Finite Group Databasebeta

← #361233

Subgroups — #361233

Order 512.

1008 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#107C211yesmaximal, characteristic
2#13921C211yesmaximal, characteristic
3#13821C211yesmaximal, characteristic
4#11292C212yesmaximal
5#11219C212yesmaximal
6#10907C212yesmaximal
7#10820C211yesmaximal, characteristic
8#10806C211yesmaximal, characteristic
9#10806C211yesmaximal, characteristic
10#10806C212yesmaximal
11#10797C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#389C2212yes—
16C2×#383—21no—
17C2×C4×C4⋊C4—21no—
18C2×C4×C4⋊C4C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C4×#107C418yescomplemented
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1443—22no—
34#1436—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39#1434—21no—
40#1434—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×#107—22no—
58C2×#107—24no—
59C2×#107—22no—
60C2×#107C2×C414yes—
61C2×#107C2×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⋊C4—44no—
72C4×C4⋊C4C2×C414yes—
73C4×C4⋊C4C2×C414yes—
74C2×#95—22no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95D411yescharacteristic
87C2×#95D412yes—
88C2×#95D411yescharacteristic
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340C2×C414yes—
96#340C2×C414yes—
97#336—24no—
98#336—22no—
99#336—42no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4C2×D412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2411yescharacteristic
111C22×C8—22no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8C22×C412yes—
126C22×C8C22×C412yes—
127C22×C8#4114yes—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
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—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4#4114yes—
160C2×C4⋊C4#4112yes—
161C2×C4⋊C4#4114yes—
162C2×C4⋊C4#4112yes—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—44no—
178C2×C42—44no—
179C2×C42—82no—
180C2×C42C22×C412yes—
181C2×C42#4112yes—
182#107—28no—
183#107—28no—
184C4×C8—28no—
185#95—28no—
186#95—24no—
187#95—24no—
188#95—22no—
189#95—42no—
190#95—44no—
191#95—44no—
192#95—42no—
193#95—44no—
194#95—44no—
195#95—42no—
196#95—81no—
197#95—81no—
198#95—82no—
199#95D814yes—
200C24C22×D411yescharacteristic
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C4×D412yes—
259C22×C4C4×D412yes—
260C22×C4C2×D814yes—
261C22×C4C2×D812yes—
262C22×C4#13512yes—
263C22×C4C23×C411yescharacteristic
264C22×C4C2×#4112yes—
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4112yes—
267C22×C4C2×#4111yescharacteristic
268C22×C4C2×#4112yes—
269C22×C4C2×#4111yescharacteristic
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C42−1+412yes—
272C22×C42−1+411yescharacteristic
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8#11714yes—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—416no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4#11714yes—
312C4⋊C4#11714yes—
313C42—22no—
314C42—22no—
315C42—24no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—48no—
321C42—48no—
322C42—88no—
323C23C2×C4×D411yescharacteristic
324C23#52512yes—
325C23#54611yescharacteristic
326C23#54712yes—
327C23#54811yescharacteristic
328C23C22×D811yescharacteristic
329C23C2×#13511yescharacteristic
330C23#57512yes—
331C23#57711yescharacteristic
332C23#57711yescharacteristic
333C23#57812yes—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4C4×D814yes—
396C2×C4C2×#11711yescharacteristic
397C2×C4C2×#11711yescharacteristic
398C2×C4C2×#11711yescharacteristic
399C2×C4C4×#4111yescharacteristic
400C2×C4#51911yescharacteristic
401C2×C4#52011yescharacteristic
402C2×C4#52912yes—
403C2×C4#52912yes—
404C2×C4#52912yes—
405C2×C4#54112yes—
406C2×C4#54212yes—
407C2×C4#55212yes—
408C2×C4#55211yescharacteristic
409C2×C4#55212yes—
410C2×C4#55312yes—
411C2×C4#55311yescharacteristic
412C2×C4#55612yes—
413C2×C4#55612yes—
414C8—24no—
415C8—42no—
416C8—82no—
417C22#229011yescharacteristic
418C22#234411yescharacteristic
419C22#235211yescharacteristic
420C22#246812yes—
421C22#246912yes—
422C22C2×C4×D811yescharacteristic
423C22#294311yescharacteristic
424C22#295011yescharacteristic
425C22#295211yescharacteristic
426C22#310012yes—
427C22#310212yes—
428C22#312012yes—
429C22#312112yes—
430C22#329411yescharacteristic
431C22#330311yescharacteristic
432C22#332712yes—
433C22#333412yes—
434C22#333612yes—
435C22#335012yes—
436C22#335012yes—
437C22#335811yescharacteristic
438C22#336011yescharacteristic
439C22#337712yes—
440C4—24no—
441C4—42no—
442C4—44no—
443C4—44no—
444C4—44no—
445C4—82no—
446C4—84no—
447C4—82no—
448C4—82no—
449C4—162no—
450C4#228311yescharacteristic
451C4#232011yescharacteristic
452C4#234711yescharacteristic
453C4#235611yescharacteristic
454C4#252412yes—
455C4#252512yes—
456C2#1389311yescharacteristic
457C2#2111311yescharacteristic
458C2#2113211yescharacteristic
459C2#2261111yescharacteristic
460C2#2262011yescharacteristic
461C2#2276111yescharacteristic
462C2#2276611yescharacteristic
463C2#2709412yes—
464C2#2710212yes—
465C2#2712612yes—
466C2#2713412yes—
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
21. C4×#107C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1072
#139212
#138212
#112922
#112192
#109072
#108202
#108062
#107972