Finite Group Databasebeta

← #190879

Subgroups — #190879

Order 512.

1005 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34123C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#8711C212yesmaximal, complemented
5#8568C212yesmaximal, complemented
6#8563C212yesmaximal, complemented
7#8312C212yesmaximal
8#8182C211yesmaximal, characteristic, complemented
9#8176C211yesmaximal, characteristic, complemented
10#8174C211yesmaximal, characteristic, complemented
11C4×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#433—21no—
14#2356C412yes—
15#2353—22no—
16#2349C412yes—
17#2346C412yes—
18#2343—22no—
19#2339C412yes—
20#2295C2211yescharacteristic
21#2295C2211yescharacteristic
22#2290—21no—
23#2290—21no—
24#2283—21no—
25#2283C2211yescharacteristic
26C42×D4C2211yescharacteristic
27C42×D4C2211yescharacteristic
28C4×#117C2211yescharacteristic
29C2×#376—21no—
30#1923—22no—
31#1916—22no—
32#1842C2212yes—
33#1832—21no—
34#1798—21no—
35#1797C2212yes—
36#1768—21no—
37#1767C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41#1744C2211yescharacteristic
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C22×#41C2×C411yescharacteristic
52C4×#41—24no—
53C4×#41—24no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2311yescharacteristic
62C2×#117—21no—
63C2×#117C2×C411yescharacteristic
64C2×#117D411yescharacteristic
65C2×#117D411yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#433—44no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×#97—21no—
79C2×C4×C8—21no—
80C2×C4×C8—22no—
81C2×C4×C8C2312yes—
82C2×C4×C8C2312yes—
83#391C2×C412yes—
84#391C2×C412yes—
85#389C2×C412yes—
86#389C2×C412yes—
87#388—22no—
88#388—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#386—22no—
94#386—22no—
95#385—22no—
96#385—22no—
97#383—21no—
98#383—22no—
99#382—22no—
100#382C2×C412yes—
101#381—21no—
102#381—22no—
103#380—22no—
104#380C2×C412yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4C2311yescharacteristic
112C4×C4⋊C4C2311yescharacteristic
113C4×C22⋊C4—21no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4D412yes—
120#376—21no—
121#376—21no—
122#376—21no—
123#376—22no—
124#376—22no—
125#376D411yescharacteristic
126#376D411yescharacteristic
127C2×#95—22no—
128C2×#95—22no—
129C43C2311yescharacteristic
130C43C2311yescharacteristic
131#336—22no—
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137Q8⋊C8—24no—
138Q8⋊C8—24no—
139D4⋊C8—24no—
140D4⋊C8—24no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—24no—
144C2×#41—42no—
145C22×Q8C22×C411yescharacteristic
146C22×D4—21no—
147C22×D4C22×C411yescharacteristic
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8#4114yes—
164C4×D4—28no—
165C4×D4—28no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—22no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4#4114yes—
174#117—28no—
175#117—22no—
176#117—22no—
177#117—24no—
178#117—22no—
179#117—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C42—28no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—44no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C22×C412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2411yescharacteristic
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—44no—
228C4⋊C8—44no—
229C4⋊C8—44no—
230C22⋊C8—44no—
231C22⋊C8—44no—
232#97—42no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—44no—
238C4×C8—42no—
239#95—22no—
240#95—22no—
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#95—21no—
251#95—21no—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C24—21no—
257C24C2×C22⋊C411yescharacteristic
258#41—48no—
259#41—44no—
260#41—44no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8#11712yes—
265C2×D4—24no—
266C2×D4—21no—
267C2×D4—22no—
268C2×D4—44no—
269C2×D4#11712yes—
270C2×D4#13114yes—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—24no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4#11712yes—
335C4⋊C4#11712yes—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—82no—
344C42—24no—
345C42—24no—
346C42—21no—
347C42—21no—
348C42—22no—
349C42—24no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—21no—
357C42—21no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—41no—
362C42—41no—
363C42—44no—
364C42—44no—
365C42#13514yes—
366C42AGL(1,ℤ/8)12yes—
367C42#13712yes—
368C42C2×#4112yes—
369C42C2×#4112yes—
370C23—24no—
371C23—22no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—22no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—42no—
380C23C22×C22⋊C411yescharacteristic
381Q8—24no—
382Q8—41no—
383Q8—41no—
384D4—22no—
385D4—24no—
386D4—48no—
387D4#43314yescomplemented
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—48no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—88no—
429C2×C4—88no—
430C2×C4—84no—
431C2×C4#41614yes—
432C2×C4#41714yes—
433C2×C4C2×#11711yescharacteristic
434C2×C4C2×#11711yescharacteristic
435C2×C4#52512yes—
436C2×C4#52512yes—
437C2×C4C2×#13112yes—
438C2×C4#56812yes—
439C2×C4C2×#13512yes—
440C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
441C2×C4C2×#13711yescharacteristic
442C8—44no—
443C8—82no—
444C8—82no—
445C8—82no—
446C22—24no—
447C22—42no—
448C22—42no—
449C22—44no—
450C22—44no—
451C22—41no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—41no—
456C22#228011yescharacteristic
457C22#288012yes—
458C22C2×#41611yescharacteristic
459C22C2×#41711yescharacteristic
460C22#290112yes—
461C4—22no—
462C4—24no—
463C4—24no—
464C4—41no—
465C4—42no—
466C4—41no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—41no—
471C4—42no—
472C4—82no—
473C4—82no—
474C4C2×#43311yescharacteristic
475C4#292111yescharacteristic
476C4#292412yes—
477C4#310612yes—
478C4#310712yes—
479C2—44no—
480C2—81no—
481C2#1750812yes—
482C2#1751212yes—
483C2#2071611yescharacteristic
484C2#2073711yescharacteristic
485C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21430C2semidirect
3. #20812C2semidirect
4. #8711C2semidirect
5. #8568C2semidirect
6. #8563C2semidirect
8. #8182C2semidirect
9. #8176C2semidirect
10. #8174C2semidirect
11. C4×D4⋊C8C2semidirect
387. D4#433semidirect

Maximal subgroups

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

subgroupindex
#341232
#214302
#208122
#87112
#85682
#85632
#83122
#81822
#81762
#81742
C4×D4⋊C82