Finite Group Databasebeta

← #417353

Subgroups — #417353

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1972C211yesmaximal, characteristic
2C2×#1944C211yesmaximal, characteristic, complemented
3C2×#1936C212yesmaximal, complemented
4#14700C212yesmaximal, complemented
5#11378C212yesmaximal, complemented
6#11123C212yesmaximal, complemented
7#11121C212yesmaximal, complemented
8#11022C212yesmaximal
9#10693C211yesmaximal, characteristic, complemented
10C23×D8C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#106C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#97C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C2×#390C2212yes—
20C2×#390C2212yes—
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×#388—24no—
24C2×C43C2211yescharacteristic
25#1972—24no—
26#1944—24no—
27#1936—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1437—24no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C23×D4C2312yes—
37C22×D8—24no—
38C22×D8—22no—
39C22×D8—24no—
40C22×D8—41no—
41C22×D8—41no—
42C23×C8C2312yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—24no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—24no—
52C22×C22⋊C4—22no—
53C22×C42D412yes—
54C22×C42D412yes—
55C22×C42C2312yes—
56C22×C42C2311yescharacteristic
57C2×#107—22no—
58C2×#107—41no—
59C2×#107—41no—
60C2×#106—22no—
61C2×#106—24no—
62C2×#106—22no—
63C2×#106—24no—
64C2×#102—24no—
65C2×#102—24no—
66C2×#102—24no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—42no—
72C2×#102—42no—
73C2×#97—41no—
74C2×#97—41no—
75C2×#97—41no—
76C2×#97—41no—
77C2×#97D414yes—
78C2×#97D414yes—
79#390—24no—
80#390—24no—
81#390—24no—
82#390—24no—
83#390—42no—
84#390—42no—
85#389—24no—
86#389—24no—
87#388—416no—
88C2×#95—24no—
89C2×#95—24no—
90C2×#95—22no—
91C43—41no—
92C43—41no—
93#340—24no—
94#340—24no—
95#340—24no—
96#336—24no—
97#336—24no—
98C25—22no—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4#4114yes—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D414yes—
119C23×C4C2411yescharacteristic
120C2×D8—28no—
121C2×D8—48no—
122C2×D8—48no—
123C2×D8—44no—
124C2×D8—44no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8C2×D414yes—
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C22×C8#4114yes—
141C22×C8#4114yes—
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—48no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4#4114yes—
162C2×C22⋊C4—216no—
163C2×C22⋊C4—48no—
164C2×C22⋊C4—48no—
165C2×C22⋊C4—48no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—48no—
168C2×C22⋊C4—48no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—48no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—28no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—44no—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188#107—44no—
189#106—28no—
190#106—44no—
191#102—216no—
192#102—48no—
193#102—48no—
194#97—44no—
195#97—44no—
196#95—48no—
197#95—44no—
198#95—48no—
199C24—24no—
200C24—42no—
201C24—42no—
202C24—42no—
203C24—42no—
204C24—42no—
205C24—42no—
206C24—44no—
207C24—44no—
208C24—44no—
209C24—44no—
210C24C22×D411yescharacteristic
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4#12712yes—
267C22×C4#12712yes—
268C22×C4AGL(1,ℤ/8)18yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C2×#4112yes—
272C22×C4C2×#4112yes—
273C22×C4C2×#4112yes—
274C22×C4C2×#4112yes—
275C22×C4H(5,𝔽2)11yescharacteristic
276C22×C4H(5,𝔽2)11yescharacteristic
277C22×C4H(5,𝔽2)11yescharacteristic
278C22×C42−1+411yescharacteristic
279D8—416no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8#12414yes—
296C2×C8#12414yes—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—88no—
310C22⋊C4—432no—
311C22⋊C4—88no—
312C22⋊C4—88no—
313C22⋊C4—816no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—48no—
319C42—48no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—42no—
324C23—44no—
325C23—44no—
326C23—48no—
327C23—48no—
328C23—44no—
329C23—82no—
330C23—82no—
331C23—82no—
332C23—84no—
333C23—84no—
334C23—82no—
335C23—82no—
336C23—82no—
337C23—82no—
338C23—82no—
339C23—84no—
340C23—84no—
341C23—84no—
342C23—84no—
343C23—84no—
344C23—84no—
345C23—84no—
346C23—84no—
347C23#52512yes—
348C23C2×#12711yescharacteristic
349C23#53212yes—
350C23#53411yescharacteristic
351C23#53711yescharacteristic
352C23C2×AGL(1,ℤ/8)14yes—
353C23#57514yes—
354D4—44no—
355D4—44no—
356D4—44no—
357D4—44no—
358D4—82no—
359D4—82no—
360D4—82no—
361D4—82no—
362C2×C4—28no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—44no—
379C2×C4—84no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4#52511yescharacteristic
393C2×C4#52511yescharacteristic
394C2×C4#52512yes—
395C2×C4C2×#12411yescharacteristic
396C2×C4C2×#12411yescharacteristic
397C2×C4#53212yes—
398C2×C4#53411yescharacteristic
399C2×C4#53411yescharacteristic
400C2×C4#53711yescharacteristic
401C2×C4#53711yescharacteristic
402C2×C4#55112yes—
403C2×C4#55112yes—
404C2×C4#55212yes—
405C2×C4#55312yes—
406C2×C4#55312yes—
407C2×C4#55312yes—
408C2×C4#55612yes—
409C2×C4#55912yes—
410C8—24no—
411C8—24no—
412C8—44no—
413C22—88no—
414C22—88no—
415C22—84no—
416C22—84no—
417C22—84no—
418C22—164no—
419C22—164no—
420C22—162no—
421C22—162no—
422C22—162no—
423C22—162no—
424C22#239711yescharacteristic
425C22#260112yes—
426C22#260212yes—
427C22#265611yescharacteristic
428C22#285811yescharacteristic
429C22#311414yes—
430C22#311514yes—
431C22#315112yes—
432C22#315312yes—
433C22#317114yes—
434C22#318014yes—
435C22#319212yes—
436C22#319712yes—
437C22#325312yes—
438C22#325612yes—
439C4—48no—
440C4—42no—
441C4—42no—
442C4—84no—
443C4—162no—
444C4—162no—
445C4#256111yescharacteristic
446C4#256211yescharacteristic
447C4#259111yescharacteristic
448C4#259211yescharacteristic
449C4#260112yes—
450C4#260212yes—
451C2—164no—
452C2#1525011yescharacteristic
453C2#2380612yes—
454C2#2944514yes—
455C2#2947614yes—
456C2#3024711yescharacteristic
457C2#3024911yescharacteristic
458C2#3325112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1944C2semidirect
3. C2×#1936C2semidirect
4. #14700C2semidirect
5. #11378C2semidirect
6. #11123C2semidirect
7. #11121C2semidirect
9. #10693C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19722
C2×#19442
C2×#19362
#147002
#113782
#111232
#111212
#110222
#106932