Finite Group Databasebeta

← #469900

Subgroups — #469900

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15135C211yesmaximal, characteristic
2#13924C211yesmaximal, characteristic
3#11378C212yesmaximal, complemented
4#11267C212yesmaximal, complemented
5#11227C212yesmaximal, complemented
6#11220C212yesmaximal, complemented
7#11095C212yesmaximal
8#11042C211yesmaximal, characteristic
9#10730C212yesmaximal
10C23×C4⋊C4C2211yescharacteristic
11C22×#106C2212yes—
12C22×C4⋊C8C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C2×#399—21no—
15C2×#389C2212yes—
16C2×#384C2211yescharacteristic
17C2×#380C2212yes—
18C2×#380C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2211yescharacteristic
29#1445—21no—
30#1445—21no—
31#1445—21no—
32#1444—21no—
33#1439—21no—
34C4⋊#95C2212yes—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2212yes—
37C22⋊#95—21no—
38C22⋊#95—21no—
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—41no—
49C22×C4⋊C4—41no—
50C22×C4⋊C4—41no—
51C22×C4⋊C4—41no—
52C22×C4⋊C4Q814yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D411yescharacteristic
60C22×C22⋊C4D411yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—42no—
66C2×C4⋊C8—41no—
67C2×C4⋊C8—42no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—42no—
71#399—44no—
72#389—24no—
73#389—42no—
74#384—41no—
75#384—41no—
76#380—24no—
77#380—42no—
78#380—41no—
79#380—41no—
80C4×C22⋊C4—41no—
81C4×C22⋊C4—41no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D411yescharacteristic
92C2×#95D411yescharacteristic
93#340—42no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#336—24no—
101#336—41no—
102C25C2×D411yescharacteristic
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—82no—
119C23×C4—82no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×Q812yes—
126C23×C4#4112yes—
127C23×C4C2411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8#4114yes—
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—24no—
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—22no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4C2×Q814yes—
180C2×C4⋊C4#4114yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4SD1614yes—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201#106—48no—
202C4⋊C8—44no—
203C22⋊C8—48no—
204#95—41no—
205#95—41no—
206#95—44no—
207#95—44no—
208#95—42no—
209#95—42no—
210#95—42no—
211#95—82no—
212#95—82no—
213#95—82no—
214#95—82no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—22no—
219C24—21no—
220C24—21no—
221C24—22no—
222C24—22no—
223C24C22⋊Q812yes—
224C24C2×SD1612yes—
225C24#13712yes—
226C24C22×D411yescharacteristic
227C22×C4—21no—
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—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—48no—
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×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—88no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4—88no—
299C22×C4C22⋊Q812yes—
300C22×C4C2×SD1612yes—
301C22×C4C2×SD1612yes—
302C22×C4#13712yes—
303C22×C4C22×Q811yescharacteristic
304C22×C4C2×#4112yes—
305C22×C4C2×#4112yes—
306C22×C4C2×#4111yescharacteristic
307C22×C4H(5,𝔽2)11yescharacteristic
308C22×C4H(5,𝔽2)11yescharacteristic
309C22×C4H(5,𝔽2)11yescharacteristic
310C22×C4H(5,𝔽2)11yescharacteristic
311C22×C4H(5,𝔽2)12yes—
312C22×C42−1+412yes—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—48no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—88no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—88no—
350C22⋊C4—84no—
351C42—42no—
352C42—42no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C23—22no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—24no—
365C23—22no—
366C23—22no—
367C23—24no—
368C23—24no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—42no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—42no—
379C23#47514yes—
380C23C2×C22⋊Q811yescharacteristic
381C23#53411yescharacteristic
382C23#53511yescharacteristic
383C23#54612yes—
384C23#54712yes—
385C23C22×SD1611yescharacteristic
386C23C2×#13711yescharacteristic
387C23C2×#13711yescharacteristic
388C23#57511yescharacteristic
389C23#57612yes—
390C23#57712yes—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—816no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4#53312yes—
452C2×C4#53811yescharacteristic
453C2×C4#53811yescharacteristic
454C2×C4#53912yes—
455C2×C4#53912yes—
456C2×C4#54012yes—
457C2×C4#54012yes—
458C2×C4#54311yescharacteristic
459C2×C4#54311yescharacteristic
460C2×C4#54412yes—
461C2×C4#55112yes—
462C2×C4#55112yes—
463C2×C4#55212yes—
464C2×C4#55911yescharacteristic
465C2×C4#55911yescharacteristic
466C2×C4#56011yescharacteristic
467C2×C4#56212yes—
468C2×C4#56311yescharacteristic
469C8—44no—
470C8—82no—
471C22—28no—
472C22—24no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22—42no—
481C22—42no—
482C22#243812yes—
483C22#262611yescharacteristic
484C22#270612yes—
485C22#270911yescharacteristic
486C22#272411yescharacteristic
487C22C2×#47511yescharacteristic
488C22#307811yescharacteristic
489C22#308112yes—
490C22#318711yescharacteristic
491C22#319511yescharacteristic
492C22#319911yescharacteristic
493C22#320011yescharacteristic
494C22#320611yescharacteristic
495C22#321511yescharacteristic
496C22#322412yes—
497C22#329512yes—
498C22#330912yes—
499C22#332012yes—
500C22#333112yes—
501C22#333512yes—
502C22#334612yes—
503C22#334612yes—
504C22#335112yes—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—84no—
510C4—81no—
511C4—84no—
512C4—162no—
513C4—162no—
514C4#260711yescharacteristic
515C4#266712yes—
516C4#274312yes—
517C4#275211yescharacteristic
518C4#277111yescharacteristic
519C4#284111yescharacteristic
520C2—44no—
521C2#1666811yescharacteristic
522C2#2599212yes—
523C2#2601812yes—
524C2#2995111yescharacteristic
525C2#2995911yescharacteristic
526C2#3193412yes—
527C2#3194712yes—
528C2#3202811yescharacteristic
529C2#3203711yescharacteristic
530C2#3245911yescharacteristic
531C2#3246711yescharacteristic
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. #11378C2semidirect
4. #11267C2semidirect
5. #11227C2semidirect
6. #11220C2semidirect

Maximal subgroups

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

subgroupindex
#151352
#139242
#113782
#112672
#112272
#112202
#110952
#110422
#107302