Finite Group Databasebeta

← #188413

Subgroups — #188413

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34128C211yesmaximal, characteristic
2#21974C211yesmaximal, characteristic
3#20809C211yesmaximal, characteristic
4C2×#1487C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic
6#8570C212yesmaximal, complemented
7#8568C212yesmaximal, complemented
8#8566C212yesmaximal, complemented
9#8564C212yesmaximal, complemented
10#7899C211yesmaximal, characteristic
11#7894C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#2356—21no—
18#2356—21no—
19#2356C411yescharacteristic
20#2356C411yescharacteristic
21#2351C411yescharacteristic
22#2351C411yescharacteristic
23#2344—22no—
24#2344C412yes—
25Q8×C22⋊C4C412yes—
26#2295—22no—
27#2295C2212yes—
28#2283—21no—
29#2283C2211yescharacteristic
30C42×Q8C2212yes—
31C4×#117—21no—
32C4×#117C2211yescharacteristic
33#1939—22no—
34#1851—21no—
35#1851—21no—
36#1843—21no—
37#1842C2212yes—
38#1832—21no—
39#1831C2212yes—
40#1798—22no—
41C4×C4⋊C8C2212yes—
42C4×C4⋊C8C2212yes—
43#1487—24no—
44#1479—24no—
45C2×Q8⋊C8C2211yescharacteristic
46C2×Q8⋊C8C2211yescharacteristic
47C2×Q8⋊C8C2211yescharacteristic
48C2×Q8⋊C8C2211yescharacteristic
49C2×Q8⋊C8C2212yes—
50C2×Q8⋊C8C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C23×Q8C2×C411yescharacteristic
56C22×M16—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D414yes—
66C2×C4×Q8Q814yes—
67C2×C4×Q8C2311yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×#117—22no—
70C2×#117D412yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#431—22no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#431—22no—
81#423—22no—
82#423—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2311yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2312yes—
100#391—22no—
101#391—22no—
102#391C2×C412yes—
103#391C2×C412yes—
104#389—22no—
105#389—22no—
106#389C2×C412yes—
107#389C2×C412yes—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#383—21no—
115#383—21no—
116#383—21no—
117#382—21no—
118#382—21no—
119#382C2×C411yescharacteristic
120#382C2×C411yescharacteristic
121#381—21no—
122#381—22no—
123#380—22no—
124#380C2×C412yes—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4C2×C412yes—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4D412yes—
136C4×C4⋊C4C2312yes—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4C2×C412yes—
140C4×C22⋊C4D411yescharacteristic
141C4×C22⋊C4D411yescharacteristic
142#376—22no—
143#376—21no—
144#376—21no—
145#376—21no—
146#376D411yescharacteristic
147#376D411yescharacteristic
148C43—22no—
149C43C2312yes—
150#336—22no—
151#336—22no—
152#336—22no—
153#336—22no—
154Q8⋊C8—22no—
155Q8⋊C8—22no—
156Q8⋊C8—22no—
157Q8⋊C8—22no—
158Q8⋊C8—24no—
159Q8⋊C8—24no—
160C22×Q8—21no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8C4⋊C414yes—
167C22×Q8C22×C411yescharacteristic
168C22×Q8C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C2×M16—42no—
175C2×M16—82no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8C4⋊C414yes—
197C4×Q8C4⋊C414yes—
198C4×Q8C2×D412yes—
199C4×Q8C2×D412yes—
200C4×Q8C2×Q812yes—
201C4×Q8C2×Q812yes—
202#117—44no—
203#117—48no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C4⋊C414yes—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42C22×C412yes—
238C2×C42C22×C412yes—
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×Q812yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250C2×C42C2411yescharacteristic
251C4⋊C8—22no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—22no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—44no—
258C4⋊C8—44no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262C4⋊C8—82no—
263C4⋊C8—82no—
264C22⋊C8—42no—
265C22⋊C8—42no—
266C22⋊C8—42no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—22no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—22no—
273C4×C8—42no—
274#95—21no—
275#95—21no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—21no—
284#95—21no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95C22⋊C411yescharacteristic
289#95C22⋊C411yescharacteristic
290#95C22⋊C412yes—
291#95C22⋊C412yes—
292#95C22⋊C411yescharacteristic
293#95C22⋊C411yescharacteristic
294C24C2×C22⋊C411yescharacteristic
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—24no—
300C2×Q8—28no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8C2×C4⋊C412yes—
311C2×Q8C2×C4⋊C412yes—
312C2×Q8#13112yes—
313C2×Q8#13112yes—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C4⋊C412yes—
340C22×C4#11712yes—
341C22×C4#12112yes—
342C22×C4#12112yes—
343C22×C4C22⋊Q812yes—
344C22×C4C22⋊Q812yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348M16—84no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4C2×C4⋊C412yes—
383C4⋊C4C2×C4⋊C412yes—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C22⋊C4—44no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—24no—
400C42—24no—
401C42—24no—
402C42—42no—
403C42—44no—
404C42—44no—
405C42—44no—
406C42—42no—
407C42—42no—
408C42—44no—
409C42C2×C4⋊C412yes—
410C42#11712yes—
411C42#12112yes—
412C42#12112yes—
413C42C22⋊Q812yes—
414C42C22⋊Q812yes—
415C42C22×D411yescharacteristic
416C42C22×Q811yescharacteristic
417C42C2×#4111yescharacteristic
418C42C2×#4111yescharacteristic
419C23—21no—
420C23—21no—
421C23—22no—
422C23—22no—
423C23#38012yes—
424C23C22×C22⋊C411yescharacteristic
425Q8—22no—
426Q8—22no—
427Q8—22no—
428Q8—22no—
429Q8—22no—
430Q8—48no—
431Q8—41no—
432Q8—41no—
433Q8#42414yes—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4#38012yes—
478C2×C4#38012yes—
479C2×C4#38012yes—
480C2×C4#42114yes—
481C2×C4C22×C4⋊C411yescharacteristic
482C2×C4C2×#11711yescharacteristic
483C2×C4C2×#12111yescharacteristic
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×C22⋊Q811yescharacteristic
486C2×C4C2×C22⋊Q811yescharacteristic
487C2×C4C2×#13111yescharacteristic
488C2×C4C2×#13111yescharacteristic
489C2×C4#56811yescharacteristic
490C2×C4#56811yescharacteristic
491C2×C4#57712yes—
492C2×C4#57812yes—
493C8—81no—
494C8—81no—
495C8—82no—
496C8—81no—
497C8—82no—
498C8—81no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—42no—
503C22—42no—
504C22C2×#38011yescharacteristic
505C22#288011yescharacteristic
506C22#288011yescharacteristic
507C22#289812yes—
508C22C2×#42112yes—
509C4—24no—
510C4—22no—
511C4—24no—
512C4—22no—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—82no—
524C4C2×#42411yescharacteristic
525C4#290511yescharacteristic
526C4#292411yescharacteristic
527C4#292411yescharacteristic
528C4#304311yescharacteristic
529C4#304411yescharacteristic
530C4#308212yes—
531C2—42no—
532C2#1752412yes—
533C2#1752512yes—
534C2#2062811yescharacteristic
535C2#2065111yescharacteristic
536C2#2065411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8570C2semidirect
7. #8568C2semidirect
8. #8566C2semidirect
9. #8564C2semidirect

Maximal subgroups

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

subgroupindex
#341282
#219742
#208092
C2×#14872
C2×#14792
#85702
#85682
#85662
#85642
#78992
#78942