Finite Group Databasebeta

← #442743

Subgroups — #442743

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15135C211yesmaximal, characteristic
2#14130C211yesmaximal, characteristic
3#11220C212yesmaximal, complemented
4#11220C212yesmaximal, complemented
5#11207C212yesmaximal, complemented
6#11095C212yesmaximal
7#10921C211yesmaximal, characteristic
8#10880C212yesmaximal, complemented
9#10730C212yesmaximal
10C23×C4⋊C4C2211yescharacteristic
11C22×#106C2212yes—
12C22×C22⋊C8C2212yes—
13C22×C4×C8C2211yescharacteristic
14C2×#399—21no—
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#385—21no—
18C2×#384C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C2×#380C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31#1445—21no—
32#1444—21no—
33#1444—21no—
34#1442—21no—
35#1439—21no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C22⋊#95—21no—
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—24no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—41no—
50C22×C4⋊C4—41no—
51C22×C4⋊C4—42no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—42no—
65C2×C22⋊C8—42no—
66C2×C22⋊C8—42no—
67C2×C22⋊C8Q818yes—
68C2×C4×C8—41no—
69C2×C4×C8—42no—
70#399—44no—
71#389—24no—
72#389—24no—
73#389—42no—
74#389—42no—
75#385—44no—
76#384—41no—
77#384—41no—
78#380—24no—
79#380—24no—
80#380—42no—
81C4×C22⋊C4—41no—
82C4×C22⋊C4—41no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D411yescharacteristic
92C2×#95D411yescharacteristic
93#340—24no—
94#340—24no—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#336—41no—
101#336—42no—
102C25C2×D411yescharacteristic
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—44no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—82no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×Q814yes—
125C23×C4C2411yescharacteristic
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—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8C2×Q814yes—
139C22×C8C2×Q814yes—
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—48no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4#4114yes—
178C2×C4⋊C4#4114yes—
179C2×C4⋊C4#4114yes—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4SD1614yes—
191C2×C22⋊C4SD1614yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200#106—48no—
201C22⋊C8—48no—
202C4×C8—44no—
203#95—24no—
204#95—41no—
205#95—41no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—42no—
210#95—44no—
211#95—42no—
212#95—82no—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—22no—
217C24—21no—
218C24—21no—
219C24—22no—
220C24—22no—
221C24C4⋊Q812yes—
222C24C2×SD1612yes—
223C24C2×SD1612yes—
224C24C22×D411yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
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—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—48no—
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×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—88no—
291C22×C4—88no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4C4⋊Q812yes—
295C22×C4C2×SD1612yes—
296C22×C4C2×SD1612yes—
297C22×C4C2×SD1612yes—
298C22×C4C2×SD1612yes—
299C22×C4C22×Q812yes—
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302C22×C4C2×#4112yes—
303C22×C4H(5,𝔽2)11yescharacteristic
304C22×C4H(5,𝔽2)12yes—
305C22×C4H(5,𝔽2)12yes—
306C22×C4H(5,𝔽2)11yescharacteristic
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—48no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—88no—
339C4⋊C4—88no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—48no—
345C22⋊C4—88no—
346C42—42no—
347C42—42no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—24no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23#49814yes—
375C23#52511yescharacteristic
376C23C2×C4⋊Q811yescharacteristic
377C23#53411yescharacteristic
378C23#54612yes—
379C23#54612yes—
380C23C22×SD1611yescharacteristic
381C23C22×SD1611yescharacteristic
382C23#57511yescharacteristic
383C23#57511yescharacteristic
384C23#57612yes—
385C23#57812yes—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—816no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
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#52912yes—
445C2×C4#53312yes—
446C2×C4#53812yes—
447C2×C4#54012yes—
448C2×C4#54012yes—
449C2×C4#54312yes—
450C2×C4#55112yes—
451C2×C4#55111yescharacteristic
452C2×C4#55312yes—
453C2×C4#55412yes—
454C2×C4#55412yes—
455C2×C4#55412yes—
456C2×C4#55412yes—
457C2×C4#55911yescharacteristic
458C2×C4#56011yescharacteristic
459C2×C4#56011yescharacteristic
460C8—44no—
461C8—44no—
462C22—28no—
463C22—24no—
464C22—42no—
465C22—42no—
466C22—44no—
467C22—44no—
468C22—42no—
469C22—44no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22#245712yes—
474C22#254912yes—
475C22#258311yescharacteristic
476C22#269711yescharacteristic
477C22#270911yescharacteristic
478C22#310111yescharacteristic
479C22#310311yescharacteristic
480C22#312412yes—
481C22C2×#49811yescharacteristic
482C22#316412yes—
483C22#316611yescharacteristic
484C22#318711yescharacteristic
485C22#318711yescharacteristic
486C22#319911yescharacteristic
487C22#320211yescharacteristic
488C22#329512yes—
489C22#329512yes—
490C22#329812yes—
491C22#329812yes—
492C22#330912yes—
493C22#331512yes—
494C22#332212yes—
495C22#332712yes—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—84no—
502C4—84no—
503C4—84no—
504C4—162no—
505C4#252212yes—
506C4#258912yes—
507C4#274812yes—
508C4#278012yes—
509C2—44no—
510C2#1573011yescharacteristic
511C2#2665812yes—
512C2#2666012yes—
513C2#2877112yes—
514C2#2878612yes—
515C2#2929711yescharacteristic
516C2#2931011yescharacteristic
517C2#3166711yescharacteristic
518C2#3166811yescharacteristic
519C2#3202711yescharacteristic
520C2#3202811yescharacteristic
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. #11220C2semidirect
4. #11220C2semidirect
5. #11207C2semidirect
8. #10880C2semidirect

Maximal subgroups

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

subgroupindex
#151352
#141302
#112202
#112072
#110952
#109212
#108802
#107302