Finite Group Databasebeta

← #164253

Subgroups — #164253

Order 512.

777 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36480C211yesmaximal, characteristic
2#22636C211yesmaximal, characteristic
3#20368C211yesmaximal, characteristic
4#7709C212yesmaximal, complemented
5#7693C212yesmaximal, complemented
6#7688C211yesmaximal, characteristic
7#7665C211yesmaximal, characteristic
8#7638C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic
10#7588C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic
12C2×#405—21no—
13#2863C2212yes—
14#2857C2211yescharacteristic
15#2605—22no—
16#2567C411yescharacteristic
17#2551C411yescharacteristic
18#2509—21no—
19#2497C412yes—
20#2492C412yes—
21#2465—22no—
22#2439—21no—
23#2369C411yescharacteristic
24#2365—21no—
25#2355C411yescharacteristic
26#2347—21no—
27C4×C4⋊Q8C2212yes—
28C4×C22⋊Q8C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1939—21no—
31#1924—22no—
32#1831C2212yes—
33#1797C2211yescharacteristic
34#1770—21no—
35C4×C22⋊C8C2211yescharacteristic
36#1746—22no—
37#1745C2212yes—
38#1745C2212yes—
39#1744C2212yes—
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C22×M16—21no—
54C23×C8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C412yes—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×C4×Q8—22no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×#117D411yescharacteristic
63C2×#117D411yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69#405—28no—
70C2×#97—22no—
71C2×#97—22no—
72C2×C4×C8—22no—
73C2×C4×C8—21no—
74#400—22no—
75#400—22no—
76#396—22no—
77#395—22no—
78#395—22no—
79#393—22no—
80#391—22no—
81#391—21no—
82#391—22no—
83#391—22no—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2311yescharacteristic
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389—21no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2311yescharacteristic
96#389C2312yes—
97#388—22no—
98#388—22no—
99#387—21no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—21no—
106#384C2×C412yes—
107#384C2×C411yescharacteristic
108#383—22no—
109#382—21no—
110#380—21no—
111#380C2×C412yes—
112#380C2×C411yescharacteristic
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4C2×C412yes—
126C4×C22⋊C4D412yes—
127#376D412yes—
128C43C2312yes—
129#336—22no—
130#336—22no—
131#336—21no—
132#336—21no—
133#336—22no—
134#336—22no—
135#336C2312yes—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141C22×Q8—22no—
142C22×Q8—21no—
143C22×Q8C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—42no—
150C22×C8—24no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—42no—
158C22×C8—42no—
159C4⋊Q8—24no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—44no—
163C22⋊Q8—22no—
164C22⋊Q8—41no—
165C22⋊Q8—81no—
166C22⋊Q8—81no—
167C4×Q8—42no—
168C4×Q8—82no—
169C4×Q8—82no—
170#117—22no—
171#117—22no—
172#117—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—42no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C412yes—
206C2×C42C22×C412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2411yescharacteristic
214C4⋊C8—42no—
215C4⋊C8—82no—
216C22⋊C8—42no—
217C22⋊C8—81no—
218#97—28no—
219#97—44no—
220C4×C8—28no—
221C4×C8—42no—
222#95—21no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—21no—
235#95—22no—
236#95—21no—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241C24C2×C22⋊C411yescharacteristic
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—44no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—44no—
249C2×Q8—82no—
250C2×Q8—81no—
251C2×Q8—81no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4#13712yes—
276C22×C4#13712yes—
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280M16—48no—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—48no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—41no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—81no—
323C4⋊C4#10414yes—
324C4⋊C4#13112yes—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—42no—
328C22⋊C4—81no—
329C22⋊C4—82no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—24no—
339C42—42no—
340C42—41no—
341C42—41no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—82no—
346C42—82no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23#41912yes—
354C23C22×C22⋊C411yescharacteristic
355Q8—88no—
356Q8—81no—
357Q8—81no—
358Q8—162no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4#41012yes—
407C2×C4#41912yes—
408C2×C4C2×#10414yes—
409C2×C4C2×#10412yes—
410C2×C4C2×#13111yescharacteristic
411C2×C4#56811yescharacteristic
412C2×C4C2×#13711yescharacteristic
413C2×C4C2×#13711yescharacteristic
414C2×C4#57611yescharacteristic
415C2×C4#57711yescharacteristic
416C8—42no—
417C8—42no—
418C8—42no—
419C8—44no—
420C8—81no—
421C8—82no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22#288011yescharacteristic
430C22C2×#41011yescharacteristic
431C22#288911yescharacteristic
432C22C2×#41911yescharacteristic
433C22#289711yescharacteristic
434C22C22×#10411yescharacteristic
435C22#290111yescharacteristic
436C4—21no—
437C4—21no—
438C4—24no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—41no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—41no—
448C4—42no—
449C4—81no—
450C4—82no—
451C4—81no—
452C4—82no—
453C4—81no—
454C4#150614yes—
455C2—42no—
456C2#1742011yescharacteristic
457C2#1751411yescharacteristic
458C2#1751711yescharacteristic
459C2C2×#150611yescharacteristic
460C2#1761011yescharacteristic
461C2#1765411yescharacteristic
462C2#1776111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7709C2semidirect
5. #7693C2semidirect
8. #7638C2semidirect
10. #7588C2semidirect

Maximal subgroups

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

subgroupindex
#364802
#226362
#203682
#77092
#76932
#76882
#76652
#76382
#76172
#75882
#75662