Finite Group Databasebeta

← #316211

Subgroups — #316211

Order 512.

800 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27678C211yesmaximal, characteristic
2#24299C211yesmaximal, characteristic
3C8×#121C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic, complemented
5#10601C212yesmaximal, complemented
6#10481C212yesmaximal, complemented
7#9567C212yesmaximal, complemented
8#9215C212yesmaximal, complemented
9#8725C211yesmaximal, characteristic, complemented
10#8596C211yesmaximal, characteristic, complemented
11#8342C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#476C2212yes—
14C2×C8×D4—21no—
15C2×C8×D4C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C4×#121C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#2100—21no—
20#2094—21no—
21#2058—21no—
22#2023—21no—
23#1990—21no—
24#1990—21no—
25#1990C2212yes—
26#1990C2212yes—
27#1987C2212yes—
28#1954C2211yescharacteristic
29#1948C2212yes—
30#1939C2211yescharacteristic
31#1923—21no—
32#1874—21no—
33#1874—21no—
34#1869—21no—
35#1853—21no—
36#1853—21no—
37#1802C2212yes—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42C8×C22⋊C4—21no—
43C2×#340—21no—
44C2×#340—21no—
45C2×#335C2212yes—
46C2×#333C2212yes—
47C2×#333C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C23×Q8—21no—
52C23×C8—21no—
53C23×C8D411yescharacteristic
54C23×C8D411yescharacteristic
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8C2312yes—
61C2×#121D411yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4C2311yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C42C2311yescharacteristic
70#476—24no—
71#476—42no—
72C8×D4—42no—
73C8×D4—44no—
74C2×#106—22no—
75C2×#106—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2312yes—
81C2×#102—22no—
82C2×#102—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—42no—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#390D411yescharacteristic
96#389D411yescharacteristic
97#388—21no—
98#386—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384C2312yes—
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#340—22no—
113#340—22no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#336—21no—
119#335—42no—
120#333—24no—
121#333—42no—
122C22×Q8—22no—
123C22×Q8—44no—
124C22×Q8—41no—
125C22×Q8—82no—
126C22×Q8—82no—
127C22×D4—21no—
128C22×D4C2×D411yescharacteristic
129C23×C4—21no—
130C23×C4—41no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4111yescharacteristic
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—41no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—42no—
147C22×C8SD1614yes—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C4⋊Q8—24no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158C4⋊Q8—42no—
159C4⋊Q8#4114yes—
160#121—22no—
161#121Q1614yes—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—42no—
165C4×D4—42no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42C2411yescharacteristic
199#106—44no—
200#106—82no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—42no—
208#102—44no—
209#102—82no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214C22⋊C8—42no—
215C22⋊C8—42no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—41no—
219C4×C8—41no—
220C4×C8—42no—
221#95—21no—
222#95—21no—
223#95—21no—
224C24—21no—
225C24#12111yescharacteristic
226C2×Q8—24no—
227C2×Q8—44no—
228C2×Q8—42no—
229C2×Q8—816no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—21no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4C2×Q1612yes—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4C2×SD1612yes—
271C22×C4C2×Q1612yes—
272C22×C4#13512yes—
273C22×C4#13712yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—84no—
302C2×C8C2×SD1612yes—
303C2×C8C2×SD1612yes—
304C4⋊C4—21no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4C2×Q1612yes—
323C22⋊C4—24no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C42—21no—
329C42—21no—
330C42—22no—
331C42—22no—
332C42—41no—
333C42—41no—
334C42C2×#4112yes—
335C42H(5,𝔽2)12yes—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23#46112yes—
348C23#46712yes—
349C23#47212yes—
350C23C2×#12111yescharacteristic
351Q8—84no—
352Q8—84no—
353Q8—88no—
354D4—24no—
355D4—48no—
356D4—42no—
357C2×C4—28no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—88no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—164no—
393C2×C4#46112yes—
394C2×C4#46712yes—
395C2×C4#47112yes—
396C2×C4#52511yescharacteristic
397C2×C4#53411yescharacteristic
398C2×C4#54612yes—
399C2×C4#55112yes—
400C2×C4C22×SD1611yescharacteristic
401C2×C4C22×Q1611yescharacteristic
402C2×C4C2×#13511yescharacteristic
403C2×C4C2×#13711yescharacteristic
404C2×C4#57511yescharacteristic
405C2×C4#57511yescharacteristic
406C2×C4#57712yes—
407C8—22no—
408C8—22no—
409C8—41no—
410C8—41no—
411C8—81no—
412C8—81no—
413C8—82no—
414C8—82no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22#167514yes—
429C22#253411yescharacteristic
430C22C2×#46111yescharacteristic
431C22#304011yescharacteristic
432C22C2×#46711yescharacteristic
433C22#305711yescharacteristic
434C22C2×#47111yescharacteristic
435C22C2×#47211yescharacteristic
436C4—22no—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—42no—
441C4—81no—
442C4—84no—
443C4—81no—
444C4—162no—
445C4#310311yescharacteristic
446C4#311711yescharacteristic
447C4#318711yescharacteristic
448C4#319611yescharacteristic
449C4#329912yes—
450C4#330112yes—
451C2—24no—
452C2—82no—
453C2C2×#167511yescharacteristic
454C2#1919111yescharacteristic
455C2#1927711yescharacteristic
456C2#1936211yescharacteristic
457C2#2809511yescharacteristic
458C2#2811511yescharacteristic
459C2#2813411yescharacteristic
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
4. C2×#1990C2semidirect
5. #10601C2semidirect
6. #10481C2semidirect
7. #9567C2semidirect
8. #9215C2semidirect
9. #8725C2semidirect
10. #8596C2semidirect
11. #8342C2semidirect

Maximal subgroups

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

subgroupindex
#276782
#242992
C8×#1212
C2×#19902
#106012
#104812
#95672
#92152
#87252
#85962
#83422