Finite Group Databasebeta

← #409289

Subgroups — #409289

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14340C211yesmaximal, characteristic
2#13795C211yesmaximal, characteristic
3#11372C212yesmaximal, complemented
4#11316C212yesmaximal, complemented
5#11315C212yesmaximal, complemented
6#11267C212yesmaximal, complemented
7#10731C212yesmaximal
8#10730C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C2×#384C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#380C2212yes—
17C2×#340C2212yes—
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#336C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2211yescharacteristic
28#1441—22no—
29#1439—22no—
30C4⋊#95—22no—
31C4⋊#95C2212yes—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C22⋊#95—22no—
35C24×C4—21no—
36C24×C4C2311yescharacteristic
37C23×C8C2312yes—
38C23×C8C2312yes—
39C2×#117—22no—
40C2×#117—24no—
41C2×#117—24no—
42C2×#117—41no—
43C2×#117—41no—
44C2×#117—41no—
45C2×#117—41no—
46C2×#117—41no—
47C2×#117—41no—
48C2×#117Q814yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—41no—
54C22×C4⋊C4—41no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D412yes—
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69#384—41no—
70#384—41no—
71#382—24no—
72#380—24no—
73#380—24no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82#340—24no—
83#340—24no—
84#340—42no—
85#340—42no—
86#340—42no—
87#340—42no—
88#340—42no—
89#340—42no—
90#336—24no—
91#336—41no—
92#336—41no—
93C25C2×D411yescharacteristic
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—82no—
108C23×C4—82no—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×Q812yes—
113C23×C4#4112yes—
114C23×C4#4114yes—
115C23×C4#4114yes—
116C23×C4C2411yescharacteristic
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×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131#117—48no—
132#117—48no—
133#117—44no—
134#117—44no—
135#117—44no—
136#117—44no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
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—48no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C2×Q814yes—
171C2×C4⋊C4#4114yes—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4—82no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—24no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42#4114yes—
192C22⋊C8—48no—
193#95—44no—
194#95—44no—
195#95—82no—
196#95—82no—
197#95—82no—
198#95—82no—
199#95—82no—
200#95—82no—
201C24—21no—
202C24—21no—
203C24—21no—
204C24—21no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—22no—
209C24C22⋊Q812yes—
210C24#12312yes—
211C24#12412yes—
212C24C22×D411yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—48no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
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—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4—88no—
276C22×C4—88no—
277C22×C4C22⋊Q812yes—
278C22×C4#12314yes—
279C22×C4#12312yes—
280C22×C4#12412yes—
281C22×C4#12518yes—
282C22×C4#12618yes—
283C22×C4C22×Q811yescharacteristic
284C22×C4C2×#4112yes—
285C22×C4C2×#4112yes—
286C22×C4C2×#4111yescharacteristic
287C22×C4C2×#4112yes—
288C22×C4C2×#4112yes—
289C22×C4H(5,𝔽2)11yescharacteristic
290C22×C4H(5,𝔽2)11yescharacteristic
291C22×C4H(5,𝔽2)11yescharacteristic
292C22×C4H(5,𝔽2)11yescharacteristic
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—48no—
312C4⋊C4—48no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—88no—
326C22⋊C4—48no—
327C22⋊C4—48no—
328C22⋊C4—88no—
329C42—24no—
330C42—24no—
331C42—48no—
332C42—42no—
333C42—42no—
334C23—24no—
335C23—24no—
336C23—24no—
337C23—24no—
338C23—22no—
339C23—22no—
340C23—24no—
341C23—24no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23#40014yes—
357C23C2×C22⋊Q811yescharacteristic
358C23C2×#12311yescharacteristic
359C23C2×#12311yescharacteristic
360C23#52511yescharacteristic
361C23C2×#12411yescharacteristic
362C23#53411yescharacteristic
363C23#53511yescharacteristic
364C23#57612yes—
365C23#57712yes—
366C23#57712yes—
367C23#57812yes—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—28no—
377C2×C4—28no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—816no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4#40014yes—
422C2×C4C2×#12512yes—
423C2×C4C2×#12612yes—
424C2×C4#52912yes—
425C2×C4#53312yes—
426C2×C4#53811yescharacteristic
427C2×C4#53811yescharacteristic
428C2×C4#54311yescharacteristic
429C2×C4#54311yescharacteristic
430C2×C4#55112yes—
431C2×C4#55112yes—
432C2×C4#55112yes—
433C2×C4#55112yes—
434C2×C4#55312yes—
435C2×C4#55312yes—
436C2×C4#55312yes—
437C2×C4#55312yes—
438C8—82no—
439C8—82no—
440C22—24no—
441C22—28no—
442C22—44no—
443C22—44no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—44no—
451C22C2×#40011yescharacteristic
452C22#248211yescharacteristic
453C22#248311yescharacteristic
454C22#250411yescharacteristic
455C22#259111yescharacteristic
456C22#262611yescharacteristic
457C22#285511yescharacteristic
458C22#308312yes—
459C22#308412yes—
460C22#309612yes—
461C22#309712yes—
462C22#309812yes—
463C22#309912yes—
464C22#312211yescharacteristic
465C22#312311yescharacteristic
466C22#312611yescharacteristic
467C22#312711yescharacteristic
468C22#313612yes—
469C22#313712yes—
470C22#319812yes—
471C22#320112yes—
472C22#322412yes—
473C22#322512yes—
474C4—24no—
475C4—44no—
476C4—84no—
477C4—84no—
478C4—82no—
479C4—162no—
480C4—162no—
481C4#251012yes—
482C4#251012yes—
483C4#256912yes—
484C4#256912yes—
485C2—44no—
486C2#1501511yescharacteristic
487C2#2374812yes—
488C2#2759111yescharacteristic
489C2#2759211yescharacteristic
490C2#2760612yes—
491C2#2798412yes—
492C2#2934011yescharacteristic
493C2#2934111yescharacteristic
494C2#2996112yes—
495C2#3315611yescharacteristic
496C2#3315711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11372C2semidirect
4. #11316C2semidirect
5. #11315C2semidirect
6. #11267C2semidirect

Maximal subgroups

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

subgroupindex
#143402
#137952
#113722
#113162
#113152
#112672
#107312
#107302
#107132