Finite Group Databasebeta

← #200522

Subgroups — #200522

Order 512.

1156 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#34953C211yesmaximal, characteristic
3C2×#1801C211yesmaximal, characteristic
4#17749C212yesmaximal
5#9402C212yesmaximal, complemented
6#9400C212yesmaximal, complemented
7#9394C212yesmaximal, complemented
8#9392C212yesmaximal, complemented
9#8097C212yesmaximal
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2984—24no—
13#2983—22no—
14#2983C2212yes—
15#2982C2212yes—
16Q8×M16—24no—
17#2920—22no—
18#2920C2212yes—
19C2×#423C2212yes—
20C22×#101—21no—
21#2584—22no—
22#2583C414yes—
23#2567C2211yescharacteristic
24#2567C2211yescharacteristic
25#2546C2211yescharacteristic
26#2546C2211yescharacteristic
27#2502—21no—
28#2434—21no—
29#2399C414yes—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32#2296—22no—
33C4×#117C2211yescharacteristic
34#1801—21no—
35#1801—21no—
36#1801C2212yes—
37#1801C2212yes—
38C2×#343—21no—
39#1559—24no—
40#1550—24no—
41#1548C2214yes—
42#1546C2214yes—
43#1303C2214yes—
44#1303C2214yes—
45C23×Q8D411yescharacteristic
46C22×M16—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D412yes—
68C2×C4×Q8C2311yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×#117—21no—
71C2×#117—21no—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4D411yescharacteristic
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76#446—216no—
77#446—28no—
78C8×Q8—28no—
79#432—24no—
80#432—24no—
81#432—22no—
82#432D414yes—
83#431—24no—
84#431—24no—
85#431—22no—
86#431C2314yes—
87#423—24no—
88#423—22no—
89#423—22no—
90#423—22no—
91#423D414yes—
92#423C2314yes—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×#101—22no—
96C2×#101—22no—
97C2×#101—44no—
98#407—22no—
99C4×M16—24no—
100C4×M16—22no—
101#398—24no—
102#398C2×C414yes—
103#398C2×C414yes—
104#395—24no—
105#395C2×C414yes—
106#395C2×C414yes—
107#391—21no—
108#391—21no—
109#391—21no—
110#391—21no—
111#389—24no—
112#389—24no—
113#389—22no—
114#389—22no—
115#389—22no—
116#389—22no—
117#385—24no—
118#385—24no—
119#384—21no—
120#384—21no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4C2×C412yes—
128#376C2×C412yes—
129C43—21no—
130#343—22no—
131#343—22no—
132#333—28no—
133#332—28no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—42no—
140C22×Q8C22⋊C412yes—
141C22×Q8C22×C412yes—
142C22×Q8C2×D411yescharacteristic
143C22×Q8C2×D411yescharacteristic
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—24no—
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—44no—
153C2×M16—44no—
154C22×C8—42no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—44no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162#125—42no—
163#125—42no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—24no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—42no—
183C4×Q8—42no—
184#117—28no—
185#117—41no—
186#117—41no—
187#117—41no—
188#117—41no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42C22⋊C412yes—
224C2×C42C22⋊C412yes—
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4112yes—
233C2×C42#4112yes—
234C2×C42C2411yescharacteristic
235C4⋊C8—28no—
236C4⋊C8—28no—
237C4⋊C8—24no—
238C4⋊C8—24no—
239C4⋊C8—24no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8C2×D418yes—
243#101—44no—
244#101—44no—
245C22⋊C8—24no—
246C22⋊C8—24no—
247#97—28no—
248#97—24no—
249C4×C8—28no—
250C4×C8—24no—
251#95—22no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95C22×C412yes—
257#95C22×C412yes—
258C24C2×C22⋊C411yescharacteristic
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—42no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—82no—
278C2×Q8AGL(1,ℤ/8)12yes—
279C2×Q8#13712yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—81no—
295C22×C4—82no—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C4×D412yes—
304C22×C4C4×D412yes—
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12112yes—
308C22×C4#12112yes—
309C22×C4#12311yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—48no—
315M16—44no—
316M16—44no—
317M16—84no—
318C2×C8—28no—
319C2×C8—44no—
320C2×C8—84no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—48no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C22⋊C4—28no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C42—28no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—82no—
360C42C4×D412yes—
361C42C4×D412yes—
362C42#12012yes—
363C42#12112yes—
364C42#12112yes—
365C42#12312yes—
366C42C22×D412yes—
367C42C2×#4112yes—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—41no—
372C23#38611yescharacteristic
373C23#38611yescharacteristic
374C23C22×C22⋊C411yescharacteristic
375Q8—24no—
376Q8—24no—
377Q8—48no—
378Q8—42no—
379Q8—42no—
380Q8—84no—
381Q8—82no—
382Q8—82no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—82no—
422C2×C4#38611yescharacteristic
423C2×C4#38612yes—
424C2×C4#38612yes—
425C2×C4#38611yescharacteristic
426C2×C4#41714yes—
427C2×C4#45614yes—
428C2×C4C2×C4×D412yes—
429C2×C4C2×#12011yescharacteristic
430C2×C4C2×#12112yes—
431C2×C4C2×#12311yescharacteristic
432C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
433C2×C4C2×#13711yescharacteristic
434C2×C4#57512yes—
435C8—48no—
436C8—84no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—42no—
445C22—42no—
446C22#188612yes—
447C22C2×#38611yescharacteristic
448C22C2×#41711yescharacteristic
449C22#289611yescharacteristic
450C4—22no—
451C4—22no—
452C4—21no—
453C4—21no—
454C4—22no—
455C4—24no—
456C4—44no—
457C4—44no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—82no—
462C4—82no—
463C4#188612yes—
464C4C2×#45612yes—
465C4#305512yes—
466C4#305612yes—
467C2—21no—
468C2—21no—
469C2—22no—
470C2—41no—
471C2C2×#188611yescharacteristic
472C2#2097111yescharacteristic
473C2#2101711yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9402C2semidirect
6. #9400C2semidirect
7. #9394C2semidirect
8. #9392C2semidirect

Maximal subgroups

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

subgroupindex
#602482
#349532
C2×#18012
#177492
#94022
#94002
#93942
#93922
#80972