Finite Group Databasebeta

← #274690

Subgroups — #274690

Order 512.

791 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32906C211yesmaximal, characteristic
2#27948C211yesmaximal, characteristic
3Q8×C22⋊C8C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#10620C212yesmaximal, complemented
6#10479C212yesmaximal, complemented
7#9218C212yesmaximal, complemented
8#9131C211yesmaximal, characteristic
9#8830C211yesmaximal, characteristic
10#8725C211yesmaximal, characteristic
11#8643C212yesmaximal, complemented
12C22×C4×Q8C2211yescharacteristic
13C2×#462C2211yescharacteristic
14C2×#462C2211yescharacteristic
15C2×#451—21no—
16C2×C8×Q8C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#103—21no—
19#2857C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#2084—22no—
22#1990C2212yes—
23#1984—22no—
24#1984C2212yes—
25#1878—21no—
26#1878—21no—
27#1872—21no—
28#1853—21no—
29#1853—21no—
30#1844C2212yes—
31#1803—21no—
32#1803—21no—
33#1803C2212yes—
34#1803C2211yescharacteristic
35#1803C2212yes—
36#1803C2211yescharacteristic
37#1797—21no—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40C4×C22⋊C8—21no—
41C4×C22⋊C8C2211yescharacteristic
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8D411yescharacteristic
50C22×Q16—21no—
51C22×Q16—21no—
52C23×C8—21no—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C22⋊Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#462—24no—
71#462—42no—
72#451—44no—
73C8×Q8—24no—
74C8×Q8—42no—
75C2×#106—22no—
76C2×#106—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—21no—
87C2×#103—21no—
88C2×#103—21no—
89C2×#103—21no—
90C2×#103—44no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8D411yescharacteristic
94C2×C22⋊C8D411yescharacteristic
95C2×C4×C8—22no—
96C2×C4×C8C2312yes—
97#391—21no—
98#389—21no—
99#389—21no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384C2311yescharacteristic
104#384C2312yes—
105#380—21no—
106#380—21no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—21no—
112#340—21no—
113#336—22no—
114#336—22no—
115#336—21no—
116#336—21no—
117#336D412yes—
118#334—42no—
119#332—42no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8SD1614yes—
128C22×Q8C2×D411yescharacteristic
129C22×Q8C2×D411yescharacteristic
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4#4111yescharacteristic
136C2×Q16—42no—
137C2×Q16—42no—
138C2×Q16—42no—
139C2×Q16—42no—
140C2×Q16—42no—
141C2×Q16—42no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8—42no—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C22⋊Q8—42no—
154C22⋊Q8—42no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—44no—
161C4×Q8—41no—
162C4×Q8—41no—
163C4×Q8—44no—
164C4×Q8#4114yes—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C22⋊C4—41no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—44no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42C2411yescharacteristic
203#106—82no—
204#106—82no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—44no—
212C4⋊C8—42no—
213#103—42no—
214#103—44no—
215#103—42no—
216#103—42no—
217#103—44no—
218#103—42no—
219C22⋊C8—22no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222C22⋊C8Q1614yescomplemented
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—44no—
226#95—41no—
227#95—41no—
228C24#12111yescharacteristic
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—24no—
234C2×Q8—28no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8#12312yes—
251C2×Q8C2×SD1612yes—
252C2×Q8C2×SD1612yes—
253C2×Q8AGL(1,ℤ/8)12yes—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—81no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4C2×SD1612yes—
283C22×C4C2×Q1612yes—
284C22×C4#13512yes—
285C22×C4#13712yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C2×#4111yescharacteristic
289Q16—84no—
290Q16—84no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—84no—
309C2×C8C2×Q1614yes—
310C4⋊C4—21no—
311C4⋊C4—24no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—81no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4#12312yes—
334C22⋊C4—82no—
335C42—21no—
336C42—21no—
337C42—24no—
338C42—22no—
339C42—22no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42C2×#4112yes—
344C42H(5,𝔽2)12yes—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23#46212yes—
350C23#46512yes—
351C23#47112yes—
352C23C2×#12111yescharacteristic
353Q8—22no—
354Q8—22no—
355Q8—22no—
356Q8—22no—
357Q8—22no—
358Q8—48no—
359Q8—41no—
360Q8—41no—
361Q8—81no—
362Q8—81no—
363Q8—81no—
364Q8—81no—
365Q8#48114yes—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—84no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—81no—
405C2×C4—164no—
406C2×C4#46212yes—
407C2×C4#46512yes—
408C2×C4C2×#12311yescharacteristic
409C2×C4#53411yescharacteristic
410C2×C4#54612yes—
411C2×C4#55112yes—
412C2×C4C22×SD1611yescharacteristic
413C2×C4C22×Q1611yescharacteristic
414C2×C4C2×#13511yescharacteristic
415C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
416C2×C4C2×#13711yescharacteristic
417C2×C4#57511yescharacteristic
418C2×C4#57712yes—
419C8—24no—
420C8—42no—
421C8—81no—
422C8—81no—
423C8—82no—
424C8—82no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22#167214yes—
432C22#241211yescharacteristic
433C22C2×#46211yescharacteristic
434C22#304311yescharacteristic
435C22C2×#46511yescharacteristic
436C22#305511yescharacteristic
437C22C2×#47111yescharacteristic
438C22#306911yescharacteristic
439C4—22no—
440C4—22no—
441C4—24no—
442C4—22no—
443C4—42no—
444C4—41no—
445C4—42no—
446C4—81no—
447C4—162no—
448C4—161no—
449C4—161no—
450C4C2×#48111yescharacteristic
451C4#309111yescharacteristic
452C4#318711yescharacteristic
453C4#318811yescharacteristic
454C4#329912yes—
455C4#331612yes—
456C2—24no—
457C2C2×#167211yescharacteristic
458C2#1917211yescharacteristic
459C2#1939711yescharacteristic
460C2#1946811yescharacteristic
461C2#2453211yescharacteristic
462C2#2455311yescharacteristic
463C2#2457111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
5. #10620C2semidirect
6. #10479C2semidirect
7. #9218C2semidirect
11. #8643C2semidirect
222. C22⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#329062
#279482
Q8×C22⋊C82
C2×#18032
#106202
#104792
#92182
#91312
#88302
#87252
#86432