Finite Group Databasebeta

← #314690

Subgroups — #314690

Order 512.

640 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27679C212yesmaximal
2C8×#121C211yesmaximal, characteristic
3#20817C211yesmaximal, characteristic, complemented
4#10652C211yesmaximal, characteristic, complemented
5#10650C211yesmaximal, characteristic, complemented
6#10616C211yesmaximal, characteristic, complemented
7#10616C211yesmaximal, characteristic, complemented
8#9616C212yesmaximal, complemented
9#8727C212yesmaximal, complemented
10#8648C212yesmaximal, complemented
11#8596C211yesmaximal, characteristic, complemented
12C2×C8×D4—21no—
13C2×C8×D4C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C4×#121C2211yescharacteristic
16#2285C2212yes—
17#2101—22no—
18#2092—22no—
19#1954C2211yescharacteristic
20#1948C2212yes—
21#1947C2212yes—
22#1939C2211yescharacteristic
23#1923—21no—
24#1875—22no—
25#1869—21no—
26#1854—22no—
27#1845C2211yescharacteristic
28#1845C2211yescharacteristic
29#1834C2212yes—
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×#335C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1277C2211yescharacteristic
46#1277C2211yescharacteristic
47C23×C8—21no—
48C23×C8D411yescharacteristic
49C23×C8D411yescharacteristic
50C2×#121D411yescharacteristic
51C2×C4×D4—21no—
52C2×C4×D4C2311yescharacteristic
53C2×#117—22no—
54C22×C42C2311yescharacteristic
55C8×D4—24no—
56C8×D4—44no—
57C2×#107—21no—
58C2×#107—21no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#107—22no—
62C2×#106—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×#102—22no—
69C2×#102—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C4×C8—24no—
76C2×C4×C8—21no—
77C2×C4×C8D411yescharacteristic
78C2×C4×C8D411yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#390D411yescharacteristic
82#389D411yescharacteristic
83#388—21no—
84#386—21no—
85#384C2312yes—
86#384C2312yes—
87#382—22no—
88#380—22no—
89#380—22no—
90#380D411yescharacteristic
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2311yescharacteristic
94C4×C22⋊C4—21no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#336—21no—
102#336—21no—
103#336—22no—
104#336—21no—
105#335—24no—
106C22×D4—21no—
107C22×D4C2×D411yescharacteristic
108C23×C4—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8—42no—
123C22×C8—42no—
124C22×C8—42no—
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129#121—22no—
130#121Q1614yes—
131C4×D4—22no—
132C4×D4—22no—
133C4×D4—42no—
134C4×D4—42no—
135#117—82no—
136#117—82no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C2×D411yescharacteristic
145C2×C4⋊C4C2×D411yescharacteristic
146C2×C4⋊C4C2×D411yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—42no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—21no—
156C2×C42—42no—
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42C2411yescharacteristic
161#107—42no—
162#107—42no—
163#107—42no—
164#107—42no—
165#107—82no—
166#106—82no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—42no—
174#102—82no—
175#102—82no—
176C22⋊C8—42no—
177C22⋊C8—44no—
178C22⋊C8—42no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—42no—
185C4×C8#4114yes—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—42no—
190#95—42no—
191C24—21no—
192C24#12111yescharacteristic
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—21no—
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4C2×Q1612yes—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4C2×Q1612yes—
226C22×C4#13512yes—
227C22×C4C22×D411yescharacteristic
228C22×C4C22×D411yescharacteristic
229C22×C4C2×#4111yescharacteristic
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—84no—
253C2×C8#12312yes—
254C2×C8#12312yes—
255C2×C8#13512yes—
256C2×C8AGL(1,ℤ/8)12yes—
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4C2×Q1612yes—
272C22⋊C4—24no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—84no—
278C42—21no—
279C42—21no—
280C42—22no—
281C42—22no—
282C42—41no—
283C42—41no—
284C42—84no—
285C42C2×#4112yes—
286C422−1+412yes—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23#46712yes—
301C23C2×#12111yescharacteristic
302D4—24no—
303D4—48no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—44no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4#46712yes—
342C2×C4C2×#12311yescharacteristic
343C2×C4#53611yescharacteristic
344C2×C4#54712yes—
345C2×C4#55212yes—
346C2×C4C22×Q1611yescharacteristic
347C2×C4C2×#13511yescharacteristic
348C2×C4C2×#13511yescharacteristic
349C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
350C2×C4#57511yescharacteristic
351C2×C4#57511yescharacteristic
352C2×C4#57612yes—
353C8—22no—
354C8—22no—
355C8—22no—
356C8—22no—
357C8—82no—
358C8—82no—
359C8—81no—
360C8—81no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22#248611yescharacteristic
376C22#303511yescharacteristic
377C22#303611yescharacteristic
378C22C2×#46711yescharacteristic
379C22#305811yescharacteristic
380C22#306711yescharacteristic
381C22#306711yescharacteristic
382C4—22no—
383C4—22no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—81no—
388C4—81no—
389C4—162no—
390C4—162no—
391C4#309111yescharacteristic
392C4#309411yescharacteristic
393C4#322811yescharacteristic
394C4#324011yescharacteristic
395C4#333812yes—
396C4#333912yes—
397C2—42no—
398C2—82no—
399C2#1912611yescharacteristic
400C2#1922511yescharacteristic
401C2#1925611yescharacteristic
402C2#1933511yescharacteristic
403C2#2765011yescharacteristic
404C2#2766211yescharacteristic
405C2#2767311yescharacteristic
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. #10652C2semidirect
5. #10650C2semidirect
6. #10616C2semidirect
7. #10616C2semidirect
8. #9616C2semidirect
9. #8727C2semidirect
10. #8648C2semidirect
11. #8596C2semidirect

Maximal subgroups

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

subgroupindex
#276792
C8×#1212
#208172
#106522
#106502
#106162
#96162
#87272
#86482
#85962