Finite Group Databasebeta

← #274108

Subgroups — #274108

Order 512.

796 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#33106C211yesmaximal, characteristic
2#25903C211yesmaximal, characteristic
3#22599C211yesmaximal, characteristic, complemented
4C2×#1987C211yesmaximal, characteristic, complemented
5#10589C212yesmaximal, complemented
6#10539C212yesmaximal, complemented
7#9735C212yesmaximal, complemented
8#9568C212yesmaximal, complemented
9#9133C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic, complemented
11#8342C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#486C2212yes—
14C2×#480—21no—
15C2×#431—21no—
16C22×C4⋊C8C2211yescharacteristic
17#2856C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#2023—22no—
20#1990C2212yes—
21#1987—21no—
22#1987—21no—
23#1987C2212yes—
24#1987C2212yes—
25#1982C2211yescharacteristic
26#1982C2211yescharacteristic
27#1947C2211yescharacteristic
28#1947C2211yescharacteristic
29#1939C2212yes—
30#1925—21no—
31#1925—21no—
32#1916—22no—
33#1874—21no—
34#1874—21no—
35#1874—21no—
36#1817—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C8×C4⋊C4C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C2×#340—21no—
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#333C2212yes—
45C2×#333C2212yes—
46C2×#331C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8—21no—
50C23×C8—21no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8C2312yes—
57C2×#127C2311yescharacteristic
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#486—24no—
68#486—42no—
69#480—44no—
70#431—44no—
71C2×#107—21no—
72C2×#107—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—42no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×#102—22no—
81C2×#102—22no—
82C2×#102—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8D411yescharacteristic
89C2×C22⋊C8D411yescharacteristic
90C2×C4×C8—22no—
91C2×C4×C8C2312yes—
92#390—21no—
93#390—21no—
94#390—21no—
95#384C2311yescharacteristic
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4D411yescharacteristic
99C4×C4⋊C4D411yescharacteristic
100C4×C4⋊C4C2311yescharacteristic
101C4×C4⋊C4C2311yescharacteristic
102C2×#95—21no—
103#340—22no—
104#340—22no—
105#340—21no—
106#340—21no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#336—22no—
112#336—22no—
113#336—21no—
114#336—21no—
115#336D412yes—
116#334—42no—
117#333—24no—
118#333—42no—
119#331—42no—
120C22×Q8—22no—
121C22×Q8—44no—
122C22×Q8—41no—
123C22×Q8—82no—
124C22×Q8—82no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—41no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4111yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C4⋊Q8—24no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—44no—
149C4⋊Q8—44no—
150C4⋊Q8—42no—
151C4⋊Q8#4114yes—
152#127—22no—
153#127—22no—
154#121—42no—
155#121—42no—
156#121—42no—
157#121—42no—
158#121—82no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4SD1614yes—
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2×D411yescharacteristic
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42C2411yescharacteristic
194#107—42no—
195#107—42no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—42no—
203C4⋊C8—44no—
204#102—44no—
205#102—44no—
206#102—42no—
207#102—42no—
208#102—42no—
209#102—42no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—44no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221C24—41no—
222C24#12111yescharacteristic
223C2×Q8—24no—
224C2×Q8—44no—
225C2×Q8—42no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×Q8—816no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—81no—
243C2×D4—81no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4—82no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—81no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4C2×SD1612yes—
275C22×C4#13512yes—
276C22×C4#13512yes—
277C22×C4AGL(1,ℤ/8)12yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—84no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—81no—
318C4⋊C4C2×SD1612yes—
319C4⋊C4C2×SD1612yes—
320C4⋊C4#13712yes—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C42—21no—
325C42—21no—
326C42—22no—
327C42—22no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—42no—
333C42—42no—
334C42C2×#4112yes—
335C42H(5,𝔽2)12yes—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23#46312yes—
348C23#46512yes—
349C23#47212yes—
350C23C2×#12111yescharacteristic
351Q8—84no—
352Q8—84no—
353Q8—88no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—82no—
359D4—82no—
360D4—162no—
361D4—164no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—44no—
381C2×C4—48no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—84no—
393C2×C4—88no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4#46312yes—
400C2×C4#46512yes—
401C2×C4#52511yescharacteristic
402C2×C4#53511yescharacteristic
403C2×C4#54612yes—
404C2×C4#55112yes—
405C2×C4C22×SD1611yescharacteristic
406C2×C4C2×#13511yescharacteristic
407C2×C4C2×#13511yescharacteristic
408C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
409C2×C4C2×#13711yescharacteristic
410C2×C4#57511yescharacteristic
411C2×C4#57812yes—
412C8—42no—
413C8—42no—
414C8—81no—
415C8—81no—
416C8—82no—
417C8—82no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22#167714yes—
432C22#241111yescharacteristic
433C22C2×#46311yescharacteristic
434C22#304411yescharacteristic
435C22C2×#46511yescharacteristic
436C22#305611yescharacteristic
437C22C2×#47211yescharacteristic
438C22#306811yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—42no—
443C4—42no—
444C4—41no—
445C4—42no—
446C4—84no—
447C4—81no—
448C4—161no—
449C4#310311yescharacteristic
450C4#310811yescharacteristic
451C4#320611yescharacteristic
452C4#321111yescharacteristic
453C4#330012yes—
454C4#330812yes—
455C2—24no—
456C2—321no—
457C2C2×#167711yescharacteristic
458C2#1916811yescharacteristic
459C2#1940311yescharacteristic
460C2#1947711yescharacteristic
461C2#2448111yescharacteristic
462C2#2450311yescharacteristic
463C2#2452011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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. #22599C2semidirect
4. C2×#1987C2semidirect
5. #10589C2semidirect
6. #10539C2semidirect
7. #9735C2semidirect
8. #9568C2semidirect
10. #8830C2semidirect
11. #8342C2semidirect

Maximal subgroups

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

subgroupindex
#331062
#259032
#225992
C2×#19872
#105892
#105392
#97352
#95682
#91332
#88302
#83422