Finite Group Databasebeta

← #361239

Subgroups — #361239

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#107C211yesmaximal, characteristic
2#13936C211yesmaximal, characteristic
3#13818C211yesmaximal, characteristic
4#11293C212yesmaximal
5#11219C212yesmaximal
6#10907C212yesmaximal
7#10820C211yesmaximal, characteristic
8#10807C211yesmaximal, characteristic
9#10807C211yesmaximal, characteristic
10#10807C212yesmaximal
11#10797C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#389C2212yes—
16C2×C4×C4⋊C4C2212yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C4×#107C418yescomplemented
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#336C2211yescharacteristic
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1443—22no—
32#1436—22no—
33C4⋊#95—21no—
34C4⋊#95C2212yes—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38#1434—21no—
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×#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—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—24no—
69C4×C4⋊C4—42no—
70C4×C4⋊C4C2×C414yes—
71C4×C4⋊C4C2×C414yes—
72C2×#95—22no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95D411yescharacteristic
85C2×#95D412yes—
86C2×#95D411yescharacteristic
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—42no—
93#340C2×C414yes—
94#340C2×C414yes—
95#336—24no—
96#336—22no—
97#336—42no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2411yescharacteristic
109C22×C8—22no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8C22×C412yes—
124C22×C8C22×C412yes—
125C22×C8#4114yes—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
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—28no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4#4114yes—
158C2×C4⋊C4#4112yes—
159C2×C4⋊C4#4114yes—
160C2×C4⋊C4#4112yes—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—44no—
172C2×C42—44no—
173C2×C42—82no—
174C2×C42—82no—
175C2×C42—82no—
176C2×C42C22×C412yes—
177C2×C42#4112yes—
178#107—28no—
179#107—28no—
180C4×C8—28no—
181#95—28no—
182#95—24no—
183#95—24no—
184#95—22no—
185#95—42no—
186#95—44no—
187#95—44no—
188#95—42no—
189#95—42no—
190#95—82no—
191#95—82no—
192#95—81no—
193#95—81no—
194#95—82no—
195#95Q1614yes—
196C24C22×D411yescharacteristic
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C4×D412yes—
251C22×C4C4×D412yes—
252C22×C4C2×Q1614yes—
253C22×C4C2×Q1612yes—
254C22×C4#13512yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C2×#4112yes—
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4112yes—
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4112yes—
261C22×C4C2×#4111yescharacteristic
262C22×C4H(5,𝔽2)11yescharacteristic
263C22×C42−1+412yes—
264C22×C42−1+411yescharacteristic
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8#11714yes—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—88no—
303C4⋊C4#11714yes—
304C4⋊C4#11714yes—
305C42—22no—
306C42—22no—
307C42—24no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—84no—
313C42—84no—
314C42—88no—
315C23C2×C4×D411yescharacteristic
316C23#52512yes—
317C23#54611yescharacteristic
318C23#54712yes—
319C23#54811yescharacteristic
320C23C22×Q1611yescharacteristic
321C23C2×#13511yescharacteristic
322C23#57512yes—
323C23#57612yes—
324C23#57711yescharacteristic
325C23#57711yescharacteristic
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—44no—
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—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4C4×Q1614yes—
384C2×C4C2×#11711yescharacteristic
385C2×C4C2×#11711yescharacteristic
386C2×C4C2×#11711yescharacteristic
387C2×C4C4×#4111yescharacteristic
388C2×C4#51911yescharacteristic
389C2×C4#52011yescharacteristic
390C2×C4#52912yes—
391C2×C4#52912yes—
392C2×C4#52912yes—
393C2×C4#54112yes—
394C2×C4#54212yes—
395C2×C4#55212yes—
396C2×C4#55211yescharacteristic
397C2×C4#55212yes—
398C2×C4#55312yes—
399C2×C4#55311yescharacteristic
400C2×C4#55612yes—
401C2×C4#55612yes—
402C8—24no—
403C8—42no—
404C8—82no—
405C22#229011yescharacteristic
406C22#234411yescharacteristic
407C22#235211yescharacteristic
408C22#246812yes—
409C22#246912yes—
410C22C2×C4×Q1611yescharacteristic
411C22#294311yescharacteristic
412C22#294811yescharacteristic
413C22#295211yescharacteristic
414C22#310212yes—
415C22#310412yes—
416C22#311812yes—
417C22#311912yes—
418C22#329911yescharacteristic
419C22#330411yescharacteristic
420C22#332212yes—
421C22#333812yes—
422C22#333912yes—
423C22#334812yes—
424C22#334812yes—
425C22#336311yescharacteristic
426C22#336511yescharacteristic
427C22#337012yes—
428C4—24no—
429C4—42no—
430C4—44no—
431C4—44no—
432C4—44no—
433C4—82no—
434C4—82no—
435C4—84no—
436C4—82no—
437C4—162no—
438C4#228311yescharacteristic
439C4#232011yescharacteristic
440C4#234711yescharacteristic
441C4#235611yescharacteristic
442C4#252412yes—
443C4#252512yes—
444C2#1389311yescharacteristic
445C2#2111911yescharacteristic
446C2#2113411yescharacteristic
447C2#2261211yescharacteristic
448C2#2261911yescharacteristic
449C2#2276411yescharacteristic
450C2#2277111yescharacteristic
451C2#2710012yes—
452C2#2710812yes—
453C2#2713212yes—
454C2#2714012yes—
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
19. 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
#139362
#138182
#112932
#112192
#109072
#108202
#108072
#107972