Finite Group Databasebeta

← #114166

Subgroups — #114166

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3139C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic
6#10851C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3139—22no—
11#3139—22no—
12#3139—22no—
13#3139C414yescomplemented
14C2×#491—22no—
15C2×#487—21no—
16C2×#487—21no—
17C2×#431C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C2×#419—22no—
20C2×C4×M16C2211yescharacteristic
21#2285C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1990—22no—
24#1988—22no—
25#1988—22no—
26#1988—22no—
27#1987—21no—
28#1981—21no—
29#1892—22no—
30#1823—22no—
31#1803—22no—
32C2×#340—22no—
33C2×#339—22no—
34#1503—22no—
35#1503—22no—
36C2×#328—22no—
37#1283—22no—
38C23×Q8—21no—
39C22×M16D412yes—
40C23×C8D412yes—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533C2×C412yes—
48#533D412yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D412yes—
62C2×C4⋊Q8D412yes—
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#491—28no—
76#491—28no—
77#487—24no—
78#487—24no—
79#487—24no—
80#487—24no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92#419—28no—
93#419—28no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×C22⋊C8D412yes—
98C4×M16—21no—
99C4×M16—21no—
100C4×M16C2×C412yes—
101C4×M16D412yes—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105#384D412yes—
106#382—21no—
107#380—21no—
108#380—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D412yes—
115C2×#95—21no—
116#340—24no—
117#340—28no—
118#340—28no—
119#339—44no—
120#328—28no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C23×C4—41no—
134C23×C4—41no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16C22⋊C414yes—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—44no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8C22⋊C414yes—
172C4⋊Q8C22⋊C412yes—
173C4⋊Q8C22⋊C412yes—
174#125—22no—
175#125—22no—
176#125—41no—
177#125—41no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194#117—22no—
195#117—22no—
196#117—41no—
197#117—41no—
198#117—42no—
199#117—42no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C42—22no—
228C2×C42—41no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C4⋊C8—41no—
246C4⋊C8—41no—
247C4⋊C8—41no—
248C4⋊C8—41no—
249C4⋊C8—41no—
250C4⋊C8—41no—
251#103—216no—
252#103—216no—
253#103—84no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256#97—22no—
257#97—22no—
258C4×C8—22no—
259C4×C8—22no—
260C4×C8—41no—
261C4×C8—41no—
262#95—22no—
263#95—41no—
264#95—41no—
265#95—41no—
266C24#12711yescharacteristic
267C2×Q8—24no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—82no—
286C2×Q8—82no—
287C2×Q8—82no—
288C2×Q8—82no—
289C2×Q8—82no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—82no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—81no—
307C22×C4—81no—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4#12011yescharacteristic
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12411yescharacteristic
326C22×C4#13512yes—
327C22×C4#13512yes—
328C22×C4AGL(1,ℤ/8)12yes—
329C22×C4#13712yes—
330M16—24no—
331M16—24no—
332M16—42no—
333M16—42no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—44no—
342C2×C8—82no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—81no—
373C22⋊C4—81no—
374C42—42no—
375C42—42no—
376C42—44no—
377C42—82no—
378C42—82no—
379C42C2×C22⋊C411yescharacteristic
380C42C2×C22⋊C411yescharacteristic
381C42C4×D411yescharacteristic
382C42C4×D411yescharacteristic
383C42#12011yescharacteristic
384C42#12011yescharacteristic
385C42#12111yescharacteristic
386C42#12111yescharacteristic
387C42#12111yescharacteristic
388C42#12111yescharacteristic
389C42#12311yescharacteristic
390C42#12311yescharacteristic
391C42#12311yescharacteristic
392C42#12311yescharacteristic
393C42#12411yescharacteristic
394C42#12711yescharacteristic
395C23—22no—
396C23—22no—
397C23#39011yescharacteristic
398C23#39211yescharacteristic
399C23#39211yescharacteristic
400C23#49511yescharacteristic
401C23#49511yescharacteristic
402C23#49611yescharacteristic
403C23#49711yescharacteristic
404Q8—44no—
405Q8—48no—
406Q8—82no—
407Q8—82no—
408Q8—84no—
409Q8—81no—
410Q8—81no—
411Q8—84no—
412Q8—84no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—82no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4#37911yescharacteristic
446C2×C4#38611yescharacteristic
447C2×C4#38611yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4#38611yescharacteristic
450C2×C4#39011yescharacteristic
451C2×C4#39211yescharacteristic
452C2×C4#39211yescharacteristic
453C2×C4#39411yescharacteristic
454C2×C4#39411yescharacteristic
455C2×C4#39411yescharacteristic
456C2×C4#39411yescharacteristic
457C2×C4#41612yes—
458C2×C4#41712yes—
459C2×C4#44912yes—
460C2×C4#44912yes—
461C2×C4#45212yes—
462C2×C4#45212yes—
463C2×C4#45512yes—
464C2×C4#45612yes—
465C2×C4#46312yes—
466C2×C4#46312yes—
467C2×C4#46412yes—
468C2×C4#46412yes—
469C2×C4#48812yes—
470C2×C4#48912yes—
471C8—44no—
472C8—82no—
473C8—82no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22#143811yescharacteristic
483C22#195811yescharacteristic
484C22#195911yescharacteristic
485C22#200712yes—
486C22#200912yes—
487C22#201211yescharacteristic
488C22#201211yescharacteristic
489C22#201511yescharacteristic
490C22#201511yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—84no—
498C4—84no—
499C4—82no—
500C4—82no—
501C4—162no—
502C4#178812yes—
503C4#188212yes—
504C4#198312yes—
505C4#200411yescharacteristic
506C4#200511yescharacteristic
507C4#204212yes—
508C4#204212yes—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2#1020512yes—
514C2#1020712yes—
515C2#1100911yescharacteristic
516C2#1101711yescharacteristic
517C2#1103111yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3139C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31392
#218852
#218332
C2×#19882
C2×#15032
#108512
C2×#12832