Finite Group Databasebeta

← #409290

Subgroups — #409290

Order 512.

1216 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#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—21no—
29#1441—21no—
30#1439—21no—
31#1439—21no—
32C4⋊#95—21no—
33C4⋊#95—21no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C22⋊#95—21no—
38C22⋊#95—21no—
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#117—24no—
44C2×#117—24no—
45C2×#117—24no—
46C2×#117—24no—
47C2×#117—41no—
48C2×#117—41no—
49C2×#117—41no—
50C2×#117—41no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—41no—
56C22×C4⋊C4—41no—
57C22×C4⋊C4—41no—
58C22×C4⋊C4—41no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4Q814yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4D411yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76#384—41no—
77#384—41no—
78#382—24no—
79#380—24no—
80#380—24no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90#340—24no—
91#340—24no—
92#340—42no—
93#340—42no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#336—24no—
99#336—41no—
100#336—41no—
101C25C2×D411yescharacteristic
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—82no—
116C23×C4—82no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×Q812yes—
123C23×C4#4112yes—
124C23×C4#4114yes—
125C23×C4#4114yes—
126C23×C4C2411yescharacteristic
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×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141#117—48no—
142#117—48no—
143#117—48no—
144#117—48no—
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—24no—
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—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4C2×Q814yes—
185C2×C4⋊C4#4114yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42#4114yes—
209C22⋊C8—48no—
210#95—42no—
211#95—42no—
212#95—44no—
213#95—82no—
214#95—82no—
215#95—82no—
216#95—82no—
217#95—82no—
218#95—82no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24C22⋊Q812yes—
228C24#12312yes—
229C24#12412yes—
230C24C22×D411yescharacteristic
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—48no—
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—42no—
296C22×C4—88no—
297C22×C4—84no—
298C22×C4—84no—
299C22×C4—88no—
300C22×C4C22⋊Q812yes—
301C22×C4#12312yes—
302C22×C4#12312yes—
303C22×C4#12312yes—
304C22×C4#12412yes—
305C22×C4#12518yes—
306C22×C4#12618yes—
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4112yes—
309C22×C4C2×#4112yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4112yes—
312C22×C4C2×#4112yes—
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C4H(5,𝔽2)11yescharacteristic
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C4H(5,𝔽2)11yescharacteristic
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—44no—
336C4⋊C4—44no—
337C4⋊C4—48no—
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—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—48no—
355C22⋊C4—88no—
356C42—24no—
357C42—24no—
358C42—48no—
359C42—42no—
360C42—42no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—22no—
366C23—22no—
367C23—24no—
368C23—24no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23#40014yes—
384C23C2×C22⋊Q811yescharacteristic
385C23C2×#12311yescharacteristic
386C23C2×#12311yescharacteristic
387C23#52511yescharacteristic
388C23C2×#12411yescharacteristic
389C23#53411yescharacteristic
390C23#53511yescharacteristic
391C23#57612yes—
392C23#57712yes—
393C23#57712yes—
394C23#57812yes—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—816no—
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—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#40014yes—
455C2×C4C2×#12512yes—
456C2×C4C2×#12612yes—
457C2×C4#52912yes—
458C2×C4#53312yes—
459C2×C4#53811yescharacteristic
460C2×C4#53811yescharacteristic
461C2×C4#54311yescharacteristic
462C2×C4#54311yescharacteristic
463C2×C4#55112yes—
464C2×C4#55112yes—
465C2×C4#55112yes—
466C2×C4#55112yes—
467C2×C4#55312yes—
468C2×C4#55312yes—
469C2×C4#55312yes—
470C2×C4#55312yes—
471C8—82no—
472C8—82no—
473C22—24no—
474C22—28no—
475C22—42no—
476C22—42no—
477C22—44no—
478C22—44no—
479C22—42no—
480C22—42no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22C2×#40011yescharacteristic
485C22#248211yescharacteristic
486C22#248311yescharacteristic
487C22#250411yescharacteristic
488C22#259111yescharacteristic
489C22#262611yescharacteristic
490C22#285511yescharacteristic
491C22#308112yes—
492C22#308212yes—
493C22#309812yes—
494C22#309812yes—
495C22#309912yes—
496C22#309912yes—
497C22#311912yes—
498C22#312012yes—
499C22#313612yes—
500C22#313712yes—
501C22#319911yescharacteristic
502C22#320011yescharacteristic
503C22#320011yescharacteristic
504C22#320211yescharacteristic
505C22#322412yes—
506C22#322512yes—
507C4—24no—
508C4—42no—
509C4—42no—
510C4—84no—
511C4—84no—
512C4—81no—
513C4—81no—
514C4—162no—
515C4—162no—
516C4#251012yes—
517C4#251012yes—
518C4#256912yes—
519C4#256912yes—
520C2—44no—
521C2#1501511yescharacteristic
522C2#2374912yes—
523C2#2759312yes—
524C2#2760811yescharacteristic
525C2#2760911yescharacteristic
526C2#2798412yes—
527C2#2934311yescharacteristic
528C2#2934411yescharacteristic
529C2#2995911yescharacteristic
530C2#2996011yescharacteristic
531C2#3315812yes—
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