristic
424C2#1754511yescharacteristic
425C2C2×#151811yescharacteristic
426C2#1759611yescharacteristic
427C2#2179411yescharacteristic
428C2#2180411yescharacteristic
429C2#2181511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #2868C4semidirect
9. #2868C4semidirect
10. #2862C4semidirect
19. C4×C4⋊Q8C4semidirect
20. C4×C4⋊Q8C4semidirect

Maximal subgroups

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

subgroupindex
#366452
#207772
C4×#3822
#97112
#96792
#83332