Finite Group Databasebeta

← #480910

Subgroups — #480910

Order 512.

825 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38081C211yesmaximal, characteristic
2#17417C212yesmaximal
3#11568C212yesmaximal, complemented
4#11556C212yesmaximal, complemented
5#11551C212yesmaximal, complemented
6#11536C211yesmaximal, characteristic, complemented
7#11510C211yesmaximal, characteristic, complemented
8#11506C211yesmaximal, characteristic, complemented
9#11482C211yesmaximal, characteristic, complemented
10#11434C211yesmaximal, characteristic, complemented
11C2×#1319C211yesmaximal, characteristic, complemented
12C2×#552C2211yescharacteristic
13C2×D4×Q8C411yescharacteristic
14#2924—22no—
15#2857C411yescharacteristic
16#2713—22no—
17#2689—22no—
18#2642C411yescharacteristic
19#2621C411yescharacteristic
20#2565C411yescharacteristic
21#2546C2211yescharacteristic
22#2503C2211yescharacteristic
23#2499C2211yescharacteristic
24#2438—22no—
25#2434C411yescharacteristic
26#2434C411yescharacteristic
27#2419—22no—
28#2327C2211yescharacteristic
29#2310C2211yescharacteristic
30C2×#391C2211yescharacteristic
31#2290C411yescharacteristic
32#1473—22no—
33#1470—22no—
34#1468—22no—
35#1467C2212yes—
36#1464C2212yes—
37#1463C2212yes—
38#1461C2212yes—
39#1458C2212yes—
40C2×#324C2212yes—
41#1324C2212yes—
42#1324C2212yes—
43#1319C2211yescharacteristic
44#1319C2211yescharacteristic
45#1319C2212yes—
46#1319C2212yes—
47#1319C2212yes—
48#1319C2212yes—
49#1319C2211yescharacteristic
50#1319C2211yescharacteristic
51C23×Q8C2×C411yescharacteristic
52C23×Q8C2×C411yescharacteristic
53#552—42no—
54#552—42no—
55D4×Q8—42no—
56D4×Q8—42no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×#126D412yes—
63C2×#125C2×C411yescharacteristic
64C2×#125D412yes—
65C2×#123D412yes—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C22⋊Q8C2×C411yescharacteristic
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117D412yes—
78C2×#117C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C42C2×C411yescharacteristic
81#433—42no—
82#433—42no—
83#432—22no—
84#432—22no—
85#431—22no—
86#431—22no—
87#424—42no—
88#407—22no—
89C2×C22⋊C8—22no—
90#405—42no—
91#400—22no—
92#400—22no—
93#400—22no—
94#398—22no—
95#398—22no—
96#398—22no—
97#397—22no—
98#397—22no—
99#395—22no—
100#395—22no—
101#391—22no—
102#391—22no—
103#391C2×C411yescharacteristic
104#391C2×C411yescharacteristic
105#391C2×C411yescharacteristic
106#391C2×C411yescharacteristic
107#391C2×C411yescharacteristic
108#391C2×C411yescharacteristic
109#391C2×C411yescharacteristic
110#391C2311yescharacteristic
111#391C2311yescharacteristic
112#391C2311yescharacteristic
113#389—22no—
114#389—22no—
115#388—21no—
116#387—22no—
117#387—21no—
118#387—22no—
119#387—22no—
120#387—22no—
121#386—21no—
122#385—22no—
123#385—22no—
124#385—21no—
125#384C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#382C2311yescharacteristic
128#381—22no—
129#381—22no—
130#380—22no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C22⋊C4C2×C411yescharacteristic
133C4×C22⋊C4C2×C411yescharacteristic
134#376C2×C411yescharacteristic
135C2×#95—22no—
136#324—22no—
137#324—22no—
138#324—22no—
139#324—22no—
140#324C2312yes—
141#324C2312yes—
142#324C2312yes—
143#324C2312yes—
144#323—22no—
145#323—22no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—41no—
151C22×Q8—41no—
152C22×Q8—42no—
153C22×Q8—41no—
154C22×Q8—41no—
155C22×Q8—81no—
156C22×Q8—81no—
157C22×Q8C22×C411yescharacteristic
158C22×Q8C22×C411yescharacteristic
159C22×Q8C22×C411yescharacteristic
160C22×Q8C22×C411yescharacteristic
161C22×D4C22⋊C411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D412yes—
168C2×M16—42no—
169C22×C8—42no—
170C4⋊Q8—22no—
171C4⋊Q8—41no—
172C4⋊Q8—42no—
173C4⋊Q8—44no—
174C4⋊Q8—81no—
175C4⋊Q8—81no—
176#126—42no—
177#126—42no—
178#125—42no—
179#125—42no—
180#125—42no—
181#123—42no—
182#123—42no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C22⋊Q8—44no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C4×Q8—42no—
193C4×Q8—42no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—42no—
197C4×D4—42no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—22no—
202#117—42no—
203#117—81no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C22⋊C411yescharacteristic
235C2×C22⋊C4C22⋊C412yes—
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C2×D412yes—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—81no—
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258#97—42no—
259#97—42no—
260C4×C8—42no—
261C4×C8—42no—
262#95—22no—
263#95—22no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279C24C2×C22⋊C411yescharacteristic
280C24C2×C22⋊C411yescharacteristic
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8—81no—
301C2×Q8—81no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—82no—
305C2×Q8—82no—
306C2×Q8—82no—
307C2×Q8—82no—
308C2×Q8#13112yes—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—42no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—81no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D412yes—
351M16—84no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—84no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—81no—
392C22⋊C4#9912yes—
393C22⋊C4#9912yes—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—21no—
400C42—21no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—42no—
405C42—81no—
406C23—21no—
407C23—21no—
408C23—22no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23C2×#9911yescharacteristic
416C23C2×#9911yescharacteristic
417C23C22×C22⋊C411yescharacteristic
418Q8—42no—
419Q8—42no—
420Q8—81no—
421Q8—81no—
422Q8—84no—
423Q8—81no—
424Q8—81no—
425Q8—81no—
426Q8—81no—
427Q8—84no—
428D4—42no—
429D4—42no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4C2×#9912yes—
485C2×C4C2×#9912yes—
486C2×C4C2×#13111yescharacteristic
487C2×C4#56811yescharacteristic
488C8—82no—
489C8—82no—
490C8—82no—
491C8—82no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—81no—
505C22—81no—
506C22#212912yes—
507C22#288011yescharacteristic
508C22C22×#9911yescharacteristic
509C22#288811yescharacteristic
510C4—21no—
511C4—21no—
512C4—22no—
513C4—22no—
514C4—42no—
515C4—41no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—41no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—81no—
528C2—21no—
529C2—21no—
530C2—22no—
531C2—41no—
532C2—81no—
533C2#1741111yescharacteristic
534C2C2×#212911yescharacteristic
535C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#552 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11568C2semidirect
4. #11556C2semidirect
5. #11551C2semidirect
6. #11536C2semidirect
7. #11510C2semidirect
8. #11506C2semidirect
9. #11482C2semidirect
10. #11434C2semidirect
11. C2×#1319C2semidirect

Maximal subgroups

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

subgroupindex
#380812
#174172
#115682
#115562
#115512
#115362
#115102
#115062
#114822
#114342
C2×#13192