Finite Group Databasebeta

← #163966

Subgroups — #163966

Order 512.

1090 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#34268C211yesmaximal, characteristic
2#22471C212yesmaximal
3#7648C212yesmaximal, complemented
4#7634C212yesmaximal, complemented
5#7623C212yesmaximal, complemented
6#7617C212yesmaximal, complemented
7#7595C211yesmaximal, characteristic, complemented
8#7579C212yesmaximal, complemented
9C2×#1272C211yesmaximal, characteristic, complemented
10C2×#536C411yescharacteristic
11C2×#519C411yescharacteristic
12C2×C4×#41C2211yescharacteristic
13C2×#435—22no—
14#2367—22no—
15#2365—22no—
16#2359—22no—
17#2359—22no—
18#2323C412yes—
19#2323C412yes—
20#2311C411yescharacteristic
21C4×C22⋊Q8C2212yes—
22C4×C22⋊Q8C2212yes—
23#2296C2211yescharacteristic
24C2×#391C2211yescharacteristic
25#2274C411yescharacteristic
26#1985—22no—
27#1870—22no—
28#1869—22no—
29#1817C2212yes—
30#1817C2212yes—
31#1797C2212yes—
32#1749C2212yes—
33#1728C2212yes—
34C2×#336C2212yes—
35#1272C2212yes—
36#1272C2211yescharacteristic
37#1272C2211yescharacteristic
38#1272C2211yescharacteristic
39#1272C2211yescharacteristic
40#1272C2212yes—
41#1271C2212yes—
42#1271C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45C22×#41C2×C411yescharacteristic
46C23×Q8C2×C411yescharacteristic
47C22×M16—22no—
48C23×C8—22no—
49#536—24no—
50#536—24no—
51C2×C4⋊Q8—21no—
52C2×#124—21no—
53C2×#123—22no—
54C2×C22⋊Q8C2×C412yes—
55C2×C22⋊Q8C2×C412yes—
56#519—44no—
57C4×#41—44no—
58C2×C4×Q8—22no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8D412yes—
62C2×C4×D4D412yes—
63C2×C4×D4D412yes—
64C2×C4×D4D412yes—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C412yes—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#435—48no—
75C2×C4⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×#97—22no—
81#391—21no—
82#391—22no—
83#391C2×C411yescharacteristic
84#391C2×C411yescharacteristic
85#391C2311yescharacteristic
86#391C2311yescharacteristic
87#390—21no—
88#389—22no—
89#389—22no—
90#389C2×C412yes—
91#389C2312yes—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#386—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#380C2×C412yes—
105#380C2×C412yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116C2×#95—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336C2312yes—
122#336C2312yes—
123#336C2312yes—
124#336C2312yes—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41—42no—
128C2×#41—42no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8C22⋊C411yescharacteristic
134C22×Q8M1614yescomplemented
135C22×Q8C22×C411yescharacteristic
136C22×Q8C22×C411yescharacteristic
137C22×D4C22⋊C412yes—
138C22×D4C22⋊C411yescharacteristic
139C23×C4—21no—
140C23×C4—21no—
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C2×M16—44no—
147C2×M16—44no—
148C2×M16—82no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—44no—
154C22×C8—44no—
155C22×C8—82no—
156C4⋊Q8—28no—
157#124—28no—
158#123—216no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C4×Q8—44no—
164C4×Q8—44no—
165C4×Q8—44no—
166C4×Q8—48no—
167C4×D4—28no—
168C4×D4—44no—
169C4×D4—44no—
170#117—22no—
171#117—24no—
172#117—44no—
173#117—44no—
174#117—44no—
175#117—42no—
176#117M1614yescomplemented
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C412yes—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C22⋊C411yescharacteristic
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C411yescharacteristic
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—44no—
209C2×C42—42no—
210C2×C42C22×C412yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—44no—
217C22⋊C8—44no—
218C22⋊C8—44no—
219C22⋊C8—44no—
220C22⋊C8—82no—
221#97—44no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232C24C2×C22⋊C412yes—
233C24C2×C22⋊C411yescharacteristic
234#41—48no—
235#41—48no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—24no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—48no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—48no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8C2×M1612yes—
251C2×Q8C2×M1612yes—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—22no—
256C2×D4—41no—
257C2×D4—41no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
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×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—82no—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×M1612yes—
293C22×C4C2×M1612yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D412yes—
296M16—88no—
297C2×C8—24no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—88no—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—82no—
325C4⋊C4C2×M1614yes—
326C22⋊C4—28no—
327C22⋊C4—28no—
328C22⋊C4—24no—
329C22⋊C4—22no—
330C22⋊C4—24no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4#9914yes—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—41no—
340C42—41no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—44no—
345C42—44no—
346C42—82no—
347C42—82no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23#40712yes—
358C23C2×#9912yes—
359C23#41312yes—
360C23C22×C22⋊C411yescharacteristic
361Q8—42no—
362Q8—42no—
363Q8—42no—
364Q8—42no—
365Q8—48no—
366Q8—44no—
367D4—44no—
368D4—48no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—48no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—162no—
414C2×C4#40712yes—
415C2×C4C2×#9914yes—
416C2×C4C22×M1611yescharacteristic
417C2×C4C22×M1611yescharacteristic
418C2×C4#57712yes—
419C2×C4#57812yes—
420C8—44no—
421C8—82no—
422C8—82no—
423C8—82no—
424C22—22no—
425C22—24no—
426C22—22no—
427C22—24no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22#146514yes—
436C22C2×#40711yescharacteristic
437C22C22×#9911yescharacteristic
438C22C2×#41311yescharacteristic
439C22#289812yes—
440C22#290112yes—
441C4—22no—
442C4—22no—
443C4—24no—
444C4—21no—
445C4—21no—
446C4—22no—
447C4—44no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—81no—
454C4—81no—
455C4—82no—
456C2—24no—
457C2—82no—
458C2C2×#146511yescharacteristic
459C2#1745412yes—
460C2#1760812yes—
461C2#1768611yescharacteristic
462C2#1778011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7648C2semidirect
4. #7634C2semidirect
5. #7623C2semidirect
6. #7617C2semidirect
7. #7595C2semidirect
8. #7579C2semidirect
9. C2×#1272C2semidirect
134. C22×Q8M16semidirect
176. #117M16semidirect

Maximal subgroups

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

subgroupindex
#342682
#224712
#76482
#76342
#76232
#76172
#75952
#75792
C2×#12722