Finite Group Databasebeta

← #196277

Subgroups — #196277

Order 512.

1113 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic, complemented
2#35246C211yesmaximal, characteristic
3#17465C212yesmaximal
4C2×#1461C211yesmaximal, characteristic, complemented
5#9106C212yesmaximal, complemented
6#9104C212yesmaximal, complemented
7#9102C212yesmaximal, complemented
8#9100C212yesmaximal, complemented
9#7858C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2984—24no—
13#2983—22no—
14#2983C2212yes—
15#2982C2212yes—
16Q8×M16—24no—
17#2920—22no—
18#2920C2212yes—
19C2×#431C2212yes—
20#2853—22no—
21#2637—22no—
22#2599—22no—
23#2579—22no—
24#2565C411yescharacteristic
25#2565C411yescharacteristic
26#2545C411yescharacteristic
27#2545C411yescharacteristic
28#2504C411yescharacteristic
29#2502C411yescharacteristic
30#2436C411yescharacteristic
31#2434C411yescharacteristic
32#2327C2211yescharacteristic
33#2327C2211yescharacteristic
34#2326C2211yescharacteristic
35#2326C2211yescharacteristic
36#2283C2211yescharacteristic
37#1483—24no—
38#1482—24no—
39#1479C2214yes—
40#1478C2214yes—
41#1461—22no—
42#1461C2212yes—
43#1461C2212yes—
44C2×#324—22no—
45#1274C2214yes—
46#1274C2214yes—
47C23×Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50#529—22no—
51#529—22no—
52#529—22no—
53#529—22no—
54C2×#126—21no—
55C2×#125C2×C411yescharacteristic
56C2×#124C2×C411yescharacteristic
57C2×#124C2×C411yescharacteristic
58C2×#124C2×C411yescharacteristic
59C2×#123—21no—
60C2×C22⋊Q8C2×C411yescharacteristic
61C2×#121C2×C411yescharacteristic
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D412yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8C2311yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4D412yes—
78C22×C42D412yes—
79C22×C42C2311yescharacteristic
80#446—216no—
81#446—28no—
82C8×Q8—28no—
83#432—24no—
84#432—24no—
85#432—22no—
86#432—22no—
87#431—24no—
88#431—24no—
89#431—22no—
90#431—22no—
91#431—22no—
92#431D414yes—
93#431C2314yes—
94#431C2314yes—
95#423—24no—
96#423—22no—
97C2×C4⋊C8—22no—
98#407—22no—
99C2×C22⋊C8—22no—
100C4×M16—24no—
101C4×M16D414yes—
102C2×C4×C8—22no—
103#398—22no—
104#398—22no—
105#398—22no—
106#397—22no—
107#396—22no—
108#396—22no—
109#395—22no—
110#393—22no—
111#391C2×C411yescharacteristic
112#391C2×C411yescharacteristic
113#391C2×C411yescharacteristic
114#391C2×C411yescharacteristic
115#391C2×C411yescharacteristic
116#391C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#390C2×C411yescharacteristic
119#388—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#387—22no—
124#387—22no—
125#387—22no—
126#387—22no—
127#387—22no—
128#387—22no—
129#386—22no—
130#386—22no—
131#385—22no—
132#385—22no—
133#385—22no—
134#385—22no—
135#384C2×C411yescharacteristic
136#384C2×C411yescharacteristic
137#383—21no—
138#382—21no—
139#382C2×C411yescharacteristic
140#382C2×C411yescharacteristic
141#381—21no—
142#380—21no—
143C4×C4⋊C4C2×C411yescharacteristic
144C4×C4⋊C4C2×C411yescharacteristic
145C4×C22⋊C4C2×C411yescharacteristic
146C4×C22⋊C4C2×C411yescharacteristic
147#376C2311yescharacteristic
148#376C2311yescharacteristic
149Q8⋊C8—28no—
150D4⋊C8—28no—
151#324—24no—
152#324—24no—
153C22×Q8—24no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—42no—
157C22×Q8C22⋊C412yes—
158C22×Q8C22×C411yescharacteristic
159C22×Q8C22×C411yescharacteristic
160C22×Q8C22×C411yescharacteristic
161C22×D4—21no—
162C22×D4C22×C411yescharacteristic
163C23×C4—21no—
164C23×C4C22⋊C412yes—
165C23×C4C22⋊C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×M16—24no—
170C22×C8—42no—
171C4⋊Q8—24no—
172C4⋊Q8—42no—
173#126—42no—
174#126—42no—
175#125—22no—
176#125—22no—
177#124—24no—
178#124—22no—
179#124—22no—
180#124—42no—
181#123—41no—
182#123—41no—
183#123—41no—
184#123—41no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187#121—22no—
188#121—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—42no—
206C4×D4—42no—
207#117—42no—
208#117—42no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42C22⋊C412yes—
249C2×C42C22⋊C412yes—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42C2411yescharacteristic
264C4⋊C8—28no—
265C4⋊C8—28no—
266C4⋊C8—24no—
267C4⋊C8—24no—
268C4⋊C8—24no—
269C4⋊C8—42no—
270C22⋊C8—24no—
271C22⋊C8—24no—
272C22⋊C8—24no—
273C22⋊C8—42no—
274#97—28no—
275#97—24no—
276C4×C8—28no—
277C4×C8—24no—
278C4×C8—42no—
279C4×C8C2×D418yes—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296C24—21no—
297C24C2×C22⋊C411yescharacteristic
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—42no—
309C2×Q8—48no—
310C2×Q8—48no—
311C2×Q8#13114yes—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—44no—
316C2×D4—41no—
317C2×D4—41no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—81no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4#12311yescharacteristic
352C22×C4#12311yescharacteristic
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356M16—48no—
357C2×C8—28no—
358C2×C8—44no—
359C2×C8—84no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—44no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C22⋊C4—44no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—42no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—42no—
392C22⋊C4—44no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—42no—
408C42C4×D412yes—
409C42C4×D412yes—
410C42#12012yes—
411C42#12112yes—
412C42#12112yes—
413C42#12312yes—
414C42C22×D412yes—
415C42C2×#4112yes—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—41no—
420C23—42no—
421C23—42no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23#38611yescharacteristic
426C23#38611yescharacteristic
427C23C22×C22⋊C411yescharacteristic
428Q8—28no—
429Q8—48no—
430Q8—44no—
431Q8—44no—
432D4—44no—
433D4—82no—
434D4—82no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—84no—
481C2×C4—82no—
482C2×C4#38611yescharacteristic
483C2×C4#38611yescharacteristic
484C2×C4#38612yes—
485C2×C4#38612yes—
486C2×C4#40814yes—
487C2×C4C2×C4×D412yes—
488C2×C4C2×#12011yescharacteristic
489C2×C4C2×#12112yes—
490C2×C4C2×#12311yescharacteristic
491C2×C4C2×#13112yes—
492C2×C4#56812yes—
493C8—48no—
494C8—84no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—42no—
504C22—82no—
505C22—82no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22#191212yes—
510C22C2×#38611yescharacteristic
511C22C2×#40811yescharacteristic
512C22#287911yescharacteristic
513C4—22no—
514C4—22no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—22no—
520C4—24no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4#293314yes—
532C2—21no—
533C2—21no—
534C2—22no—
535C2—41no—
536C2—161no—
537C2#2089711yescharacteristic
538C2#2100911yescharacteristic
539C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60248C2semidirect
4. C2×#1461C2semidirect
5. #9106C2semidirect
6. #9104C2semidirect
7. #9102C2semidirect
8. #9100C2semidirect
9. #7858C2semidirect

Maximal subgroups

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

subgroupindex
#602482
#352462
#174652
C2×#14612
#91062
#91042
#91022
#91002
#78582