Finite Group Databasebeta

← #224783

Subgroups — #224783

Order 512.

986 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#556C211yesmaximal, characteristic
2#22722C211yesmaximal, characteristic, complemented
3C4×#446C211yesmaximal, characteristic, complemented
4#9676C212yesmaximal
5#9673C211yesmaximal, characteristic, complemented
6#9646C212yesmaximal
7#9637C211yesmaximal, characteristic, complemented
8#8564C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic
10#8173C212yesmaximal, complemented
11C4×D4⋊C8C211yesmaximal, characteristic
12C2×#556C411yescharacteristic
13C2×#446—21no—
14C2×#446—21no—
15#2373—22no—
16#2367C412yes—
17#2365C411yescharacteristic
18#2362—21no—
19#2356C412yes—
20#2355C411yescharacteristic
21#2354C411yescharacteristic
22#2347—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×#126—22no—
25C4×#124C2212yes—
26C4×C22⋊Q8—21no—
27C42×Q8C2212yes—
28C42×Q8C2211yescharacteristic
29C42×D4C2211yescharacteristic
30C4×#117—21no—
31#1940—22no—
32#1939—21no—
33#1842C2211yescharacteristic
34#1842C2212yes—
35#1771—22no—
36#1770—21no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41C4×#97C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×D4⋊C8C2211yescharacteristic
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2212yes—
50#1279C2212yes—
51#1279C2211yescharacteristic
52#1279C2211yescharacteristic
53#556—24no—
54#556—24no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—22no—
57C2×#124C2×C412yes—
58C2×C22⋊Q8—21no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8C2312yes—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117—21no—
69C2×#117—21no—
70C22×C42—21no—
71#446—28no—
72#446—44no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2311yescharacteristic
82C2×#97—22no—
83C2×#97—22no—
84C2×#97—22no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2312yes—
90C2×C4×C8C2311yescharacteristic
91#391—21no—
92#391C2×C412yes—
93#391C2×C412yes—
94#390C2×C412yes—
95#389—21no—
96#389—21no—
97#389—21no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#385—22no—
108#384C2×C411yescharacteristic
109#383—22no—
110#382C2×C412yes—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136#376—22no—
137C43—22no—
138C43C2311yescharacteristic
139C43C2312yes—
140#336—22no—
141#336—21no—
142#336—22no—
143#336—21no—
144Q8⋊C8—24no—
145Q8⋊C8—24no—
146Q8⋊C8—42no—
147Q8⋊C8—42no—
148D4⋊C8—22no—
149D4⋊C8—22no—
150D4⋊C8—42no—
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C22×C412yes—
153C22×D4C22×C411yescharacteristic
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C4⋊Q8—22no—
164C4⋊Q8M1614yes—
165#126—216no—
166#124—24no—
167#124—24no—
168C22⋊Q8—28no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—24no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—24no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—42no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—41no—
189C4×D4M1614yes—
190#117—28no—
191#117—44no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C412yes—
228C2×C42C22×C412yes—
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2411yescharacteristic
239C4⋊C8—24no—
240C4⋊C8—28no—
241C4⋊C8—21no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—44no—
247C4⋊C8—42no—
248C4⋊C8—82no—
249C4⋊C8—81no—
250C4⋊C8#4112yes—
251#97—28no—
252#97—44no—
253#97—44no—
254C4×C8—24no—
255C4×C8—21no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—42no—
261C4×C8—42no—
262C4×C8#4112yes—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—21no—
267#95—21no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—21no—
274#95C22⋊C412yes—
275#95C22⋊C412yes—
276#95C22⋊C412yes—
277C24—21no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8C2×M1614yes—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4C2×M1612yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—81no—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—24no—
326C2×C8—22no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—42no—
339C4⋊C4—28no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—81no—
358C4⋊C4#10414yes—
359C4⋊C4C2×M1612yes—
360C4⋊C4#13112yes—
361C22⋊C4—28no—
362C22⋊C4—28no—
363C22⋊C4—24no—
364C22⋊C4—42no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—44no—
368C22⋊C4—82no—
369C42—28no—
370C42—22no—
371C42—22no—
372C42—21no—
373C42—22no—
374C42—21no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—21no—
380C42—22no—
381C42—21no—
382C42—22no—
383C42—24no—
384C42—41no—
385C42—42no—
386C42—44no—
387C42—42no—
388C42—42no—
389C42—41no—
390C42—41no—
391C42—42no—
392C42—42no—
393C42—44no—
394C42—82no—
395C42C2×M1612yes—
396C42C2×M1612yes—
397C42AGL(1,ℤ/8)12yes—
398C42#13712yes—
399C42C2×#4111yescharacteristic
400C42C2×#4111yescharacteristic
401C42H(5,𝔽2)11yescharacteristic
402C422−1+411yescharacteristic
403C23—22no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23C22×C22⋊C411yescharacteristic
411Q8—24no—
412Q8—24no—
413Q8—42no—
414Q8—42no—
415Q8—44no—
416D4—22no—
417D4—22no—
418D4—41no—
419D4—41no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—48no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—84no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4#40712yes—
462C2×C4#40712yes—
463C2×C4#41712yes—
464C2×C4#41712yes—
465C2×C4C2×#10414yes—
466C2×C4C2×#10412yes—
467C2×C4C4×#4111yescharacteristic
468C2×C4#52011yescharacteristic
469C2×C4#54811yescharacteristic
470C2×C4#54811yescharacteristic
471C2×C4#55011yescharacteristic
472C2×C4#55011yescharacteristic
473C2×C4C22×M1611yescharacteristic
474C2×C4C22×M1611yescharacteristic
475C2×C4C2×#13111yescharacteristic
476C2×C4#56811yescharacteristic
477C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
478C2×C4C2×#13711yescharacteristic
479C2×C4#57711yescharacteristic
480C2×C4#57811yescharacteristic
481C8—24no—
482C8—42no—
483C8—44no—
484C8—42no—
485C8—81no—
486C8—82no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22#234111yescharacteristic
495C22C2×#40711yescharacteristic
496C22#288011yescharacteristic
497C22C2×#41711yescharacteristic
498C22#289811yescharacteristic
499C22C22×#10411yescharacteristic
500C22#290111yescharacteristic
501C4—21no—
502C4—21no—
503C4—22no—
504C4—22no—
505C4—24no—
506C4—44no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—41no—
513C4—81no—
514C4—82no—
515C4#148914yes—
516C4#296511yescharacteristic
517C4#297011yescharacteristic
518C4#298911yescharacteristic
519C4#299011yescharacteristic
520C4#336711yescharacteristic
521C4#336911yescharacteristic
522C4#339711yescharacteristic
523C4#339811yescharacteristic
524C2—82no—
525C2C2×#148911yescharacteristic
526C2#1745411yescharacteristic
527C2#1751211yescharacteristic
528C2#1751911yescharacteristic
529C2#2253211yescharacteristic
530C2#2254411yescharacteristic
531C2#2255011yescharacteristic
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×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22722C2semidirect
3. C4×#446C2semidirect
5. #9673C2semidirect
7. #9637C2semidirect
8. #8564C2semidirect
10. #8173C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5562
#227222
C4×#4462
#96762
#96732
#96462
#96372
#85642
#85632
#81732
C4×D4⋊C82