Finite Group Databasebeta

← #314668

Subgroups — #314668

Order 512.

640 conjugacy classes of subgroups.

The subgroup lattice has 441 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#10656C212yesmaximal, complemented
5#10622C212yesmaximal, complemented
6#9615C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#8727C212yesmaximal, complemented
9#8644C211yesmaximal, characteristic, complemented
10#8644C211yesmaximal, characteristic, 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×#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
55C8×D4—22no—
56C8×D4—22no—
57C8×D4—44no—
58C2×#107—21no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#107—21no—
62C2×#107—22no—
63C2×#106—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×#102—22no—
70C2×#102—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C4×C8—22no—
77C2×C4×C8—22no—
78C2×C4×C8—21no—
79C2×C4×C8D411yescharacteristic
80C2×C4×C8D411yescharacteristic
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83#390D411yescharacteristic
84#389D411yescharacteristic
85#388—21no—
86#386—21no—
87#384C2312yes—
88#384C2312yes—
89#382—22no—
90#380—22no—
91#380—22no—
92#380D411yescharacteristic
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2311yescharacteristic
96C4×C22⋊C4—21no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#336—21no—
104#336—21no—
105#336—22no—
106#336—21no—
107#335—22no—
108#335—22no—
109C22×D4—21no—
110C22×D4C2×D411yescharacteristic
111C23×C4—21no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4#4111yescharacteristic
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8—42no—
126C22×C8—42no—
127C22×C8—42no—
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132#121—22no—
133#121Q1614yes—
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—42no—
139C4×D4—42no—
140#117—82no—
141#117—82no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4C2×D411yescharacteristic
150C2×C4⋊C4C2×D411yescharacteristic
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—42no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—42no—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42C2411yescharacteristic
168#107—42no—
169#107—42no—
170#107—42no—
171#107—42no—
172#107—82no—
173#106—82no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—42no—
183#102—82no—
184#102—82no—
185C22⋊C8—42no—
186C22⋊C8—42no—
187C22⋊C8—42no—
188C22⋊C8—44no—
189C22⋊C8—42no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—21no—
193C4×C8—21no—
194C4×C8—42no—
195C4×C8#4112yes—
196C4×C8#4112yes—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—42no—
201#95—42no—
202C24—21no—
203C24#12111yescharacteristic
204C2×D4—24no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4C2×Q1612yes—
213C22×C4—21no—
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—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4C2×Q1612yes—
242C22×C4#13512yes—
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C2×#4111yescharacteristic
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—84no—
269C2×C8#12312yes—
270C2×C8#12312yes—
271C2×C8#13512yes—
272C2×C8AGL(1,ℤ/8)12yes—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—42no—
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—82no—
288C4⋊C4—82no—
289C4⋊C4C2×Q1612yes—
290C22⋊C4—24no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—84no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—41no—
304C42—41no—
305C42—84no—
306C42C2×#4111yescharacteristic
307C42C2×#4111yescharacteristic
308C422−1+411yescharacteristic
309C422−1+411yescharacteristic
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#46712yes—
324C23C2×#12111yescharacteristic
325D4—22no—
326D4—22no—
327D4—48no—
328D4—41no—
329D4—41no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
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—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4#46712yes—
370C2×C4C2×#12311yescharacteristic
371C2×C4#53611yescharacteristic
372C2×C4#54711yescharacteristic
373C2×C4#54711yescharacteristic
374C2×C4#55211yescharacteristic
375C2×C4#55211yescharacteristic
376C2×C4C22×Q1611yescharacteristic
377C2×C4C2×#13511yescharacteristic
378C2×C4C2×#13511yescharacteristic
379C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
380C2×C4#57511yescharacteristic
381C2×C4#57511yescharacteristic
382C2×C4#57611yescharacteristic
383C2×C4#57611yescharacteristic
384C8—22no—
385C8—41no—
386C8—41no—
387C8—82no—
388C8—82no—
389C8—81no—
390C8—81no—
391C8#48214yes—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#248611yescharacteristic
407C22#303511yescharacteristic
408C22#303611yescharacteristic
409C22C2×#46711yescharacteristic
410C22#305811yescharacteristic
411C22#306711yescharacteristic
412C22#306711yescharacteristic
413C4—22no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—42no—
418C4—41no—
419C4—81no—
420C4—81no—
421C4—162no—
422C4—162no—
423C4C2×#48211yescharacteristic
424C4#309311yescharacteristic
425C4#323311yescharacteristic
426C4#323511yescharacteristic
427C4#333811yescharacteristic
428C4#333811yescharacteristic
429C4#333911yescharacteristic
430C4#333911yescharacteristic
431C2—42no—
432C2—82no—
433C2#1912611yescharacteristic
434C2#1921911yescharacteristic
435C2#1925611yescharacteristic
436C2#1932911yescharacteristic
437C2#2764611yescharacteristic
438C2#2766211yescharacteristic
439C2#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. #10656C2semidirect
5. #10622C2semidirect
6. #9615C2semidirect
7. #9614C2semidirect
8. #8727C2semidirect
9. #8644C2semidirect
10. #8644C2semidirect
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
#106562
#106222
#96152
#96142
#87272
#86442
#85962