Finite Group Databasebeta

← #453313

Subgroups — #453313

Order 512.

1324 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#1919C212yesmaximal
3#13926C211yesmaximal, characteristic
4#11384C212yesmaximal
5#11380C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11220C212yesmaximal
8#11207C212yesmaximal
9#10908C212yesmaximal
10C22×C4⋊Q8—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#106C2212yes—
14#2546—28no—
15C2×#398—21no—
16C2×#398—21no—
17C2×#391—21no—
18C2×#391—21no—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1919—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1441—21no—
37#1441—21no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C23×Q8—21no—
42C23×Q8D411yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—28no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—44no—
51C2×C4⋊Q8—44no—
52C2×C4×Q8—24no—
53C2×C4×Q8—24no—
54C2×C4×Q8—41no—
55C2×C4×Q8—41no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C42—21no—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#106—24no—
74C2×#106—24no—
75C2×#106Q818yescomplemented
76#398—28no—
77#398—28no—
78#391—28no—
79#391—44no—
80#389—24no—
81#389—24no—
82#389—24no—
83#389D418yes—
84#384—41no—
85#384—41no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—24no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—42no—
102#340—42no—
103#336—24no—
104#336—24no—
105#336—41no—
106#336—41no—
107C22×Q8—24no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8SD1614yes—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8C2×Q814yes—
146C4⋊Q8—48no—
147C4⋊Q8—48no—
148C4⋊Q8—432no—
149C4×Q8—48no—
150C4×Q8—48no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×Q814yes—
196C2×C4⋊C4C2×Q814yes—
197C2×C4⋊C4#4114yes—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42C2×D414yes—
215C2×C42#4114yes—
216#106—216no—
217#95—24no—
218#95—28no—
219#95—28no—
220#95—44no—
221#95—42no—
222#95—44no—
223#95—82no—
224#95—82no—
225C24C22×D411yescharacteristic
226C2×Q8—28no—
227C2×Q8—28no—
228C2×Q8—44no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—44no—
233C2×Q8—48no—
234C2×Q8—48no—
235C2×Q8—48no—
236C2×Q8—84no—
237C2×Q8—84no—
238C2×Q8—84no—
239C2×Q8—84no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—24no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4#12012yes—
296C22×C4#12012yes—
297C22×C4#12312yes—
298C22×C4#12312yes—
299C22×C4C2×SD1612yes—
300C22×C4C2×SD1612yes—
301C22×C4C2×SD1612yes—
302C22×C4AGL(1,ℤ/8)12yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4C22×D412yes—
305C22×C4C22×Q812yes—
306C22×C4C2×#4112yes—
307C22×C4C2×#4112yes—
308C22×C4H(5,𝔽2)11yescharacteristic
309C22×C4H(5,𝔽2)11yescharacteristic
310C22×C42−1+412yes—
311C22×C42−1+412yes—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—48no—
337C4⋊C4—416no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—416no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C4⋊C4C22⋊Q818yes—
350C42—24no—
351C42—24no—
352C42—24no—
353C42—24no—
354C42—48no—
355C42—42no—
356C42—42no—
357C42—84no—
358C23C2×#12011yescharacteristic
359C23C2×#12311yescharacteristic
360C23#53411yescharacteristic
361C23#54812yes—
362C23D4×Q812yes—
363C23C22×SD1611yescharacteristic
364C23C2×AGL(1,ℤ/8)11yescharacteristic
365C23C2×AGL(1,ℤ/8)11yescharacteristic
366C23#57511yescharacteristic
367C23#57712yes—
368C23#57812yes—
369Q8—416no—
370Q8—816no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—28no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—41no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#45014yes—
434C2×C4#48114yes—
435C2×C4C2×C22⋊Q812yes—
436C2×C4#53312yes—
437C2×C4#53612yes—
438C2×C4#54112yes—
439C2×C4#54212yes—
440C2×C4#54412yes—
441C2×C4#54612yes—
442C2×C4#54612yes—
443C2×C4#54812yes—
444C2×C4D4×Q812yes—
445C2×C4#55511yescharacteristic
446C2×C4#55612yes—
447C2×C4#55712yes—
448C2×C4Q8211yescharacteristic
449C2×C4#56311yescharacteristic
450C2×C4#56311yescharacteristic
451C8—44no—
452C8—82no—
453C22#240511yescharacteristic
454C22#240912yes—
455C22#244612yes—
456C22#273111yescharacteristic
457C22#273411yescharacteristic
458C22C2×#45011yescharacteristic
459C22#300611yescharacteristic
460C22#301312yes—
461C22C2×#48111yescharacteristic
462C22#309411yescharacteristic
463C22#309712yes—
464C22#318711yescharacteristic
465C22#319211yescharacteristic
466C22#320011yescharacteristic
467C22#320211yescharacteristic
468C22#335912yes—
469C22#336812yes—
470C22#337312yes—
471C22#337712yes—
472C22Q8×SD1612yes—
473C22#338612yes—
474C22#338612yes—
475C22#338812yes—
476C4—24no—
477C4—24no—
478C4—48no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—84no—
483C4—84no—
484C4—81no—
485C4—162no—
486C4#243912yes—
487C4#245112yes—
488C4#271312yes—
489C4#271612yes—
490C2#1609311yescharacteristic
491C2#2400511yescharacteristic
492C2#2401611yescharacteristic
493C2#2433012yes—
494C2#2434612yes—
495C2#2627812yes—
496C2#2629212yes—
497C2#3263011yescharacteristic
498C2#3264611yescharacteristic
499C2#3268311yescharacteristic
500C2#3269311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
75. C2×#106Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#19192
#139262
#113842
#113802
#112932
#112202
#112072
#109082