Finite Group Databasebeta

← #160888

Subgroups — #160888

Order 512.

1002 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
1C4×#533C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3C2×#1767C211yesmaximal, characteristic
4#7630C214yesmaximal, complemented
5#7625C212yesmaximal, complemented
6#7624C212yesmaximal, complemented
7#7589C214yesmaximal
8C2×#533C411yescharacteristic
9C22×C4⋊C8—21no—
10C22×#97—21no—
11C4×C4⋊Q8C411yescharacteristic
12C4×C4⋊Q8C411yescharacteristic
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⋊Q8C2212yes—
21#2296—21no—
22#2295—21no—
23#2295C2212yes—
24#2285C411yescharacteristic
25C42×Q8—21no—
26C42×Q8C412yes—
27C4×#117—21no—
28C2×C43C2211yescharacteristic
29#1839C2212yes—
30#1831—21no—
31#1817—21no—
32#1797C2212yes—
33#1767—21no—
34#1767—21no—
35#1767—21no—
36#1767C2212yes—
37#1767C2212yes—
38#1749—21no—
39#1745C2212yes—
40C2×#336C2212yes—
41#1272C2218yes—
42#1271C2218yes—
43C23×C8—22no—
44#533—28no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2×C412yes—
56C2×#117—21no—
57C2×#117—21no—
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C4⋊C8—28no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—22no—
71C2×#97—22no—
72C2×#97—22no—
73C2×#97—44no—
74#391—21no—
75#391—21no—
76#391C2×C412yes—
77#391C2312yes—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389C2×C412yes—
85#389C2312yes—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#385—22no—
91#385—22no—
92#384C2×C411yescharacteristic
93#384C2×C411yescharacteristic
94#382—21no—
95#380—21no—
96#380C2×C412yes—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115#376—22no—
116C43—22no—
117C43C2×C411yescharacteristic
118C43C2×C411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43C2311yescharacteristic
124C43C2311yescharacteristic
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336C2314yes—
130#336C2314yes—
131C22×Q8—21no—
132C22×Q8C22×C412yes—
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C22×C8—28no—
141C22×C8—24no—
142C22×C8—24no—
143C22×C8—44no—
144C22×C8—44no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147#125—28no—
148C22⋊Q8—28no—
149C22⋊Q8—24no—
150C22⋊Q8M1618yes—
151C4×Q8—28no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—44no—
155C4×Q8—44no—
156C4×Q8—44no—
157C4×Q8—42no—
158C4×Q8—42no—
159#117—28no—
160#117—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C42—24no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2411yescharacteristic
221C4⋊C8—28no—
222C4⋊C8—28no—
223C4⋊C8—44no—
224C22⋊C8—44no—
225#97—44no—
226#97—44no—
227#97—44no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238C24C2×C22⋊C411yescharacteristic
239C2×Q8—24no—
240C2×Q8—22no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8C2×M1614yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4#10414yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
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×M1614yes—
282C22×C4#13714yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C2×C8—28no—
287C2×C8—416no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—44no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—28no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4C2×M1614yes—
311C22⋊C4—28no—
312C22⋊C4—44no—
313C42—28no—
314C42—22no—
315C42—22no—
316C42—21no—
317C42—22no—
318C42—22no—
319C42—21no—
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—22no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—44no—
343C42—44no—
344C42—44no—
345C42—44no—
346C42—44no—
347C42AGL(1,ℤ/8)12yes—
348C42#13712yes—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23#40712yes—
354C23#41912yes—
355C23C2×#10412yes—
356C23C22×C22⋊C411yescharacteristic
357Q8—44no—
358Q8—48no—
359Q8—44no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—28no—
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—21no—
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—28no—
388C2×C4—21no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4#40712yes—
411C2×C4#41012yes—
412C2×C4#41312yes—
413C2×C4#41712yes—
414C2×C4#41712yes—
415C2×C4#41912yes—
416C2×C4C2×#10412yes—
417C2×C4C2×#10412yes—
418C2×C4#42112yes—
419C2×C4C22×M1612yes—
420C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
421C2×C4C2×#13712yes—
422C2×C4C2×#13711yescharacteristic
423C8—48no—
424C8—84no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22#149114yes—
432C22C2×#40711yescharacteristic
433C22C2×#41011yescharacteristic
434C22C2×#41311yescharacteristic
435C22C2×#41911yescharacteristic
436C22C2×#41711yescharacteristic
437C22C22×#10411yescharacteristic
438C22C2×#42111yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—21no—
444C4—21no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—41no—
453C4—44no—
454C4—41no—
455C4—44no—
456C4—82no—
457C4—82no—
458C2—24no—
459C2#1736311yescharacteristic
460C2C2×#149111yescharacteristic
461C2#1745111yescharacteristic
462C2#1754611yescharacteristic
463C2#1759611yescharacteristic
464C2#1764611yescharacteristic
465C2#1774911yescharacteristic
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 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7630C2semidirect
5. #7625C2semidirect
6. #7624C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
#207782
C2×#17672
#76302
#76252
#76242
#75892