Finite Group Databasebeta

← #314667

Subgroups — #314667

Order 512.

660 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27646C212yesmaximal
2C8×#121C211yesmaximal, characteristic
3#20817C211yesmaximal, characteristic, complemented
4#10655C212yesmaximal, complemented
5#10621C212yesmaximal, complemented
6#9615C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#8726C212yesmaximal, complemented
9#8644C211yesmaximal, characteristic, complemented
10#8644C211yesmaximal, characteristic, complemented
11#8596C211yesmaximal, characteristic, complemented
12C2×#482—22no—
13C2×C8×D4—21no—
14C2×C8×D4C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C4×#121C2211yescharacteristic
17#2285C2212yes—
18#2085—22no—
19#2073—22no—
20#1954C2211yescharacteristic
21#1948C2212yes—
22#1947C2212yes—
23#1939C2211yescharacteristic
24#1923—21no—
25#1869—21no—
26#1845C2211yescharacteristic
27#1845C2211yescharacteristic
28#1834C2212yes—
29#1825—22no—
30#1808—21no—
31#1808—21no—
32#1802C2212yes—
33#1802C2212yes—
34#1797C2211yescharacteristic
35#1797C2212yes—
36C4×#107—21no—
37C4×#107—21no—
38C8×C4⋊C4C2211yescharacteristic
39C8×C22⋊C4—21no—
40C2×#335C2211yescharacteristic
41C2×#335C2211yescharacteristic
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1296C2212yes—
46#1277C2212yes—
47C23×C8—21no—
48C23×C8D411yescharacteristic
49C23×C8D411yescharacteristic
50C2×#121D411yescharacteristic
51C2×C4×D4—21no—
52C2×C4×D4C2311yescharacteristic
53C2×#117—22no—
54C22×C42C2311yescharacteristic
55#482—48no—
56C8×D4—22no—
57C8×D4—22no—
58C8×D4—44no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107—22no—
64C2×#106—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×#102—22no—
71C2×#102—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8—21no—
80C2×C4×C8D411yescharacteristic
81C2×C4×C8D411yescharacteristic
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2311yescharacteristic
84#390D411yescharacteristic
85#389D411yescharacteristic
86#388—21no—
87#386—21no—
88#384C2312yes—
89#384C2312yes—
90#382—22no—
91#380—22no—
92#380—22no—
93#380D411yescharacteristic
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4C2311yescharacteristic
97C4×C22⋊C4—21no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#336—21no—
105#336—21no—
106#336—22no—
107#336—21no—
108#335—22no—
109#335—22no—
110C22×D4—21no—
111C22×D4C2×D411yescharacteristic
112C23×C4—21no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4#4111yescharacteristic
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8—42no—
127C22×C8—42no—
128C22×C8—42no—
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133#121—22no—
134#121D814yes—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—21no—
139C4×D4—42no—
140C4×D4—42no—
141#117—44no—
142#117—44no—
143C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—42no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—42no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42C2411yescharacteristic
169#107—42no—
170#107—42no—
171#107—42no—
172#107—42no—
173#107—44no—
174#106—44no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—42no—
184#102—44no—
185#102—44no—
186C22⋊C8—42no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189C22⋊C8—44no—
190C22⋊C8—42no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—21no—
194C4×C8—21no—
195C4×C8—42no—
196C4×C8#4112yes—
197C4×C8#4112yes—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—42no—
202#95—42no—
203C24—21no—
204C24#12111yescharacteristic
205C2×D4—24no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4C2×D812yes—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
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#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4C2×D812yes—
243C22×C4#13512yes—
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D411yescharacteristic
246C22×C4C2×#4111yescharacteristic
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—84no—
270C2×C8#12312yes—
271C2×C8#12312yes—
272C2×C8#13512yes—
273C2×C8AGL(1,ℤ/8)12yes—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4C2×D812yes—
291C22⋊C4—24no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—84no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—41no—
305C42—41no—
306C42—84no—
307C42C2×#4111yescharacteristic
308C42C2×#4111yescharacteristic
309C422−1+411yescharacteristic
310C422−1+411yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23#46612yes—
325C23C2×#12111yescharacteristic
326D4—22no—
327D4—22no—
328D4—48no—
329D4—41no—
330D4—41no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—42no—
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—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4#46612yes—
371C2×C4C2×#12311yescharacteristic
372C2×C4#53611yescharacteristic
373C2×C4#54711yescharacteristic
374C2×C4#54711yescharacteristic
375C2×C4#55211yescharacteristic
376C2×C4#55211yescharacteristic
377C2×C4C22×D811yescharacteristic
378C2×C4C2×#13511yescharacteristic
379C2×C4C2×#13511yescharacteristic
380C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
381C2×C4#57511yescharacteristic
382C2×C4#57511yescharacteristic
383C2×C4#57811yescharacteristic
384C2×C4#57811yescharacteristic
385C8—22no—
386C8—41no—
387C8—41no—
388C8—82no—
389C8—82no—
390C8—81no—
391C8—81no—
392C8#48214yescomplemented
393C22—22no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22#248611yescharacteristic
408C22#303511yescharacteristic
409C22#303611yescharacteristic
410C22C2×#46611yescharacteristic
411C22#305911yescharacteristic
412C22#306711yescharacteristic
413C22#306711yescharacteristic
414C4—22no—
415C4—21no—
416C4—21no—
417C4—41no—
418C4—42no—
419C4—41no—
420C4—81no—
421C4—81no—
422C4—162no—
423C4—162no—
424C4C2×#48211yescharacteristic
425C4#309311yescharacteristic
426C4#323311yescharacteristic
427C4#323511yescharacteristic
428C4#333411yescharacteristic
429C4#333411yescharacteristic
430C4#333611yescharacteristic
431C4#333611yescharacteristic
432C2—42no—
433C2—82no—
434C2#1912511yescharacteristic
435C2#1922011yescharacteristic
436C2#1925511yescharacteristic
437C2#1933011yescharacteristic
438C2#2764611yescharacteristic
439C2#2766111yescharacteristic
440C2#2767411yescharacteristic
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
3. #20817C2semidirect
4. #10655C2semidirect
5. #10621C2semidirect
6. #9615C2semidirect
7. #9614C2semidirect
8. #8726C2semidirect
9. #8644C2semidirect
10. #8644C2semidirect
11. #8596C2semidirect
392. C8#482semidirect

Maximal subgroups

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

subgroupindex
#276462
C8×#1212
#208172
#106552
#106212
#96152
#96142
#87262
#86442
#85962