Finite Group Databasebeta

← #475943

Subgroups — #475943

Order 512.

898 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1941C211yesmaximal, characteristic
2C2×#1939C211yesmaximal, characteristic
3#15200C211yesmaximal, characteristic
4#15156C211yesmaximal, characteristic
5#11378C212yesmaximal
6#11257C212yesmaximal
7#11231C212yesmaximal
8#11231C212yesmaximal
9#11228C211yesmaximal, characteristic
10#11227C211yesmaximal, characteristic
11#10861C211yesmaximal, characteristic
12C22×#106C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#400—21no—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1941—24no—
23#1939—24no—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32#1445—21no—
33#1443—22no—
34#1442—22no—
35#1442—21no—
36#1441—21no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×C8C2312yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58C2×#106—22no—
59C2×#106—22no—
60C2×#106—22no—
61C2×#106—24no—
62C2×#106—42no—
63C2×#106—42no—
64C2×#106—42no—
65C2×#106Q814yes—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—42no—
71C2×C4×C8—42no—
72C2×C4×C8D414yes—
73#400—44no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95D412yes—
95#340—22no—
96#340—22no—
97#340—24no—
98#340—42no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336—24no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2411yescharacteristic
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8C2×D412yes—
127C22×C8C2×D412yes—
128C22×C8C2×Q812yes—
129C22×C8#4112yes—
130C22×C8#4114yes—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4SD1614yes—
171C2×C4⋊C4C2×Q812yes—
172C2×C4⋊C4C2×Q812yes—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42C2×D412yes—
182C2×C42#4112yes—
183#106—28no—
184#106—44no—
185#106—48no—
186C4⋊C8—44no—
187C4⋊C8—48no—
188C4×C8—44no—
189#95—24no—
190#95—44no—
191#95—44no—
192#95—42no—
193#95—42no—
194#95—44no—
195#95—42no—
196#95—44no—
197#95—42no—
198#95—82no—
199#95—81no—
200#95—81no—
201#95—82no—
202#95—82no—
203#95—82no—
204C24C22×D411yescharacteristic
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—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4#12112yes—
261C22×C4#12112yes—
262C22×C4C2×SD1614yes—
263C22×C4C2×SD1612yes—
264C22×C4#13512yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C22×Q811yescharacteristic
267C22×C4C2×#4111yescharacteristic
268C22×C4C2×#4112yes—
269C22×C4C2×#4111yescharacteristic
270C22×C4H(5,𝔽2)12yes—
271C22×C4H(5,𝔽2)11yescharacteristic
272C22×C42−1+412yes—
273C22×C42−1+412yes—
274C22×C42−1+411yescharacteristic
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8C22⋊Q814yes—
291C2×C8#12414yes—
292C4⋊C4—28no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—44no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C42—22no—
319C42—22no—
320C42—24no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42—42no—
325C23C2×#12111yescharacteristic
326C23#53712yes—
327C23#54712yes—
328C23D4×Q811yescharacteristic
329C23#55011yescharacteristic
330C23C22×SD1611yescharacteristic
331C23C2×#13511yescharacteristic
332C23#57512yes—
333C23#57712yes—
334C23#57711yescharacteristic
335C23#57711yescharacteristic
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—24no—
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—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4#46514yes—
396C2×C4C2×C22⋊Q811yescharacteristic
397C2×C4C2×#12411yescharacteristic
398C2×C4#53411yescharacteristic
399C2×C4#53611yescharacteristic
400C2×C4#53611yescharacteristic
401C2×C4#53711yescharacteristic
402C2×C4#53812yes—
403C2×C4#53912yes—
404C2×C4#53912yes—
405C2×C4#53912yes—
406C2×C4#54412yes—
407C2×C4#55411yescharacteristic
408C2×C4#55612yes—
409C2×C4#55612yes—
410C2×C4#55611yescharacteristic
411C2×C4#55712yes—
412C2×C4#56312yes—
413C2×C4#56412yes—
414C8—24no—
415C8—24no—
416C8—82no—
417C22#253811yescharacteristic
418C22#254711yescharacteristic
419C22#261811yescharacteristic
420C22#272612yes—
421C22#272712yes—
422C22C2×#46511yescharacteristic
423C22#305411yescharacteristic
424C22#305711yescharacteristic
425C22#306011yescharacteristic
426C22#324612yes—
427C22#325012yes—
428C22#326012yes—
429C22#326312yes—
430C22#333512yes—
431C22#334012yes—
432C22#335112yes—
433C22#335112yes—
434C22Q8×SD1611yescharacteristic
435C22#338111yescharacteristic
436C22#338812yes—
437C22#339211yescharacteristic
438C22#339411yescharacteristic
439C22#340112yes—
440C4—24no—
441C4—44no—
442C4—42no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4—162no—
447C4—162no—
448C4—161no—
449C4—161no—
450C4#256811yescharacteristic
451C4#262711yescharacteristic
452C4#263911yescharacteristic
453C4#280512yes—
454C4#281012yes—
455C4#282211yescharacteristic
456C2#1707411yescharacteristic
457C2#2826411yescharacteristic
458C2#2827011yescharacteristic
459C2#2870611yescharacteristic
460C2#2871011yescharacteristic
461C2#2977411yescharacteristic
462C2#2978111yescharacteristic
463C2#3250512yes—
464C2#3250712yes—
465C2#3252512yes—
466C2#3252712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#19412
C2×#19392
#152002
#151562
#113782
#112572
#112312
#112282
#112272
#108612