Finite Group Databasebeta

← #6528573

Subgroups — #6528573

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2C2×#2570C211yesmaximal, characteristic
3#29335C212yesmaximal
4#29329C212yesmaximal
5#27983C212yesmaximal
6#27972C212yesmaximal
7#27762C218yesmaximal, complemented
8#27746C218yesmaximal, complemented
9#21866C212yesmaximal
10#21394C212yesmaximal
11#20534C211yesmaximal, characteristic
12C2×#533C2212yes—
13C2×#432C2212yes—
14#2857C2212yes—
15#2857C2212yes—
16#2659—21no—
17#2659—21no—
18#2584—24no—
19#2570—24no—
20#2570C2218yes—
21#2567C2214yes—
22#2546C2214yes—
23C2×#400—21no—
24C2×#398—21no—
25#2316—21no—
26#2316—21no—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#387—21no—
30C2×#387—21no—
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#2100C2214yes—
37#2099C2214yes—
38#2074C2218yes—
39#2072C2218yes—
40#2043C2214yes—
41#2043C2214yes—
42#1993C2214yes—
43#1988C2214yes—
44#1981C2214yes—
45#1960C2214yes—
46#1960C2214yes—
47#1956C2214yes—
48#1935C2218yes—
49#1933C2218yes—
50#1872C2214yes—
51#1872C2214yes—
52#1872C2214yes—
53#1872C2214yes—
54#1836C2214yes—
55#1834C2214yes—
56#1805C2218yes—
57#1771C2218yes—
58#1763C2218yes—
59#1749C2212yes—
60C4×C22⋊C8C2212yes—
61#533—28no—
62#533—28no—
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×#125—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8C2314yes—
70C2×C4×Q8—22no—
71C2×C4×Q8C2314yes—
72C2×#117—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#432—28no—
88C2×C4⋊C8C2314yes—
89C2×#103C2318yes—
90C2×#103C2318yes—
91C2×#103C2318yes—
92C2×#103C2318yes—
93C2×C22⋊C8C2314yes—
94C2×#97C2314yes—
95C2×#97C2314yes—
96C2×C4×C8C2314yes—
97#400—28no—
98#398—28no—
99#398—24no—
100#398—24no—
101#395—24no—
102#395—24no—
103#391C2314yes—
104#391C2314yes—
105#389—24no—
106#389—24no—
107#389—24no—
108#389—24no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389—22no—
114#389—22no—
115#389C2314yes—
116#389C2314yes—
117#387—28no—
118#387—28no—
119#384—22no—
120#384C2312yes—
121#384C2312yes—
122#384C2314yes—
123#382—22no—
124#382—22no—
125#382C2314yes—
126#382C2314yes—
127#380—22no—
128#380—22no—
129#380—22no—
130#380—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4D414yes—
133C4×C4⋊C4C2314yes—
134C4×C4⋊C4C2314yes—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95D411yescharacteristic
143C2×#95D411yescharacteristic
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#336C2314yes—
149#336C2314yes—
150C22×Q8—22no—
151C22×Q8C2414yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8C2418yes—
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170#125—48no—
171C22⋊Q8—28no—
172C22⋊Q8—28no—
173C22⋊Q8—48no—
174C4×Q8—28no—
175C4×Q8—28no—
176C4×Q8—48no—
177#117—48no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2412yes—
212C2×C4⋊C4C2412yes—
213C2×C4⋊C4C2412yes—
214C2×C4⋊C4C2412yes—
215C2×C4⋊C4C2414yes—
216C2×C22⋊C4—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—82no—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42#4118yes—
226C2×C42C2412yes—
227C2×C42C2412yes—
228C2×C42C2412yes—
229C4⋊C8—28no—
230#103—48no—
231#103—48no—
232#103—48no—
233#103—48no—
234C22⋊C8—28no—
235#97—28no—
236#97—44no—
237C4×C8—44no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95C2×D414yes—
250#95C2×D414yes—
251C24C22×D411yescharacteristic
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—28no—
255C2×Q8—28no—
256C2×Q8—44no—
257C2×Q8—82no—
258C2×Q8—82no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4#12414yes—
294C22×C4#12414yes—
295C22×C4#13714yes—
296C22×C4#13714yes—
297C22×C4C22×D412yes—
298C22×C4C22×D412yes—
299C22×C4C22×D412yes—
300C22×C4C22×D412yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D412yes—
303C22×C4C22×D412yes—
304C22×C4C2×#4114yes—
305C22×C4C2511yescharacteristic
306C2×C8—28no—
307C2×C8—28no—
308C2×C8—28no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—48no—
325C4⋊C4—44no—
326C4⋊C4—48no—
327C4⋊C4—48no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—48no—
339C42—24no—
340C42—24no—
341C42—42no—
342C42—42no—
343C42—48no—
344C42—84no—
345C42#12418yes—
346C42C2×#4114yes—
347C42C2×#4114yes—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23C2×#12412yes—
353C23C2×#13712yes—
354C23C2×#13712yes—
355C23C23×D411yescharacteristic
356Q8—48no—
357Q8—48no—
358Q8—88no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—28no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—28no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4C2×#12414yes—
405C2×C4C2×#12412yes—
406C2×C4C2×#12414yes—
407C2×C4C2×#13714yes—
408C2×C4C2×#13714yes—
409C2×C4C22×#4112yes—
410C2×C4#58512yes—
411C2×C4#58512yes—
412C8—48no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22#313914yes—
420C22C22×#12411yescharacteristic
421C22#348511yescharacteristic
422C22#348711yescharacteristic
423C22C22×#13711yescharacteristic
424C22C22×#13711yescharacteristic
425C22#358511yescharacteristic
426C22#358611yescharacteristic
427C4—24no—
428C4—24no—
429C4—22no—
430C4—24no—
431C4—24no—
432C4—42no—
433C4—42no—
434C4—82no—
435C4—84no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4—84no—
440C4#351214yes—
441C2—24no—
442C2#3564911yescharacteristic
443C2C2×#313911yescharacteristic
444C2#6062111yescharacteristic
445C2#6170511yescharacteristic
446C2#6170811yescharacteristic
447C2#6188911yescharacteristic
448C2#6189211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #27762C2semidirect
8. #27746C2semidirect

Maximal subgroups

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

subgroupindex
#351732
C2×#25702
#293352
#293292
#279832
#279722
#277622
#277462
#218662
#213942
#205342