Finite Group Databasebeta

← #215443

Subgroups — #215443

Order 512.

970 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2#21912C211yesmaximal, characteristic
3#13949C211yesmaximal, characteristic
4#9714C212yesmaximal
5#9711C212yesmaximal
6#9679C212yesmaximal
7#9679C212yesmaximal
8#8333C212yesmaximal
9#8333C212yesmaximal
10C2×#533C2212yes—
11#2857C2212yes—
12#2857C2212yes—
13#2659—21no—
14#2659—21no—
15#2584—24no—
16#2567C414yescomplemented
17#2546C414yescomplemented
18#2316—21no—
19#2316—21no—
20C2×#387—21no—
21C2×C4×C4⋊C4C2211yescharacteristic
22#1916—24no—
23#1817C2212yes—
24#1797C2212yes—
25C4×C22⋊C8C2212yes—
26C2×#328C2214yes—
27C2×#328C2214yes—
28C2×#328C2214yes—
29C2×#328C2214yes—
30#1442—21no—
31#1441—21no—
32#1436—21no—
33C4⋊#95C2211yescharacteristic
34C4⋊#95C2211yescharacteristic
35#1434C2211yescharacteristic
36#1434C2211yescharacteristic
37#1434C2211yescharacteristic
38C4×#95C2211yescharacteristic
39#533—44no—
40#533—44no—
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×#125—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8C2314yes—
48C2×C4×Q8—22no—
49C2×C4×Q8C2×C414yes—
50C2×#117—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×C4⋊C8—22no—
60C2×C22⋊C8C2314yes—
61C2×C4×C8—22no—
62#398—24no—
63#398—24no—
64#395—24no—
65#395—24no—
66#391C2×C414yes—
67#391C2×C414yes—
68#389—24no—
69#389—24no—
70#389—24no—
71#389—24no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#387—44no—
77#384C2×C412yes—
78#384C2×C412yes—
79#380—22no—
80#380—22no—
81#380—22no—
82#380—22no—
83C4×C4⋊C4C2×C414yes—
84C4×C4⋊C4D414yes—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95D411yescharacteristic
95C2×#95D411yescharacteristic
96C2×#95C2311yescharacteristic
97C2×#95C2311yescharacteristic
98C2×#95C2311yescharacteristic
99C2×#95C2311yescharacteristic
100#336—22no—
101#336—24no—
102#336—22no—
103#328—44no—
104#328—44no—
105#328—44no—
106#328—44no—
107C22×Q8—22no—
108C22×Q8C22×C414yes—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C22×C8—24no—
121C4⋊Q8—44no—
122C4⋊Q8—44no—
123C4⋊Q8—82no—
124C4⋊Q8—82no—
125#125—48no—
126C22⋊Q8—24no—
127C22⋊Q8—24no—
128C22⋊Q8—24no—
129C22⋊Q8—24no—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132C4×Q8—44no—
133C4×Q8—44no—
134C4×Q8—48no—
135#117—44no—
136#117—44no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
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—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—81no—
157C2×C4⋊C4—81no—
158C2×C4⋊C4C22⋊C414yes—
159C2×C4⋊C4C22⋊C414yes—
160C2×C4⋊C4SD1614yes—
161C2×C4⋊C4Q1614yes—
162C2×C4⋊C4C22×C414yes—
163C2×C22⋊C4—24no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—24no—
167C2×C42—42no—
168C2×C42—42no—
169C2×C42—81no—
170C2×C42—81no—
171C2×C42—81no—
172C2×C42C22×C412yes—
173C2×C42C22×C412yes—
174C2×C42C22×C412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C4⋊C8—82no—
179C22⋊C8—24no—
180C22⋊C8—24no—
181C4×C8—82no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—42no—
191#95—42no—
192#95—42no—
193#95—42no—
194#95—42no—
195#95—42no—
196#95—42no—
197#95—42no—
198#95—81no—
199#95C22⋊C414yes—
200#95C22⋊C414yes—
201C24C22×D411yescharacteristic
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—44no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—82no—
208C2×Q8—82no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×SD1612yes—
266C22×C4C2×SD1612yes—
267C22×C4C2×SD1612yes—
268C22×C4C2×Q1612yes—
269C22×C4C2×Q1612yes—
270C22×C4C2×Q1612yes—
271C22×C4#13712yes—
272C22×C4#13712yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4H(5,𝔽2)12yes—
276C22×C4H(5,𝔽2)12yes—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—44no—
280C4⋊C4—28no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—84no—
298C4⋊C4—84no—
299C4⋊C4#103116yes—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C42—24no—
303C42—24no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—48no—
309C42—84no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23C22×C22⋊C411yescharacteristic
315C23#53511yescharacteristic
316C23#53511yescharacteristic
317C23C22×SD1611yescharacteristic
318C23C22×Q1611yescharacteristic
319C23C2×#13711yescharacteristic
320C23C2×#13711yescharacteristic
321Q8—84no—
322Q8—84no—
323Q8—88no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
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—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4C2×#10314yes—
393C2×C4C2×#10314yes—
394C2×C4C2×#10314yes—
395C2×C4#41914yes—
396C2×C4#51812yes—
397C2×C4#53512yes—
398C8—84no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22#230911yescharacteristic
412C22#288311yescharacteristic
413C22#288411yescharacteristic
414C22C22×#10311yescharacteristic
415C22C2×#41911yescharacteristic
416C22#290111yescharacteristic
417C22#290111yescharacteristic
418C22#320611yescharacteristic
419C22#320611yescharacteristic
420C22#320911yescharacteristic
421C22#320911yescharacteristic
422C4—22no—
423C4—22no—
424C4—24no—
425C4—24no—
426C4—24no—
427C4—24no—
428C4—42no—
429C4—42no—
430C4—82no—
431C4—82no—
432C4—84no—
433C4—84no—
434C4—161no—
435C4—161no—
436C2—21no—
437C2—21no—
438C2—21no—
439C2—21no—
440C2#1761411yescharacteristic
441C2#1761911yescharacteristic
442C2#1762111yescharacteristic
443C2#1763211yescharacteristic
444C2#2179611yescharacteristic
445C2#2180411yescharacteristic
446C2#2181711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2567C4semidirect
17. #2546C4semidirect

Maximal subgroups

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

subgroupindex
#351732
#219122
#139492
#97142
#97112
#96792
#83332