Finite Group Databasebeta

← #224319

Subgroups — #224319

Order 512.

1132 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×Q8C211yesmaximal, characteristic
2#22722C211yesmaximal, characteristic, complemented
3#21339C211yesmaximal, characteristic, complemented
4#9874C212yesmaximal, complemented
5#9870C211yesmaximal, characteristic, complemented
6#9869C212yesmaximal, complemented
7#9865C211yesmaximal, characteristic, complemented
8#8563C211yesmaximal, characteristic
9#8563C212yesmaximal
10#8174C212yesmaximal
11C4×D4⋊C8C211yesmaximal, characteristic
12C2×D4×Q8C411yescharacteristic
13C22×C4×Q8—21no—
14C2×#446—21no—
15#2365C412yes—
16#2365C411yescharacteristic
17#2360—22no—
18#2360—21no—
19Q8×C4⋊C4C411yescharacteristic
20D4×C4⋊C4C412yes—
21D4×C4⋊C4C411yescharacteristic
22Q8×C22⋊C4—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2212yes—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—22no—
27C42×Q8C2211yescharacteristic
28C42×D4C2211yescharacteristic
29C42×D4C2212yes—
30#1984—21no—
31#1940—22no—
32#1939—21no—
33#1939—21no—
34#1916—22no—
35#1842C2211yescharacteristic
36#1842C2212yes—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41#1744C2212yes—
42C42×C8C2211yescharacteristic
43C2×D4⋊C8C2211yescharacteristic
44C2×D4⋊C8C2212yes—
45C2×D4⋊C8C2211yescharacteristic
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53C23×Q8—21no—
54D4×Q8—24no—
55D4×Q8—24no—
56C2×C4⋊Q8C2×C412yes—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—22no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—22no—
65C2×C4×Q8—44no—
66C2×C4×Q8—44no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×D4C2×C412yes—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C412yes—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C42—21no—
78C22×C42—22no—
79#446—44no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2311yescharacteristic
89C2×#97—22no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2312yes—
95C2×C4×C8C2311yescharacteristic
96#391—21no—
97#391—22no—
98#391—21no—
99#391—22no—
100#389—21no—
101#389—22no—
102#389—21no—
103#389—22no—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2×C412yes—
107#387—22no—
108#387—22no—
109#387—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#384C2×C412yes—
114#384C2×C411yescharacteristic
115#380—21no—
116#380—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D412yes—
129C4×C4⋊C4C2312yes—
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—22no—
137C43C2311yescharacteristic
138C43C2312yes—
139#336—21no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—21no—
145D4⋊C8—22no—
146D4⋊C8—24no—
147D4⋊C8Q814yescomplemented
148C22×Q8—28no—
149C22×Q8—22no—
150C22×Q8—21no—
151C22×Q8—44no—
152C22×Q8—44no—
153C22×Q8C22⋊C411yescharacteristic
154C22×D4C22×C411yescharacteristic
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—22no—
169C4⋊Q8M1614yes—
170C22⋊Q8—28no—
171C22⋊Q8—216no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—42no—
175C4×Q8—44no—
176C4×Q8—48no—
177C4×Q8—44no—
178C4×Q8—42no—
179C4×Q8—44no—
180C4×Q8—44no—
181C4×Q8—42no—
182C4×Q8D814yes—
183C4×Q8SD1614yes—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—22no—
187C4×D4—24no—
188C4×D4—22no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—41no—
192C4×D4—42no—
193C4×D4M1614yes—
194C4×D4C2×Q812yes—
195C4×D4#4112yes—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—42no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42—42no—
233C2×C42C22×C412yes—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C412yes—
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2411yescharacteristic
246C4⋊C8—21no—
247C4⋊C8—22no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—22no—
251C4⋊C8—42no—
252C4⋊C8—44no—
253C4⋊C8—42no—
254C4⋊C8—44no—
255C4⋊C8—82no—
256C4⋊C8—81no—
257C4⋊C8C2×Q812yescomplemented
258#97—44no—
259C4×C8—21no—
260C4×C8—22no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—22no—
264C4×C8—42no—
265C4×C8—42no—
266C4×C8—42no—
267C4×C8C2×Q812yescomplemented
268#95—22no—
269#95—21no—
270#95—21no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—21no—
276#95—22no—
277#95C22⋊C412yes—
278#95C22⋊C412yes—
279#95C22⋊C412yes—
280C24—21no—
281C2×Q8—28no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—22no—
285C2×Q8—44no—
286C2×Q8—432no—
287C2×Q8—44no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8#10214yes—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4C4×Q812yes—
295C2×D4C2×M1612yes—
296C2×D4#13112yes—
297C22×C4—28no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—81no—
319C22×C4—82no—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—42no—
349C4⋊C4—28no—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—42no—
367C4⋊C4—81no—
368C4⋊C4—82no—
369C4⋊C4#10218yes—
370C4⋊C4#10214yes—
371C4⋊C4C4×Q812yes—
372C4⋊C4C2×M1614yes—
373C4⋊C4C2×M1612yes—
374C22⋊C4—28no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—24no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C42—22no—
381C42—22no—
382C42—21no—
383C42—22no—
384C42—21no—
385C42—22no—
386C42—21no—
387C42—22no—
388C42—21no—
389C42—22no—
390C42—22no—
391C42—41no—
392C42—42no—
393C42—42no—
394C42—41no—
395C42—42no—
396C42—42no—
397C42—41no—
398C42—42no—
399C42—44no—
400C42—44no—
401C42—42no—
402C42C2×M1612yes—
403C42C2×M1612yes—
404C42C2×D812yes—
405C42C2×D814yes—
406C42C2×SD1614yes—
407C42C2×SD1612yes—
408C42C22×Q811yescharacteristic
409C42C2×#4111yescharacteristic
410C422−1+411yescharacteristic
411C422−1+411yescharacteristic
412C23—22no—
413C23—22no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23C22×C22⋊C411yescharacteristic
420Q8—28no—
421Q8—416no—
422Q8—48no—
423D4—22no—
424D4—22no—
425D4—22no—
426D4#44614yescomplemented
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—416no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—48no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—84no—
468C2×C4—88no—
469C2×C4#40712yes—
470C2×C4#40712yes—
471C2×C4C2×#10212yes—
472C2×C4C2×#10214yes—
473C2×C4C2×#10214yes—
474C2×C4C2×#10212yes—
475C2×C4#42112yes—
476C2×C4C2×C4×Q811yescharacteristic
477C2×C4#51911yescharacteristic
478C2×C4#54811yescharacteristic
479C2×C4#54811yescharacteristic
480C2×C4D4×Q811yescharacteristic
481C2×C4D4×Q811yescharacteristic
482C2×C4C22×M1611yescharacteristic
483C2×C4C22×M1611yescharacteristic
484C2×C4C2×#13111yescharacteristic
485C2×C4#56811yescharacteristic
486C2×C4C22×D811yescharacteristic
487C2×C4C22×SD1611yescharacteristic
488C2×C4#57711yescharacteristic
489C2×C4#57811yescharacteristic
490C8—42no—
491C8—42no—
492C8—42no—
493C8—82no—
494C8—81no—
495C8—82no—
496C22—24no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22Q8×C22⋊C411yescharacteristic
504C22C2×#40711yescharacteristic
505C22#288011yescharacteristic
506C22C22×#10211yescharacteristic
507C22#289811yescharacteristic
508C22C2×#42111yescharacteristic
509C22#290111yescharacteristic
510C4—21no—
511C4—21no—
512C4—24no—
513C4—22no—
514C4—22no—
515C4—24no—
516C4—44no—
517C4—42no—
518C4—42no—
519C4—41no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—81no—
524C4#148614yes—
525C4C2×#44611yescharacteristic
526C4Q8×M1611yescharacteristic
527C4#298311yescharacteristic
528C4#298411yescharacteristic
529C4#335811yescharacteristic
530C4#335911yescharacteristic
531C4Q8×D811yescharacteristic
532C4Q8×SD1611yescharacteristic
533C2—44no—
534C2C2×#148611yescharacteristic
535C2#1745411yescharacteristic
536C2#1750711yescharacteristic
537C2#1752311yescharacteristic
538C2#2251011yescharacteristic
539C2Q8×#10211yescharacteristic
540C2#2252911yescharacteristic
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. #21339C2semidirect
4. #9874C2semidirect
5. #9870C2semidirect
6. #9869C2semidirect
7. #9865C2semidirect
147. D4⋊C8Q8semidirect
257. C4⋊C8C2×Q8semidirect
267. C4×C8C2×Q8semidirect
426. D4#446semidirect

Maximal subgroups

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

subgroupindex
C4×D4×Q82
#227222
#213392
#98742
#98702
#98692
#98652
#85632
#81742
C4×D4⋊C82