Finite Group Databasebeta

← #224580

Subgroups — #224580

Order 512.

967 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36599C211yesmaximal, characteristic
2#22625C211yesmaximal, characteristic
3#13818C211yesmaximal, characteristic
4#9711C211yesmaximal, characteristic
5#9711C212yesmaximal
6#9679C211yesmaximal, characteristic
7#9679C212yesmaximal
8#8578C212yesmaximal
9#8572C211yesmaximal, characteristic
10C4×#328C212yesmaximal
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×#389—21no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1939—21no—
32#1916—22no—
33#1797C2211yescharacteristic
34#1797C2212yes—
35C8×C4⋊C4—21no—
36C4×C22⋊C8C2211yescharacteristic
37C4×C22⋊C8C2212yes—
38C2×#328C2211yescharacteristic
39C2×#328C2212yes—
40C2×#328C2211yescharacteristic
41C2×#328C2212yes—
42C2×#328C2212yes—
43C2×#328C2212yes—
44C2×#328C2212yes—
45#1436—22no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C4×#95C2212yes—
49C4×#95C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2212yes—
52#1294C2211yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C412yes—
55C2×#125—21no—
56C2×C22⋊Q8C2311yescharacteristic
57C2×C22⋊Q8C2312yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C412yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42—22no—
66C22×C42C2311yescharacteristic
67C22×C42C2312yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2312yes—
71C2×C22⋊C8C2311yescharacteristic
72C2×C22⋊C8C2311yescharacteristic
73C2×C4×C8—21no—
74C2×C4×C8—22no—
75#400—22no—
76#400—22no—
77#400—22no—
78#398—22no—
79#398—22no—
80#398—22no—
81#391C2×C412yes—
82#391C2×C412yes—
83#391C2×C412yes—
84#391C2×C411yescharacteristic
85#391C2×C412yes—
86#389—28no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—22no—
93#389—22no—
94#389—21no—
95#389C2×C412yes—
96#389C2×C411yescharacteristic
97#387—22no—
98#387—22no—
99#387—21no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—21no—
106#385—22no—
107#384C2×C411yescharacteristic
108#384C2×C412yes—
109#382—21no—
110#380—21no—
111#380—22no—
112#380—22no—
113#380—21no—
114#380C2311yescharacteristic
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—44no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—22no—
129C4×C22⋊C4—21no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95D412yes—
135C2×#95C2312yes—
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138#336—21no—
139#336—22no—
140#336—21no—
141#336—22no—
142#328—24no—
143#328—22no—
144#328—24no—
145#328—22no—
146#328—42no—
147#328—42no—
148#328—42no—
149C22×Q8C22×C411yescharacteristic
150C22×Q8C22×C412yes—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2411yescharacteristic
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C4⋊Q8—42no—
163C4⋊Q8—82no—
164#125—44no—
165C22⋊Q8—21no—
166C22⋊Q8—22no—
167C22⋊Q8—21no—
168C22⋊Q8—22no—
169C22⋊Q8—21no—
170C22⋊Q8—22no—
171C22⋊Q8—21no—
172C22⋊Q8—22no—
173C4×Q8—42no—
174C4×Q8—82no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4SD1614yes—
203C2×C4⋊C4Q1614yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C42—24no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—44no—
220C2×C42—44no—
221C2×C42—82no—
222C2×C42—82no—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C412yes—
225C2×C42C22×C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C4⋊C8—42no—
232C4⋊C8—82no—
233C22⋊C8—21no—
234C22⋊C8—22no—
235C22⋊C8—21no—
236C22⋊C8—22no—
237C22⋊C8—21no—
238C22⋊C8#4112yes—
239C4×C8—42no—
240C4×C8—82no—
241#95—28no—
242#95—21no—
243#95—21no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—21no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—21no—
257#95—21no—
258#95—22no—
259#95—21no—
260#95—22no—
261#95—21no—
262#95—42no—
263#95—44no—
264#95—81no—
265#95C22⋊C412yes—
266#95C22⋊C412yes—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269#95#4112yes—
270C24C22×D411yescharacteristic
271C2×Q8—21no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—41no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×SD1612yes—
329C22×C4C2×SD1614yes—
330C22×C4C2×Q1612yes—
331C22×C4C2×Q1614yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C42−1+411yescharacteristic
338C2×C8—22no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—48no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—82no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—88no—
373C4⋊C4#10318yes—
374C4⋊C4#10314yes—
375C4⋊C4#10314yes—
376C4⋊C4#10414yes—
377C22⋊C4—42no—
378C22⋊C4—41no—
379C22⋊C4—42no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C42—22no—
384C42—22no—
385C42—42no—
386C42—42no—
387C42—41no—
388C42—41no—
389C42—42no—
390C42—42no—
391C42—48no—
392C42—42no—
393C42—42no—
394C42—82no—
395C42—82no—
396C42—88no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23C22×C22⋊C411yescharacteristic
402C23#54811yescharacteristic
403C23#55011yescharacteristic
404C23C22×SD1611yescharacteristic
405C23C22×Q1611yescharacteristic
406C23#57711yescharacteristic
407C23#57811yescharacteristic
408Q8—81no—
409Q8—82no—
410Q8—81no—
411Q8—82no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4#41012yes—
485C2×C4C2×#10312yes—
486C2×C4C2×#10314yes—
487C2×C4C2×#10314yes—
488C2×C4C2×#10312yes—
489C2×C4C2×#10414yes—
490C2×C4C2×#10412yes—
491C2×C4C4×#4111yescharacteristic
492C2×C4#52011yescharacteristic
493C2×C4#54811yescharacteristic
494C2×C4#55011yescharacteristic
495C8—42no—
496C8—81no—
497C8—82no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22#234111yescharacteristic
511C22C2×#41011yescharacteristic
512C22#288311yescharacteristic
513C22C22×#10311yescharacteristic
514C22#289811yescharacteristic
515C22C22×#10411yescharacteristic
516C22#290111yescharacteristic
517C22#335911yescharacteristic
518C22#336311yescharacteristic
519C22#339211yescharacteristic
520C22#339511yescharacteristic
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—22no—
526C4—24no—
527C4—24no—
528C4—24no—
529C4—44no—
530C4—42no—
531C4—41no—
532C4—41no—
533C4—82no—
534C4—84no—
535C4—81no—
536C4—82no—
537C4—81no—
538C4—82no—
539C4—81no—
540C4—81no—
541C4#150314yes—
542C2—21no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2C2×#150311yescharacteristic
547C2#1754811yescharacteristic
548C2#1760511yescharacteristic
549C2#1761411yescharacteristic
550C2#2253511yescharacteristic
551C2#2254211yescharacteristic
552C2#2255011yescharacteristic
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
#226252
#138182
#97112
#96792
#85782
#85722
C4×#3282