Finite Group Databasebeta

← #160913

Subgroups — #160913

Order 512.

930 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34324C211yesmaximal, characteristic
2#22700C211yesmaximal, characteristic
3#20568C211yesmaximal, characteristic
4#7634C212yesmaximal, complemented
5#7632C212yesmaximal, complemented
6#7617C212yesmaximal, complemented
7#7613C212yesmaximal, complemented
8#7588C211yesmaximal, characteristic, complemented
9#7577C211yesmaximal, characteristic, complemented
10#7570C211yesmaximal, characteristic, complemented
11#7562C211yesmaximal, characteristic, complemented
12C2×#556C411yescharacteristic
13C2×#436—21no—
14C2×#436—21no—
15#2373—22no—
16#2367—21no—
17#2365C411yescharacteristic
18#2362C412yes—
19#2355—21no—
20#2354C411yescharacteristic
21Q8×C4⋊C4C411yescharacteristic
22#2346C412yes—
23#2334—22no—
24#2326—21no—
25C4×C4⋊Q8C2211yescharacteristic
26C4×C22⋊Q8C2212yes—
27#2296C2212yes—
28#2295C2211yescharacteristic
29#2276—21no—
30C42×D4C2211yescharacteristic
31#1939—21no—
32#1851—22no—
33#1831C2211yescharacteristic
34#1797C2212yes—
35#1770—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1753—22no—
39#1749C2212yes—
40#1744C2211yescharacteristic
41#1728C2212yes—
42#1728C2212yes—
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53C22×M16—22no—
54C22×M16—22no—
55#556—24no—
56#556—24no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#126—22no—
59C2×#124—21no—
60C2×C22⋊Q8C2×C412yes—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C412yes—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×#117C2×C412yes—
70C2×#117C2×C412yes—
71C22×C42C2312yes—
72#436—28no—
73#436—44no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×#97—22no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#391—21no—
85#391—21no—
86#391C2×C411yescharacteristic
87#391C2311yescharacteristic
88#390—21no—
89#389—21no—
90#389—22no—
91#389C2×C411yescharacteristic
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2312yes—
95#389C2311yescharacteristic
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#387—21no—
102#387—22no—
103#387—21no—
104#387—22no—
105#387—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#384—21no—
110#384C2×C411yescharacteristic
111#382—22no—
112#380C2×C412yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—22no—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
131#376—21no—
132#376—22no—
133C43C2311yescharacteristic
134#336—21no—
135#336—22no—
136#336—22no—
137#336—21no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2311yescharacteristic
144C22×Q8—21no—
145C22×Q8C22×C411yescharacteristic
146C22×D4C22⋊C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C2×M16—28no—
151C2×M16—44no—
152C2×M16—44no—
153C2×M16—82no—
154C2×M16—82no—
155C2×M16—82no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C4⋊Q8—22no—
163C4⋊Q8M1614yescomplemented
164#126—216no—
165#124—28no—
166C22⋊Q8—24no—
167C22⋊Q8—24no—
168C4×Q8—28no—
169C4×Q8—22no—
170C4×Q8—44no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—82no—
174C4×Q8M1614yescomplemented
175C4×D4—24no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180#117—24no—
181#117—24no—
182#117—82no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—82no—
217C2×C42C22×C412yes—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2411yescharacteristic
227C4⋊C8—24no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8—82no—
231C22⋊C8—28no—
232C22⋊C8—44no—
233#97—82no—
234C4×C8—24no—
235C4×C8—42no—
236C4×C8—81no—
237#95—21no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—22no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—22no—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253C24C2×C22⋊C412yes—
254C2×Q8—21no—
255C2×Q8—24no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8C2×M1612yes—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292M16—88no—
293M16—88no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—24no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—42no—
312C4⋊C4—28no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—81no—
329C4⋊C4—81no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4#10414yes—
333C4⋊C4C2×M1612yes—
334C4⋊C4C2×M1614yes—
335C22⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C42—28no—
343C42—21no—
344C42—21no—
345C42—24no—
346C42—22no—
347C42—42no—
348C42—41no—
349C42—42no—
350C42—41no—
351C42—42no—
352C42—41no—
353C42—44no—
354C42—44no—
355C42—42no—
356C42—82no—
357C42—82no—
358C42C2×M1612yes—
359C42C2×M1612yes—
360C42AGL(1,ℤ/8)12yes—
361C42#13712yes—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23C22×C22⋊C411yescharacteristic
370Q8—42no—
371Q8—42no—
372Q8—48no—
373D4—44no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4#40712yes—
420C2×C4#40712yes—
421C2×C4#41712yes—
422C2×C4#41712yes—
423C2×C4C2×#10414yes—
424C2×C4C2×#10412yes—
425C2×C4C22×M1611yescharacteristic
426C2×C4C22×M1611yescharacteristic
427C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
428C2×C4C2×#13711yescharacteristic
429C2×C4#57711yescharacteristic
430C2×C4#57811yescharacteristic
431C8—42no—
432C8—44no—
433C8—42no—
434C8—42no—
435C8—81no—
436C8—82no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22C2×#40711yescharacteristic
445C22#288311yescharacteristic
446C22#288911yescharacteristic
447C22C2×#41711yescharacteristic
448C22#289811yescharacteristic
449C22C22×#10411yescharacteristic
450C22#290111yescharacteristic
451C4—22no—
452C4—22no—
453C4—21no—
454C4—21no—
455C4—24no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—44no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—41no—
466C4—81no—
467C4—82no—
468C4—81no—
469C4#148914yes—
470C2—82no—
471C2#1737011yescharacteristic
472C2C2×#148911yescharacteristic
473C2#1745411yescharacteristic
474C2#1754811yescharacteristic
475C2#1763011yescharacteristic
476C2#1765411yescharacteristic
477C2#1776311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7634C2semidirect
5. #7632C2semidirect
6. #7617C2semidirect
7. #7613C2semidirect
8. #7588C2semidirect
9. #7577C2semidirect
10. #7570C2semidirect
11. #7562C2semidirect
163. C4⋊Q8M16semidirect
174. C4×Q8M16semidirect

Maximal subgroups

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

subgroupindex
#343242
#227002
#205682
#76342
#76322
#76172
#76132
#75882
#75772
#75702
#75622