Finite Group Databasebeta

← #115500

Subgroups — #115500

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2C2×#1986C211yesmaximal, characteristic
3#21303C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6C2×#1303C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#3104—22no—
11#3104—22no—
12#3104—22no—
13#3104—22no—
14C2×#484—21no—
15C2×#477C412yescomplemented
16C2×#462C412yescomplemented
17C2×#451—21no—
18C2×C4×Q16—22no—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×#103—21no—
22C2×C4×M16C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1986—22no—
26#1986—22no—
27#1985—22no—
28#1984—22no—
29C4×#103—22no—
30C4×#103—22no—
31C2×#340—21no—
32C2×#339—22no—
33#1728—22no—
34#1493—22no—
35#1493—22no—
36C2×Q8⋊C8—22no—
37#1303—22no—
38#1278—22no—
39C23×Q8D411yescharacteristic
40C22×Q16—22no—
41C22×M16—21no—
42C22×M16D412yes—
43C23×C8—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533D412yes—
51#533D412yes—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8C2×C412yes—
70C2×C4×Q8D412yes—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#484—24no—
79#484—24no—
80#477—24no—
81#477—24no—
82#462—24no—
83#462—24no—
84#451—24no—
85#451—24no—
86C4×Q16—28no—
87C4×Q16—28no—
88#431—21no—
89#431—21no—
90#431D412yes—
91#431D412yes—
92C2×#107—22no—
93#423—21no—
94#423—21no—
95#423—21no—
96#423—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2×C412yes—
99C2×C4⋊C8D412yes—
100C2×#103—28no—
101C2×#103—24no—
102C2×#103—22no—
103C2×#103—22no—
104C2×C22⋊C8—21no—
105C4×M16—21no—
106C4×M16—21no—
107C4×M16—21no—
108C4×M16—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8D412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4D412yes—
122C2×#95—21no—
123#340—24no—
124#339—28no—
125#336—28no—
126Q8⋊C8—28no—
127C22×Q8—21no—
128C22×Q8—24no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—42no—
136C22×Q8C22⋊C412yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—41no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4#4111yescharacteristic
143C2×Q16—28no—
144C2×Q16—44no—
145C2×Q16—44no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—41no—
152C2×M16—41no—
153C2×M16—42no—
154C2×M16—42no—
155C2×M16Q1614yescomplemented
156C22×C8—24no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—42no—
162C22×C8—41no—
163C22×C8—41no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C22⋊C412yes—
171C4⋊Q8C22⋊C412yes—
172#125—22no—
173#125—22no—
174#125—41no—
175#125—41no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208#117—22no—
209#117—22no—
210#117—41no—
211#117—41no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4C22⋊C412yes—
234C2×C22⋊C4—41no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—41no—
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42#4111yescharacteristic
257C2×C42#4111yescharacteristic
258#107—28no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8C22⋊C414yes—
265#103—216no—
266#103—28no—
267#103—28no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270#97—41no—
271#97—41no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—22no—
275C4×C8—22no—
276C4×C8—41no—
277C4×C8—41no—
278C4×C8—41no—
279C4×C8—41no—
280#95—22no—
281#95—41no—
282C24#12111yescharacteristic
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—44no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—44no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8#10414yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—44no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4#12011yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4#12011yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4#12111yescharacteristic
333C22×C4#12111yescharacteristic
334C22×C4#12111yescharacteristic
335C22×C4#12311yescharacteristic
336C22×C4#12311yescharacteristic
337C22×C4#12311yescharacteristic
338C22×C4#12411yescharacteristic
339C22×C4#12711yescharacteristic
340C22×C4C2×Q1612yes—
341C22×C4#13512yes—
342C22×C4#13712yes—
343C22×C4#13712yes—
344Q16—416no—
345M16—24no—
346M16—24no—
347M16—42no—
348M16—42no—
349M16—82no—
350M16—81no—
351M16—81no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—44no—
356C2×C8—42no—
357C2×C8—44no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—44no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—44no—
397C42—81no—
398C42—81no—
399C42C2×C22⋊C411yescharacteristic
400C42C2×C22⋊C411yescharacteristic
401C42C4×D411yescharacteristic
402C42C4×D411yescharacteristic
403C42#12011yescharacteristic
404C42#12011yescharacteristic
405C42#12011yescharacteristic
406C42#12111yescharacteristic
407C42#12111yescharacteristic
408C42#12111yescharacteristic
409C42#12111yescharacteristic
410C42#12111yescharacteristic
411C42#12111yescharacteristic
412C42#12311yescharacteristic
413C42#12411yescharacteristic
414C42#12711yescharacteristic
415C23—22no—
416C23—22no—
417C23#38611yescharacteristic
418C23#39211yescharacteristic
419C23#39411yescharacteristic
420C23#46212yes—
421C23#46412yes—
422Q8—24no—
423Q8—24no—
424Q8—44no—
425Q8—44no—
426Q8—44no—
427Q8—42no—
428Q8—42no—
429Q8—44no—
430Q8—82no—
431Q8—81no—
432Q8—81no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—82no—
457C2×C4—88no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4#37911yescharacteristic
464C2×C4#38611yescharacteristic
465C2×C4#38611yescharacteristic
466C2×C4#38611yescharacteristic
467C2×C4#39011yescharacteristic
468C2×C4#39011yescharacteristic
469C2×C4#39211yescharacteristic
470C2×C4#39211yescharacteristic
471C2×C4#39211yescharacteristic
472C2×C4#39411yescharacteristic
473C2×C4#39411yescharacteristic
474C2×C4#39411yescharacteristic
475C2×C4C2×#10412yes—
476C2×C4#42112yes—
477C2×C4C4×Q1612yes—
478C2×C4#44112yes—
479C2×C4#45112yes—
480C2×C4#45112yes—
481C2×C4#45212yes—
482C2×C4#45212yes—
483C2×C4#46212yes—
484C2×C4#46412yes—
485C2×C4#46712yes—
486C2×C4#47012yes—
487C2×C4#49411yescharacteristic
488C2×C4#49511yescharacteristic
489C2×C4#49711yescharacteristic
490C2×C4#49711yescharacteristic
491C8—44no—
492C8—44no—
493C8—44no—
494C8—82no—
495C22—21no—
496C22—21no—
497C22—24no—
498C22—24no—
499C22—21no—
500C22—21no—
501C22#143811yescharacteristic
502C22#187112yes—
503C22#187812yes—
504C22#188112yes—
505C22#201511yescharacteristic
506C22#201611yescharacteristic
507C22#205612yes—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—44no—
515C4—84no—
516C4—82no—
517C4—82no—
518C4—82no—
519C4#178912yes—
520C4#187812yes—
521C4#189611yescharacteristic
522C4#189711yescharacteristic
523C4#196911yescharacteristic
524C4#197111yescharacteristic
525C4#200112yes—
526C4#201511yescharacteristic
527C4#201611yescharacteristic
528C2—21no—
529C2—22no—
530C2—21no—
531C2#899014yes—
532C2#1097111yescharacteristic
533C2#1101811yescharacteristic
534C2#1102411yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. C2×#477C4semidirect
16. C2×#462C4semidirect
155. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
C2×#19862
#213032
C2×C4×#1032
C2×#14932
C2×#13032
C2×#12782