Finite Group Databasebeta

← #362399

Subgroups — #362399

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#107C211yesmaximal, characteristic
2#13936C211yesmaximal, characteristic
3#13933C211yesmaximal, characteristic
4#11379C212yesmaximal
5#11219C212yesmaximal
6#10881C212yesmaximal
7#10821C212yesmaximal
8#10820C211yesmaximal, characteristic
9#10807C211yesmaximal, characteristic
10#10807C211yesmaximal, characteristic
11#10696C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#97C2212yes—
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
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—
30#1443—22no—
31#1442—22no—
32C4⋊#95—21no—
33C4⋊#95C2211yescharacteristic
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38#1434—21no—
39#1434—21no—
40#1434—21no—
41C4×#95C2211yescharacteristic
42C23×C8C2312yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C42—21no—
53C22×C42—21no—
54C22×C42C2312yes—
55C22×C42C2311yescharacteristic
56C2×#107—24no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#107C2×C414yes—
60C2×#107C2×C414yes—
61C2×#97—24no—
62C2×#97—24no—
63C2×#97—42no—
64C2×C4×C8—24no—
65C2×C4×C8C2×C414yes—
66#389—24no—
67#389—42no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2×C414yes—
71C4×C4⋊C4C2×C414yes—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95D411yescharacteristic
86C2×#95D411yescharacteristic
87C2×#95D412yes—
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—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2411yescharacteristic
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8C22×C412yes—
122C22×C8C22×C412yes—
123C22×C8#4114yes—
124C22×C8#4114yes—
125C22×C8#4114yes—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
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—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4#4112yes—
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—82no—
172C2×C42—82no—
173C2×C42C22×C412yes—
174C2×C42#4112yes—
175#107—28no—
176#107—28no—
177#97—48no—
178C4×C8—28no—
179#95—24no—
180#95—24no—
181#95—22no—
182#95—22no—
183#95—42no—
184#95—44no—
185#95—44no—
186#95—42no—
187#95—42no—
188#95—81no—
189#95—81no—
190#95—82no—
191#95—82no—
192#95—82no—
193#95—82no—
194#95—82no—
195C24C22×D411yescharacteristic
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
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—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
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×C4#13712yes—
253C22×C4#13712yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C2×#4112yes—
256C22×C4C2×#4112yes—
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4112yes—
260C22×C4C2×#4111yescharacteristic
261C22×C4H(5,𝔽2)11yescharacteristic
262C22×C42−1+412yes—
263C22×C42−1+411yescharacteristic
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8#11714yes—
280C2×C8#11714yes—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—88no—
303C4⋊C4#11714yes—
304C42—22no—
305C42—22no—
306C42—24no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—84no—
312C42—84no—
313C23C2×C4×D411yescharacteristic
314C23#53212yes—
315C23#54611yescharacteristic
316C23#54712yes—
317C23#54811yescharacteristic
318C23C2×#13711yescharacteristic
319C23C2×#13711yescharacteristic
320C23#57512yes—
321C23#57611yescharacteristic
322C23#57612yes—
323C23#57811yescharacteristic
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
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—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
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#44114yes—
382C2×C4C2×#11711yescharacteristic
383C2×C4C2×#11711yescharacteristic
384C2×C4C2×#11711yescharacteristic
385C2×C4C4×#4111yescharacteristic
386C2×C4#51911yescharacteristic
387C2×C4#52011yescharacteristic
388C2×C4#52912yes—
389C2×C4#52912yes—
390C2×C4#52912yes—
391C2×C4#54112yes—
392C2×C4#54212yes—
393C2×C4#55112yes—
394C2×C4#55212yes—
395C2×C4#55212yes—
396C2×C4#55212yes—
397C2×C4#55211yescharacteristic
398C2×C4#55311yescharacteristic
399C2×C4#55612yes—
400C8—24no—
401C8—24no—
402C8—44no—
403C22#229311yescharacteristic
404C22#234411yescharacteristic
405C22#235211yescharacteristic
406C22#251912yes—
407C22#252112yes—
408C22C2×#44111yescharacteristic
409C22#294311yescharacteristic
410C22#294811yescharacteristic
411C22#295111yescharacteristic
412C22#317212yes—
413C22#317212yes—
414C22#317512yes—
415C22#317612yes—
416C22#332212yes—
417C22#332811yescharacteristic
418C22#333311yescharacteristic
419C22#334612yes—
420C22#334612yes—
421C22#334712yes—
422C22#334912yes—
423C22#337012yes—
424C22#337411yescharacteristic
425C22#337511yescharacteristic
426C4—24no—
427C4—42no—
428C4—44no—
429C4—82no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4—82no—
434C4—162no—
435C4—162no—
436C4#228311yescharacteristic
437C4#232011yescharacteristic
438C4#234311yescharacteristic
439C4#235611yescharacteristic
440C4#251512yes—
441C4#252412yes—
442C2#1391411yescharacteristic
443C2#2127811yescharacteristic
444C2#2128211yescharacteristic
445C2#2261611yescharacteristic
446C2#2261811yescharacteristic
447C2#2276311yescharacteristic
448C2#2276411yescharacteristic
449C2#2804012yes—
450C2#2804112yes—
451C2#2807212yes—
452C2#2807312yes—
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
#139332
#113792
#112192
#108812
#108212
#108202
#108072
#106962