Finite Group Databasebeta

← #219931

Subgroups — #219931

Order 512.

1082 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36131C211yesmaximal, characteristic
2#20491C212yesmaximal
3C2×#1526C211yesmaximal, characteristic, complemented
4C22×#330C211yesmaximal, characteristic, complemented
5#9686C212yesmaximal, complemented
6#9122C212yesmaximal, complemented
7#9121C212yesmaximal, complemented
8#8584C211yesmaximal, characteristic, complemented
9#8338C211yesmaximal, characteristic, complemented
10#8335C211yesmaximal, characteristic, complemented
11C4×#330C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C2×#435—22no—
15C2×C4×M16C2212yes—
16C22×#97C2212yes—
17#2859C2211yescharacteristic
18#2851C2211yescharacteristic
19#2851C2211yescharacteristic
20#2556—22no—
21#2550C412yes—
22#2462—22no—
23#2449—22no—
24#2441C412yes—
25#2440C412yes—
26#2360—22no—
27#2352C412yes—
28C4×#125C2211yescharacteristic
29C42×D4C2211yescharacteristic
30#1771—22no—
31#1767C2212yes—
32#1753—22no—
33C8×C22⋊C4—22no—
34#1749C2212yes—
35#1749C2212yes—
36C4×#97C2212yes—
37#1526—24no—
38C2×#330—24no—
39C2×#330C2211yescharacteristic
40C2×#330C2211yescharacteristic
41C2×#330C2211yescharacteristic
42C2×#330C2212yes—
43C2×#330C2212yes—
44C2×#330C2212yes—
45C2×#330C2212yes—
46C2×#330C2211yescharacteristic
47#1295C2212yes—
48#1295C2212yes—
49C22×M16—22no—
50C23×C8—22no—
51#533—22no—
52#533—22no—
53#533—42no—
54C2×C4⋊Q8C2×C412yes—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—42no—
59C2×#125C2×C412yes—
60C2×#125C2311yescharacteristic
61C2×#125C2311yescharacteristic
62C2×#125C2311yescharacteristic
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D412yes—
68C2×#117C2×C412yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2×C412yes—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#435—216no—
74C2×C4⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C4×M16—24no—
78C4×M16—42no—
79C2×#97—22no—
80C2×#97—24no—
81C2×#97—22no—
82C2×#97—42no—
83C2×#97C2312yes—
84C2×#97C2312yes—
85C2×#97C2312yes—
86C2×#97C2312yes—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89#399—22no—
90#398—22no—
91#398—22no—
92#396—22no—
93#395—22no—
94#395—22no—
95#391—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#388—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#386—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#384C2×C412yes—
114#382C2×C412yes—
115#382C2311yescharacteristic
116#380C2×C412yes—
117#380C2×C412yes—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D412yes—
122C4×C22⋊C4—22no—
123C4×C22⋊C4C2×C411yescharacteristic
124C4×C22⋊C4C2×C411yescharacteristic
125C4×C22⋊C4D412yes—
126C4×C22⋊C4D412yes—
127C43C2311yescharacteristic
128#336—22no—
129#336—22no—
130#336—22no—
131#330—22no—
132#330—22no—
133#330—22no—
134#330—24no—
135#330—24no—
136#330—42no—
137#330—42no—
138#330D414yescomplemented
139C22×Q8—22no—
140C22×D4C22⋊C411yescharacteristic
141C23×C4—22no—
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C2×M16—28no—
147C2×M16—44no—
148C2×M16—44no—
149C22×C8—22no—
150C22×C8—28no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—44no—
155C22×C8—44no—
156C4⋊Q8—24no—
157C4⋊Q8—42no—
158#125—24no—
159#125—21no—
160#125—21no—
161#125—21no—
162#125—21no—
163#125—22no—
164#125—22no—
165#125—22no—
166#125—42no—
167#125—44no—
168#125—44no—
169#125C2×D412yes—
170#125#4112yes—
171C22⋊Q8—44no—
172C22⋊Q8—44no—
173C22⋊Q8—48no—
174C4×Q8—44no—
175C4×Q8—44no—
176C4×D4—24no—
177C4×D4—28no—
178#117—24no—
179#117—42no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4C22⋊C411yescharacteristic
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C411yescharacteristic
208C2×C22⋊C4C22⋊C412yes—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2411yescharacteristic
226C4⋊C8—28no—
227C22⋊C8—28no—
228C22⋊C8—28no—
229#97—28no—
230#97—22no—
231#97—22no—
232#97—22no—
233#97—22no—
234#97—22no—
235#97—22no—
236#97—22no—
237#97—44no—
238#97C2×D414yes—
239C4×C8—44no—
240C4×C8—44no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253C24C2×C22⋊C411yescharacteristic
254C24C2×C22⋊C411yescharacteristic
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—82no—
258C2×D4—22no—
259C2×D4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—82no—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D412yes—
292M16—416no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—416no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4C4×D414yes—
330C22⋊C4—22no—
331C22⋊C4—24no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—82no—
335C22⋊C4#10114yes—
336C22⋊C4#10418yes—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—24no—
347C42—42no—
348C42—41no—
349C42—44no—
350C42—82no—
351C42C22×D411yescharacteristic
352C42C2×#4111yescharacteristic
353C42H(5,𝔽2)12yes—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—22no—
361C23—22no—
362C23C2×#10112yes—
363C23C2×#10414yes—
364C23C22×C22⋊C411yescharacteristic
365Q8—88no—
366D4—44no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—88no—
407C2×C4C2×#10112yes—
408C2×C4C2×#10112yes—
409C2×C4C2×#10414yes—
410C2×C4C2×#10414yes—
411C2×C4C2×C4×D411yescharacteristic
412C2×C4#51811yescharacteristic
413C2×C4D4212yes—
414C2×C4#54612yes—
415C8—28no—
416C8—44no—
417C8—44no—
418C8—44no—
419C22—22no—
420C22—22no—
421C22—24no—
422C22—24no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—41no—
427C22—42no—
428C22#152614yescomplemented
429C22D4×C22⋊C411yescharacteristic
430C22C22×#10111yescharacteristic
431C22#288911yescharacteristic
432C22C22×#10412yes—
433C22#290112yes—
434C4—22no—
435C4—21no—
436C4—21no—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—24no—
441C4—24no—
442C4—41no—
443C4—42no—
444C4—44no—
445C4—41no—
446C4—82no—
447C4—82no—
448C4—82no—
449C2—24no—
450C2—81no—
451C2C2×#152611yescharacteristic
452C2#1765312yes—
453C2#1768911yescharacteristic
454C2D4×#10111yescharacteristic
455C2D4×#10412yes—
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 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1526C2semidirect
4. C22×#330C2semidirect
5. #9686C2semidirect
6. #9122C2semidirect
7. #9121C2semidirect
8. #8584C2semidirect
9. #8338C2semidirect
10. #8335C2semidirect
11. C4×#330C2semidirect
138. #330D4semidirect
428. C22#1526semidirect

Maximal subgroups

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

subgroupindex
#361312
#204912
C2×#15262
C22×#3302
#96862
#91222
#91212
#85842
#83382
#83352
C4×#3302