Finite Group Databasebeta

← #164239

Subgroups — #164239

Order 512.

1021 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36275C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic
3#20378C211yesmaximal, characteristic
4#7701C212yesmaximal, complemented
5#7688C211yesmaximal, characteristic, complemented
6#7675C212yesmaximal, complemented
7#7665C211yesmaximal, characteristic, complemented
8#7627C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic, complemented
10#7573C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic, complemented
12C2×#554C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#446—21no—
15C2×#408—21no—
16C2×#405—21no—
17#2857C2211yescharacteristic
18#2855C2212yes—
19#2582C411yescharacteristic
20#2567C411yescharacteristic
21#2487C412yes—
22#2453—21no—
23#2449—22no—
24#2441—21no—
25#2438—22no—
26#2417C412yes—
27D4×C4⋊C4—21no—
28D4×C4⋊C4C411yescharacteristic
29#2339—21no—
30C4×C22⋊Q8C2211yescharacteristic
31C2×#389C2212yes—
32#1955—22no—
33#1954—21no—
34#1797C2212yes—
35#1797C2211yescharacteristic
36#1750—22no—
37#1749C2212yes—
38#1749C2212yes—
39C4×C22⋊C8C2211yescharacteristic
40C4×C22⋊C8C2212yes—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2212yes—
52#1271C2212yes—
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56#554—44no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#125—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×C22⋊Q8C2×C412yes—
63C4×#41—44no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C412yes—
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D412yes—
73C2×#117D411yescharacteristic
74C2×#117D412yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2×C412yes—
77C22×C4⋊C4C2×C412yes—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#446—44no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83#408—28no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—21no—
90#405—44no—
91C2×#97—22no—
92C2×#97—22no—
93C2×C4×C8—22no—
94C2×C4×C8—21no—
95#400—22no—
96#399—22no—
97#398—22no—
98#398—22no—
99#396—22no—
100#395—22no—
101#391—21no—
102#391—21no—
103#391—21no—
104#391C2×C412yes—
105#391C2311yescharacteristic
106#389—21no—
107#389—21no—
108#389—21no—
109#389—22no—
110#389—22no—
111#389C2×C412yes—
112#389C2×C412yes—
113#389C2×C412yes—
114#389C2312yes—
115#389C2311yescharacteristic
116#387—22no—
117#387—22no—
118#387—21no—
119#386—22no—
120#386—22no—
121#385—22no—
122#385—21no—
123#385—22no—
124#385—22no—
125#384—21no—
126#384C2×C411yescharacteristic
127#381—22no—
128#380—21no—
129#380—22no—
130#380C2×C412yes—
131#380C2×C412yes—
132#380C2×C411yescharacteristic
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C22⋊C4—21no—
136C2×#95—22no—
137#336—21no—
138#336—21no—
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2312yes—
144#336C2311yescharacteristic
145C2×#41—24no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—42no—
149C22×Q8—21no—
150C22×Q8C22⋊C411yescharacteristic
151C22×Q8C22×C411yescharacteristic
152C22×D4C22⋊C412yes—
153C22×D4C22⋊C411yescharacteristic
154C23×C4—22no—
155C23×C4—22no—
156C23×C4C22×C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C2×M16—24no—
163C2×M16—42no—
164C2×M16—42no—
165C22×C8—24no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—42no—
173C22×C8—42no—
174C4⋊Q8—42no—
175#125—44no—
176C22⋊Q8—22no—
177C22⋊Q8—48no—
178C22⋊Q8—41no—
179C22⋊Q8—44no—
180C22⋊Q8—81no—
181C22⋊Q8—81no—
182C4×Q8—24no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×D4—24no—
186C4×D4—44no—
187C4×D4—42no—
188C4×D4—44no—
189C4×D4—44no—
190C4×D4—44no—
191#117—24no—
192#117—44no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22⋊C412yes—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—44no—
231C2×C42—42no—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2411yescharacteristic
239C4⋊C8—42no—
240C4⋊C8—44no—
241C22⋊C8—28no—
242C22⋊C8—28no—
243C22⋊C8—44no—
244C22⋊C8—42no—
245C22⋊C8—82no—
246C22⋊C8—81no—
247#97—44no—
248#97—44no—
249C4×C8—44no—
250C4×C8—42no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—21no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—21no—
261#95—22no—
262#95—21no—
263#95—21no—
264C24C2×C22⋊C411yescharacteristic
265C24C2×C22⋊C412yes—
266#41—48no—
267#41—48no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—81no—
277C2×Q8—81no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—42no—
283C2×D4#10414yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4#13712yes—
325C22×C4#13712yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329M16—48no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—48no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—42no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—44no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—81no—
373C4⋊C4#9914yes—
374C4⋊C4#13112yes—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—81no—
382C22⋊C4—82no—
383C42—24no—
384C42—22no—
385C42—24no—
386C42—22no—
387C42—41no—
388C42—42no—
389C42—41no—
390C42—42no—
391C42—42no—
392C23—21no—
393C23—22no—
394C23—21no—
395C23—22no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—21no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23—22no—
404C23#41912yes—
405C23C2×#10414yes—
406C23C22×C22⋊C411yescharacteristic
407Q8—44no—
408Q8—81no—
409Q8—81no—
410Q8—162no—
411D4—28no—
412D4—48no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—88no—
464C2×C4—164no—
465C2×C4C2×#9914yes—
466C2×C4C2×#9912yes—
467C2×C4#41912yes—
468C2×C4C2×#10412yes—
469C2×C4C2×#13111yescharacteristic
470C2×C4#56811yescharacteristic
471C2×C4C2×#13711yescharacteristic
472C2×C4C2×#13711yescharacteristic
473C2×C4#57611yescharacteristic
474C2×C4#57711yescharacteristic
475C8—42no—
476C8—42no—
477C8—42no—
478C8—44no—
479C8—81no—
480C8—82no—
481C22—22no—
482C22—22no—
483C22—24no—
484C22—22no—
485C22—24no—
486C22—24no—
487C22—42no—
488C22—41no—
489C22—42no—
490C22—42no—
491C22—41no—
492C22—42no—
493C22—41no—
494C22—41no—
495C22#288011yescharacteristic
496C22C22×#9911yescharacteristic
497C22#288911yescharacteristic
498C22C2×#41911yescharacteristic
499C22#289711yescharacteristic
500C22C22×#10411yescharacteristic
501C22#290111yescharacteristic
502C4—21no—
503C4—21no—
504C4—24no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—41no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—41no—
514C4—81no—
515C4—82no—
516C4—81no—
517C4—82no—
518C4—81no—
519C4#150814yes—
520C2—42no—
521C2—44no—
522C2#1741611yescharacteristic
523C2#1751411yescharacteristic
524C2#1751711yescharacteristic
525C2C2×#150811yescharacteristic
526C2#1761111yescharacteristic
527C2#1765311yescharacteristic
528C2#1776111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7701C2semidirect
5. #7688C2semidirect
6. #7675C2semidirect
7. #7665C2semidirect
8. #7627C2semidirect
9. #7617C2semidirect
10. #7573C2semidirect
11. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#362752
#225322
#203782
#77012
#76882
#76752
#76652
#76272
#76172
#75732
#75662