Finite Group Databasebeta

← #470090

Subgroups — #470090

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2075C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic
3#14697C212yesmaximal
4#11328C212yesmaximal, complemented
5#11315C212yesmaximal, complemented
6#11292C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11199C212yesmaximal
9#11096C211yesmaximal, characteristic
10C23×C42C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#385—22no—
18C2×#380C2212yes—
19C2×#380C2212yes—
20#2075—24no—
21#1817—24no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1436—22no—
33#1436—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C22⋊#95—22no—
37C24×C4D411yescharacteristic
38C24×C4D411yescharacteristic
39C24×C4C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2311yescharacteristic
42C23×C8C2311yescharacteristic
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C42—22no—
52C22×C42—22no—
53C22×C42—41no—
54C22×C42—41no—
55C22×C42—41no—
56C22×C42—41no—
57C22×C42D412yes—
58C22×C42C2312yes—
59C2×#107—22no—
60C2×#107—24no—
61C2×#107—41no—
62C2×#107—41no—
63C2×#107—41no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8—41no—
67C2×C22⋊C8—41no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8D414yes—
70#389—42no—
71#389—42no—
72#389—42no—
73#389—42no—
74#385—48no—
75#380—24no—
76#380—24no—
77#380—24no—
78#380—24no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87#340—22no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—41no—
93#340—42no—
94#340—42no—
95#336—24no—
96#336—24no—
97C25C2×D411yescharacteristic
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—24no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—42no—
115C23×C4—41no—
116C23×C4—41no—
117C23×C4—41no—
118C23×C4—41no—
119C23×C4—41no—
120C23×C4—41no—
121C23×C4D814yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4#4112yes—
127C23×C4C2411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
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×C8C2×D412yes—
144C22×C8C2×D412yes—
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—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4#4114yes—
174C2×C4⋊C4#4114yes—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C42—24no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42—44no—
191C2×C42—44no—
192C2×C42—44no—
193C2×C42—44no—
194#107—44no—
195#107—44no—
196C22⋊C8—44no—
197C22⋊C8—44no—
198#95—28no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206C24—22no—
207C24—21no—
208C24—21no—
209C24—22no—
210C24—22no—
211C24—22no—
212C24—21no—
213C24—21no—
214C24#12112yes—
215C24C2×D812yes—
216C24#13512yes—
217C24C22×D411yescharacteristic
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—28no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
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#12112yes—
293C22×C4C2×D814yes—
294C22×C4#13714yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4112yes—
298C22×C4C2×#4112yes—
299C22×C4H(5,𝔽2)11yescharacteristic
300C22×C4H(5,𝔽2)11yescharacteristic
301C22×C4H(5,𝔽2)11yescharacteristic
302C22×C4H(5,𝔽2)11yescharacteristic
303C22×C42−1+412yes—
304C22×C42−1+412yes—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
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—42no—
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—88no—
339C4⋊C4—88no—
340C22⋊C4—416no—
341C22⋊C4—88no—
342C42—48no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—42no—
347C23—24no—
348C23—24no—
349C23—22no—
350C23—24no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—24no—
356C23—24no—
357C23—22no—
358C23—24no—
359C23—41no—
360C23—42no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—42no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23#46614yes—
370C23C2×#12111yescharacteristic
371C23#52512yes—
372C23#53411yescharacteristic
373C23#53411yescharacteristic
374C23#53612yes—
375C23C22×D811yescharacteristic
376C23C2×#13511yescharacteristic
377C23C2×#13712yes—
378C23#57512yes—
379C23#57511yescharacteristic
380C23#57511yescharacteristic
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—416no—
390C2×C4—416no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
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#53411yescharacteristic
437C2×C4#53511yescharacteristic
438C2×C4#53811yescharacteristic
439C2×C4#53811yescharacteristic
440C2×C4#53912yes—
441C2×C4#53912yes—
442C2×C4#54212yes—
443C2×C4#54212yes—
444C2×C4#54712yes—
445C2×C4#54712yes—
446C2×C4#55111yescharacteristic
447C2×C4#55212yes—
448C2×C4#55212yes—
449C2×C4#55312yes—
450C2×C4#55911yescharacteristic
451C2×C4#56311yescharacteristic
452C2×C4#56312yes—
453C2×C4#56311yescharacteristic
454C8—42no—
455C8—42no—
456C8—82no—
457C22—28no—
458C22—24no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—44no—
463C22—42no—
464C22—44no—
465C22—44no—
466C22—42no—
467C22—42no—
468C22#241312yes—
469C22#260011yescharacteristic
470C22#261711yescharacteristic
471C22#263812yes—
472C22#281711yescharacteristic
473C22C2×#46611yescharacteristic
474C22#305411yescharacteristic
475C22#305612yes—
476C22#310012yes—
477C22#310812yes—
478C22#311112yes—
479C22#311612yes—
480C22#318611yescharacteristic
481C22#318611yescharacteristic
482C22#319111yescharacteristic
483C22#319111yescharacteristic
484C22#319511yescharacteristic
485C22#319612yes—
486C22#319711yescharacteristic
487C22#322612yes—
488C22#323212yes—
489C22#323712yes—
490C22#324012yes—
491C4—44no—
492C4—42no—
493C4—44no—
494C4—84no—
495C4—84no—
496C4—162no—
497C4—162no—
498C4#271711yescharacteristic
499C4#271812yes—
500C4#272411yescharacteristic
501C4#276411yescharacteristic
502C4#276512yes—
503C4#277111yescharacteristic
504C2—44no—
505C2#1667511yescharacteristic
506C2#2460012yes—
507C2#2461112yes—
508C2#2940811yescharacteristic
509C2#2941611yescharacteristic
510C2#2973011yescharacteristic
511C2#2974311yescharacteristic
512C2#3016512yes—
513C2#3017012yes—
514C2#3278611yescharacteristic
515C2#3279111yescharacteristic
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
4. #11328C2semidirect
5. #11315C2semidirect
6. #11292C2semidirect
7. #11219C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
C2×#18172
#146972
#113282
#113152
#112922
#112192
#111992
#110962