Finite Group Databasebeta

← #278218

Subgroups — #278218

Order 512.

707 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#24843C212yesmaximal
3#21487C211yesmaximal, characteristic
4C2×#1695C211yesmaximal, characteristic, complemented
5C2×#1566C211yesmaximal, characteristic, complemented
6#10589C212yesmaximal, complemented
7#10480C212yesmaximal, complemented
8#9739C211yesmaximal, characteristic, complemented
9#9620C211yesmaximal, characteristic, complemented
10#9135C212yesmaximal, complemented
11#8652C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#476C2212yes—
14C2×#436C2211yescharacteristic
15C2×#435—21no—
16C2×#423C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2857C2212yes—
19C4×#121C2211yescharacteristic
20#2088—22no—
21#2067—22no—
22#1990C2212yes—
23#1987C2212yes—
24#1954C2211yescharacteristic
25#1947C2212yes—
26#1942—21no—
27#1942—21no—
28#1939C2211yescharacteristic
29#1923—21no—
30#1895—22no—
31#1869—21no—
32#1845C2211yescharacteristic
33#1809—22no—
34#1808—21no—
35#1808—21no—
36#1802C2212yes—
37#1797C2211yescharacteristic
38#1770C2211yescharacteristic
39#1753—21no—
40#1695—24no—
41#1566—24no—
42C2×#334C2211yescharacteristic
43C2×#334C2212yes—
44C2×#334C2211yescharacteristic
45C2×#333C2212yes—
46C2×#333C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×M16—22no—
50C22×M16—21no—
51C23×C8D411yescharacteristic
52C23×C8D411yescharacteristic
53C2×C4⋊Q8C2312yes—
54C2×C22⋊Q8—22no—
55C2×#121D411yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4C2311yescharacteristic
58C22×C42C2311yescharacteristic
59#500—22no—
60#500—22no—
61#476—24no—
62#476—42no—
63#436—22no—
64#436—22no—
65#435—44no—
66C2×#107—21no—
67C2×#107—21no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—22no—
71C2×#106—22no—
72#423—24no—
73#423—42no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×#102—22no—
81C2×#102—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×#97—21no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8D411yescharacteristic
90C2×C4×C8D411yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#391—22no—
94#390D411yescharacteristic
95#389—22no—
96#389—22no—
97#389D411yescharacteristic
98#388—21no—
99#386—21no—
100#384C2312yes—
101#380—22no—
102#380—22no—
103#380D411yescharacteristic
104C4×C4⋊C4C2311yescharacteristic
105C4×C22⋊C4—21no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#336—21no—
113#336—21no—
114#336—21no—
115#334—24no—
116#334—22no—
117#334D414yescomplemented
118#333—24no—
119#333—42no—
120C22×Q8—22no—
121C22×D4—21no—
122C22×D4C2×D411yescharacteristic
123C23×C4—21no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4#4111yescharacteristic
127C2×M16—44no—
128C2×M16—44no—
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C2×M16—82no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—41no—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C22⋊Q8—84no—
152#121—24no—
153C4×D4—21no—
154C4×D4—21no—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4—42no—
158C4×D4—42no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—42no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42C2411yescharacteristic
184#107—42no—
185#107—42no—
186#107—42no—
187#107—42no—
188#107—82no—
189#106—44no—
190C4⋊C8—21no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—22no—
196C4⋊C8—21no—
197C4⋊C8—44no—
198C4⋊C8—42no—
199C4⋊C8C2×D414yes—
200#102—44no—
201#102—82no—
202C22⋊C8—42no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206#97—42no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8C2×D412yes—
212C4×C8#4112yes—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—42no—
217#95—42no—
218C24—21no—
219C24#12111yescharacteristic
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#13714yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C2×#4111yescharacteristic
262M16—88no—
263M16—84no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—24no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—44no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—44no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8#12112yes—
286C2×C8#12112yes—
287C2×C8AGL(1,ℤ/8)12yes—
288C2×C8#13712yes—
289C2×C8#13712yes—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—82no—
304C4⋊C4—84no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—84no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—41no—
319C42—41no—
320C42C22×D411yescharacteristic
321C42C2×#4111yescharacteristic
322C42H(5,𝔽2)11yescharacteristic
323C422−1+411yescharacteristic
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23#47012yes—
338C23C2×#12111yescharacteristic
339Q8—162no—
340Q8—162no—
341D4—42no—
342D4—42no—
343D4—84no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4#47012yes—
381C2×C4#47212yes—
382C2×C4C2×#12111yescharacteristic
383C2×C4#53711yescharacteristic
384C2×C4D4211yescharacteristic
385C2×C4#54711yescharacteristic
386C2×C4#54811yescharacteristic
387C2×C4#55311yescharacteristic
388C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
389C2×C4C2×#13712yes—
390C2×C4C2×#13711yescharacteristic
391C2×C4C2×#13711yescharacteristic
392C2×C4#57511yescharacteristic
393C2×C4#57712yes—
394C8—22no—
395C8—22no—
396C8—44no—
397C8—41no—
398C8—81no—
399C8—82no—
400C8—81no—
401C8#46814yes—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22#241611yescharacteristic
417C22#303611yescharacteristic
418C22#303911yescharacteristic
419C22C2×#47011yescharacteristic
420C22#306011yescharacteristic
421C22C2×#47211yescharacteristic
422C22#306711yescharacteristic
423C4—22no—
424C4—21no—
425C4—21no—
426C4—22no—
427C4—41no—
428C4—81no—
429C4—81no—
430C4—81no—
431C4—162no—
432C4—162no—
433C4C2×#46811yescharacteristic
434C4#305611yescharacteristic
435C4#325211yescharacteristic
436C4#325411yescharacteristic
437C4#326612yes—
438C4#334412yes—
439C2—42no—
440C2—161no—
441C2#1928111yescharacteristic
442C2#1929711yescharacteristic
443C2#1933711yescharacteristic
444C2#1936011yescharacteristic
445C2#2475911yescharacteristic
446C2#2478311yescharacteristic
447C2#2480311yescharacteristic
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
1. #33209C2semidirect
4. C2×#1695C2semidirect
5. C2×#1566C2semidirect
6. #10589C2semidirect
7. #10480C2semidirect
8. #9739C2semidirect
9. #9620C2semidirect
10. #9135C2semidirect
11. #8652C2semidirect
117. #334D4semidirect

Maximal subgroups

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

subgroupindex
#332092
#248432
#214872
C2×#16952
C2×#15662
#105892
#104802
#97392
#96202
#91352
#86522