Finite Group Databasebeta

← #215641

Subgroups — #215641

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2569C211yesmaximal, characteristic
2#21904C212yesmaximal
3#9881C212yesmaximal, complemented
4#9881C212yesmaximal, complemented
5#9880C212yesmaximal, complemented
6#9880C212yesmaximal, complemented
7#8704C211yesmaximal, characteristic
8#8704C211yesmaximal, characteristic
9#8698C212yesmaximal
10#2569—22no—
11#2569—22no—
12#2569—22no—
13#2569—22no—
14C2×#400C412yes—
15C2×#398C412yes—
16C2×#387C2212yes—
17C2×#387C2212yes—
18C2×#385C412yes—
19C2×#380C412yes—
20C2×#380C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C22×#95C2211yescharacteristic
23#1924—22no—
24#1924—22no—
25#1923—22no—
26#1923—22no—
27#1797C2212yes—
28#1797C2212yes—
29#1779C2212yes—
30#1779C2212yes—
31#1749C2212yes—
32C4×C22⋊C8C2212yes—
33#1280C2211yescharacteristic
34#1280C2211yescharacteristic
35#1280C2212yes—
36#1280C2212yes—
37#1280C2211yescharacteristic
38#1280C2211yescharacteristic
39#1280C2212yes—
40#1280C2212yes—
41#1280C2212yes—
42#1280C2212yes—
43C24×C4C2×C412yes—
44C24×C4C2311yescharacteristic
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C412yes—
50C22×C22⋊C4C2×C412yes—
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C42C2312yes—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8C2312yes—
61C2×C22⋊C8C2312yes—
62C2×C22⋊C8C2312yes—
63C2×C22⋊C8C2312yes—
64C2×#97—22no—
65C2×C4×C8—22no—
66#400—22no—
67#400—22no—
68#400—22no—
69#400—22no—
70#398—22no—
71#398—22no—
72#398—22no—
73#398—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#385—22no—
83#385—22no—
84#385—22no—
85#385—22no—
86#380—22no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92C4×C22⋊C4D412yes—
93C4×C22⋊C4D412yes—
94C4×C22⋊C4D412yes—
95C4×C22⋊C4D412yes—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C412yes—
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105C2×#95C2312yes—
106C2×#95C2311yescharacteristic
107C2×#95C2311yescharacteristic
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112C25C22×C411yescharacteristic
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—42no—
120C23×C4—44no—
121C23×C4—82no—
122C23×C4—82no—
123C23×C4C22×C412yes—
124C23×C4C22×C412yes—
125C23×C4C22×C412yes—
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4C22⋊C412yes—
160C2×C22⋊C4C22⋊C412yes—
161C2×C22⋊C4C22⋊C411yescharacteristic
162C2×C22⋊C4C22⋊C411yescharacteristic
163C2×C22⋊C4C22⋊C411yescharacteristic
164C2×C22⋊C4C22⋊C411yescharacteristic
165C2×C42—82no—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C4⋊C8—82no—
171C4⋊C8—82no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—82no—
181C22⋊C8—82no—
182C22⋊C8—82no—
183C22⋊C8—82no—
184#97—82no—
185C4×C8—82no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95C22⋊C412yes—
201#95C22⋊C412yes—
202#95C22⋊C412yes—
203#95C22⋊C412yes—
204C24—21no—
205C24—21no—
206C24—22no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—42no—
211C24C2×C22⋊C411yescharacteristic
212C24C2×C22⋊C411yescharacteristic
213C24C23×C411yescharacteristic
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—84no—
262C22×C4—88no—
263C22×C4—88no—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4#13114yes—
270C22×C4C22×D412yes—
271C22×C42−1+411yescharacteristic
272C22×C42−1+411yescharacteristic
273C22×C42−1+412yes—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—42no—
308C23—22no—
309C23—21no—
310C23—21no—
311C23—24no—
312C23—24no—
313C23—21no—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—21no—
320C23—21no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23#41312yes—
331C23#41312yes—
332C23C22×C22⋊C411yescharacteristic
333C23#51911yescharacteristic
334C23#51911yescharacteristic
335C23C2×#13112yes—
336C23#56812yes—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—88no—
382C2×C4—88no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4#53612yes—
390C2×C4#53612yes—
391C8—82no—
392C8—82no—
393C8—82no—
394C8—82no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—24no—
404C22—24no—
405C22—22no—
406C22—22no—
407C22—48no—
408C22—44no—
409C22—44no—
410C22—42no—
411C22—42no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22#231011yescharacteristic
417C22#288012yes—
418C22#288312yes—
419C22C2×#41311yescharacteristic
420C22C2×#41311yescharacteristic
421C22#298512yes—
422C22#298611yescharacteristic
423C22#298711yescharacteristic
424C4—22no—
425C4—22no—
426C4—42no—
427C4—42no—
428C4—44no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—82no—
433C4—82no—
434C4—82no—
435C2—22no—
436C2—21no—
437C2—21no—
438C2—42no—
439C2—42no—
440C2#1742212yes—
441C2#1742712yes—
442C2#2182411yescharacteristic
443C2#2182711yescharacteristic
444C2#2183011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9881C2semidirect
4. #9881C2semidirect
5. #9880C2semidirect
6. #9880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25692
#219042
#98812
#98802
#87042
#86982