Finite Group Databasebeta

← #416984

Subgroups — #416984

Order 512.

1060 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
1C2×#1991C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic
3C2×#1835C211yesmaximal, characteristic
4C2×C4×C4⋊C8C211yesmaximal, characteristic
5#14709C211yesmaximal, characteristic
6#14707C211yesmaximal, characteristic
7#11316C212yesmaximal
8#11315C212yesmaximal
9#11228C212yesmaximal
10#11227C212yesmaximal
11#10842C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4×C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C43C2211yescharacteristic
22#1991—24no—
23#1991—24no—
24#1835—24no—
25C4×C4⋊C8—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1436—22no—
36#1436—22no—
37#1436—22no—
38#1436—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125Q814yes—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C42D412yes—
61C22×C42D412yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C4⋊C8—24no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—24no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8D414yes—
74#389—24no—
75#389—24no—
76#382—22no—
77#382—22no—
78#382—22no—
79#382—22no—
80#382—22no—
81#382D414yes—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C43—22no—
94C43—22no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336—24no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2411yescharacteristic
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—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8C2×D414yes—
128C22×C8#4114yes—
129#125—28no—
130#125—28no—
131#125—44no—
132#125—44no—
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—28no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
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—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×Q814yes—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—24no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×Q812yes—
184C2×C42#4114yes—
185C2×C42#4112yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188C4⋊C8—216no—
189C4×C8—28no—
190C4×C8—44no—
191#95—28no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—42no—
198#95—42no—
199#95—44no—
200#95—44no—
201#95—44no—
202C24C22×D411yescharacteristic
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—22no—
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—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4#12114yes—
255C22×C4#12114yes—
256C22×C4C22⋊Q814yes—
257C22×C4#12412yes—
258C22×C4#12412yes—
259C22×C4#13514yes—
260C22×C4#13514yes—
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×Q811yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4112yes—
269C22×C42−1+412yes—
270C22×C42−1+412yes—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—416no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C42—216no—
305C42—24no—
306C42—24no—
307C42—48no—
308C42—42no—
309C42—42no—
310C42#12114yes—
311C42C22⋊Q814yes—
312C42C22⋊Q814yes—
313C42#12614yes—
314C23C2×#12111yescharacteristic
315C23C2×#12111yescharacteristic
316C23C2×C22⋊Q811yescharacteristic
317C23C2×#12411yescharacteristic
318C23#53211yescharacteristic
319C23#53611yescharacteristic
320C23#53611yescharacteristic
321C23C2×#13512yes—
322C23C2×#13512yes—
323C23#57512yes—
324C23#57512yes—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4#39618yes—
382C2×C4#48814yes—
383C2×C4C2×#12111yescharacteristic
384C2×C4C2×C22⋊Q811yescharacteristic
385C2×C4C2×C22⋊Q811yescharacteristic
386C2×C4#52512yes—
387C2×C4C2×#12611yescharacteristic
388C2×C4#52912yes—
389C2×C4#53612yes—
390C2×C4#54112yes—
391C2×C4#54712yes—
392C2×C4#54812yes—
393C2×C4#55212yes—
394C2×C4#55612yes—
395C2×C4#55612yes—
396C2×C4#55612yes—
397C2×C4#55712yes—
398C2×C4#55712yes—
399C8—44no—
400C8—44no—
401C22C2×#39611yescharacteristic
402C22#254511yescharacteristic
403C22#254711yescharacteristic
404C22#256511yescharacteristic
405C22#256811yescharacteristic
406C22#262111yescharacteristic
407C22#285011yescharacteristic
408C22#303512yes—
409C22#303512yes—
410C22#305012yes—
411C22#305412yes—
412C22#307612yes—
413C22#307612yes—
414C22C2×#48811yescharacteristic
415C22#313111yescharacteristic
416C22#314011yescharacteristic
417C22#314211yescharacteristic
418C22#316812yes—
419C22#318312yes—
420C22#323212yes—
421C22#323212yes—
422C22#323312yes—
423C22#323312yes—
424C4—24no—
425C4—28no—
426C4—42no—
427C4—44no—
428C4—82no—
429C4—82no—
430C4—84no—
431C4—161no—
432C4—161no—
433C4—162no—
434C4#241512yes—
435C4#245312yes—
436C4#251412yes—
437C4#258112yes—
438C2#1523711yescharacteristic
439C2#2360912yes—
440C2#2865712yes—
441C2#2872011yescharacteristic
442C2#2872111yescharacteristic
443C2#2918012yes—
444C2#2922111yescharacteristic
445C2#2922211yescharacteristic
446C2#2987912yes—
447C2#3304411yescharacteristic
448C2#3305011yescharacteristic
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×#19912
C2×#18352
C2×C4×C4⋊C82
#147092
#147072
#113162
#113152
#112282
#112272
#108422