Finite Group Databasebeta

← #208422

Subgroups — #208422

Order 512.

926 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3C4×#383C211yesmaximal, characteristic
4#9908C212yesmaximal
5#9711C212yesmaximal
6#9153C214yesmaximal
7#8217C214yesmaximal
8#2864—21no—
9#2863—21no—
10#2863C412yescomplemented
11#2862C411yescharacteristic, complemented
12#2859C411yescharacteristic, complemented
13#2857C2211yescharacteristic
14#2857C2211yescharacteristic
15#2855—21no—
16#2855C2212yes—
17#2851—21no—
18C4×C4⋊Q8C411yescharacteristic, complemented
19C4×C4⋊Q8C411yescharacteristic, complemented
20C4×#125—21no—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8C2212yes—
23#2295—21no—
24#2295C412yescomplemented
25#2283—21no—
26C2×#383C2212yes—
27C2×C43C2211yescharacteristic
28#1839—21no—
29#1831—21no—
30#1817C2212yes—
31#1797C2212yes—
32#1767—21no—
33#1749C2212yes—
34#1745—21no—
35C2×#328C2218yes—
36#1434—21no—
37#1434—21no—
38#1434C2212yes—
39C4×#95—21no—
40C4×#95—21no—
41C4×#95C2212yes—
42#1294C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2312yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C22×C42—22no—
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—22no—
59C2×C22⋊C8C2314yes—
60C2×#97—22no—
61#391—21no—
62#391—21no—
63#391—21no—
64#391—21no—
65#391C2×C412yes—
66#391C2×C412yes—
67#391C2×C412yes—
68#391C2×C412yes—
69#389—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#385—22no—
90#385—22no—
91#384C2×C411yescharacteristic
92#384C2×C411yescharacteristic
93#383—22no—
94#383—28no—
95#382—21no—
96#382—21no—
97#382—21no—
98#381—22no—
99#380—21no—
100#380—21no—
101#380—21no—
102#380C2312yes—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2×C412yes—
109C4×C22⋊C4—22no—
110#376—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95C2314yes—
117C43C2×C411yescharacteristic
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120C43C2×C411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#328—216no—
130C22×Q8—21no—
131C22×Q8C22×C412yes—
132C23×C4—24no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—24no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143#125—44no—
144C22⋊Q8—28no—
145C22⋊Q8—44no—
146C4×Q8—44no—
147C4×Q8—44no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42—44no—
176C2×C42—44no—
177C2×C42—44no—
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C411yescharacteristic
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×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C4⋊C8—44no—
207C22⋊C8—28no—
208#97—44no—
209#95—28no—
210#95—28no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—24no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95#4118yes—
230C24C22×D411yescharacteristic
231C2×Q8—24no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C22×C4—22no—
238C22×C4—22no—
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—28no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
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×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4#13714yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C2×#4114yes—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—44no—
286C4⋊C4—24no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C22⋊C4—44no—
303C22⋊C4—44no—
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—22no—
318C42—22no—
319C42—22no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—44no—
333C42—44no—
334C42—88no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23C22×C22⋊C411yescharacteristic
340C23#52512yes—
341C23C2×#13712yes—
342C23#57512yes—
343Q8—84no—
344Q8—84no—
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—28no—
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—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4#41012yes—
406C2×C4#41012yes—
407C2×C4#41912yes—
408C2×C4#41912yes—
409C2×C4#42112yes—
410C2×C4#42112yes—
411C2×C4C4×#4112yes—
412C2×C4#53212yes—
413C8—84no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22#229711yescharacteristic
421C22C2×#41011yescharacteristic
422C22C2×#41011yescharacteristic
423C22C2×#41911yescharacteristic
424C22#289611yescharacteristic
425C22C2×#42111yescharacteristic
426C22C2×#42111yescharacteristic
427C22#311614yes—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—88no—
442C4—82no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4#151514yes—
447C2—24no—
448C2C2×#151511yescharacteristic
449C2#1756211yescharacteristic
450C2#1757911yescharacteristic
451C2#1759411yescharacteristic
452C2#2139211yescharacteristic
453C2#2141811yescharacteristic
454C2#2142111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2863C4semidirect
11. #2862C4semidirect
12. #2859C4semidirect
18. C4×C4⋊Q8C4semidirect
19. C4×C4⋊Q8C4semidirect
24. #2295C4semidirect

Maximal subgroups

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

subgroupindex
#366352
#207782
C4×#3832
#99082
#97112
#91532
#82172