Finite Group Databasebeta

← #435818

Subgroups — #435818

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2077C212yesmaximal
2C2×#1852C212yesmaximal
3#15150C211yesmaximal, characteristic
4#13904C211yesmaximal, characteristic
5#11378C212yesmaximal, complemented
6#11377C212yesmaximal, complemented
7#11324C212yesmaximal, complemented
8#10922C211yesmaximal, characteristic
9#10863C212yesmaximal, complemented
10C23×C4⋊C4C2211yescharacteristic
11C22×#107C2212yes—
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×C22⋊C8C2212yes—
15C22×#97C2211yescharacteristic
16C2×#397—22no—
17C2×#384C2211yescharacteristic
18C2×#381—22no—
19C2×#380C2212yes—
20C2×#380C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22#2077—28no—
23#1852—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31#1445—22no—
32#1445—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C24×C4—21no—
38C24×C4C2311yescharacteristic
39C23×C8C2312yes—
40C23×C8C2312yes—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—41no—
46C22×C4⋊C4—41no—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4D412yes—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#107—24no—
57C2×#107—24no—
58C2×#107—24no—
59C2×#107—24no—
60C2×#107—42no—
61C2×#107—42no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—24no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8D418yes—
68C2×#97—24no—
69C2×#97—41no—
70#397—48no—
71#384—41no—
72#384—41no—
73#381—48no—
74#380—24no—
75#380—24no—
76#380—24no—
77C4×C22⋊C4—41no—
78C4×C22⋊C4—41no—
79C2×#95—24no—
80C2×#95—22no—
81C2×#95—24no—
82C2×#95—24no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95D412yes—
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#336—24no—
92#336—41no—
93C25C2×D411yescharacteristic
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—42no—
107C23×C4—44no—
108C23×C4—42no—
109C23×C4—44no—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D414yes—
113C23×C4C2×D412yes—
114C23×C4C2411yescharacteristic
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8C2×D414yes—
126C22×C8C2×D414yes—
127C22×C8#4114yes—
128C22×C8#4114yes—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
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—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—48no—
154C2×C4⋊C4—48no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4#4114yes—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—48no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—22no—
176#107—48no—
177#107—48no—
178#106—48no—
179C22⋊C8—216no—
180#97—44no—
181#95—42no—
182#95—48no—
183#95—44no—
184#95—48no—
185#95—42no—
186#95—44no—
187#95—48no—
188C24—21no—
189C24—21no—
190C24—22no—
191C24—22no—
192C24—22no—
193C24—24no—
194C24#12712yes—
195C24AGL(1,ℤ/8)14yes—
196C24C22×D411yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—48no—
228C22×C4—416no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—84no—
249C22×C4—88no—
250C22×C4—84no—
251C22×C4#12712yes—
252C22×C4AGL(1,ℤ/8)14yes—
253C22×C4C22×D412yes—
254C22×C4C2×#4112yes—
255C22×C4C2×#4112yes—
256C22×C4C2×#4112yes—
257C22×C4H(5,𝔽2)11yescharacteristic
258C22×C4H(5,𝔽2)11yescharacteristic
259C22×C42−1+412yes—
260C22×C42−1+412yes—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8#12318yes—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—48no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—88no—
293C22⋊C4—48no—
294C22⋊C4—84no—
295C22⋊C4—88no—
296C42—42no—
297C42—42no—
298C42—41no—
299C42—41no—
300C42—41no—
301C42—41no—
302C23—28no—
303C23—28no—
304C23—24no—
305C23—22no—
306C23—24no—
307C23—24no—
308C23—24no—
309C23—22no—
310C23—44no—
311C23—44no—
312C23—42no—
313C23—42no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23#49614yes—
318C23#52511yescharacteristic
319C23C2×#12711yescharacteristic
320C23#53411yescharacteristic
321C23#54712yes—
322C23#54712yes—
323C23C2×AGL(1,ℤ/8)12yes—
324C23C2×AGL(1,ℤ/8)12yes—
325C23#57814yes—
326C2×C4—28no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—816no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4C2×#12312yes—
367C2×C4#52912yes—
368C2×C4#53612yes—
369C2×C4#53912yes—
370C2×C4#54112yes—
371C2×C4#54212yes—
372C2×C4#54612yes—
373C2×C4#54612yes—
374C2×C4#54712yes—
375C2×C4#54712yes—
376C2×C4#55111yescharacteristic
377C2×C4#55212yes—
378C2×C4#55612yes—
379C2×C4#55911yescharacteristic
380C2×C4#56311yescharacteristic
381C2×C4#56311yescharacteristic
382C8—28no—
383C8—44no—
384C22—28no—
385C22—24no—
386C22—44no—
387C22—44no—
388C22—48no—
389C22—42no—
390C22—42no—
391C22—44no—
392C22#247812yes—
393C22#250112yes—
394C22#253911yescharacteristic
395C22#271711yescharacteristic
396C22#272411yescharacteristic
397C22#310912yes—
398C22#312712yes—
399C22C2×#49611yescharacteristic
400C22#315111yescharacteristic
401C22#315612yes—
402C22#319212yes—
403C22#320212yes—
404C22#334314yes—
405C22#334314yes—
406C22#334314yes—
407C22#335314yes—
408C4—42no—
409C4—44no—
410C4—82no—
411C4—84no—
412C4—162no—
413C4—162no—
414C4—162no—
415C4#241912yes—
416C4#244712yes—
417C4#268912yes—
418C4#269812yes—
419C2—44no—
420C2#1553911yescharacteristic
421C2#2747514yes—
422C2#2793912yes—
423C2#2794412yes—
424C2#2831011yescharacteristic
425C2#2832211yescharacteristic
426C2#3227112yes—
427C2#3246912yes—
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
5. #11378C2semidirect
6. #11377C2semidirect
7. #11324C2semidirect
9. #10863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20772
C2×#18522
#151502
#139042
#113782
#113772
#113242
#109222
#108632