Finite Group Databasebeta

← #475678

Subgroups — #475678

Order 512.

1106 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2100C211yesmaximal, characteristic, complemented
2C2×#2094C211yesmaximal, characteristic
3C2×#1934C211yesmaximal, characteristic
4C2×#1845C211yesmaximal, characteristic, complemented
5#15244C211yesmaximal, characteristic
6#15197C211yesmaximal, characteristic, complemented
7#11325C212yesmaximal, complemented
8#11325C212yesmaximal, complemented
9#11123C212yesmaximal
10#11123C212yesmaximal
11#10797C211yesmaximal, characteristic, complemented
12C22×#107C2212yes—
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2212yes—
21C2×#389C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2212yes—
25#2100—24no—
26#2094—24no—
27#1934—24no—
28#1845—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36#1442—22no—
37#1441—22no—
38#1439—22no—
39#1439—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—24no—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62C2×#107—24no—
63C2×#107—24no—
64C2×#107—42no—
65C2×#107—42no—
66C2×#107—42no—
67C2×#107—42no—
68C2×#106—24no—
69C2×#106—42no—
70C2×#106—42no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—24no—
74C2×#102—22no—
75C2×#102—24no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102Q814yes—
79C2×C4×C8—24no—
80C2×C4×C8D414yes—
81#390—22no—
82#390—22no—
83#390—42no—
84#390—42no—
85#389—22no—
86#389—22no—
87#384—22no—
88#384—22no—
89#384—22no—
90#384—22no—
91#382—42no—
92#382—42no—
93C2×#95—24no—
94C2×#95—21no—
95C2×#95—24no—
96C2×#95—24no—
97C2×#95D412yes—
98C2×#95D412yes—
99#340—24no—
100#340—24no—
101#340—22no—
102#340—22no—
103#336—24no—
104#336—24no—
105#336D414yes—
106C25—21no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—21no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4C2×Q812yes—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4C2×D414yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2411yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8C2×D414yes—
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C22×C8C2×Q812yes—
141C22×C8#4112yes—
142C22×C8#4114yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
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—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4C2×Q812yes—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—84no—
185C2×C22⋊C4—84no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196#107—48no—
197#107—48no—
198#106—48no—
199#102—28no—
200#102—48no—
201#102—44no—
202C4×C8—28no—
203#95—24no—
204#95—24no—
205#95—48no—
206#95—42no—
207#95—84no—
208#95—84no—
209C24—22no—
210C24—41no—
211C24—42no—
212C24—42no—
213C24—41no—
214C24—42no—
215C24—42no—
216C24—42no—
217C24—42no—
218C24C22×D411yescharacteristic
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—41no—
224C2×D4—42no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4#12712yes—
272C22×C4#12712yes—
273C22×C4#13514yes—
274C22×C4#13714yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×Q811yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4112yes—
280C22×C4H(5,𝔽2)12yes—
281C22×C4H(5,𝔽2)12yes—
282C22×C42−1+411yescharacteristic
283C22×C42−1+411yescharacteristic
284C22×C42−1+412yes—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8C22⋊Q814yes—
301C2×C8#12314yes—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C4⋊C4—84no—
325C22⋊C4—816no—
326C22⋊C4—84no—
327C22⋊C4—84no—
328C42—22no—
329C42—22no—
330C42—24no—
331C42—42no—
332C42—42no—
333C42—42no—
334C42—42no—
335C23—44no—
336C23—44no—
337C23—44no—
338C23—42no—
339C23—84no—
340C23—82no—
341C23—84no—
342C23—82no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—84no—
349C23—84no—
350C23—82no—
351C23—81no—
352C23C2×#12711yescharacteristic
353C23#53512yes—
354C23#54712yes—
355C23#54711yescharacteristic
356C23D4×Q811yescharacteristic
357C23C2×#13512yes—
358C23C2×#13712yes—
359C23#57612yes—
360C23#57612yes—
361D4—42no—
362D4—44no—
363D4—42no—
364D4—81no—
365D4—81no—
366D4—82no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—84no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—81no—
403C2×C4#49514yes—
404C2×C4#49714yes—
405C2×C4C2×C22⋊Q811yescharacteristic
406C2×C4C2×#12311yescharacteristic
407C2×C4#53511yescharacteristic
408C2×C4#53511yescharacteristic
409C2×C4#53512yes—
410C2×C4#53611yescharacteristic
411C2×C4#53611yescharacteristic
412C2×C4#53912yes—
413C2×C4#53912yes—
414C2×C4#54212yes—
415C2×C4#54412yes—
416C2×C4#54711yescharacteristic
417C2×C4#54712yes—
418C2×C4D4×Q811yescharacteristic
419C2×C4#55312yes—
420C2×C4#55412yes—
421C2×C4#56312yes—
422C2×C4#56312yes—
423C8—24no—
424C8—24no—
425C8—44no—
426C22—88no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—164no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22#250111yescharacteristic
436C22#254211yescharacteristic
437C22#265711yescharacteristic
438C22#271812yes—
439C22#272012yes—
440C22C2×#49511yescharacteristic
441C22C2×#49711yescharacteristic
442C22#315511yescharacteristic
443C22#315511yescharacteristic
444C22#321014yes—
445C22#322014yes—
446C22#333914yes—
447C22#334112yes—
448C22#334614yes—
449C22#335712yes—
450C22#338012yes—
451C22#338512yes—
452C4—24no—
453C4—24no—
454C4—84no—
455C4—82no—
456C4—161no—
457C4—161no—
458C4—162no—
459C4—162no—
460C4#248711yescharacteristic
461C4#262311yescharacteristic
462C4#262811yescharacteristic
463C4#271812yes—
464C4#272012yes—
465C4#282111yescharacteristic
466C2—162no—
467C2#1706411yescharacteristic
468C2#2793211yescharacteristic
469C2#2793811yescharacteristic
470C2#2852811yescharacteristic
471C2#2853911yescharacteristic
472C2#3026111yescharacteristic
473C2#3027011yescharacteristic
474C2#3228314yes—
475C2#3234714yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2100C2semidirect
4. C2×#1845C2semidirect
6. #15197C2semidirect
7. #11325C2semidirect
8. #11325C2semidirect
11. #10797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21002
C2×#20942
C2×#19342
C2×#18452
#152442
#151972
#113252
#111232
#107972