Finite Group Databasebeta

← #492996

Subgroups — #492996

Order 512.

1257 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34266C211yesmaximal, characteristic
2C2×#2123C212yesmaximal
3#20397C211yesmaximal, characteristic
4#20393C211yesmaximal, characteristic
5#12621C214yesmaximal, complemented
6C4×#352C214yesmaximal, complemented
7#12573C212yesmaximal
8C2×#536C2211yescharacteristic
9C2×#520C2211yescharacteristic
10C22×C4×Q8C411yescharacteristic, complemented
11C2×#413C2212yes—
12C2×#410C2212yes—
13#2873—21no—
14#2367—22no—
15#2365—22no—
16#2362—21no—
17#2362—21no—
18#2360—21no—
19#2360—21no—
20#2322C412yescomplemented
21#2322C2212yes—
22#2310C411yescharacteristic, complemented
23C4×#123C412yescomplemented
24C4×#123C2212yes—
25#2297C2211yescharacteristic
26#2296C411yescharacteristic, complemented
27#2274—21no—
28#2274C411yescharacteristic, complemented
29#2123C4116yescomplemented
30C2×#352C2218yes—
31C4×#101—22no—
32C4×#100C2214yes—
33C4×#99—22no—
34C4×#99C2214yes—
35C2×#343—21no—
36#1740—21no—
37#1390C2214yes—
38#1390C2214yes—
39C22×#41C2311yescharacteristic
40C23×Q8C2×C411yescharacteristic
41C22×M16—22no—
42#536—24no—
43#536C2×C418yes—
44C2×C4⋊Q8—21no—
45C2×#124—21no—
46C2×#123C2×C412yes—
47C2×#123C2312yes—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50#520—44no—
51C2×C4×Q8—21no—
52C2×C4×Q8—22no—
53C2×C4×Q8—21no—
54C2×C4×Q8—44no—
55C2×C4×Q8D412yes—
56C2×C4×D4C2×C412yes—
57C2×C4×D4D412yes—
58C2×C4×D4C2312yes—
59C2×#117—22no—
60C2×#117C2×C411yescharacteristic
61C2×#117C2×C411yescharacteristic
62C2×#117C2×C411yescharacteristic
63C2×#117D411yescharacteristic
64C2×#117D411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C412yes—
71#413—24no—
72#413C2×C418yes—
73C2×#101—22no—
74C2×#100C2314yes—
75#410—24no—
76#410C2×C418yes—
77C2×#99—22no—
78C2×#99C2314yes—
79#405—44no—
80C4×M16—24no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391—22no—
85#390—21no—
86#389—22no—
87#389—21no—
88#389—21no—
89#389—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#386C2×C411yescharacteristic
99#386C2×C411yescharacteristic
100#386C2311yescharacteristic
101#386C2311yescharacteristic
102#385—22no—
103#385—22no—
104#385C2×C412yes—
105#385C2×C412yes—
106#381C2×C412yes—
107#381C2×C412yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—44no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—24no—
120C4×C22⋊C4C2×C412yes—
121C4×C22⋊C4C2×C412yes—
122#352C2×C4132yescomplemented
123#343—24no—
124#342—24no—
125C2×#41—42no—
126C2×#41—42no—
127C2×#41C22×C414yes—
128C2×#41#4114yes—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8C4214yescomplemented
135C22×Q8C2×D411yescharacteristic
136C22×D4C22×C411yescharacteristic
137C22×D4C2×D411yescharacteristic
138C22×D4C2411yescharacteristic
139C23×C4C22⋊C412yes—
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C412yes—
142C23×C4C22×C412yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C2×M16—28no—
149C2×M16—24no—
150C2×M16—44no—
151C2×M16—42no—
152C22×C8—42no—
153C4⋊Q8—28no—
154#124—28no—
155#123—24no—
156#123—24no—
157#123C4218yescomplemented
158#123C22×C418yes—
159C22⋊Q8—28no—
160C22⋊Q8—28no—
161C4×Q8—44no—
162C4×Q8—44no—
163C4×Q8—48no—
164C4×Q8—44no—
165C4×D4—44no—
166C4×D4—44no—
167C4×D4—44no—
168#117—22no—
169#117—24no—
170#117—28no—
171#117—42no—
172#117—42no—
173#117—44no—
174#117—42no—
175#117C4214yescomplemented
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22×C412yes—
205C2×C22⋊C4C22×C411yescharacteristic
206C2×C22⋊C4C22×C411yescharacteristic
207C2×C22⋊C4C22×C412yes—
208C2×C22⋊C4C2×D412yes—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—44no—
220C2×C42—44no—
221C2×C42C22⋊C412yes—
222C2×C42C22⋊C412yes—
223C2×C42C2×D412yes—
224#101—28no—
225#100—24no—
226#100C22×C418yes—
227#99—28no—
228#99—24no—
229#99C22×C418yes—
230#97—44no—
231C4×C8—44no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95—44no—
245C24C2×C22⋊C411yescharacteristic
246C24C23×C411yescharacteristic
247C24C22×D411yescharacteristic
248#41—44no—
249#41—44no—
250#41—44no—
251#41—44no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—48no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—48no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8C4×D414yes—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4C2×C4212yes—
270C2×D4#11712yes—
271C2×D4C4×D414yes—
272C2×D4C23×C412yes—
273C2×D4C2×#4112yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—41no—
296C22×C4—82no—
297C22×C4—81no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4C2×C4212yes—
301C22×C4C2×C4214yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4#11712yes—
310C22×C4C4×D414yes—
311C22×C4C4×D414yes—
312C22×C4C22×D411yescharacteristic
313M16—28no—
314M16—44no—
315M16—48no—
316C2×C8—44no—
317C2×C8—48no—
318C4⋊C4—28no—
319C4⋊C4—24no—
320C4⋊C4—28no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—42no—
330C4⋊C4—48no—
331C4⋊C4—44no—
332C4⋊C4—82no—
333C4⋊C4#9914yes—
334C22⋊C4—22no—
335C22⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—24no—
338C22⋊C4—28no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—81no—
347C22⋊C4—81no—
348C22⋊C4C2×C4214yes—
349C42—24no—
350C42—24no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—44no—
355C42—48no—
356C42—44no—
357C42—42no—
358C42—42no—
359C42—82no—
360C42—82no—
361C42—164no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23C4×C22⋊C412yes—
375C23C22×C4211yescharacteristic
376C23C22×C22⋊C411yescharacteristic
377C23C2×#11711yescharacteristic
378C23C2×C4×D412yes—
379Q8—44no—
380Q8—48no—
381Q8—48no—
382D4—42no—
383D4—42no—
384D4—44no—
385D4—42no—
386D4—42no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—82no—
434C2×C4—162no—
435C2×C4C4×C22⋊C412yes—
436C2×C4C4×C22⋊C412yes—
437C2×C4C4×C22⋊C412yes—
438C2×C4C2×#9914yes—
439C2×C4C2×#9912yes—
440C2×C4#41012yes—
441C2×C4C2×C4×D412yes—
442C8—48no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—21no—
448C22—22no—
449C22—41no—
450C22—42no—
451C22—42no—
452C22—41no—
453C22—42no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—81no—
461C22—81no—
462C22C2×C4×C22⋊C411yescharacteristic
463C22C22×#9911yescharacteristic
464C22C2×#41011yescharacteristic
465C4—24no—
466C4—22no—
467C4—22no—
468C4—22no—
469C4—44no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—44no—
475C4—81no—
476C4—81no—
477C4—88no—
478C4—81no—
479C4C4×#9914yes—
480C2—21no—
481C2—21no—
482C2—42no—
483C2—41no—
484C2—41no—
485C2—42no—
486C2C2×C4×#9911yescharacteristic
487C2#3334911yescharacteristic
488C2#3335011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC23C22×C42
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #12621C2semidirect
6. C4×#352C2semidirect
10. C22×C4×Q8C4semidirect
20. #2322C4semidirect
22. #2310C4semidirect
23. C4×#123C4semidirect
26. #2296C4semidirect
28. #2274C4semidirect
29. #2123C4semidirect
122. #352C2×C4semidirect
134. C22×Q8C42semidirect
157. #123C42semidirect
175. #117C42semidirect

Maximal subgroups

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

subgroupindex
#342662
C2×#21232
#203972
#203932
#126212
C4×#3522
#125732