Finite Group Databasebeta

← #430474

Subgroups — #430474

Order 512.

1380 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
1C2×#1935C212yesmaximal, complemented
2C2×#1873C212yesmaximal
3#15178C211yesmaximal, characteristic
4#14129C211yesmaximal, characteristic
5#11316C212yesmaximal, complemented
6#11067C212yesmaximal, complemented
7#11010C212yesmaximal, complemented
8#10970C212yesmaximal
9#10922C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×#117C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2211yescharacteristic
19C2×#391C2212yes—
20C2×#389C2212yes—
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23#1935—28no—
24#1873—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31#1444—22no—
32#1439—22no—
33#1439—22no—
34#1436—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C23×Q8C2312yes—
38C24×C4C2311yescharacteristic
39C23×C8C2312yes—
40C23×C8C2312yes—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—42no—
43C2×C4⋊Q8—41no—
44C2×C4⋊Q8—41no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—42no—
50C2×C22⋊Q8—42no—
51C2×#117—24no—
52C2×#117—42no—
53C2×#117—41no—
54C2×#117—41no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D412yes—
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—42no—
83C2×#97—24no—
84C2×#97—41no—
85#391—24no—
86#391—42no—
87#389—24no—
88#389—42no—
89#380—24no—
90C4×C22⋊C4—41no—
91C4×C22⋊C4—41no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—42no—
103#340D418yes—
104#336—42no—
105#336—41no—
106C25C2×D411yescharacteristic
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8#4114yes—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4#4114yes—
136C23×C4C2411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8C2×D414yes—
151C4⋊Q8—44no—
152C4⋊Q8—48no—
153C4⋊Q8—44no—
154C22⋊Q8—216no—
155C22⋊Q8—48no—
156C22⋊Q8—48no—
157C22⋊Q8—48no—
158#117—44no—
159#117—48no—
160#117—44no—
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—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4C2×D414yes—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4#4114yes—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—82no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—44no—
207C2×C42—42no—
208#103—216no—
209#103—48no—
210#103—48no—
211#103—48no—
212C22⋊C8—48no—
213#97—44no—
214#95—24no—
215#95—42no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—82no—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24#12412yes—
231C24C22×D411yescharacteristic
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4#12412yes—
322C22×C4#13714yes—
323C22×C4C22×D412yes—
324C22×C4C2×#4112yes—
325C22×C4C2×#4112yes—
326C22×C4C2×#4112yes—
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C4H(5,𝔽2)12yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C4H(5,𝔽2)11yescharacteristic
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—416no—
352C4⋊C4—416no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—88no—
360C22⋊C4—48no—
361C22⋊C4—416no—
362C22⋊C4—48no—
363C22⋊C4—84no—
364C42—48no—
365C42—42no—
366C42—42no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C23—22no—
372C23—22no—
373C23—24no—
374C23—22no—
375C23—24no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23#52511yescharacteristic
393C23C2×#12411yescharacteristic
394C23#53411yescharacteristic
395C23D4212yes—
396C23#54612yes—
397C23C2×#13712yes—
398C23#57512yes—
399C23#57712yes—
400C23#57812yes—
401Q8—44no—
402Q8—44no—
403Q8—44no—
404Q8—44no—
405Q8—82no—
406Q8—82no—
407Q8—82no—
408Q8—82no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
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—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4#52512yes—
461C2×C4#52512yes—
462C2×C4#53512yes—
463C2×C4#53812yes—
464C2×C4#53812yes—
465C2×C4#53812yes—
466C2×C4D4211yescharacteristic
467C2×C4#54612yes—
468C2×C4#55012yes—
469C2×C4#55111yescharacteristic
470C2×C4#55112yes—
471C2×C4#55312yes—
472C2×C4#55312yes—
473C2×C4#55312yes—
474C2×C4#56011yescharacteristic
475C2×C4#56011yescharacteristic
476C8—44no—
477C8—82no—
478C22—42no—
479C22—44no—
480C22—44no—
481C22—42no—
482C22—42no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22#245612yes—
488C22#247212yes—
489C22#259111yescharacteristic
490C22#266911yescharacteristic
491C22#269711yescharacteristic
492C22#311612yes—
493C22#312311yescharacteristic
494C22#312711yescharacteristic
495C22#313511yescharacteristic
496C22#313712yes—
497C22#313911yescharacteristic
498C22#319612yes—
499C22#320112yes—
500C22#326712yes—
501C22#326912yes—
502C22#327612yes—
503C22#327812yes—
504C22#331212yes—
505C22#331312yes—
506C22#332012yes—
507C22#332112yes—
508C4—42no—
509C4—44no—
510C4—84no—
511C4—82no—
512C4—84no—
513C4—84no—
514C4—84no—
515C4#245612yes—
516C4#259812yes—
517C4#271412yes—
518C4#277812yes—
519C2—82no—
520C2#1542011yescharacteristic
521C2#2663912yes—
522C2#2664212yes—
523C2#2726512yes—
524C2#2728312yes—
525C2#2933811yescharacteristic
526C2#2934111yescharacteristic
527C2#3051412yes—
528C2#3167712yes—
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
1. C2×#1935C2semidirect
5. #11316C2semidirect
6. #11067C2semidirect
7. #11010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19352
C2×#18732
#151782
#141292
#113162
#110672
#110102
#109702
#109222