Finite Group Databasebeta

← #190712

Subgroups — #190712

Order 512.

936 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3#20777C211yesmaximal, characteristic
4#8726C212yesmaximal
5#8594C212yesmaximal, complemented
6#8593C212yesmaximal, complemented
7#8343C212yesmaximal
8#8237C212yesmaximal, complemented
9#8235C212yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C4×C4⋊Q8C2211yescharacteristic
12C4×C4⋊Q8C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15C4×#125—21no—
16C4×#125—21no—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8—21no—
19C4×C22⋊Q8C412yes—
20C4×C22⋊Q8C412yes—
21#2296—21no—
22#2295—21no—
23#2295C412yes—
24#2285C2211yescharacteristic
25C42×Q8—21no—
26C42×Q8C412yes—
27C4×#117—21no—
28C2×C43C2211yescharacteristic
29#1839—21no—
30#1839—21no—
31#1831C2212yes—
32#1831C2212yes—
33#1817—21no—
34#1817—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39#1749—21no—
40C4×C22⋊C8—21no—
41#1745—21no—
42#1744—21no—
43C2×#333C2214yes—
44C2×#332C2214yes—
45#1296C2214yes—
46#1296C2214yes—
47#533—24no—
48#533—24no—
49C2×C4⋊Q8C2311yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8C2×C412yes—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C412yes—
60C2×#117—21no—
61C2×#117—21no—
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8C2314yes—
70C2×C4⋊C8C2314yes—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—22no—
74C2×C4×C8—22no—
75#391—21no—
76#391—21no—
77#391C2×C412yes—
78#391C2×C412yes—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389C2×C412yes—
86#389C2×C412yes—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#385—22no—
92#385—22no—
93#384C2311yescharacteristic
94#384C2311yescharacteristic
95#382—21no—
96#380—21no—
97#380C2×C412yes—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116#376—22no—
117C43—22no—
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43C2311yescharacteristic
125C43C2311yescharacteristic
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#333—28no—
135#332—28no—
136C22×Q8—21no—
137C22×Q8C22×C412yes—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C22×C8—24no—
146C22×C8—24no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8#4114yes—
150C4⋊Q8#4114yes—
151#125—28no—
152C22⋊Q8—28no—
153C22⋊Q8—24no—
154C22⋊Q8—24no—
155C4×Q8—28no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—44no—
159C4×Q8—44no—
160C4×Q8—44no—
161C4×Q8—44no—
162#117—28no—
163#117—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42—22no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224C4⋊C8—24no—
225C4⋊C8—24no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228C22⋊C8—44no—
229C22⋊C8—44no—
230#97—44no—
231C4×C8—44no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242C24C2×C22⋊C411yescharacteristic
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8#11714yes—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4#13514yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—44no—
295C2×C8—44no—
296C4⋊C4—28no—
297C4⋊C4—28no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4#11714yes—
313C22⋊C4—28no—
314C22⋊C4—44no—
315C42—28no—
316C42—21no—
317C42—21no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—44no—
344C42—44no—
345C42—44no—
346C42—44no—
347C42—44no—
348C42#13714yes—
349C42C2×#4112yes—
350C42C2×#4112yes—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23#41612yes—
358C23C22×C22⋊C411yescharacteristic
359Q8—44no—
360Q8—44no—
361Q8—48no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—28no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
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—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4#41312yes—
413C2×C4#41312yes—
414C2×C4#41612yes—
415C2×C4#41912yes—
416C2×C4#41912yes—
417C2×C4C2×#11711yescharacteristic
418C2×C4C2×#11711yescharacteristic
419C2×C4#52512yes—
420C2×C4#52512yes—
421C2×C4C2×#13512yes—
422C2×C4C2×#13712yes—
423C2×C4#57512yes—
424C2×C4#57512yes—
425C8—84no—
426C8—84no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22#228011yescharacteristic
435C22C2×#41311yescharacteristic
436C22C2×#41311yescharacteristic
437C22C2×#41611yescharacteristic
438C22C2×#41911yescharacteristic
439C22#289611yescharacteristic
440C22#289611yescharacteristic
441C4—22no—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—21no—
446C4—21no—
447C4—22no—
448C4—41no—
449C4—44no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—44no—
456C4—82no—
457C4—82no—
458C4#310814yes—
459C4#311114yes—
460C2—42no—
461C2#1771311yescharacteristic
462C2#1771611yescharacteristic
463C2#1774711yescharacteristic
464C2#1774811yescharacteristic
465C2#2072811yescharacteristic
466C2#2073511yescharacteristic
467C2#2074211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8594C2semidirect
6. #8593C2semidirect
8. #8237C2semidirect
9. #8235C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
#207782
#207772
#87262
#85942
#85932
#83432
#82372
#82352