Finite Group Databasebeta

← #403055

Subgroups — #403055

Order 512.

1088 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2026C211yesmaximal, characteristic
2C2×C4×C4⋊C8C211yesmaximal, characteristic
3#14709C211yesmaximal, characteristic
4#14703C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic
6#11268C211yesmaximal, characteristic
7#11227C212yesmaximal
8#11077C212yesmaximal
9#11052C211yesmaximal, characteristic
10#10926C212yesmaximal
11#10820C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×C4×C8C2212yes—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×C43C2211yescharacteristic
23#2026—24no—
24C4×C4⋊C8—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33#1436—22no—
34#1436—22no—
35#1436—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40#1434—22no—
41C23×Q8C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#125—24no—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—42no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C42D412yes—
56C22×C42D412yes—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—24no—
61C2×C4⋊C8—24no—
62C2×C4⋊C8—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—42no—
67C2×#103—42no—
68C2×#103—42no—
69C2×C4×C8—24no—
70C2×C4×C8—24no—
71C2×C4×C8—42no—
72#391—22no—
73#391—22no—
74#391—22no—
75#391—24no—
76#391Q814yes—
77#389—24no—
78#382—42no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C43—22no—
92C43—22no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—42no—
98#340—42no—
99#336—22no—
100#336D414yes—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8C2×Q812yes—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—22no—
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—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8C2×D414yes—
135C22×C8#4114yes—
136#125—48no—
137#125—48no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4C2×Q812yes—
166C2×C4⋊C4#4114yes—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—42no—
182C2×C42C2×D412yes—
183C2×C42C2×Q812yes—
184C2×C42#4112yes—
185C2×C42#4112yes—
186C2×C42#4114yes—
187C4⋊C8—216no—
188#103—48no—
189#103—48no—
190C4×C8—48no—
191#95—28no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—42no—
197#95—42no—
198#95—42no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203C24C22×D411yescharacteristic
204C2×Q8—24no—
205C2×Q8—22no—
206C2×Q8—24no—
207C2×Q8—22no—
208C2×Q8—42no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
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#12114yes—
274C22×C4C22⋊Q814yes—
275C22×C4#12412yes—
276C22×C4#12412yes—
277C22×C4#13514yes—
278C22×C4#13514yes—
279C22×C4#13714yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×Q811yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4C2×#4112yes—
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C22×C4C2×#4112yes—
287C22×C4H(5,𝔽2)11yescharacteristic
288C22×C4H(5,𝔽2)12yes—
289C22×C42−1+411yescharacteristic
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—416no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—84no—
320C42—216no—
321C42—24no—
322C42—24no—
323C42—22no—
324C42—24no—
325C42—22no—
326C42—24no—
327C42—22no—
328C42—22no—
329C42—48no—
330C23C2×#12111yescharacteristic
331C23C2×C22⋊Q811yescharacteristic
332C23#52511yescharacteristic
333C23C2×#12411yescharacteristic
334C23#53211yescharacteristic
335C23#53511yescharacteristic
336C23#53711yescharacteristic
337C23C2×#13512yes—
338C23C2×#13512yes—
339C23C2×#13712yes—
340C23#57512yes—
341Q8—44no—
342Q8—44no—
343Q8—82no—
344Q8—82no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4#47918yes—
404C2×C4#48814yes—
405C2×C4#52512yes—
406C2×C4#52912yes—
407C2×C4#52912yes—
408C2×C4#53312yes—
409C2×C4#53812yes—
410C2×C4#54212yes—
411C2×C4#54611yescharacteristic
412C2×C4#54612yes—
413C2×C4D4×Q811yescharacteristic
414C2×C4#55112yes—
415C2×C4#55211yescharacteristic
416C2×C4#55411yescharacteristic
417C2×C4#55412yes—
418C2×C4#55512yes—
419C2×C4#55612yes—
420C2×C4#55912yes—
421C8—44no—
422C8—44no—
423C22#239611yescharacteristic
424C22#241111yescharacteristic
425C22#241711yescharacteristic
426C22#256311yescharacteristic
427C22#259211yescharacteristic
428C22#266211yescharacteristic
429C22#285011yescharacteristic
430C22#305012yes—
431C22#305612yes—
432C22C2×#47912yes—
433C22#307612yes—
434C22#310212yes—
435C22#310712yes—
436C22C2×#48811yescharacteristic
437C22#313111yescharacteristic
438C22#313911yescharacteristic
439C22#314211yescharacteristic
440C22#316812yes—
441C22#317212yes—
442C22#321012yes—
443C22#321212yes—
444C22#324812yes—
445C22#325012yes—
446C4—28no—
447C4—24no—
448C4—42no—
449C4—44no—
450C4—82no—
451C4—84no—
452C4—82no—
453C4—161no—
454C4—162no—
455C4—161no—
456C4#245912yes—
457C4#246512yes—
458C4#253212yes—
459C4#258912yes—
460C2#1486611yescharacteristic
461C2#2377912yes—
462C2#2450512yes—
463C2#2484012yes—
464C2#2914011yescharacteristic
465C2#2914411yescharacteristic
466C2#2934911yescharacteristic
467C2#2935111yescharacteristic
468C2#3034112yes—
469C2#3304411yescharacteristic
470C2#3305911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20262
C2×C4×C4⋊C82
#147092
#147032
#113852
#112682
#112272
#110772
#110522
#109262
#108202