Finite Group Databasebeta

← #415565

Subgroups — #415565

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic
2C2×#1989C212yesmaximal
3#14709C211yesmaximal, characteristic
4#11385C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11292C212yesmaximal
8#11268C211yesmaximal, characteristic
9#11268C211yesmaximal, characteristic
10#10850C212yesmaximal
11#10832C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C4×#125—28no—
16C2×#389C2212yes—
17C2×#387—22no—
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×C4×C4⋊C4—22no—
21C2×C4×C4⋊C4C2212yes—
22C2×C43C2211yescharacteristic
23#1989—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1436—22no—
36#1436—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#125—24no—
44C2×#125—24no—
45C2×#125—24no—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125—24no—
49C2×#125Q814yes—
50C2×#125Q814yes—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C42—22no—
62C22×C42D412yes—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—24no—
70#389—24no—
71#389—24no—
72#387—216no—
73#382—24no—
74#382—24no—
75C4×C4⋊C4—216no—
76C4×C4⋊C4—24no—
77C4×C4⋊C4D418yes—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—21no—
87C43—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#336—22no—
97#336—22no—
98#336—42no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2411yescharacteristic
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124#125—216no—
125#125—216no—
126#125—216no—
127#125—48no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4C2×D414yes—
163C2×C4⋊C4C2×D414yes—
164C2×C4⋊C4C2×Q812yes—
165C2×C4⋊C4C2×Q812yes—
166C2×C4⋊C4C2×Q814yes—
167C2×C4⋊C4#4114yes—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—28no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—44no—
182C2×C42—44no—
183C2×C42C2×D414yes—
184C2×C42C2×Q812yes—
185C2×C42C2×Q812yes—
186C2×C42#4112yes—
187C2×C42#4114yes—
188C2×C42#4112yes—
189C4⋊C8—48no—
190#95—28no—
191#95—28no—
192#95—28no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—42no—
197#95—42no—
198#95—44no—
199C24C22×D411yescharacteristic
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4#12012yes—
251C22×C4#12012yes—
252C22×C4C22⋊Q814yes—
253C22×C4C22⋊Q812yes—
254C22×C4C22⋊Q812yes—
255C22×C4#12412yes—
256C22×C4#12412yes—
257C22×C4#13514yes—
258C22×C4C22×D412yes—
259C22×C4C22×Q811yescharacteristic
260C22×C4C22×Q811yescharacteristic
261C22×C4C2×#4112yes—
262C22×C4C2×#4112yes—
263C22×C4C2×#4111yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4H(5,𝔽2)11yescharacteristic
266C22×C42−1+411yescharacteristic
267C22×C42−1+412yes—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C4⋊C4—232no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—416no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—48no—
296C4⋊C4—48no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4C22⋊Q818yes—
302C42—24no—
303C42—24no—
304C42—48no—
305C42—48no—
306C42—416no—
307C42—42no—
308C42—42no—
309C42C22⋊Q818yes—
310C42#12418yes—
311C23C2×#12011yescharacteristic
312C23C2×C22⋊Q811yescharacteristic
313C23C2×C22⋊Q811yescharacteristic
314C23#52511yescharacteristic
315C23C2×#12411yescharacteristic
316C23#53611yescharacteristic
317C23#53711yescharacteristic
318C23C2×#13512yes—
319C23#57512yes—
320C23#57512yes—
321C23#57512yes—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—28no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4#39314yes—
379C2×C4#39314yes—
380C2×C4C2×C22⋊Q812yes—
381C2×C4C2×C22⋊Q812yes—
382C2×C4#52512yes—
383C2×C4C2×#12412yes—
384C2×C4#53612yes—
385C2×C4#53712yes—
386C2×C4#55612yes—
387C2×C4#55612yes—
388C2×C4#55612yes—
389C2×C4#55711yescharacteristic
390C2×C4#55712yes—
391C2×C4#55711yescharacteristic
392C2×C4#55712yes—
393C2×C4Q8211yescharacteristic
394C2×C4#55912yes—
395C2×C4#55911yescharacteristic
396C8—82no—
397C8—82no—
398C22C2×#39311yescharacteristic
399C22#256811yescharacteristic
400C22#258111yescharacteristic
401C22#258211yescharacteristic
402C22#259211yescharacteristic
403C22#261611yescharacteristic
404C22#284811yescharacteristic
405C22#300212yes—
406C22#300512yes—
407C22#307612yes—
408C22#307612yes—
409C22#308012yes—
410C22#308012yes—
411C22#310212yes—
412C22#311112yes—
413C22#313111yescharacteristic
414C22#313511yescharacteristic
415C22#314211yescharacteristic
416C22#314211yescharacteristic
417C22#323212yes—
418C22#324012yes—
419C22#325012yes—
420C22#325612yes—
421C4—24no—
422C4—24no—
423C4—44no—
424C4—48no—
425C4—44no—
426C4—82no—
427C4—82no—
428C4—84no—
429C4—161no—
430C4—161no—
431C4#256812yes—
432C4#258112yes—
433C4#258212yes—
434C4#259212yes—
435C2#1520111yescharacteristic
436C2#2345111yescharacteristic
437C2#2345311yescharacteristic
438C2#2922111yescharacteristic
439C2#2922711yescharacteristic
440C2#2927912yes—
441C2#2929112yes—
442C2#2935111yescharacteristic
443C2#2935311yescharacteristic
444C2#2969312yes—
445C2#3296112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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
C2×C4×#1252
C2×#19892
#147092
#113852
#112932
#112922
#112682
#108502
#108322