Finite Group Databasebeta

← #314692

Subgroups — #314692

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27677C211yesmaximal, characteristic
2#27645C211yesmaximal, characteristic
3C8×#121C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic, complemented
5#10658C212yesmaximal, complemented
6#10625C212yesmaximal, complemented
7#9613C212yesmaximal, complemented
8#8724C211yesmaximal, characteristic, complemented
9#8724C211yesmaximal, characteristic, complemented
10#8646C212yesmaximal, complemented
11#8595C211yesmaximal, characteristic, complemented
12C2×#481—21no—
13C2×C8×D4—21no—
14C2×C8×D4C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C4×#121C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#2099—21no—
20#2093—21no—
21#2084—21no—
22#2071—21no—
23#1954C2211yescharacteristic
24#1948C2212yes—
25#1947C2212yes—
26#1939C2211yescharacteristic
27#1923—21no—
28#1874—21no—
29#1869—21no—
30#1852—21no—
31#1834—21no—
32#1834—21no—
33#1834C2212yes—
34#1834C2212yes—
35#1824—21no—
36#1802C2212yes—
37#1802C2212yes—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42C8×C22⋊C4—21no—
43C2×#340—21no—
44C2×#340—21no—
45C2×#334C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C23×C8—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54C2×#121D411yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4C2311yescharacteristic
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C42C2311yescharacteristic
64#481—44no—
65C8×D4—24no—
66C8×D4—44no—
67C2×#107—22no—
68C2×#106—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×#102—22no—
75C2×#102—22no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C4×C8—24no—
83C2×C4×C8—21no—
84C2×C4×C8D411yescharacteristic
85C2×C4×C8D411yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#390D411yescharacteristic
89#389D411yescharacteristic
90#388—21no—
91#386—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384C2312yes—
99#384C2312yes—
100#380D411yescharacteristic
101C4×C4⋊C4C2311yescharacteristic
102C4×C22⋊C4—21no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116#334—24no—
117C22×D4—21no—
118C22×D4C2×D411yescharacteristic
119C23×C4—21no—
120C23×C4—41no—
121C23×C4—41no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4#4111yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8C2×D411yescharacteristic
139C22×C8C2×D411yescharacteristic
140C22×C8C2×D411yescharacteristic
141C22×C8C2×D411yescharacteristic
142#121—22no—
143#121SD1614yes—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—42no—
147C4×D4—42no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2×D411yescharacteristic
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42C2411yescharacteristic
185#107—82no—
186#106—44no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—42no—
194#102—44no—
195#102—82no—
196C22⋊C8—42no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200C22⋊C8—42no—
201C22⋊C8—42no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—42no—
206C4×C8#4114yes—
207#95—21no—
208#95—21no—
209#95—21no—
210C24—21no—
211C24#12111yescharacteristic
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—21no—
215C2×D4—22no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4C2×SD1612yes—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4C2×SD1612yes—
249C22×C4#13512yes—
250C22×C4#13512yes—
251C22×C4AGL(1,ℤ/8)12yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—84no—
280C2×C8#12312yes—
281C2×C8#12312yes—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4C2×SD1612yes—
303C22⋊C4—24no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C42—21no—
309C42—21no—
310C42—22no—
311C42—22no—
312C42—41no—
313C42—41no—
314C42C2×#4112yes—
315C422−1+412yes—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—22no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23#46312yes—
328C23#46512yes—
329C23#47312yes—
330C23C2×#12111yescharacteristic
331D4—24no—
332D4—48no—
333D4—42no—
334C2×C4—28no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—164no—
369C2×C4—164no—
370C2×C4#46312yes—
371C2×C4#46512yes—
372C2×C4#47212yes—
373C2×C4C2×#12311yescharacteristic
374C2×C4#53611yescharacteristic
375C2×C4#54712yes—
376C2×C4#55212yes—
377C2×C4C22×SD1611yescharacteristic
378C2×C4C2×#13511yescharacteristic
379C2×C4C2×#13511yescharacteristic
380C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
381C2×C4#57511yescharacteristic
382C2×C4#57511yescharacteristic
383C2×C4#57712yes—
384C8—22no—
385C8—22no—
386C8—41no—
387C8—41no—
388C8—82no—
389C8—82no—
390C8—81no—
391C8—81no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22#167914yes—
406C22#248611yescharacteristic
407C22C2×#46311yescharacteristic
408C22#304011yescharacteristic
409C22C2×#46511yescharacteristic
410C22#305711yescharacteristic
411C22C2×#47211yescharacteristic
412C22C2×#47311yescharacteristic
413C4—22no—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—42no—
418C4—81no—
419C4—81no—
420C4—162no—
421C4—162no—
422C4#309111yescharacteristic
423C4#309411yescharacteristic
424C4#323211yescharacteristic
425C4#323411yescharacteristic
426C4#333512yes—
427C4#333712yes—
428C2—24no—
429C2—82no—
430C2C2×#167911yescharacteristic
431C2#1919611yescharacteristic
432C2#1927411yescharacteristic
433C2#1936311yescharacteristic
434C2#2765111yescharacteristic
435C2#2765811yescharacteristic
436C2#2767511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1834C2semidirect
5. #10658C2semidirect
6. #10625C2semidirect
7. #9613C2semidirect
8. #8724C2semidirect
9. #8724C2semidirect
10. #8646C2semidirect
11. #8595C2semidirect

Maximal subgroups

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

subgroupindex
#276772
#276452
C8×#1212
C2×#18342
#106582
#106252
#96132
#87242
#86462
#85952