Finite Group Databasebeta

← #222983

Subgroups — #222983

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#556C211yesmaximal, characteristic
2#22650C211yesmaximal, characteristic
3C4×#436C211yesmaximal, characteristic
4#9870C211yesmaximal, characteristic, complemented
5#9865C211yesmaximal, characteristic, complemented
6#9077C212yesmaximal, complemented
7#8942C212yesmaximal, complemented
8#8708C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10C4×Q8⋊C8C211yesmaximal, characteristic, complemented
11#7903C212yesmaximal, complemented
12C2×#556C411yescharacteristic
13C2×#436—21no—
14C2×#436—21no—
15C2×#423C2212yes—
16C2×C4×M16C2212yes—
17#2373—22no—
18#2367—21no—
19#2365C411yescharacteristic
20#2362C412yes—
21#2356—21no—
22#2355C411yescharacteristic
23#2354C411yescharacteristic
24#2347C412yes—
25C4×C4⋊Q8C2211yescharacteristic
26C4×#126—22no—
27C4×#124—21no—
28C4×C22⋊Q8C2212yes—
29C42×Q8—21no—
30C42×Q8C2211yescharacteristic
31C42×D4C2211yescharacteristic
32C4×#117C2212yes—
33#1939—21no—
34#1851—22no—
35#1842C2211yescharacteristic
36#1797C2212yes—
37#1770—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1753—22no—
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×Q8⋊C8C2211yescharacteristic
45C2×Q8⋊C8C2212yes—
46#1279C2211yescharacteristic
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2211yescharacteristic
53C22×M16—22no—
54C22×M16—22no—
55#556—24no—
56#556—24no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#126—22no—
59C2×#124—21no—
60C2×C22⋊Q8C2×C412yes—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×#117C2×C412yes—
71C2×#117C2×C412yes—
72C22×C42C2312yes—
73#436—28no—
74#436—44no—
75#423—24no—
76#423—42no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2311yescharacteristic
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C4×M16—24no—
87C4×M16—42no—
88C2×#97—22no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8C2312yes—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#391—21no—
96#391—21no—
97#391C2×C412yes—
98#390—21no—
99#389—21no—
100#389C2×C412yes—
101#389C2×C412yes—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#385—22no—
112#384C2×C411yescharacteristic
113#383—22no—
114#382—21no—
115#380C2×C412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4C2312yes—
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4C2×C412yes—
138C4×C22⋊C4D412yes—
139C4×C22⋊C4D412yes—
140#376—22no—
141C43—22no—
142C43—21no—
143C43C2311yescharacteristic
144#336—22no—
145#336—21no—
146#336—22no—
147#336—21no—
148Q8⋊C8—22no—
149Q8⋊C8—22no—
150Q8⋊C8—42no—
151C22×Q8—21no—
152C22×Q8C22×C411yescharacteristic
153C22×D4C22⋊C411yescharacteristic
154C23×C4C22×C412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C2×M16—28no—
158C2×M16—44no—
159C2×M16—44no—
160C2×M16—44no—
161C2×M16—44no—
162C2×M16—82no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—21no—
169C4⋊Q8—22no—
170C4⋊Q8M1614yescomplemented
171#126—216no—
172#124—28no—
173C22⋊Q8—24no—
174C22⋊Q8—24no—
175C4×Q8—28no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—41no—
183C4×Q8—44no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8M1614yescomplemented
187C4×D4—24no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192#117—24no—
193#117—24no—
194#117—24no—
195#117—42no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C412yes—
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2411yescharacteristic
245C4⋊C8—24no—
246C4⋊C8—21no—
247C4⋊C8—22no—
248C4⋊C8—21no—
249C4⋊C8—22no—
250C4⋊C8—21no—
251C4⋊C8—44no—
252C4⋊C8—42no—
253C4⋊C8—81no—
254C4⋊C8#4112yes—
255C22⋊C8—28no—
256C22⋊C8—44no—
257#97—44no—
258C4×C8—24no—
259C4×C8—21no—
260C4×C8—22no—
261C4×C8—21no—
262C4×C8—22no—
263C4×C8—21no—
264C4×C8—42no—
265C4×C8—42no—
266C4×C8#4112yes—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—21no—
277#95—22no—
278#95C22⋊C412yes—
279#95C22⋊C412yes—
280#95C22⋊C412yes—
281C24C2×C22⋊C412yes—
282C2×Q8—21no—
283C2×Q8—24no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8C2×M1612yes—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326M16—416no—
327M16—88no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—41no—
346C4⋊C4—28no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—81no—
365C4⋊C4#10414yes—
366C4⋊C4C2×M1612yes—
367C4⋊C4C2×M1614yes—
368C4⋊C4#13112yes—
369C22⋊C4—28no—
370C22⋊C4—28no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—44no—
375C22⋊C4—82no—
376C42—28no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—24no—
388C42—22no—
389C42—41no—
390C42—44no—
391C42—42no—
392C42—41no—
393C42—41no—
394C42—42no—
395C42—42no—
396C42—41no—
397C42—41no—
398C42—44no—
399C42—42no—
400C42—81no—
401C42—82no—
402C42C2×M1612yes—
403C42C2×M1612yes—
404C42AGL(1,ℤ/8)12yes—
405C42#13712yes—
406C42C2×#4111yescharacteristic
407C42C2×#4111yescharacteristic
408C42H(5,𝔽2)11yescharacteristic
409C422−1+411yescharacteristic
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23C22×C22⋊C411yescharacteristic
418Q8—22no—
419Q8—22no—
420Q8—41no—
421Q8—41no—
422Q8—48no—
423D4—44no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4#40712yes—
470C2×C4#40712yes—
471C2×C4#41712yes—
472C2×C4#41712yes—
473C2×C4C2×#10414yes—
474C2×C4C2×#10412yes—
475C2×C4C4×#4111yescharacteristic
476C2×C4#52011yescharacteristic
477C2×C4#54611yescharacteristic
478C2×C4#54611yescharacteristic
479C2×C4#54711yescharacteristic
480C2×C4#54711yescharacteristic
481C2×C4C22×M1611yescharacteristic
482C2×C4C22×M1611yescharacteristic
483C2×C4C2×#13111yescharacteristic
484C2×C4#56811yescharacteristic
485C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
486C2×C4C2×#13711yescharacteristic
487C2×C4#57711yescharacteristic
488C2×C4#57811yescharacteristic
489C8—24no—
490C8—42no—
491C8—44no—
492C8—42no—
493C8—81no—
494C8—82no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22#233911yescharacteristic
503C22C2×#40711yescharacteristic
504C22#288011yescharacteristic
505C22C2×#41711yescharacteristic
506C22#289811yescharacteristic
507C22C22×#10411yescharacteristic
508C22#290111yescharacteristic
509C4—22no—
510C4—22no—
511C4—21no—
512C4—21no—
513C4—24no—
514C4—44no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—82no—
522C4—81no—
523C4#148914yes—
524C4#296511yescharacteristic
525C4#297111yescharacteristic
526C4#298911yescharacteristic
527C4#299111yescharacteristic
528C4#332411yescharacteristic
529C4#332811yescharacteristic
530C4#335311yescharacteristic
531C4#335611yescharacteristic
532C2—82no—
533C2C2×#148911yescharacteristic
534C2#1745411yescharacteristic
535C2#1751511yescharacteristic
536C2#1751811yescharacteristic
537C2#2247311yescharacteristic
538C2#2250411yescharacteristic
539C2#2250511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9870C2semidirect
5. #9865C2semidirect
6. #9077C2semidirect
7. #8942C2semidirect
8. #8708C2semidirect
9. #8564C2semidirect
10. C4×Q8⋊C8C2semidirect
11. #7903C2semidirect
170. C4⋊Q8M16semidirect
186. C4×Q8M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5562
#226502
C4×#4362
#98702
#98652
#90772
#89422
#87082
#85642
C4×Q8⋊C82
#79032