Finite Group Databasebeta

← #477627

Subgroups — #477627

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2076C212yesmaximal, complemented
2#15181C212yesmaximal
3#11377C211yesmaximal, characteristic, complemented
4#11328C211yesmaximal, characteristic, complemented
5#11219C212yesmaximal, complemented
6#11096C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#11096C212yesmaximal, complemented
9#10964C211yesmaximal, characteristic
10C22×C4×D4C2211yescharacteristic
11C22×#107C2212yes—
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C2×#389C2212yes—
17C2×#380C2212yes—
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×#380C2212yes—
21#2076—28no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31#1442—22no—
32#1439—22no—
33#1439—22no—
34#1438—22no—
35C4⋊#95C2212yes—
36C23×D4D411yescharacteristic
37C24×C4C2312yes—
38C23×C8C2311yescharacteristic
39C23×C8C2312yes—
40C23×C8C2311yescharacteristic
41C2×C4×D4—41no—
42C2×C4×D4—41no—
43C2×C4×D4—42no—
44C2×C4×D4—42no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4D412yes—
56C22×C42C2311yescharacteristic
57C2×#107—24no—
58C2×#107—22no—
59C2×#107—24no—
60C2×#107—42no—
61C2×#107—41no—
62C2×#107—41no—
63C2×#107—41no—
64C2×#107—42no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—42no—
70C2×C22⋊C8—42no—
71#389—42no—
72#389—42no—
73#380—24no—
74#380—24no—
75#380—42no—
76#380—42no—
77#380—42no—
78#380—42no—
79#380—42no—
80#380—42no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88#340—24no—
89#340—24no—
90#340—42no—
91#340—42no—
92#340—41no—
93#340—42no—
94#340D414yescomplemented
95#336—41no—
96#336—41no—
97C25C2×D412yes—
98C22×D4—42no—
99C22×D4—41no—
100C22×D4—42no—
101C22×D4—41no—
102C22×D4Q1614yes—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8C2×D412yes—
140C22×C8#4112yes—
141C22×C8#4114yes—
142C4×D4—44no—
143C4×D4—48no—
144C4×D4—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—22no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4C2×D414yes—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193#107—48no—
194#107—44no—
195#107—44no—
196C22⋊C8—48no—
197C22⋊C8—48no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—82no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—22no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24C2×Q1614yes—
220C24C22×D411yescharacteristic
221C2×D4—48no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—48no—
225C2×D4—44no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
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—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—42no—
297C22×C4—42no—
298C22×C4C2×Q1612yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4112yes—
302C22×C4H(5,𝔽2)11yescharacteristic
303C22×C4H(5,𝔽2)12yes—
304C22×C4H(5,𝔽2)11yescharacteristic
305C22×C4H(5,𝔽2)11yescharacteristic
306C22×C4H(5,𝔽2)12yes—
307C22×C42−1+412yes—
308C22×C42−1+411yescharacteristic
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—24no—
312C2×C8—24no—
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—41no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—41no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—44no—
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—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—88no—
344C22⋊C4—48no—
345C22⋊C4—88no—
346C22⋊C4—88no—
347C22⋊C4—88no—
348C22⋊C4—88no—
349C42—41no—
350C42—42no—
351C42—41no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—24no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23#53412yes—
388C23#53411yescharacteristic
389C23D4211yescharacteristic
390C23#54711yescharacteristic
391C23#54712yes—
392C23C22×Q1611yescharacteristic
393C23#57512yes—
394C23#57511yescharacteristic
395C23#57611yescharacteristic
396C23#57612yes—
397C23#57811yescharacteristic
398D4—416no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—41no—
418C2×C4—42no—
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—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4#53712yes—
453C2×C4#53711yescharacteristic
454C2×C4#53811yescharacteristic
455C2×C4#53912yes—
456C2×C4#54012yes—
457C2×C4#54212yes—
458C2×C4#54212yes—
459C2×C4#54612yes—
460C2×C4#55011yescharacteristic
461C2×C4#55112yes—
462C2×C4#55911yescharacteristic
463C2×C4#56011yescharacteristic
464C2×C4#56012yes—
465C2×C4H(3,𝔽4)11yescharacteristic
466C2×C4#56212yes—
467C2×C4#56211yescharacteristic
468C2×C4#56311yescharacteristic
469C2×C4#56312yes—
470C8—44no—
471C8—42no—
472C8—42no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22#267112yes—
489C22#268211yescharacteristic
490C22#271712yes—
491C22#272411yescharacteristic
492C22#281211yescharacteristic
493C22#318912yes—
494C22#318911yescharacteristic
495C22#319712yes—
496C22#319711yescharacteristic
497C22#319912yes—
498C22#319911yescharacteristic
499C22#320312yes—
500C22#320311yescharacteristic
501C22#326812yes—
502C22#327311yescharacteristic
503C22D4×Q1611yescharacteristic
504C22#333811yescharacteristic
505C22#333812yes—
506C22#334712yes—
507C22#334712yes—
508C22#334912yes—
509C22#335211yescharacteristic
510C22#335512yes—
511C4—42no—
512C4—84no—
513C4—81no—
514C4—82no—
515C4—162no—
516C4—162no—
517C4—162no—
518C4#270512yes—
519C4#271211yescharacteristic
520C4#275211yescharacteristic
521C4#282311yescharacteristic
522C4#283211yescharacteristic
523C4#283712yes—
524C2—84no—
525C2#1711811yescharacteristic
526C2#3056312yes—
527C2#3057412yes—
528C2#3119011yescharacteristic
529C2#3120711yescharacteristic
530C2#3226112yes—
531C2#3227212yes—
532C2#3246011yescharacteristic
533C2#3247011yescharacteristic
534C2#3271011yescharacteristic
535C2#3271511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2076C2semidirect
3. #11377C2semidirect
4. #11328C2semidirect
5. #11219C2semidirect
6. #11096C2semidirect
7. #11096C2semidirect
8. #11096C2semidirect
94. #340D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
#151812
#113772
#113282
#112192
#110962
#109642