Finite Group Databasebeta

← #200512

Subgroups — #200512

Order 512.

1195 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#34947C211yesmaximal, characteristic
3#20637C211yesmaximal, characteristic
4#17708C212yesmaximal, complemented
5#9404C212yesmaximal, complemented
6#9403C212yesmaximal, complemented
7#9400C212yesmaximal, complemented
8#9399C212yesmaximal, complemented
9C2×#1303C212yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#2989—24no—
13#2983C2212yes—
14#2975C2212yes—
15#2973—22no—
16#2965—24no—
17#2920—22no—
18C2×#431C2212yes—
19C2×#423C2212yes—
20C2×#413—21no—
21#2568C2211yescharacteristic
22#2567C2211yescharacteristic
23#2566—21no—
24#2565—21no—
25#2564C414yes—
26#2563C2211yescharacteristic
27#2542C2211yescharacteristic
28#2417—22no—
29#2396C414yes—
30C4×C22⋊Q8—22no—
31#2297—21no—
32#2285—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34#1801—21no—
35#1801C2212yes—
36#1800—21no—
37#1800C2212yes—
38#1742—21no—
39#1550—24no—
40#1549C2214yes—
41#1548C2214yes—
42#1542—24no—
43#1303C2214yes—
44#1303C2214yes—
45C22×#41D411yescharacteristic
46C22×M16—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8C2×C412yes—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C2×C4×Q8—21no—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×D4C2×C412yes—
66C2×C4×D4D412yes—
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117C2×C412yes—
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C42—21no—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76#446—28no—
77#436—28no—
78#435—216no—
79#432—24no—
80#432—24no—
81#431—24no—
82#431—22no—
83#431—22no—
84#431D414yes—
85#431C2314yes—
86#423—24no—
87#423—22no—
88#423—22no—
89#423—22no—
90#423C2314yes—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8D414yes—
94#413—44no—
95C2×#101—22no—
96C2×#100—22no—
97#407—24no—
98C2×C22⋊C8—22no—
99C4×M16—24no—
100C2×C4×C8—22no—
101#400C2×C414yes—
102#399C2×C414yes—
103#398—24no—
104#396—24no—
105#395C2×C414yes—
106#393C2×C414yes—
107#391—21no—
108#391—21no—
109#391—22no—
110#390—21no—
111#390—21no—
112#389—22no—
113#387—24no—
114#386—24no—
115#385—24no—
116#384—21no—
117#384—21no—
118#382—21no—
119#382—21no—
120#380—22no—
121#380—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C22⋊C4—24no—
129C2×#95C2×C412yes—
130#343—22no—
131#343—22no—
132#333—28no—
133#332—28no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—24no—
139C22×Q8—21no—
140C22×Q8C22×C412yes—
141C22×Q8C2×D411yescharacteristic
142C22×D4C22⋊C412yes—
143C22×D4C2×D411yescharacteristic
144C23×C4—22no—
145C23×C4C22⋊C412yes—
146C23×C4C22⋊C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—28no—
152C2×M16—24no—
153C2×M16—42no—
154C2×M16—44no—
155C2×M16—44no—
156C22×C8—24no—
157C22×C8—42no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162#125—42no—
163#125—42no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C22⋊Q8—48no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187#117—22no—
188#117—22no—
189#117—24no—
190#117—24no—
191#117—42no—
192#117—42no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C412yes—
214C2×C22⋊C4C22×C412yes—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42C22⋊C412yes—
226C2×C42C22⋊C412yes—
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4112yes—
234C2×C42#4112yes—
235C2×C42C2411yescharacteristic
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#100—44no—
245C22⋊C8—28no—
246C22⋊C8—24no—
247C22⋊C8—24no—
248#97—28no—
249C4×C8—24no—
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⋊C412yes—
259C24C2×C22⋊C411yescharacteristic
260#41—44no—
261#41—44no—
262#41—44no—
263#41—44no—
264C2×Q8—22no—
265C2×Q8—21no—
266C2×Q8—22no—
267C2×Q8—21no—
268C2×Q8—24no—
269C2×Q8—44no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—82no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—21no—
278C2×D4—22no—
279C2×D4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—81no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C4×D412yes—
306C22×C4C4×D412yes—
307C22×C4#12012yes—
308C22×C4#12112yes—
309C22×C4#12112yes—
310C22×C4#12312yes—
311C22×C4AGL(1,ℤ/8)12yes—
312C22×C4#13712yes—
313C22×C4#13714yes—
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317M16—44no—
318M16—44no—
319M16—88no—
320M16—84no—
321C2×C8—28no—
322C2×C8—44no—
323C2×C8—84no—
324C2×C8—84no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—48no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—84no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—48no—
346C22⋊C4—44no—
347C22⋊C4—84no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—42no—
359C42—42no—
360C42—44no—
361C42C4×D412yes—
362C42C4×D412yes—
363C42#12011yescharacteristic
364C42#12011yescharacteristic
365C42#12112yes—
366C42#12112yes—
367C42#12311yescharacteristic
368C42#12311yescharacteristic
369C42C22×D412yes—
370C42C2×#4112yes—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23#38612yes—
379C23#41712yes—
380C23#41912yes—
381C23C22×C22⋊C411yescharacteristic
382Q8—42no—
383Q8—42no—
384Q8—84no—
385Q8—82no—
386Q8—82no—
387D4—48no—
388D4—42no—
389D4—42no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—88no—
422C2×C4#38611yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4#38612yes—
425C2×C4#38611yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#41712yes—
428C2×C4#41912yes—
429C2×C4#46814yes—
430C2×C4#46814yes—
431C2×C4#47014yes—
432C2×C4#47014yes—
433C2×C4C2×C4×D412yes—
434C2×C4C2×#12011yescharacteristic
435C2×C4C2×#12112yes—
436C2×C4C2×#12311yescharacteristic
437C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
438C2×C4C2×#13712yes—
439C2×C4C2×#13711yescharacteristic
440C8—48no—
441C8—84no—
442C22—21no—
443C22—22no—
444C22—21no—
445C22—22no—
446C22—42no—
447C22—42no—
448C22—44no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—82no—
453C22#189414yes—
454C22C2×#38611yescharacteristic
455C22C2×#41911yescharacteristic
456C22C2×#41711yescharacteristic
457C4—21no—
458C4—21no—
459C4—22no—
460C4—21no—
461C4—22no—
462C4—44no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—82no—
467C4—82no—
468C4—82no—
469C4#189412yes—
470C4#189412yes—
471C4#301812yes—
472C4C2×#46812yes—
473C4C2×#47012yes—
474C2—21no—
475C2—21no—
476C2—24no—
477C2—82no—
478C2#2095211yescharacteristic
479C2C2×#189411yescharacteristic
480C2#2101711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17708C2semidirect
5. #9404C2semidirect
6. #9403C2semidirect
7. #9400C2semidirect
8. #9399C2semidirect
9. C2×#1303C2semidirect

Maximal subgroups

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

subgroupindex
#602772
#349472
#206372
#177082
#94042
#94032
#94002
#93992
C2×#13032