Finite Group Databasebeta

← #402457

Subgroups — #402457

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34182C211yesmaximal, characteristic
2#23747C212yesmaximal
3#13976C211yesmaximal, characteristic
4#11359C212yesmaximal
5#11296C212yesmaximal
6#11210C212yesmaximal
7#10763C211yesmaximal, characteristic
8#10750C212yesmaximal
9#10750C212yesmaximal
10C2×#519C2212yes—
11C22×C4×Q8C2211yescharacteristic
12C22×#101C2212yes—
13#2351—24no—
14Q8×C4⋊C4—22no—
15Q8×C4⋊C4—22no—
16#2316—22no—
17#2316C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#389C2211yescharacteristic
21#2285—22no—
22#2285C2212yes—
23C2×#384C2211yescharacteristic
24#2101—24no—
25#2100—22no—
26#2099—22no—
27C2×#343C2212yes—
28#1736C2214yes—
29#1736C2214yes—
30C2×#339C2214yes—
31C2×#339C2214yes—
32#1728C2212yes—
33#1444—22no—
34#1436—22no—
35C4⋊#95C2211yescharacteristic
36C4×#95C2211yescharacteristic
37C23×Q8C2311yescharacteristic
38C22×M16C2314yes—
39#519—24no—
40#519—24no—
41#519—44no—
42C2×C4×Q8—24no—
43C2×C4×Q8—24no—
44C2×C4×Q8—24no—
45C2×C4×Q8—24no—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×#117—24no—
49C2×#117C2314yes—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C42D412yes—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#101—24no—
61C2×#101—24no—
62C2×#101—24no—
63#391—22no—
64#391—24no—
65#391Q814yes—
66#389—28no—
67#389—28no—
68#389—24no—
69#389—24no—
70#389—22no—
71#389—22no—
72#384—22no—
73#384—22no—
74#384—24no—
75#384D412yes—
76#384D412yes—
77#382—24no—
78#382D414yes—
79#380—22no—
80#380—24no—
81#380—22no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92#343—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#339—44no—
98#339—44no—
99#336—22no—
100#336—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8C2×Q812yes—
106C22×Q8#4112yes—
107C22×Q8#4114yes—
108C22×Q8#4114yes—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2411yescharacteristic
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C22×C8—24no—
125C4×Q8—216no—
126C4×Q8—216no—
127C4×Q8—48no—
128C4×Q8—48no—
129C4×Q8—48no—
130C4×Q8—48no—
131#117—24no—
132#117—24no—
133#117—24no—
134#117—24no—
135#117—48no—
136#117—48no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4C2×D414yes—
162C2×C4⋊C4C2×D414yes—
163C2×C4⋊C4C2×Q812yes—
164C2×C4⋊C4#4114yes—
165C2×C22⋊C4—24no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—28no—
175C2×C42—24no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×Q812yes—
183#101—48no—
184#95—24no—
185#95—28no—
186#95—24no—
187#95—24no—
188#95—28no—
189#95—42no—
190#95—41no—
191#95—41no—
192#95—44no—
193#95—82no—
194C24C22×D411yescharacteristic
195C2×Q8—28no—
196C2×Q8—28no—
197C2×Q8—28no—
198C2×Q8—24no—
199C2×Q8—24no—
200C2×Q8—28no—
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8#12518yes—
204C2×Q8#12618yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4#12011yescharacteristic
248C22×C4#12012yes—
249C22×C4#12011yescharacteristic
250C22×C4C22⋊Q814yes—
251C22×C4#12314yes—
252C22×C4#12314yes—
253C22×C4#12414yes—
254C22×C4C22×D412yes—
255C22×C4C22×Q811yescharacteristic
256C22×C4C2×#4112yes—
257C22×C4C2×#4112yes—
258C22×C4C2×#4112yes—
259C22×C4C2×#4111yescharacteristic
260C22×C4H(5,𝔽2)12yes—
261C22×C42−1+412yes—
262M16—44no—
263M16—44no—
264M16—44no—
265M16—44no—
266C2×C8—44no—
267C2×C8—44no—
268C2×C8—44no—
269C4⋊C4—216no—
270C4⋊C4—28no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—44no—
277C4⋊C4—48no—
278C4⋊C4—48no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—84no—
285C4⋊C4—84no—
286C4⋊C4—84no—
287C4⋊C4—88no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C42—28no—
291C42—28no—
292C42—24no—
293C42—24no—
294C42—24no—
295C42—416no—
296C42—44no—
297C42—44no—
298C42—42no—
299C42—44no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23C2×#12011yescharacteristic
305C23C2×C22⋊Q811yescharacteristic
306C23C2×#12311yescharacteristic
307C23C2×#12311yescharacteristic
308C23C2×#12411yescharacteristic
309C23#53511yescharacteristic
310C23#53611yescharacteristic
311Q8—216no—
312Q8—216no—
313C2×C4—21no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—21no—
321C2×C4—24no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4#40018yes—
367C2×C4#45612yes—
368C2×C4#52514yes—
369C2×C4C2×#12512yes—
370C2×C4C2×#12612yes—
371C2×C4#53914yes—
372C2×C4#54614yes—
373C2×C4#54812yes—
374C2×C4D4×Q812yes—
375C2×C4#55214yes—
376C2×C4#55312yes—
377C2×C4#55412yes—
378C8—84no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22C2×#40011yescharacteristic
392C22#240911yescharacteristic
393C22#243511yescharacteristic
394C22#248711yescharacteristic
395C22#250311yescharacteristic
396C22#262311yescharacteristic
397C22#284911yescharacteristic
398C22C2×#45611yescharacteristic
399C22#301811yescharacteristic
400C22#302011yescharacteristic
401C22#302111yescharacteristic
402C4—22no—
403C4—28no—
404C4—44no—
405C4—42no—
406C4—44no—
407C4—41no—
408C4—82no—
409C4—84no—
410C4—84no—
411C4—82no—
412C4—161no—
413C4#247314yes—
414C4#248114yes—
415C2—21no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2#1484311yescharacteristic
420C2#2431511yescharacteristic
421C2#2431711yescharacteristic
422C2#2587111yescharacteristic
423C2#2587311yescharacteristic
424C2#3299211yescharacteristic
425C2#3299411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#341822
#237472
#139762
#113592
#112962
#112102
#107632
#107502