Finite Group Databasebeta

← #204721

Subgroups — #204721

Order 512.

1115 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35201C211yesmaximal, characteristic
2#21356C211yesmaximal, characteristic, complemented
3#20331C211yesmaximal, characteristic, complemented
4C2×#1531C211yesmaximal, characteristic, complemented
5C2×#1529C211yesmaximal, characteristic
6C2×#1526C211yesmaximal, characteristic, complemented
7C2×#1524C211yesmaximal, characteristic
8#9687C214yesmaximal, complemented
9#9686C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#529C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#405—22no—
14C2×C4×M16C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2861C2212yes—
18#2859C2212yes—
19#2663—22no—
20#2604—22no—
21#2568C411yescharacteristic
22#2567C411yescharacteristic
23#2563C411yescharacteristic
24#2542C411yescharacteristic
25#2500C412yes—
26#2499C412yes—
27#2415—22no—
28#2323—22no—
29C4×#117C2211yescharacteristic
30#1939—22no—
31#1851—22no—
32#1767C2214yes—
33#1726—22no—
34C4×#97C2214yes—
35#1531—24no—
36#1529—24no—
37#1526—24no—
38#1524—24no—
39C2×#330C2212yes—
40C2×#330C2212yes—
41C2×#330C2214yes—
42C2×#330C2214yes—
43C2×#329C2212yes—
44C2×#329C2212yes—
45C22×M16—21no—
46C22×M16—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533—22no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54#529—22no—
55#529—22no—
56#529—22no—
57#529—22no—
58C2×#126—21no—
59C2×#125C2311yescharacteristic
60C2×#125C2312yes—
61C2×#124C2311yescharacteristic
62C2×#123—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic
73C2×#117—21no—
74C2×#117C2×C412yes—
75C2×#117D412yes—
76C22×C42C2311yescharacteristic
77#432—22no—
78#432—22no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—21no—
81C2×C22⋊C8—21no—
82#405—216no—
83C4×M16—22no—
84C4×M16—22no—
85C4×M16—22no—
86C4×M16D414yes—
87C2×#97—21no—
88C2×#97—24no—
89C2×#97—22no—
90C2×#97D414yes—
91C2×#97C2312yes—
92C2×#97C2312yes—
93C2×#97C2312yes—
94C2×#97C2312yes—
95C2×C4×C8—21no—
96C2×C4×C8—24no—
97C2×C4×C8—21no—
98#400—22no—
99#399—22no—
100#397—24no—
101#395—22no—
102#394—24no—
103#393—22no—
104#391—22no—
105#391C2×C412yes—
106#391C2×C412yes—
107#390C2×C412yes—
108#390C2×C412yes—
109#389—22no—
110#389—24no—
111#389—24no—
112#388—24no—
113#388—24no—
114#388—24no—
115#387—24no—
116#384C2×C412yes—
117#384C2×C412yes—
118#382C2×C412yes—
119#382C2×C412yes—
120#380—22no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C22⋊C4C2×C412yes—
125C4×C22⋊C4D412yes—
126#376D412yes—
127C43C2312yes—
128#336—24no—
129#330—24no—
130#330—28no—
131#330—24no—
132#330—28no—
133#329—24no—
134#329—24no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8C22×C411yescharacteristic
138C22×D4—21no—
139C22×D4C22×C411yescharacteristic
140C23×C4—21no—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—28no—
145C2×M16—24no—
146C2×M16—44no—
147C2×M16—42no—
148C22×C8—28no—
149C22×C8—24no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158#126—42no—
159#126—42no—
160#125—21no—
161#125—21no—
162#125—22no—
163#125—22no—
164#125—21no—
165#125—21no—
166#125—22no—
167#125—22no—
168#124—21no—
169#124—21no—
170#124—21no—
171#124—21no—
172#123—42no—
173#123—42no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C22⋊Q8—48no—
181#121—22no—
182#121—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—42no—
192C4×D4—42no—
193#117—28no—
194#117—24no—
195#117—24no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4C22⋊C414yes—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42C22×C412yes—
226C2×C42C22×C412yes—
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D414yes—
229C2×C42C2×D414yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4112yes—
235C2×C42#4112yes—
236C2×C42C2411yescharacteristic
237C4⋊C8—48no—
238C4⋊C8—42no—
239C22⋊C8—42no—
240#97—216no—
241#97—24no—
242#97—22no—
243#97—22no—
244#97—22no—
245#97—22no—
246#97—22no—
247#97—22no—
248#97C2×D414yes—
249#97C2×D414yes—
250C4×C8—216no—
251C4×C8—24no—
252C4×C8—42no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95C22⋊C414yes—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260C24—21no—
261C24C2×C22⋊C411yescharacteristic
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—44no—
269C2×Q8—82no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—82no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4C2×C22⋊C414yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C414yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C4×D412yes—
293C22×C4C4×D412yes—
294C22×C4#12112yes—
295C22×C4#12112yes—
296C22×C4#12412yes—
297C22×C4#12712yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301M16—216no—
302M16—48no—
303C2×C8—24no—
304C2×C8—28no—
305C2×C8—24no—
306C2×C8—216no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—44no—
313C2×C8—42no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—48no—
336C4⋊C4—84no—
337C4⋊C4#10418yes—
338C22⋊C4—28no—
339C22⋊C4—44no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—48no—
348C22⋊C4—84no—
349C42—28no—
350C42—44no—
351C42—42no—
352C42—42no—
353C42—44no—
354C42—42no—
355C42C4×D412yes—
356C42C4×D412yes—
357C42#12112yes—
358C42#12112yes—
359C42#12412yes—
360C42#12712yes—
361C42C22×D411yescharacteristic
362C42C22×D411yescharacteristic
363C42C2×#4111yescharacteristic
364C42C2×#4111yescharacteristic
365C23—21no—
366C23—21no—
367C23—22no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23#39012yes—
375C23C22×C22⋊C411yescharacteristic
376Q8—81no—
377Q8—81no—
378Q8—84no—
379Q8—84no—
380D4—81no—
381D4—81no—
382D4—84no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—84no—
413C2×C4#39012yes—
414C2×C4#39012yes—
415C2×C4#39012yes—
416C2×C4C2×#10418yes—
417C2×C4C2×#10414yes—
418C2×C4C2×C4×D411yescharacteristic
419C2×C4C2×C4×D411yescharacteristic
420C2×C4C2×#12111yescharacteristic
421C2×C4C2×#12111yescharacteristic
422C2×C4C2×#12411yescharacteristic
423C2×C4C2×#12711yescharacteristic
424C8—28no—
425C8—28no—
426C8—44no—
427C8—82no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—42no—
432C22—42no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—82no—
437C22—82no—
438C22C2×#39011yescharacteristic
439C22#288911yescharacteristic
440C22#288911yescharacteristic
441C22C22×#10412yes—
442C22#290112yes—
443C4—21no—
444C4—21no—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—28no—
452C4—44no—
453C4—81no—
454C4—82no—
455C4—81no—
456C4—82no—
457C4—81no—
458C4#196214yes—
459C2—42no—
460C2—161no—
461C2#1765312yes—
462C2#1765412yes—
463C2C2×#196211yescharacteristic
464C2#2122011yescharacteristic
465C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21356C2semidirect
3. #20331C2semidirect
4. C2×#1531C2semidirect
6. C2×#1526C2semidirect
8. #9687C2semidirect
9. #9686C2semidirect

Maximal subgroups

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

subgroupindex
#352012
#213562
#203312
C2×#15312
C2×#15292
C2×#15262
C2×#15242
#96872
#96862