Finite Group Databasebeta

← #200397

Subgroups — #200397

Order 512.

1215 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#34947C211yesmaximal, characteristic
3#20632C211yesmaximal, characteristic
4#17575C212yesmaximal, complemented
5#9385C218yesmaximal, complemented
6#8014C212yesmaximal, complemented
7C2×#533C2211yescharacteristic
8C2×C4×#41C2211yescharacteristic
9#2989—24no—
10#2983—21no—
11#2975—21no—
12#2973C2214yes—
13#2965—24no—
14#2920—22no—
15C2×#431C2212yes—
16C2×#410—21no—
17#2568C411yescharacteristic, complemented
18#2567C411yescharacteristic, complemented
19#2566—21no—
20#2565—21no—
21#2564C2214yes—
22#2563C411yescharacteristic, complemented
23#2542C411yescharacteristic, complemented
24#2417—22no—
25#2396C414yescomplemented
26C4×C22⋊Q8—22no—
27#2297—21no—
28#2285—21no—
29#2285C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1799—22no—
32#1799C2214yes—
33#1740—21no—
34#1520C2218yes—
35#1518—24no—
36#1503—24no—
37#1294C2218yes—
38C22×#41D411yescharacteristic
39#533—22no—
40#533—22no—
41#533—22no—
42#533—22no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8C2312yes—
49C4×#41—22no—
50C4×#41—22no—
51C4×#41—22no—
52C4×#41—22no—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D412yes—
59C2×C4×D4D411yescharacteristic
60C2×C4×D4C2312yes—
61C2×#117—21no—
62C2×#117—22no—
63C2×#117C2×C412yes—
64C2×#117D411yescharacteristic
65C22×C4⋊C4—21no—
66C22×C42—21no—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69#446—28no—
70#436—28no—
71#435—216no—
72#432—24no—
73#432—24no—
74#431—24no—
75#431—22no—
76#431—22no—
77#431D414yes—
78#431D414yes—
79#423—24no—
80C2×C4⋊C8—22no—
81#410—44no—
82C2×#99—24no—
83#407—24no—
84C2×C22⋊C8C2314yes—
85C4×M16—24no—
86C2×C4×C8—22no—
87#400C2×C414yes—
88#399C2×C414yes—
89#398—24no—
90#396—24no—
91#395C2×C414yes—
92#393C2×C414yes—
93#391—21no—
94#391—21no—
95#391—22no—
96#390—21no—
97#390—21no—
98#389—22no—
99#387—24no—
100#386—24no—
101#385—24no—
102#384—21no—
103#384—21no—
104#384—22no—
105#382—21no—
106#382—21no—
107#382—22no—
108#380—22no—
109#380—24no—
110#380—22no—
111#380C2314yes—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C22⋊C4—24no—
121C2×#95C2312yes—
122#342—24no—
123#328—216no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41—22no—
128C2×#41—24no—
129C22×Q8—21no—
130C22×Q8C22⋊C411yescharacteristic
131C22×Q8C22×C412yes—
132C22×D4C22⋊C411yescharacteristic
133C22×D4C2×D412yes—
134C23×C4—22no—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C2×M16—28no—
142C22×C8—24no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147#125—42no—
148#125—42no—
149C22⋊Q8—24no—
150C22⋊Q8—24no—
151C22⋊Q8—48no—
152C22⋊Q8—42no—
153C22⋊Q8—42no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—41no—
161C4×Q8—41no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172#117—22no—
173#117—22no—
174#117—24no—
175#117—24no—
176#117—42no—
177#117—42no—
178#117—44no—
179#117—44no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22×C412yes—
205C2×C22⋊C4C2×D412yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42C22⋊C412yes—
218C2×C42C22⋊C412yes—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4112yes—
227C2×C42#4112yes—
228C4⋊C8—28no—
229C4⋊C8—24no—
230C4⋊C8—24no—
231#99—48no—
232C22⋊C8—28no—
233C22⋊C8—24no—
234C22⋊C8C2×D418yes—
235#97—28no—
236C4×C8—24no—
237C4×C8—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—22no—
243#95—24no—
244#95—42no—
245#95C22×C412yes—
246#95C22×C412yes—
247C24C2×C22⋊C412yes—
248C24C22×D411yescharacteristic
249#41—44no—
250#41—44no—
251#41—44no—
252#41—44no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—44no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—82no—
261C2×D4—24no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—81no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4AGL(1,ℤ/8)12yes—
301C22×C4#13712yes—
302C22×C4#13714yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D412yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4112yes—
307M16—88no—
308C2×C8—28no—
309C2×C8—84no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—48no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—84no—
327C4⋊C4—84no—
328C4⋊C4—84no—
329C4⋊C4—84no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—48no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—84no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—44no—
348C42—42no—
349C42—84no—
350C42C4×D412yes—
351C42C4×D412yes—
352C42C4×D412yes—
353C42C4×D412yes—
354C42#12012yes—
355C42#12011yescharacteristic
356C42#12011yescharacteristic
357C42#12311yescharacteristic
358C42#12312yes—
359C42#12311yescharacteristic
360C23—22no—
361C23—22no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23C22×C22⋊C411yescharacteristic
368C23C2×#12112yes—
369C23C2×AGL(1,ℤ/8)11yescharacteristic
370C23C2×#13712yes—
371C23C2×#13711yescharacteristic
372Q8—44no—
373Q8—84no—
374Q8—82no—
375Q8—82no—
376D4—44no—
377D4—44no—
378D4—44no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4#38611yescharacteristic
416C2×C4#38612yes—
417C2×C4#38611yescharacteristic
418C2×C4#38611yescharacteristic
419C2×C4#38612yes—
420C2×C4#38611yescharacteristic
421C2×C4#41712yes—
422C2×C4#41712yes—
423C2×C4#41912yes—
424C2×C4#41912yes—
425C2×C4#46814yes—
426C2×C4#46814yes—
427C2×C4#47014yes—
428C2×C4#47014yes—
429C2×C4C2×C4×D412yes—
430C2×C4C2×#12011yescharacteristic
431C2×C4C2×#12311yescharacteristic
432C8—48no—
433C22—21no—
434C22—22no—
435C22—21no—
436C22—22no—
437C22—41no—
438C22—41no—
439C22—42no—
440C22—44no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22C2×#38611yescharacteristic
446C22C2×#41911yescharacteristic
447C22C2×#41711yescharacteristic
448C22C2×#46812yes—
449C22C2×#47012yes—
450C4—21no—
451C4—21no—
452C4—22no—
453C4—21no—
454C4—22no—
455C4—44no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—164no—
463C4#188412yes—
464C4#189412yes—
465C4#189412yes—
466C4#189414yes—
467C2—21no—
468C2—22no—
469C2—21no—
470C2—22no—
471C2—82no—
472C2C2×#188411yescharacteristic
473C2C2×#189411yescharacteristic
474C2#2101411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17575C2semidirect
5. #9385C2semidirect
6. #8014C2semidirect
17. #2568C4semidirect
18. #2567C4semidirect
22. #2563C4semidirect
23. #2542C4semidirect
25. #2396C4semidirect

Maximal subgroups

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

subgroupindex
#602772
#349472
#206322
#175752
#93852
#80142