Finite Group Databasebeta

← #356563

Subgroups — #356563

Order 512.

1602 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#446C212yesmaximal
2C2×#2295C211yesmaximal, characteristic
3#10835C212yesmaximal
4#10835C212yesmaximal
5#10834C212yesmaximal
6#10833C212yesmaximal
7#10832C211yesmaximal, characteristic
8#10694C212yesmaximal
9#10693C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#446—28no—
12C2×#446—28no—
13C2×#446C4116yescomplemented
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2212yes—
17C22×#97C2212yes—
18C22×#97C2212yes—
19C22×C4×C8C2212yes—
20#2295—28no—
21C2×#391C412yes—
22C2×#389C412yes—
23C2×#387C414yes—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2212yes—
27C2×C43C2211yescharacteristic
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C23×Q8C2×C411yescharacteristic
39C23×C8C2312yes—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×C4×Q8C2×C418yes—
44C2×C4×Q8D418yes—
45C2×C4×Q8D414yes—
46C2×C4×Q8Q814yes—
47C22×C4⋊C4C2×C414yes—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C42C2×C412yes—
52C22×C42C2×C412yes—
53C22×C42C2311yescharacteristic
54C22×C42C2312yes—
55C22×C42C2311yescharacteristic
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58#446—264no—
59C2×C4⋊C8—24no—
60C2×C4⋊C8—24no—
61C2×C4⋊C8—24no—
62C2×C4⋊C8—24no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8—24no—
65C2×C4⋊C8C2×C418yes—
66C2×C4⋊C8C2×C418yes—
67C2×C4⋊C8C2×C418yes—
68C2×#97—24no—
69C2×#97—24no—
70C2×#97—24no—
71C2×#97—24no—
72C2×#97C2×C418yes—
73C2×#97C2×C418yes—
74C2×C4×C8—24no—
75C2×C4×C8—24no—
76C2×C4×C8C2×C418yes—
77#391—28no—
78#389—28no—
79#387—216no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—24no—
83#376—24no—
84#376—24no—
85C2×#95C2×C414yes—
86C2×#95C2×C414yes—
87C2×#95C2×C412yes—
88C2×#95C2×C414yes—
89C43—24no—
90#336—24no—
91#336—24no—
92#336—24no—
93#336—24no—
94#336—22no—
95#336—41no—
96#336—41no—
97#336—42no—
98#336—42no—
99#336Q814yescomplemented
100C22×Q8—42no—
101C22×Q8—42no—
102C22×Q8C4214yes—
103C22×Q8C22⋊C414yes—
104C22×Q8C4⋊C414yes—
105C23×C4C22×C414yes—
106C23×C4C22×C412yes—
107C23×C4C22×C412yes—
108C23×C4C22×C412yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C412yes—
112C23×C4C2411yescharacteristic
113C22×C8—22no—
114C22×C8—22no—
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×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8C22×C414yes—
136C22×C8C22×C414yes—
137C22×C8C22×C414yes—
138C22×C8C22×C414yes—
139C22×C8C2×Q814yes—
140C22×C8#4114yes—
141C4×Q8C22⋊C4132yes—
142C4×Q8C4⋊C4132yes—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C4218yes—
150C2×C4⋊C4C4214yes—
151C2×C4⋊C4C22⋊C418yes—
152C2×C4⋊C4C22⋊C414yes—
153C2×C4⋊C4C4⋊C418yes—
154C2×C4⋊C4C4⋊C414yes—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—21no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42C22×C414yes—
174C2×C42C22×C414yes—
175C2×C42C22×C414yes—
176C2×C42C2×D414yes—
177C2×C42C2×D412yes—
178C2×C42C2×D414yes—
179C2×C42C2×D414yes—
180C2×C42C2×D414yes—
181C2×C42C2×Q812yes—
182C2×C42C2×Q814yes—
183C2×C42C2×Q812yes—
184C2×C42#4112yes—
185C4⋊C8—216no—
186C4⋊C8—216no—
187C4⋊C8—216no—
188#97—216no—
189#97—216no—
190C4×C8—216no—
191#95—28no—
192#95—28no—
193#95—24no—
194#95—28no—
195C24C23×C411yescharacteristic
196C2×Q8—44no—
197C2×Q8—44no—
198C2×Q8—48no—
199C2×Q8—48no—
200C2×Q8#95116yes—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
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—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4C2×C4214yes—
229C22×C4C2×C4212yes—
230C22×C4C2×C4214yes—
231C22×C4C2×C4212yes—
232C22×C4C2×C22⋊C414yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C414yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C4⋊C414yes—
237C22×C4C2×C4⋊C412yes—
238C22×C4C2×C4⋊C414yes—
239C22×C4C2×C4⋊C412yes—
240C22×C4C4×Q814yes—
241C22×C4#13114yes—
242C22×C4C23×C412yes—
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D412yes—
245C22×C4C22×Q811yescharacteristic
246C22×C4C22×Q811yescharacteristic
247C22×C4C2×#4112yes—
248C22×C4C2×#4111yescharacteristic
249C22×C4H(5,𝔽2)11yescharacteristic
250C22×C42−1+411yescharacteristic
251C22×C42−1+412yes—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8C4×Q818yes—
280C4⋊C4—416no—
281C4⋊C4—48no—
282C4⋊C4#95132yes—
283C4⋊C4#95116yes—
284C42—24no—
285C42—24no—
286C42—48no—
287C42—48no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42C2×C22⋊C418yes—
295C42C2×C22⋊C418yes—
296C42C2×C22⋊C418yes—
297C42C2×C4⋊C418yes—
298C42C2×C4⋊C418yes—
299C42C2×C4⋊C418yes—
300C23C22×C4211yescharacteristic
301C23C22×C22⋊C411yescharacteristic
302C23C22×C4⋊C411yescharacteristic
303C23C2×C4×Q811yescharacteristic
304C23C4×#4111yescharacteristic
305C23#51911yescharacteristic
306C23#52011yescharacteristic
307C23C2×#13112yes—
308C23#56812yes—
309C23#56812yes—
310C23#56812yes—
311Q8—416no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—28no—
322C2×C4—24no—
323C2×C4—28no—
324C2×C4—24no—
325C2×C4—28no—
326C2×C4—24no—
327C2×C4—28no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4C2×#9518yes—
350C2×C4C2×#9514yes—
351C2×C4C2×#9518yes—
352C2×C4C2×#9514yes—
353C2×C4C22×C22⋊C412yes—
354C2×C4C22×C4⋊C412yes—
355C2×C4C2×C4×Q812yes—
356C2×C4C4×#4112yes—
357C2×C4#51912yes—
358C2×C4#52012yes—
359C2×C4#54812yes—
360C2×C4#54812yes—
361C2×C4#54812yes—
362C2×C4D4×Q811yescharacteristic
363C2×C4D4×Q812yes—
364C2×C4#55011yescharacteristic
365C2×C4#55012yes—
366C2×C4#55712yes—
367C2×C4#55711yescharacteristic
368C2×C4Q8211yescharacteristic
369C8—28no—
370C8—44no—
371C8—44no—
372C8—44no—
373C22C22×#9511yescharacteristic
374C22C42×Q811yescharacteristic
375C22#227611yescharacteristic
376C22Q8×C22⋊C411yescharacteristic
377C22#234111yescharacteristic
378C22Q8×C4⋊C411yescharacteristic
379C22#235111yescharacteristic
380C22#287311yescharacteristic
381C22#287311yescharacteristic
382C22C4×#13111yescharacteristic
383C22#287511yescharacteristic
384C22#287912yes—
385C22#288012yes—
386C22#290512yes—
387C22#290612yes—
388C22#296112yes—
389C22#296212yes—
390C22#296712yes—
391C22#296812yes—
392C22#298512yes—
393C22#298712yes—
394C22#299212yes—
395C22#299512yes—
396C4—24no—
397C4—28no—
398C4—24no—
399C4—28no—
400C4—44no—
401C4—44no—
402C4—84no—
403C4Q8×C22⋊C412yes—
404C4#234112yes—
405C4Q8×C4⋊C412yes—
406C4#235112yes—
407C2Q8×#9511yescharacteristic
408C2#2033511yescharacteristic
409C2#2033611yescharacteristic
410C2#2051811yescharacteristic
411C2#2051911yescharacteristic
412C2#2057511yescharacteristic
413C2#2057611yescharacteristic
414C2#2251112yes—
415C2#2253312yes—
416C2#2272312yes—
417C2#2273712yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
socle, prime radicalC24C23×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. C2×#446C4semidirect
99. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4462
C2×#22952
#108352
#108342
#108332
#108322
#106942
#106932