Finite Group Databasebeta

← #474591

Subgroups — #474591

Order 512.

1224 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14697C212yesmaximal
2#11328C212yesmaximal, complemented
3#11256C212yesmaximal, complemented
4#11219C212yesmaximal, complemented
5#11219C212yesmaximal, complemented
6#11096C211yesmaximal, characteristic
7#11096C212yesmaximal
8#11096C211yesmaximal, characteristic
9#10712C211yesmaximal, characteristic
10C23×C42C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#107C2212yes—
13C22×#107C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#385—22no—
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#336C2212yes—
29C2×#336C2212yes—
30#1436—22no—
31#1436—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C22⋊#95—22no—
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4C2311yescharacteristic
38C23×C8C2311yescharacteristic
39C23×C8C2312yes—
40C23×C8C2311yescharacteristic
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—22no—
49C22×C42—42no—
50C22×C42—42no—
51C22×C42—41no—
52C22×C42—41no—
53C22×C42D412yes—
54C22×C42C2312yes—
55C2×#107—24no—
56C2×#107—24no—
57C2×#107—41no—
58C2×#107—42no—
59C2×#107—42no—
60C2×#107—42no—
61C2×#107—41no—
62C2×C22⋊C8—41no—
63C2×C22⋊C8—41no—
64C2×C22⋊C8—41no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8Q814yes—
67C2×C22⋊C8Q814yes—
68#389—42no—
69#389—42no—
70#389—42no—
71#389—42no—
72#385—48no—
73#380—24no—
74#380—24no—
75#380—42no—
76#380—42no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85#340—24no—
86#340—24no—
87#340—42no—
88#340—42no—
89#340—41no—
90#340—42no—
91#340—41no—
92#336—42no—
93#336—42no—
94C25C2×D411yescharacteristic
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—41no—
108C23×C4—42no—
109C23×C4—41no—
110C23×C4—42no—
111C23×C4—41no—
112C23×C4—42no—
113C23×C4—41no—
114C23×C4—42no—
115C23×C4D814yes—
116C23×C4Q1614yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×Q812yes—
121C23×C4C2×Q812yes—
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8C2×Q812yes—
138C22×C8C2×Q814yes—
139C22×C8C2×Q812yes—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
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—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4#4114yes—
169C2×C4⋊C4#4114yes—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—44no—
186C2×C42—48no—
187C2×C42—44no—
188#107—44no—
189#107—48no—
190#107—44no—
191C22⋊C8—44no—
192C22⋊C8—44no—
193#95—28no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201C24—22no—
202C24—22no—
203C24—21no—
204C24—21no—
205C24—22no—
206C24—22no—
207C24—21no—
208C24—21no—
209C24C4⋊Q812yes—
210C24C2×D812yes—
211C24C2×Q1612yes—
212C24C22×D411yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4—48no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—48no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—41no—
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×C4C4⋊Q812yes—
286C22×C4C2×D814yes—
287C22×C4C2×Q1614yes—
288C22×C4C22×Q811yescharacteristic
289C22×C4C22×Q811yescharacteristic
290C22×C4C2×#4112yes—
291C22×C4C2×#4112yes—
292C22×C4H(5,𝔽2)11yescharacteristic
293C22×C4H(5,𝔽2)12yes—
294C22×C4H(5,𝔽2)12yes—
295C22×C4H(5,𝔽2)11yescharacteristic
296C22×C4H(5,𝔽2)11yescharacteristic
297C22×C42−1+411yescharacteristic
298C2×C8—22no—
299C2×C8—24no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—41no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—88no—
332C4⋊C4—88no—
333C22⋊C4—416no—
334C22⋊C4—88no—
335C42—48no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—42no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—22no—
344C23—24no—
345C23—22no—
346C23—24no—
347C23—22no—
348C23—22no—
349C23—24no—
350C23—22no—
351C23—22no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—41no—
362C23#50014yes—
363C23#52512yes—
364C23C2×C4⋊Q811yescharacteristic
365C23#53412yes—
366C23#53411yescharacteristic
367C23#53711yescharacteristic
368C23C22×D811yescharacteristic
369C23C22×Q1611yescharacteristic
370C23#57512yes—
371C23#57512yes—
372C23#57511yescharacteristic
373C23#57511yescharacteristic
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—416no—
383C2×C4—416no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
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#53812yes—
430C2×C4#53912yes—
431C2×C4#54012yes—
432C2×C4#54212yes—
433C2×C4#54311yescharacteristic
434C2×C4#54311yescharacteristic
435C2×C4#54411yescharacteristic
436C2×C4#54411yescharacteristic
437C2×C4#55111yescharacteristic
438C2×C4#55312yes—
439C2×C4#55412yes—
440C2×C4#55412yes—
441C2×C4#55412yes—
442C2×C4#55412yes—
443C2×C4#55911yescharacteristic
444C2×C4#56011yescharacteristic
445C2×C4#56012yes—
446C2×C4H(3,𝔽4)11yescharacteristic
447C8—42no—
448C8—44no—
449C8—42no—
450C22—24no—
451C22—28no—
452C22—44no—
453C22—44no—
454C22—42no—
455C22—42no—
456C22—44no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22#258312yes—
462C22#260011yescharacteristic
463C22#263012yes—
464C22#265911yescharacteristic
465C22#281211yescharacteristic
466C22#310012yes—
467C22#310412yes—
468C22#311112yes—
469C22#311112yes—
470C22C2×#50011yescharacteristic
471C22#316211yescharacteristic
472C22#316612yes—
473C22#318612yes—
474C22#318611yescharacteristic
475C22#318912yes—
476C22#318911yescharacteristic
477C22#319712yes—
478C22#319712yes—
479C22#319711yescharacteristic
480C22#319711yescharacteristic
481C22#324511yescharacteristic
482C22#324911yescharacteristic
483C22#325612yes—
484C4—42no—
485C4—44no—
486C4—44no—
487C4—84no—
488C4—84no—
489C4—162no—
490C4—162no—
491C4#274811yescharacteristic
492C4#276311yescharacteristic
493C4#278012yes—
494C4#279012yes—
495C4#280411yescharacteristic
496C4#280711yescharacteristic
497C2—44no—
498C2#1698711yescharacteristic
499C2#2929612yes—
500C2#2930512yes—
501C2#2940811yescharacteristic
502C2#2941511yescharacteristic
503C2#3000612yes—
504C2#3001212yes—
505C2#3029111yescharacteristic
506C2#3029711yescharacteristic
507C2#3270611yescharacteristic
508C2#3271011yescharacteristic
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
2. #11328C2semidirect
3. #11256C2semidirect
4. #11219C2semidirect
5. #11219C2semidirect

Maximal subgroups

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

subgroupindex
#146972
#113282
#112562
#112192
#110962
#107122