Finite Group Databasebeta

← #329795

Subgroups — #329795

Order 512.

738 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33171C211yesmaximal, characteristic
2#33113C212yesmaximal, complemented
3#21349C211yesmaximal, characteristic, complemented
4#10651C211yesmaximal, characteristic, complemented
5#10649C211yesmaximal, characteristic, complemented
6#10599C212yesmaximal, complemented
7#10591C212yesmaximal, complemented
8#10536C211yesmaximal, characteristic, complemented
9#10536C211yesmaximal, characteristic, complemented
10#9134C212yesmaximal, complemented
11#8596C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#482—22no—
14C2×#466—21no—
15C2×#466—21no—
16C2×#431C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2856C2212yes—
19C4×#117C2211yescharacteristic
20#1982C2212yes—
21#1982C2212yes—
22#1970—21no—
23#1970—21no—
24#1957C2212yes—
25#1948C2212yes—
26#1947C2212yes—
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1926—22no—
32#1925—22no—
33#1875—22no—
34#1850—21no—
35#1850—21no—
36#1845C2211yescharacteristic
37#1797C2212yes—
38C4×C4⋊C8—21no—
39C4×C4⋊C8—21no—
40C2×#335C2212yes—
41C2×#331C2212yes—
42C2×#331C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1277C2211yescharacteristic
46#1277C2211yescharacteristic
47C22×D8—21no—
48C22×D8—21no—
49C23×C8D411yescharacteristic
50C23×C8D411yescharacteristic
51C2×#127C2312yes—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#117D411yescharacteristic
56C2×#117D411yescharacteristic
57C22×C42C2311yescharacteristic
58#493—42no—
59#482—48no—
60#466—44no—
61#466—44no—
62#431—42no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—22no—
66C2×#106—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—42no—
85C2×C4×C8D411yescharacteristic
86C2×C4×C8D411yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#390—22no—
90#390—22no—
91#390—22no—
92#384C2312yes—
93C4×C4⋊C4D411yescharacteristic
94C4×C4⋊C4D411yescharacteristic
95C4×C4⋊C4C2311yescharacteristic
96C4×C4⋊C4C2311yescharacteristic
97C4×C22⋊C4—21no—
98C4×C22⋊C4—21no—
99#376—21no—
100C43—21no—
101#340—22no—
102#340—22no—
103#336—21no—
104#336—21no—
105#336—21no—
106#336—21no—
107#336—22no—
108#335—42no—
109#331—42no—
110#331—42no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—42no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4#4111yescharacteristic
118C2×D8—42no—
119C2×D8—42no—
120C2×D8—42no—
121C2×D8—42no—
122C2×D8—42no—
123C2×D8—42no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—41no—
132C22×C8—41no—
133C22×C8—41no—
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138#127—22no—
139#127—22no—
140#127—22no—
141#127—22no—
142#121—44no—
143#121—44no—
144#121—44no—
145#121—44no—
146#121—84no—
147#117—22no—
148#117—42no—
149#117D814yescomplemented
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42C2411yescharacteristic
174#107—42no—
175#107—42no—
176#107—44no—
177#106—44no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—42no—
185C4⋊C8—42no—
186C4⋊C8—42no—
187C4⋊C8—42no—
188#102—44no—
189#102—44no—
190#102—44no—
191#102—44no—
192#102—44no—
193#102—44no—
194C22⋊C8—42no—
195C22⋊C8—44no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—41no—
199C4×C8—41no—
200C4×C8—42no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205C24—42no—
206C24#12111yescharacteristic
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—82no—
222C2×D4—82no—
223C2×D4—84no—
224C2×D4—84no—
225C2×D4—84no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4C2×D812yes—
246C22×C4#13512yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C2×#4111yescharacteristic
250D8—84no—
251D8—84no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—44no—
262C2×C8—44no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8#13512yes—
276C2×C8AGL(1,ℤ/8)12yes—
277C2×C8AGL(1,ℤ/8)12yes—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—82no—
287C4⋊C4C2×D812yes—
288C4⋊C4C2×D812yes—
289C22⋊C4—24no—
290C22⋊C4—42no—
291C22⋊C4—84no—
292C22⋊C4—84no—
293C22⋊C4—84no—
294C42—21no—
295C42—21no—
296C42—22no—
297C42—22no—
298C42—24no—
299C42—42no—
300C42—42no—
301C42—41no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—42no—
306C42H(5,𝔽2)12yes—
307C422−1+412yes—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—82no—
315C23—82no—
316C23—82no—
317C23—82no—
318C23—82no—
319C23—82no—
320C23—82no—
321C23#46612yes—
322C23C2×#12111yescharacteristic
323D4—82no—
324D4—82no—
325D4—82no—
326D4—82no—
327D4—82no—
328D4—82no—
329D4—82no—
330D4—82no—
331D4—162no—
332D4—162no—
333D4—168no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—44no—
348C2×C4—41no—
349C2×C4—41no—
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—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4#46612yes—
366C2×C4#47212yes—
367C2×C4#53411yescharacteristic
368C2×C4#53611yescharacteristic
369C2×C4#53712yes—
370C2×C4#53912yes—
371C2×C4C22×D811yescharacteristic
372C2×C4C2×#13511yescharacteristic
373C2×C4C2×#13511yescharacteristic
374C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
375C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
376C2×C4#57511yescharacteristic
377C2×C4#57512yes—
378C8—22no—
379C8—22no—
380C8—22no—
381C8—41no—
382C8—81no—
383C8—81no—
384C8—82no—
385C8—82no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—162no—
394C22—162no—
395C22—162no—
396C22—162no—
397C22—162no—
398C22—162no—
399C22—162no—
400C22#261811yescharacteristic
401C22#303511yescharacteristic
402C22#303811yescharacteristic
403C22C2×#46611yescharacteristic
404C22#305411yescharacteristic
405C22C2×#47211yescharacteristic
406C22#306711yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—41no—
411C4—42no—
412C4—42no—
413C4—81no—
414C4—81no—
415C4—162no—
416C4#318811yescharacteristic
417C4#319411yescharacteristic
418C4#322811yescharacteristic
419C4#323511yescharacteristic
420C4#324512yes—
421C4#325012yes—
422C2—42no—
423C2—322no—
424C2#1912511yescharacteristic
425C2#1916011yescharacteristic
426C2#1926711yescharacteristic
427C2#1929411yescharacteristic
428C2#2975811yescharacteristic
429C2#2977011yescharacteristic
430C2#2978811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#33171C2
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
2. #33113C2semidirect
3. #21349C2semidirect
4. #10651C2semidirect
5. #10649C2semidirect
6. #10599C2semidirect
7. #10591C2semidirect
8. #10536C2semidirect
9. #10536C2semidirect
10. #9134C2semidirect
11. #8596C2semidirect
149. #117D8semidirect

Maximal subgroups

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

subgroupindex
#331712
#331132
#213492
#106512
#106492
#105992
#105912
#105362
#91342
#85962