Finite Group Databasebeta

← #204724

Subgroups — #204724

Order 512.

1003 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35201C211yesmaximal, characteristic
2#21356C211yesmaximal, characteristic, complemented
3#20367C211yesmaximal, characteristic, complemented
4C2×#1531C211yesmaximal, characteristic, complemented
5C2×#1529C211yesmaximal, characteristic
6C2×#1526C211yesmaximal, characteristic, complemented
7C2×#1524C211yesmaximal, characteristic
8#9690C214yesmaximal, complemented
9#9688C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#529C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15C22×#97C2211yescharacteristic
16#2861C2212yes—
17#2859C2212yes—
18#2663—22no—
19#2604—22no—
20#2568C411yescharacteristic
21#2567C411yescharacteristic
22#2563C411yescharacteristic
23#2542C411yescharacteristic
24#2500C412yes—
25#2499C412yes—
26#2415—22no—
27#2323—22no—
28C4×#117C2211yescharacteristic
29#1939—22no—
30#1851—22no—
31#1767C2214yes—
32#1746—22no—
33#1745C2214yes—
34#1729—22no—
35#1531—24no—
36#1529—24no—
37#1526—24no—
38#1524—24no—
39C2×#330C2214yes—
40C2×#330C2214yes—
41C2×#330C2212yes—
42C2×#330C2212yes—
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—
82C4×M16—22no—
83C4×M16—22no—
84C4×M16—22no—
85C4×M16D414yes—
86C2×#97—21no—
87C2×#97—22no—
88C2×#97D414yes—
89C2×#97C2312yes—
90C2×#97C2312yes—
91C2×#97C2312yes—
92C2×#97C2312yes—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95#400—22no—
96#399—22no—
97#397—24no—
98#395—22no—
99#394—24no—
100#393—22no—
101#391—22no—
102#391C2×C412yes—
103#391C2×C412yes—
104#390C2×C412yes—
105#390C2×C412yes—
106#389—22no—
107#389—24no—
108#389—24no—
109#388—24no—
110#388—24no—
111#388—24no—
112#387—24no—
113#384C2×C412yes—
114#384C2×C412yes—
115#382C2×C412yes—
116#382C2×C412yes—
117#380—22no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4D412yes—
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4D412yes—
123#376D412yes—
124C43C2312yes—
125#336—24no—
126#336—28no—
127#330—24no—
128#330—28no—
129#330—24no—
130#330—28no—
131#329—24no—
132#329—24no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8C22×C411yescharacteristic
136C22×D4—21no—
137C22×D4C22×C411yescharacteristic
138C23×C4—21no—
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C2×M16—24no—
143C2×M16—44no—
144C2×M16—44no—
145C2×M16—42no—
146C22×C8—24no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—44no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156#126—42no—
157#126—42no—
158#125—21no—
159#125—21no—
160#125—22no—
161#125—22no—
162#125—21no—
163#125—21no—
164#125—22no—
165#125—22no—
166#124—21no—
167#124—21no—
168#124—21no—
169#124—21no—
170#123—42no—
171#123—42no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—48no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179#121—22no—
180#121—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—42no—
190C4×D4—42no—
191#117—24no—
192#117—24no—
193#117—44no—
194#117—42no—
195#117—42no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4C22⋊C414yes—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42C22×C412yes—
224C2×C42C22×C412yes—
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D414yes—
227C2×C42C2×D414yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4112yes—
233C2×C42#4112yes—
234C2×C42C2411yescharacteristic
235C4⋊C8—48no—
236C4⋊C8—42no—
237C22⋊C8—42no—
238#97—24no—
239#97—22no—
240#97—22no—
241#97—22no—
242#97—22no—
243#97—22no—
244#97—22no—
245#97C2×D414yes—
246#97C2×D414yes—
247C4×C8—24no—
248C4×C8—42no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95C22⋊C414yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C24—21no—
257C24C2×C22⋊C411yescharacteristic
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—82no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—82no—
272C22×C4—24no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4C2×C22⋊C414yes—
284C22×C4C2×C22⋊C414yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C4×D412yes—
289C22×C4C4×D412yes—
290C22×C4#12112yes—
291C22×C4#12112yes—
292C22×C4#12412yes—
293C22×C4#12712yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297M16—48no—
298M16—48no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—44no—
303C2×C8—48no—
304C2×C8—44no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—44no—
308C2×C8—42no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—48no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—84no—
333C22⋊C4—44no—
334C22⋊C4—48no—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—84no—
344C42—44no—
345C42—42no—
346C42—42no—
347C42—44no—
348C42—44no—
349C42—42no—
350C42C4×D412yes—
351C42C4×D412yes—
352C42#12112yes—
353C42#12112yes—
354C42#12412yes—
355C42#12712yes—
356C42C22×D411yescharacteristic
357C42C22×D411yescharacteristic
358C42C2×#4111yescharacteristic
359C42C2×#4111yescharacteristic
360C23—21no—
361C23—21no—
362C23—22no—
363C23—22no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—41no—
369C23#39012yes—
370C23C22×C22⋊C411yescharacteristic
371Q8—81no—
372Q8—81no—
373Q8—84no—
374Q8—84no—
375D4—81no—
376D4—81no—
377D4—84no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—84no—
409C2×C4#39012yes—
410C2×C4#39012yes—
411C2×C4#39012yes—
412C2×C4#42114yes—
413C2×C4C2×C4×D411yescharacteristic
414C2×C4C2×C4×D411yescharacteristic
415C2×C4C2×#12111yescharacteristic
416C2×C4C2×#12111yescharacteristic
417C2×C4C2×#12411yescharacteristic
418C2×C4C2×#12711yescharacteristic
419C8—44no—
420C8—44no—
421C8—44no—
422C8—82no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—42no—
427C22—42no—
428C22—81no—
429C22—82no—
430C22—82no—
431C22—81no—
432C22—81no—
433C22C2×#39011yescharacteristic
434C22#288911yescharacteristic
435C22#288911yescharacteristic
436C22C2×#42112yes—
437C22#290112yes—
438C4—21no—
439C4—21no—
440C4—22no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—22no—
446C4—44no—
447C4—44no—
448C4—81no—
449C4—82no—
450C4—81no—
451C4—82no—
452C4—81no—
453C4#196314yes—
454C2—42no—
455C2—161no—
456C2#1768212yes—
457C2#1768312yes—
458C2C2×#196311yescharacteristic
459C2#2122011yescharacteristic
460C2#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. #20367C2semidirect
4. C2×#1531C2semidirect
6. C2×#1526C2semidirect
8. #9690C2semidirect
9. #9688C2semidirect

Maximal subgroups

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

subgroupindex
#352012
#213562
#203672
C2×#15312
C2×#15292
C2×#15262
C2×#15242
#96902
#96882