Finite Group Databasebeta

← #477214

Subgroups — #477214

Order 512.

1142 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
1C2×#2076C212yesmaximal
2C2×#2057C212yesmaximal, complemented
3#15135C211yesmaximal, characteristic
4#15125C211yesmaximal, characteristic
5#11324C212yesmaximal, complemented
6#11316C212yesmaximal, complemented
7#11293C212yesmaximal, complemented
8#11147C211yesmaximal, characteristic
9#11096C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#107C2212yes—
12C22×#107C2212yes—
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C2×#399—21no—
16C2×#393—21no—
17C2×#389C2212yes—
18C2×#384C2211yescharacteristic
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22#2076—28no—
23#2057—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1446—21no—
33#1445—21no—
34#1444—21no—
35#1442—21no—
36#1439—21no—
37#1439—21no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C23×Q8—21no—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—42no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D411yescharacteristic
60C22×C22⋊C4D411yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#107—24no—
64C2×#107—24no—
65C2×#107—24no—
66C2×#107—42no—
67C2×#107—42no—
68C2×#107—42no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—41no—
72C2×C22⋊C8—41no—
73#399—44no—
74#393—44no—
75#389—24no—
76#389—42no—
77#384—41no—
78#384—41no—
79#380—24no—
80#380—24no—
81#380—42no—
82#380—42no—
83C4×C22⋊C4—41no—
84C4×C22⋊C4—41no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95D411yescharacteristic
95C2×#95D411yescharacteristic
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—42no—
101#340D418yes—
102#336—24no—
103#336—24no—
104C25C2×D411yescharacteristic
105C22×Q8—44no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—82no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
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×C8C2×D414yes—
141C22×C8#4114yes—
142C4⋊Q8—44no—
143C4⋊Q8—48no—
144C4⋊Q8—44no—
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—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4C2×D414yes—
178C2×C4⋊C4C2×D414yes—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4Q1614yes—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199#107—48no—
200#107—48no—
201C22⋊C8—44no—
202C22⋊C8—44no—
203#95—41no—
204#95—41no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—42no—
209#95—42no—
210#95—42no—
211#95—42no—
212#95—44no—
213#95—82no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—22no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—22no—
223C24—22no—
224C24C2×Q1612yes—
225C24C22×D411yescharacteristic
226C2×Q8—48no—
227C2×Q8—88no—
228C2×Q8—84no—
229C2×Q8—88no—
230C2×Q8—84no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
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—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
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×C4C2×Q1612yes—
297C22×C4C2×Q1612yes—
298C22×C4AGL(1,ℤ/8)12yes—
299C22×C4AGL(1,ℤ/8)12yes—
300C22×C4C22×D412yes—
301C22×C4C2×#4112yes—
302C22×C4H(5,𝔽2)11yescharacteristic
303C22×C4H(5,𝔽2)12yes—
304C22×C4H(5,𝔽2)11yescharacteristic
305C22×C42−1+412yes—
306C22×C42−1+412yes—
307C22×C42−1+412yes—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—48no—
327C4⋊C4—42no—
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—42no—
339C4⋊C4—88no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—88no—
345C22⋊C4—88no—
346C42—42no—
347C42—42no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C23—24no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—24no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23#53411yescharacteristic
376C23#53611yescharacteristic
377C23#53611yescharacteristic
378C23D4212yes—
379C23#54712yes—
380C23C22×Q1611yescharacteristic
381C23C2×AGL(1,ℤ/8)11yescharacteristic
382C23C2×AGL(1,ℤ/8)11yescharacteristic
383C23#57511yescharacteristic
384C23#57612yes—
385C23#57812yes—
386Q8—816no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—42no—
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—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—44no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
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—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#53512yes—
445C2×C4#53612yes—
446C2×C4#53612yes—
447C2×C4#53912yes—
448C2×C4#54212yes—
449C2×C4#54212yes—
450C2×C4D4211yescharacteristic
451C2×C4#54712yes—
452C2×C4#54812yes—
453C2×C4#55111yescharacteristic
454C2×C4#56011yescharacteristic
455C2×C4#56311yescharacteristic
456C2×C4#56312yes—
457C2×C4#56312yes—
458C2×C4#56312yes—
459C2×C4#56312yes—
460C8—44no—
461C8—44no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—44no—
466C22—44no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—82no—
471C22—82no—
472C22—81no—
473C22#266911yescharacteristic
474C22#267412yes—
475C22#268012yes—
476C22#271711yescharacteristic
477C22#281711yescharacteristic
478C22#318911yescharacteristic
479C22#319211yescharacteristic
480C22#319911yescharacteristic
481C22#320211yescharacteristic
482C22#323011yescharacteristic
483C22#323011yescharacteristic
484C22#323511yescharacteristic
485C22#323511yescharacteristic
486C22#324312yes—
487C22#324312yes—
488C22#326412yes—
489C22#326912yes—
490C22#327412yes—
491C22D4×Q1612yes—
492C22#333812yes—
493C22#334312yes—
494C22#334712yes—
495C22#335312yes—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—84no—
501C4—84no—
502C4—81no—
503C4—162no—
504C4—162no—
505C4#267412yes—
506C4#272312yes—
507C4#282112yes—
508C4#283812yes—
509C2—82no—
510C2#1710611yescharacteristic
511C2#3050711yescharacteristic
512C2#3050911yescharacteristic
513C2#3077912yes—
514C2#3079612yes—
515C2#3108112yes—
516C2#3110012yes—
517C2#3226111yescharacteristic
518C2#3227111yescharacteristic
519C2#3279011yescharacteristic
520C2#3279211yescharacteristic
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
2. C2×#2057C2semidirect
5. #11324C2semidirect
6. #11316C2semidirect
7. #11293C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
C2×#20572
#151352
#151252
#113242
#113162
#112932
#111472
#110962