Finite Group Databasebeta

← #190966

Subgroups — #190966

Order 512.

1168 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34141C211yesmaximal, characteristic
2#21431C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#8707C212yesmaximal, complemented
5#8567C212yesmaximal, complemented
6#8563C212yesmaximal, complemented
7#8313C212yesmaximal
8#8181C211yesmaximal, characteristic, complemented
9#8179C211yesmaximal, characteristic, complemented
10#8172C211yesmaximal, characteristic, complemented
11#8170C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×C4×#41—21no—
14C2×#433—21no—
15#2367—22no—
16#2366C412yes—
17#2365C412yes—
18#2364—22no—
19#2361C412yes—
20#2361C412yes—
21#2297—21no—
22#2293C2211yescharacteristic
23#2293C2211yescharacteristic
24#2290—21no—
25#2290—21no—
26C42×D4C2211yescharacteristic
27C42×D4C2211yescharacteristic
28C4×#117C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1924—22no—
31#1917—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
42C4×#97C2211yescharacteristic
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C22×#41—21no—
52#532—24no—
53#532—24no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#127C2×C411yescharacteristic
56C2×#124—21no—
57C2×#121—21no—
58C2×#121C2×C411yescharacteristic
59C2×#121C2×C411yescharacteristic
60C4×#41—48no—
61C2×C4×Q8—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117—21no—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C4⋊C4C2×C412yes—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79#433—44no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×#97—21no—
88C2×#97—22no—
89C2×C4×C8—21no—
90C2×C4×C8C2312yes—
91C2×C4×C8C2312yes—
92#391—22no—
93#390—22no—
94#390C2×C412yes—
95#390C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#387—22no—
103#387—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#385—22no—
109#385—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4C2×C412yes—
126C4×C22⋊C4D412yes—
127#376D411yescharacteristic
128#376D411yescharacteristic
129C2×#95—21no—
130C43C2311yescharacteristic
131C43C2311yescharacteristic
132#336—22no—
133#336—21no—
134#336—21no—
135#336—22no—
136#336—22no—
137#336—22no—
138D4⋊C8—24no—
139D4⋊C8—24no—
140D4⋊C8—24no—
141D4⋊C8—24no—
142C2×#41—28no—
143C2×#41—44no—
144C2×#41—42no—
145C2×#41—42no—
146C22×Q8—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C22×C411yescharacteristic
152C22×D4C22×C411yescharacteristic
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167#127—22no—
168#127—22no—
169#124—28no—
170#121—28no—
171#121—22no—
172#121—22no—
173#121—22no—
174#121—22no—
175C4×Q8—48no—
176C4×D4—28no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—48no—
184C4×D4—48no—
185C4×D4—48no—
186C4×D4#4114yes—
187C4×D4#4114yes—
188#117—22no—
189#117—22no—
190#117—24no—
191#117—44no—
192#117—44no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C412yes—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—44no—
225C2×C42—44no—
226C2×C42—44no—
227C2×C42C22×C412yes—
228C2×C42C22×C412yes—
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2411yescharacteristic
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—44no—
244C4⋊C8—44no—
245C4⋊C8—44no—
246C22⋊C8—44no—
247C22⋊C8—44no—
248#97—42no—
249#97—44no—
250C4×C8—22no—
251C4×C8—22no—
252C4×C8—22no—
253C4×C8—22no—
254C4×C8—42no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—21no—
261#95—21no—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266C24—21no—
267C24—21no—
268C24C2×C22⋊C411yescharacteristic
269#41—416no—
270#41—416no—
271C2×Q8—24no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—82no—
291C2×D4#11712yes—
292C2×D4#11712yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C412yes—
329C22×C4AGL(1,ℤ/8)12yes—
330C22×C4#13712yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—24no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4#11712yes—
358C4⋊C4#11712yes—
359C4⋊C4#13114yes—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C42—21no—
373C42—21no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—21no—
381C42—21no—
382C42—22no—
383C42—21no—
384C42—21no—
385C42—42no—
386C42—42no—
387C42—44no—
388C42—44no—
389C42—42no—
390C42—42no—
391C42—44no—
392C42AGL(1,ℤ/8)14yes—
393C42C2×#4112yes—
394C42C2×#4112yes—
395C23—22no—
396C23—21no—
397C23—22no—
398C23—21no—
399C23—22no—
400C23—22no—
401C23—42no—
402C23—42no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—42no—
408C23—42no—
409C23#41712yes—
410C23C22×C22⋊C411yescharacteristic
411Q8—48no—
412D4—22no—
413D4—22no—
414D4—24no—
415D4—48no—
416D4—48no—
417D4—48no—
418D4—41no—
419D4—41no—
420D4—42no—
421D4—84no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—88no—
463C2×C4—88no—
464C2×C4—88no—
465C2×C4#41712yes—
466C2×C4#41814yes—
467C2×C4C2×#11711yescharacteristic
468C2×C4C2×#11711yescharacteristic
469C2×C4#52512yes—
470C2×C4#52512yes—
471C2×C4C2×#13112yes—
472C2×C4#56812yes—
473C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
474C2×C4C2×AGL(1,ℤ/8)12yes—
475C2×C4C2×#13711yescharacteristic
476C8—44no—
477C8—82no—
478C8—82no—
479C8—82no—
480C22—22no—
481C22—24no—
482C22—41no—
483C22—42no—
484C22—41no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22#228011yescharacteristic
496C22#288012yes—
497C22C2×#41811yescharacteristic
498C22C2×#41711yescharacteristic
499C22#290112yes—
500C4—22no—
501C4—22no—
502C4—24no—
503C4—42no—
504C4—42no—
505C4—41no—
506C4—42no—
507C4—44no—
508C4—42no—
509C4—42no—
510C4—41no—
511C4—82no—
512C4#292211yescharacteristic
513C4#292311yescharacteristic
514C4#292512yes—
515C4#310912yes—
516C4#311012yes—
517C2—42no—
518C2—82no—
519C2—82no—
520C2#1751112yes—
521C2#1751512yes—
522C2#2072011yescharacteristic
523C2#2073811yescharacteristic
524C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#532C22
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. #21431C2semidirect
3. #20812C2semidirect
4. #8707C2semidirect
5. #8567C2semidirect
6. #8563C2semidirect
8. #8181C2semidirect
9. #8179C2semidirect
10. #8172C2semidirect
11. #8170C2semidirect

Maximal subgroups

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

subgroupindex
#341412
#214312
#208122
#87072
#85672
#85632
#83132
#81812
#81792
#81722
#81702