Finite Group Databasebeta

← #208160

Subgroups — #208160

Order 512.

848 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
1#36635C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic
3C4×#432C211yesmaximal, characteristic
4#9899C212yesmaximal, complemented
5#9693C212yesmaximal, complemented
6#9138C212yesmaximal
7#9136C212yesmaximal
8#8237C212yesmaximal, complemented
9#8235C212yesmaximal, complemented
10C2×#432C2212yes—
11#2864—21no—
12#2863—21no—
13#2863C412yes—
14#2862C2211yescharacteristic
15#2859C2211yescharacteristic
16#2857C2211yescharacteristic
17#2857C2211yescharacteristic
18#2855—21no—
19#2855C412yes—
20#2851—21no—
21C4×C4⋊Q8C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×#125—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8C412yes—
26#2295—21no—
27#2295C412yes—
28#2283—21no—
29C2×C43C2211yescharacteristic
30#1839—21no—
31#1831C2212yes—
32#1817—21no—
33#1797C2212yes—
34#1767C2212yes—
35C4×C4⋊C8C2212yes—
36C4×C4⋊C8C2212yes—
37#1749—21no—
38C4×C22⋊C8—21no—
39C4×C22⋊C8—21no—
40#1745—21no—
41#1744—21no—
42C4×#97—21no—
43C2×#333C2214yes—
44C2×#332C2214yes—
45#1296C2214yes—
46#1296C2214yes—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8C2×C412yes—
52C2×C4×Q8—21no—
53C2×C4×Q8C2×C412yes—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42C2311yescharacteristic
61#432—28no—
62C2×C4⋊C8C2314yes—
63C2×C4⋊C8C2314yes—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×#97—22no—
67C2×#97—22no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70#391—21no—
71#391—21no—
72#391—21no—
73#391—21no—
74#391C2×C412yes—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2×C412yes—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2×C412yes—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#385—22no—
99#385—22no—
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#383—22no—
103#382—21no—
104#382—21no—
105#382—21no—
106#381—22no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380C2×C412yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C22⋊C4—22no—
118#376—22no—
119C43C2×C411yescharacteristic
120C43C2×C411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43C2311yescharacteristic
126C43C2311yescharacteristic
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#333—28no—
134#332—28no—
135C22×Q8—21no—
136C22×Q8C22×C412yes—
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C22×C8—24no—
145C22×C8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148#125—44no—
149C22⋊Q8—44no—
150C22⋊Q8—44no—
151C4×Q8—44no—
152C4×Q8—44no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2411yescharacteristic
208C4⋊C8—24no—
209C4⋊C8—24no—
210C4⋊C8—24no—
211C4⋊C8#4118yes—
212C22⋊C8—28no—
213C22⋊C8—44no—
214#97—28no—
215#97—44no—
216C4×C8—44no—
217C4×C8—44no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
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⋊C411yescharacteristic
233C2×Q8—24no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C2×C8—28no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—44no—
282C2×C8—44no—
283C2×C8—44no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—44no—
328C42—44no—
329C42#13514yes—
330C42#13714yes—
331C42C2×#4114yes—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23#41312yes—
339C23C22×C22⋊C411yescharacteristic
340Q8—84no—
341Q8—84no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
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—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4#41312yes—
392C2×C4#41612yes—
393C2×C4#41612yes—
394C2×C4#41912yes—
395C2×C4#41912yes—
396C2×C4C4×#4112yes—
397C2×C4#52512yes—
398C2×C4#53212yes—
399C2×C4C2×#13512yes—
400C2×C4C2×#13712yes—
401C2×C4#57512yes—
402C2×C4#57512yes—
403C8—48no—
404C8—84no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22#229711yescharacteristic
413C22C2×#41311yescharacteristic
414C22C2×#41311yescharacteristic
415C22C2×#41611yescharacteristic
416C22C2×#41911yescharacteristic
417C22#289611yescharacteristic
418C22#289611yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—82no—
432C4—82no—
433C4—82no—
434C4—82no—
435C4#310714yes—
436C4#318314yes—
437C2—42no—
438C2#1771311yescharacteristic
439C2#1771311yescharacteristic
440C2#1774711yescharacteristic
441C2#1774811yescharacteristic
442C2#2138911yescharacteristic
443C2#2141111yescharacteristic
444C2#2142611yescharacteristic
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 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9899C2semidirect
5. #9693C2semidirect
8. #8237C2semidirect
9. #8235C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#207772
C4×#4322
#98992
#96932
#91382
#91362
#82372
#82352