Finite Group Databasebeta

← #409285

Subgroups — #409285

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14682C211yesmaximal, characteristic
2#13756C211yesmaximal, characteristic
3#11372C212yesmaximal, complemented
4#11316C212yesmaximal, complemented
5#11315C212yesmaximal, complemented
6#11268C212yesmaximal, complemented
7#10731C211yesmaximal, characteristic
8#10731C211yesmaximal, characteristic
9#10731C211yesmaximal, characteristic
10#10731C211yesmaximal, characteristic
11#10712C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C4⋊#95C2212yes—
32C4⋊#95C2211yescharacteristic
33C4⋊#95C2212yes—
34C4⋊#95C2211yescharacteristic
35#1434—22no—
36C22⋊#95—22no—
37C22⋊#95—22no—
38C4×#95—22no—
39C24×C4D412yes—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#117—22no—
44C2×#117—24no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—42no—
48C2×#117—42no—
49C2×#117—42no—
50C2×#117Q814yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C42—21no—
62C22×C42—21no—
63C22×C42—22no—
64C22×C42—22no—
65C22×C42—24no—
66C22×C42—42no—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76#382—24no—
77#382—42no—
78#380—22no—
79#380—22no—
80#380—22no—
81#380—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#336—24no—
101#336—42no—
102C25C2×D411yescharacteristic
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—24no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×Q812yes—
124C23×C4#4112yes—
125C23×C4#4112yes—
126C23×C4#4112yes—
127C23×C4#4112yes—
128C23×C4#4112yes—
129C23×C4C2411yescharacteristic
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—22no—
142C22×C8—22no—
143C22×C8—22no—
144#117—48no—
145#117—48no—
146#117—48no—
147#117—48no—
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—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C2×Q814yes—
175C2×C4⋊C4#4114yes—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—48no—
198C2×C42—48no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42#4114yes—
205C22⋊C8—44no—
206C22⋊C8—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—42no—
214#95—42no—
215#95—44no—
216#95—44no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24C22⋊Q812yes—
226C24#12312yes—
227C24#12412yes—
228C24C22×D411yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—48no—
264C22×C4—48no—
265C22×C4—48no—
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—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4C22⋊Q812yes—
299C22×C4#12314yes—
300C22×C4#12312yes—
301C22×C4#12412yes—
302C22×C4#12514yes—
303C22×C4#12514yes—
304C22×C4#12614yes—
305C22×C4#12614yes—
306C22×C4#13514yes—
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4112yes—
314C22×C4C2×#4112yes—
315C22×C4H(5,𝔽2)12yes—
316C22×C4H(5,𝔽2)12yes—
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—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
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—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—84no—
350C22⋊C4—84no—
351C42—24no—
352C42—24no—
353C42—48no—
354C42—48no—
355C42—48no—
356C42—42no—
357C42—42no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23#40014yes—
381C23C2×C22⋊Q811yescharacteristic
382C23C2×#12311yescharacteristic
383C23C2×#12311yescharacteristic
384C23#52511yescharacteristic
385C23C2×#12411yescharacteristic
386C23#53411yescharacteristic
387C23#53511yescharacteristic
388C23C2×#13512yes—
389C23#57512yes—
390C23#57512yes—
391C23#57512yes—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—432no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
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—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4#40014yes—
452C2×C4C2×#12511yescharacteristic
453C2×C4C2×#12511yescharacteristic
454C2×C4C2×#12611yescharacteristic
455C2×C4C2×#12611yescharacteristic
456C2×C4#52912yes—
457C2×C4#53312yes—
458C2×C4#53812yes—
459C2×C4#54312yes—
460C2×C4#55112yes—
461C2×C4#55112yes—
462C2×C4#55112yes—
463C2×C4#55112yes—
464C2×C4#55312yes—
465C2×C4#55312yes—
466C2×C4#55312yes—
467C2×C4#55312yes—
468C8—82no—
469C8—82no—
470C22—24no—
471C22—24no—
472C22—24no—
473C22—44no—
474C22—42no—
475C22—42no—
476C22—44no—
477C22—42no—
478C22—42no—
479C22—44no—
480C22—44no—
481C22C2×#40011yescharacteristic
482C22#248211yescharacteristic
483C22#248311yescharacteristic
484C22#250411yescharacteristic
485C22#259111yescharacteristic
486C22#262611yescharacteristic
487C22#285511yescharacteristic
488C22#307612yes—
489C22#308012yes—
490C22#309112yes—
491C22#309112yes—
492C22#309312yes—
493C22#309312yes—
494C22#310212yes—
495C22#311112yes—
496C22#313111yescharacteristic
497C22#313511yescharacteristic
498C22#314011yescharacteristic
499C22#314011yescharacteristic
500C22#319112yes—
501C22#319712yes—
502C22#321212yes—
503C22#321912yes—
504C4—24no—
505C4—44no—
506C4—44no—
507C4—84no—
508C4—84no—
509C4—161no—
510C4—161no—
511C4—161no—
512C4—161no—
513C4#251012yes—
514C4#251012yes—
515C4#256912yes—
516C4#256912yes—
517C2—44no—
518C2#1501511yescharacteristic
519C2#2374411yescharacteristic
520C2#2374611yescharacteristic
521C2#2759012yes—
522C2#2760512yes—
523C2#2797611yescharacteristic
524C2#2797911yescharacteristic
525C2#2933311yescharacteristic
526C2#2933911yescharacteristic
527C2#2995812yes—
528C2#3315212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ 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. #11268C2semidirect

Maximal subgroups

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

subgroupindex
#146822
#137562
#113722
#113162
#113152
#112682
#107312
#107122