Finite Group Databasebeta

← #361237

Subgroups — #361237

Order 512.

986 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#106C211yesmaximal, characteristic
2#13921C211yesmaximal, characteristic
3#13818C211yesmaximal, characteristic
4#11292C212yesmaximal
5#11220C212yesmaximal
6#10908C212yesmaximal
7#10820C211yesmaximal, characteristic
8#10807C211yesmaximal, characteristic
9#10807C212yesmaximal
10#10806C211yesmaximal, characteristic
11#10797C211yesmaximal, characteristic
12C22×#106C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#389C2212yes—
16C2×#383—21no—
17C2×C4×C4⋊C4C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C4×#106C418yescomplemented
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1443—22no—
33#1436—22no—
34C4⋊#95—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39#1434—21no—
40C4×#95—21no—
41C4×#95C2211yescharacteristic
42C23×C8C2312yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C42—21no—
53C22×C42—21no—
54C22×C42—22no—
55C22×C42C2312yes—
56C22×C42C2311yescharacteristic
57C2×#106—22no—
58C2×#106—24no—
59C2×#106—22no—
60C2×#106C2×C414yes—
61C2×#106C2×C414yes—
62C2×C4×C8—24no—
63C2×C4×C8C2×C414yes—
64#389—24no—
65#389—24no—
66#383—44no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—24no—
70C4×C4⋊C4—42no—
71C4×C4⋊C4C2×C414yes—
72C4×C4⋊C4C2×C414yes—
73C2×#95—22no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95D411yescharacteristic
86C2×#95D412yes—
87C2×#95D411yescharacteristic
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—42no—
94#340C2×C414yes—
95#340C2×C414yes—
96#336—24no—
97#336—22no—
98#336—42no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2411yescharacteristic
110C22×C8—22no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8C22×C412yes—
125C22×C8C22×C412yes—
126C22×C8#4114yes—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
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—28no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4#4114yes—
159C2×C4⋊C4#4112yes—
160C2×C4⋊C4#4114yes—
161C2×C4⋊C4#4112yes—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—24no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—44no—
175C2×C42—44no—
176C2×C42—82no—
177C2×C42—82no—
178C2×C42C22×C412yes—
179C2×C42#4112yes—
180#106—28no—
181#106—28no—
182C4×C8—28no—
183#95—28no—
184#95—24no—
185#95—24no—
186#95—22no—
187#95—42no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—42no—
193#95—42no—
194#95—81no—
195#95—81no—
196#95—82no—
197#95SD1614yes—
198C24C22×D411yescharacteristic
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4C4×D412yes—
255C22×C4C4×D412yes—
256C22×C4C2×SD1614yes—
257C22×C4C2×SD1612yes—
258C22×C4#13512yes—
259C22×C4C23×C411yescharacteristic
260C22×C4C2×#4112yes—
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4112yes—
263C22×C4C2×#4111yescharacteristic
264C22×C4C2×#4112yes—
265C22×C4C2×#4111yescharacteristic
266C22×C4H(5,𝔽2)11yescharacteristic
267C22×C42−1+412yes—
268C22×C42−1+411yescharacteristic
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8#11714yes—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—88no—
307C4⋊C4#11714yes—
308C4⋊C4#11714yes—
309C42—22no—
310C42—22no—
311C42—24no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—48no—
317C42—84no—
318C42—88no—
319C23C2×C4×D411yescharacteristic
320C23#52512yes—
321C23#54611yescharacteristic
322C23#54712yes—
323C23#54811yescharacteristic
324C23C22×SD1611yescharacteristic
325C23C2×#13511yescharacteristic
326C23#57512yes—
327C23#57611yescharacteristic
328C23#57712yes—
329C23#57811yescharacteristic
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4C4×SD1614yes—
390C2×C4C2×#11711yescharacteristic
391C2×C4C2×#11711yescharacteristic
392C2×C4C2×#11711yescharacteristic
393C2×C4C4×#4111yescharacteristic
394C2×C4#51911yescharacteristic
395C2×C4#52011yescharacteristic
396C2×C4#52912yes—
397C2×C4#52912yes—
398C2×C4#52912yes—
399C2×C4#54112yes—
400C2×C4#54212yes—
401C2×C4#55212yes—
402C2×C4#55211yescharacteristic
403C2×C4#55212yes—
404C2×C4#55312yes—
405C2×C4#55311yescharacteristic
406C2×C4#55612yes—
407C2×C4#55612yes—
408C8—24no—
409C8—42no—
410C8—82no—
411C22#229011yescharacteristic
412C22#234411yescharacteristic
413C22#235211yescharacteristic
414C22#246812yes—
415C22#246912yes—
416C22C2×C4×SD1611yescharacteristic
417C22#294311yescharacteristic
418C22#294911yescharacteristic
419C22#295111yescharacteristic
420C22#310212yes—
421C22#310312yes—
422C22#311812yes—
423C22#312012yes—
424C22#329811yescharacteristic
425C22#330511yescharacteristic
426C22#332312yes—
427C22#333512yes—
428C22#333712yes—
429C22#334712yes—
430C22#334912yes—
431C22#336211yescharacteristic
432C22#336411yescharacteristic
433C22#337612yes—
434C4—24no—
435C4—42no—
436C4—44no—
437C4—44no—
438C4—44no—
439C4—82no—
440C4—82no—
441C4—84no—
442C4—82no—
443C4—162no—
444C4#228311yescharacteristic
445C4#232011yescharacteristic
446C4#234711yescharacteristic
447C4#235611yescharacteristic
448C4#252412yes—
449C4#252512yes—
450C2#1389311yescharacteristic
451C2#2111511yescharacteristic
452C2#2113011yescharacteristic
453C2#2260911yescharacteristic
454C2#2261611yescharacteristic
455C2#2276011yescharacteristic
456C2#2276711yescharacteristic
457C2#2709812yes—
458C2#2710612yes—
459C2#2713012yes—
460C2#2713812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
20. C4×#106C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1062
#139212
#138182
#112922
#112202
#109082
#108202
#108072
#108062
#107972