Finite Group Databasebeta

← #224835

Subgroups — #224835

Order 512.

839 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36599C211yesmaximal, characteristic
2#22629C211yesmaximal, characteristic
3#13936C211yesmaximal, characteristic
4#9714C212yesmaximal
5#9711C211yesmaximal, characteristic
6#9683C212yesmaximal
7#9679C211yesmaximal, characteristic
8#8574C212yesmaximal
9#8572C211yesmaximal, characteristic
10#8217C212yesmaximal
11C4×#328C211yesmaximal, characteristic
12#2857C2211yescharacteristic
13#2857C2212yes—
14#2585—22no—
15#2569—22no—
16#2552C411yescharacteristic, complemented
17#2546C411yescharacteristic, complemented
18#2511C412yescomplemented
19#2509—21no—
20#2446C412yescomplemented
21#2442—21no—
22#2370—21no—
23#2365C411yescharacteristic, complemented
24#2355C411yescharacteristic, complemented
25#2345—21no—
26C4×C22⋊Q8C2211yescharacteristic
27C4×C22⋊Q8C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1939—21no—
30#1917—22no—
31#1797C2211yescharacteristic
32#1797C2212yes—
33#1770—21no—
34#1749C2212yes—
35C4×C22⋊C8C2211yescharacteristic
36C2×#328C2211yescharacteristic
37C2×#328C2211yescharacteristic
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41C2×#328C2212yes—
42C2×#328C2212yes—
43#1443—22no—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95C2211yescharacteristic
47#1434C2212yes—
48#1434C2212yes—
49C4×#95C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C412yes—
55C2×#125—21no—
56C2×C22⋊Q8C2312yes—
57C2×C22⋊Q8C2311yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C412yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65C22×C42C2312yes—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8C2311yescharacteristic
69C2×C22⋊C8C2312yes—
70C2×C22⋊C8C2311yescharacteristic
71C2×#97—22no—
72C2×C4×C8—21no—
73#400—22no—
74#400—22no—
75#400—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#391C2×C412yes—
80#391C2×C412yes—
81#391C2×C412yes—
82#391C2×C411yescharacteristic
83#391C2×C412yes—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—22no—
90#389—22no—
91#389—21no—
92#389C2×C412yes—
93#389C2×C411yescharacteristic
94#387—22no—
95#387—22no—
96#387—21no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—21no—
103#385—22no—
104#384C2×C411yescharacteristic
105#384C2×C412yes—
106#382—21no—
107#380—21no—
108#380—22no—
109#380—22no—
110#380—21no—
111#380C2311yescharacteristic
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—22no—
125C4×C22⋊C4—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95D412yes—
133C2×#95C2311yescharacteristic
134C2×#95C2312yes—
135C2×#95C2311yescharacteristic
136#336—21no—
137#336—22no—
138#336—21no—
139#336—22no—
140#328—22no—
141#328—24no—
142#328—22no—
143#328—24no—
144#328—42no—
145#328—42no—
146#328—42no—
147C22×Q8C22×C411yescharacteristic
148C22×Q8C22×C412yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C4⋊Q8—42no—
161C4⋊Q8—82no—
162#125—44no—
163C22⋊Q8—22no—
164C22⋊Q8—21no—
165C22⋊Q8—22no—
166C22⋊Q8—21no—
167C22⋊Q8—22no—
168C22⋊Q8—21no—
169C22⋊Q8—22no—
170C22⋊Q8—21no—
171C4×Q8—42no—
172C4×Q8—82no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—82no—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C412yes—
217C2×C42C22×C412yes—
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C4⋊C8—42no—
224C4⋊C8—82no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—22no—
228C22⋊C8—21no—
229C22⋊C8—22no—
230C22⋊C8#4112yes—
231#97—82no—
232C4×C8—42no—
233#95—21no—
234#95—21no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—21no—
252#95—22no—
253#95—42no—
254#95—44no—
255#95—82no—
256#95—81no—
257#95—82no—
258#95—82no—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262#95C22⋊C412yes—
263#95#4112yes—
264C24C22×D411yescharacteristic
265C2×Q8—21no—
266C2×Q8—22no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—41no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4#13712yes—
325C22×C4#13712yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C42−1+411yescharacteristic
332C2×C8—22no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—81no—
364C4⋊C4—81no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—84no—
370C22⋊C4—42no—
371C22⋊C4—41no—
372C22⋊C4—42no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C42—22no—
377C42—22no—
378C42—42no—
379C42—42no—
380C42—41no—
381C42—41no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—82no—
387C42—82no—
388C42—84no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23C22×C22⋊C411yescharacteristic
394C23#54811yescharacteristic
395C23#55011yescharacteristic
396C23C2×#13711yescharacteristic
397C23C2×#13711yescharacteristic
398C23#57711yescharacteristic
399C23#57811yescharacteristic
400Q8—81no—
401Q8—82no—
402Q8—81no—
403Q8—82no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4#41012yes—
480C2×C4#41912yes—
481C2×C4#41912yes—
482C2×C4#42112yes—
483C2×C4C4×#4111yescharacteristic
484C2×C4#52011yescharacteristic
485C2×C4#54811yescharacteristic
486C2×C4#55011yescharacteristic
487C8—42no—
488C8—81no—
489C8—82no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22#234111yescharacteristic
503C22C2×#41011yescharacteristic
504C22#288311yescharacteristic
505C22C2×#41911yescharacteristic
506C22#289811yescharacteristic
507C22C2×#42111yescharacteristic
508C22#290111yescharacteristic
509C22#337211yescharacteristic
510C22#337411yescharacteristic
511C22#339811yescharacteristic
512C22#340011yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—22no—
518C4—42no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—42no—
523C4—42no—
524C4—82no—
525C4—82no—
526C4—81no—
527C4—82no—
528C4—81no—
529C4—82no—
530C4—81no—
531C4—81no—
532C4—162no—
533C4#151514yes—
534C2—21no—
535C2—21no—
536C2—21no—
537C2—21no—
538C2C2×#151511yescharacteristic
539C2#1760011yescharacteristic
540C2#1760511yescharacteristic
541C2#1762711yescharacteristic
542C2#2253511yescharacteristic
543C2#2254711yescharacteristic
544C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2552C4semidirect
17. #2546C4semidirect
18. #2511C4semidirect
20. #2446C4semidirect
23. #2365C4semidirect
24. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#365992
#226292
#139362
#97142
#97112
#96832
#96792
#85742
#85722
#82172
C4×#3282