Finite Group Databasebeta

← #479345

Subgroups — #479345

Order 512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15195C211yesmaximal, characteristic
2#15125C211yesmaximal, characteristic
3#11384C212yesmaximal, complemented
4#11377C212yesmaximal, complemented
5#11267C212yesmaximal, complemented
6#11256C212yesmaximal, complemented
7#11200C212yesmaximal
8#11147C211yesmaximal, characteristic
9#11096C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#107C2212yes—
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C2×#393—21no—
15C2×#384C2211yescharacteristic
16C2×#380C2212yes—
17C2×#380C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28#1446—21no—
29#1442—21no—
30#1442—21no—
31#1442—21no—
32#1439—21no—
33#1439—21no—
34#1439—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C23×Q8—21no—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—42no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4D411yescharacteristic
57C22×C22⋊C4D411yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#107—24no—
61C2×#107—42no—
62C2×#107—42no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—41no—
65C2×C22⋊C8—42no—
66C2×C22⋊C8—41no—
67#393—44no—
68#384—41no—
69#384—41no—
70#380—24no—
71#380—42no—
72#380—42no—
73#380—42no—
74C4×C22⋊C4—41no—
75C4×C22⋊C4—41no—
76C2×#95—22no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95D411yescharacteristic
87C2×#95D411yescharacteristic
88#340—24no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340—42no—
94#340—42no—
95#336—24no—
96#336—42no—
97C25C2×D411yescharacteristic
98C22×Q8—44no—
99C22×Q8—42no—
100C22×Q8—42no—
101C22×Q8—42no—
102C22×Q8—82no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
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—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8#4114yes—
135C4⋊Q8—44no—
136C4⋊Q8—48no—
137C4⋊Q8—44no—
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—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
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—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4#4114yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—82no—
177C2×C22⋊C4—82no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4Q1614yes—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188#107—48no—
189C22⋊C8—44no—
190C22⋊C8—44no—
191#95—41no—
192#95—41no—
193#95—44no—
194#95—42no—
195#95—42no—
196#95—44no—
197#95—42no—
198#95—42no—
199#95—44no—
200#95—82no—
201#95—82no—
202#95—82no—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—22no—
207C24—22no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—22no—
212C24—22no—
213C24C2×Q1612yes—
214C24C22×D411yescharacteristic
215C2×Q8—48no—
216C2×Q8—88no—
217C2×Q8—84no—
218C2×Q8—88no—
219C2×Q8—84no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—42no—
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—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
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—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4C2×Q1612yes—
286C22×C4C2×Q1612yes—
287C22×C4AGL(1,ℤ/8)12yes—
288C22×C4#13712yes—
289C22×C4C2×#4112yes—
290C22×C4C2×#4112yes—
291C22×C4H(5,𝔽2)12yes—
292C22×C4H(5,𝔽2)11yescharacteristic
293C22×C4H(5,𝔽2)12yes—
294C22×C4H(5,𝔽2)11yescharacteristic
295C22×C42−1+412yes—
296C22×C42−1+412yes—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—48no—
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—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—84no—
329C4⋊C4—84no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—88no—
333C22⋊C4—88no—
334C42—42no—
335C42—42no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C23—24no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—24no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—41no—
355C23—42no—
356C23—42no—
357C23—41no—
358C23—41no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23#53411yescharacteristic
364C23#53511yescharacteristic
365C23#53611yescharacteristic
366C23#54612yes—
367C23#54712yes—
368C23C22×Q1611yescharacteristic
369C23C2×AGL(1,ℤ/8)11yescharacteristic
370C23C2×#13711yescharacteristic
371C23#57511yescharacteristic
372C23#57612yes—
373C23#57712yes—
374Q8—816no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—44no—
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—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4#53812yes—
433C2×C4#53812yes—
434C2×C4#54112yes—
435C2×C4#54212yes—
436C2×C4#54212yes—
437C2×C4#54212yes—
438C2×C4#55111yescharacteristic
439C2×C4#55111yescharacteristic
440C2×C4#55212yes—
441C2×C4#55912yes—
442C2×C4H(3,𝔽4)11yescharacteristic
443C2×C4#56212yes—
444C2×C4#56212yes—
445C2×C4#56212yes—
446C2×C4#56311yescharacteristic
447C2×C4#56312yes—
448C8—44no—
449C8—82no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—44no—
454C22—44no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—82no—
460C22—82no—
461C22#268411yescharacteristic
462C22#269812yes—
463C22#271012yes—
464C22#271711yescharacteristic
465C22#281611yescharacteristic
466C22#318911yescharacteristic
467C22#319311yescharacteristic
468C22#319911yescharacteristic
469C22#320011yescharacteristic
470C22#320911yescharacteristic
471C22#321411yescharacteristic
472C22#322412yes—
473C22#323011yescharacteristic
474C22#323711yescharacteristic
475C22#324112yes—
476C22#329912yes—
477C22#331412yes—
478C22#332112yes—
479C22#332512yes—
480C22#333812yes—
481C22#334512yes—
482C22#334712yes—
483C22#335612yes—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—84no—
489C4—81no—
490C4—162no—
491C4—162no—
492C4—162no—
493C4#275112yes—
494C4#275412yes—
495C4#283112yes—
496C4#283212yes—
497C2—82no—
498C2#1720111yescharacteristic
499C2#3126311yescharacteristic
500C2#3127011yescharacteristic
501C2#3169212yes—
502C2#3171512yes—
503C2#3205212yes—
504C2#3206712yes—
505C2#3226111yescharacteristic
506C2#3226911yescharacteristic
507C2#3277411yescharacteristic
508C2#3277911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11384C2semidirect
4. #11377C2semidirect
5. #11267C2semidirect
6. #11256C2semidirect

Maximal subgroups

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

subgroupindex
#151952
#151252
#113842
#113772
#112672
#112562
#112002
#111472
#110962