Finite Group Databasebeta

← #196955

Subgroups — #196955

Order 512.

1394 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35311C211yesmaximal, characteristic
2C2×#2343C211yesmaximal, characteristic
3#20685C211yesmaximal, characteristic
4C2×#1511C211yesmaximal, characteristic
5C22×#328C211yesmaximal, characteristic
6#9153C212yesmaximal, complemented
7#9152C212yesmaximal, complemented
8#9145C212yesmaximal, complemented
9#9144C212yesmaximal, complemented
10#8024C211yesmaximal, characteristic
11#8020C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C2×#407C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2855C2212yes—
17#2855C2212yes—
18#2625C414yescomplemented
19#2503—21no—
20#2502—21no—
21#2499—22no—
22#2484—22no—
23#2483—21no—
24#2483—21no—
25#2343—24no—
26#2343—24no—
27#2317C412yescomplemented
28#2310C412yescomplemented
29C2×#387—21no—
30C2×#387—21no—
31C2×#385—21no—
32C2×#383C2211yescharacteristic
33C2×#382C2211yescharacteristic
34#2280C2211yescharacteristic
35C2×#381C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4—21no—
38C2×#376C2211yescharacteristic
39C22×#95C2211yescharacteristic
40#1818—22no—
41#1817C2212yes—
42#1817C2212yes—
43#1729—22no—
44#1511—24no—
45C2×#328—24no—
46C2×#328C2212yes—
47C2×#328C2212yes—
48C2×#328C2212yes—
49C2×#328C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54C23×Q8C2×C411yescharacteristic
55C24×C4—21no—
56C24×C4C2311yescharacteristic
57C22×M16—21no—
58C23×C8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8C2×C412yes—
66C2×C22⋊Q8C2312yes—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117C2×C412yes—
72C2×#117C2×C412yes—
73C2×#117C2×C412yes—
74C2×#117D412yes—
75C2×#117D412yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#407—22no—
91#407—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8C2312yes—
97C2×C22⋊C8C2312yes—
98#400—24no—
99#398—24no—
100#397—24no—
101#397—24no—
102#391—22no—
103#391—22no—
104#391C2×C414yescomplemented
105#391C2×C414yescomplemented
106#389C2×C414yescomplemented
107#389C2×C414yescomplemented
108#387—28no—
109#387—28no—
110#387—24no—
111#387—24no—
112#385—28no—
113#385—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#384—22no—
118#383—22no—
119#383—22no—
120#382—22no—
121#382—22no—
122#382—22no—
123#381—22no—
124#381—22no—
125#381—22no—
126#381—21no—
127#381—22no—
128#381—22no—
129#381D414yes—
130#380C2×C412yes—
131#380C2312yes—
132#379—21no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C22⋊C4—28no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138#376—22no—
139#376—22no—
140C2×#95—22no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—21no—
147C2×#95D414yes—
148C2×#95C2311yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152#336—24no—
153#336—24no—
154#328—24no—
155#328—24no—
156#328—24no—
157#328—24no—
158C25C2×D411yescharacteristic
159C22×Q8—21no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—44no—
163C22×Q8C22×C412yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—24no—
170C23×C4—21no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—82no—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4#4112yes—
187C23×C4#4112yes—
188C23×C4C2411yescharacteristic
189C2×M16—44no—
190C2×M16—42no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—44no—
194C22×C8—42no—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204C22⋊Q8—42no—
205#117—24no—
206#117—24no—
207#117—24no—
208#117—24no—
209#117—24no—
210#117—24no—
211#117—24no—
212#117—24no—
213#117—44no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4C22⋊C412yes—
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22⋊C412yes—
234C2×C4⋊C4C22⋊C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C414yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4C22⋊C412yes—
251C2×C22⋊C4C22⋊C412yes—
252C2×C22⋊C4C22⋊C412yes—
253C2×C22⋊C4C22⋊C412yes—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—44no—
266C2×C42—82no—
267C2×C42C22×C412yes—
268C2×C42C22×C412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—22no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—22no—
295#95C2×D412yes—
296#95C2×D412yes—
297#95C2×D412yes—
298#95C2×D412yes—
299C24—22no—
300C24—22no—
301C24—24no—
302C24—42no—
303C24C2×C22⋊C411yescharacteristic
304C24C2×C22⋊C411yescharacteristic
305C24#12012yes—
306C24#12312yes—
307C24C22×D411yescharacteristic
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—42no—
313C2×Q8—84no—
314C2×Q8—84no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—48no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—48no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—82no—
364C22×C4—84no—
365C22×C4—816no—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C4×D412yes—
373C22×C4C4×D412yes—
374C22×C4C4×D412yes—
375C22×C4C4×D412yes—
376C22×C4#12012yes—
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4#12112yes—
381C22×C4#12312yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388M16—84no—
389C2×C8—44no—
390C2×C8—44no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—84no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—44no—
401C4⋊C4—48no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—48no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—82no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—48no—
430C42—44no—
431C42—84no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—42no—
445C23—42no—
446C23—48no—
447C23—48no—
448C23#38612yes—
449C23#38612yes—
450C23#41014yes—
451C23C22×C22⋊C411yescharacteristic
452C23C2×#12011yescharacteristic
453C23C2×#12311yescharacteristic
454C23#57712yes—
455C23#57812yes—
456Q8—82no—
457Q8—82no—
458Q8—84no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—816no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4#38612yes—
522C2×C4#38612yes—
523C2×C4C2×C4×D411yescharacteristic
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C2×#12111yescharacteristic
526C2×C4C2×#12111yescharacteristic
527C8—82no—
528C8—82no—
529C22—22no—
530C22—24no—
531C22—24no—
532C22—28no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—21no—
537C22—21no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—44no—
542C22—44no—
543C22—48no—
544C22—44no—
545C22#189914yes—
546C22C2×#38611yescharacteristic
547C22C2×#41012yes—
548C22#289812yes—
549C22#290112yes—
550C22#300911yescharacteristic
551C22#301111yescharacteristic
552C22#309712yes—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—22no—
558C4—22no—
559C4—44no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—84no—
568C4—82no—
569C4—82no—
570C2—21no—
571C2—21no—
572C2—22no—
573C2—24no—
574C2—42no—
575C2#1760914yes—
576C2#2091311yescharacteristic
577C2#2098011yescharacteristic
578C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9153C2semidirect
7. #9152C2semidirect
8. #9145C2semidirect
9. #9144C2semidirect
18. #2625C4semidirect
27. #2317C4semidirect
28. #2310C4semidirect
104. #391C2×C4semidirect
105. #391C2×C4semidirect
106. #389C2×C4semidirect
107. #389C2×C4semidirect

Maximal subgroups

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

subgroupindex
#353112
C2×#23432
#206852
C2×#15112
C22×#3282
#91532
#91522
#91452
#91442
#80242
#80202