Finite Group Databasebeta

← #197549

Subgroups — #197549

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic
3#17578C212yesmaximal
4C2×#1463C211yesmaximal, characteristic
5#9202C214yesmaximal, complemented
6#9200C214yesmaximal, complemented
7#7983C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11#2627—21no—
12#2627—21no—
13#2627—22no—
14#2627C414yescomplemented
15#2627C2212yes—
16#2627C2212yes—
17C2×#400C412yescomplemented
18C2×#398—22no—
19C2×#395C412yescomplemented
20#2345—24no—
21#2344—24no—
22#2316C2212yes—
23#2313—21no—
24#2313C2212yes—
25C2×#385—21no—
26#2280—21no—
27#2280—21no—
28#2280C2212yes—
29C2×#381—21no—
30C2×#380C2211yescharacteristic
31#2274—21no—
32#1520C2218yes—
33#1511—24no—
34#1508—24no—
35#1463—22no—
36#1463—22no—
37C2×#324C2212yes—
38C2×#324C2212yes—
39#1289C2218yes—
40C23×Q8C2×C411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8C2312yes—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74#407—22no—
75#407—22no—
76#407—22no—
77#407—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8C2314yes—
80#400—22no—
81#400—22no—
82#400C2×C414yes—
83#400C2×C414yes—
84#398—24no—
85#398—24no—
86#398—24no—
87#398—24no—
88#395—22no—
89#395—22no—
90#395C2×C414yes—
91#395C2×C414yes—
92#389—28no—
93#387—28no—
94#387—28no—
95#385—24no—
96#385—22no—
97#385—22no—
98#381—22no—
99#381—24no—
100#381—22no—
101#381—22no—
102#381—22no—
103#381D414yes—
104#380—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—21no—
110#380—21no—
111#380C2×C412yes—
112#380C2312yes—
113#380C2314yes—
114#379—22no—
115C4×C4⋊C4—28no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4D414yes—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95C2×C412yes—
126#328—216no—
127#324—24no—
128#324—24no—
129C25C2×D411yescharacteristic
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—82no—
134C22×Q8C22×C412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—42no—
148C23×C4—82no—
149C23×C4—82no—
150C23×C4C22⋊C412yes—
151C23×C4C22⋊C412yes—
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4#4112yes—
166C23×C4#4112yes—
167C23×C4C2411yescharacteristic
168C2×M16—42no—
169C2×M16—42no—
170C2×M16—84no—
171C22×C8—24no—
172C22⋊Q8—24no—
173C22⋊Q8—24no—
174C22⋊Q8—24no—
175C22⋊Q8—24no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180#117—44no—
181#117—44no—
182#117—84no—
183#117—84no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—84no—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C22⋊C8—24no—
241C22⋊C8—24no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—24no—
256#95C22×C414yes—
257#95C2×D418yes—
258C24—22no—
259C24—22no—
260C24—42no—
261C24—42no—
262C24C2×C22⋊C411yescharacteristic
263C24C2×C22⋊C411yescharacteristic
264C24#12011yescharacteristic
265C24#12011yescharacteristic
266C24#12311yescharacteristic
267C24#12311yescharacteristic
268C24C22×D411yescharacteristic
269C2×Q8—24no—
270C2×Q8—42no—
271C2×Q8—84no—
272C2×Q8—82no—
273C2×Q8—82no—
274C2×Q8—84no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—28no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—82no—
324C22×C4—81no—
325C22×C4—82no—
326C22×C4—81no—
327C22×C4—82no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C4×D412yes—
340C22×C4C4×D412yes—
341C22×C4C4×D412yes—
342C22×C4C4×D412yes—
343C22×C4#12012yes—
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#12312yes—
349C22×C4#13714yes—
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D412yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C2×#4112yes—
354M16—84no—
355M16—84no—
356C2×C8—44no—
357C2×C8—84no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—48no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C22⋊C4—48no—
378C22⋊C4—44no—
379C22⋊C4—48no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C42—48no—
388C42—84no—
389C42—84no—
390C23—21no—
391C23—21no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—21no—
397C23—21no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—48no—
413C23—48no—
414C23#38611yescharacteristic
415C23#38611yescharacteristic
416C23#38611yescharacteristic
417C23#38611yescharacteristic
418C23C22×C22⋊C411yescharacteristic
419C23C2×#12011yescharacteristic
420C23C2×#12311yescharacteristic
421C23C2×#13712yes—
422C23#57512yes—
423Q8—84no—
424Q8—84no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—88no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4#38612yes—
482C2×C4#38612yes—
483C2×C4#41914yes—
484C2×C4C2×C4×D412yes—
485C2×C4C2×#12112yes—
486C8—84no—
487C22—24no—
488C22—24no—
489C22—21no—
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—21no—
503C22—21no—
504C22—21no—
505C22—42no—
506C22—42no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—42no—
512C22—42no—
513C22#191212yes—
514C22#191312yes—
515C22C2×#38611yescharacteristic
516C22C2×#41911yescharacteristic
517C22#289611yescharacteristic
518C22#300712yes—
519C22#309512yes—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—24no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—88no—
529C4—84no—
530C4—82no—
531C4—82no—
532C2—21no—
533C2—22no—
534C2—21no—
535C2—22no—
536C2—22no—
537C2—22no—
538C2—41no—
539C2—41no—
540C2#2091111yescharacteristic
541C2C2×#191211yescharacteristic
542C2C2×#191311yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9202C2semidirect
6. #9200C2semidirect
14. #2627C4semidirect
17. C2×#400C4semidirect
19. C2×#395C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
#341782
#175782
C2×#14632
#92022
#92002
#79832