Finite Group Databasebeta

← #114100

Subgroups — #114100

Order 512.

938 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3C2×#1981C211yesmaximal, characteristic
4C2×#1546C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6C2×#1301C211yesmaximal, characteristic
7#7908C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3229—22no—
11#3229—22no—
12#3229—22no—
13#3229C414yescomplemented
14C2×#498—21no—
15C2×#487—21no—
16C2×#483—22no—
17C2×#475—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#103—22no—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1989—22no—
25#1988—21no—
26#1987—21no—
27#1981—22no—
28#1981—22no—
29#1736—22no—
30#1546—22no—
31#1546—22no—
32C2×#332—22no—
33#1481—22no—
34#1481—22no—
35C2×Q8⋊C8—22no—
36#1301—22no—
37#1280—22no—
38C23×Q8—21no—
39C22×M16—21no—
40C23×C8—21no—
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⋊Q8D412yes—
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D411yescharacteristic
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D412yes—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#498—24no—
75#498—24no—
76#487—24no—
77#487—24no—
78#483—28no—
79#483—28no—
80#475—28no—
81#475—28no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431—21no—
87#431C2×C412yes—
88#431D412yes—
89#431D412yes—
90C2×#106—24no—
91#423—21no—
92#423—21no—
93#423—21no—
94#423—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8D412yes—
104C2×C4⋊C8D412yes—
105C2×#103—24no—
106C2×#103—28no—
107C2×#103—44no—
108C2×#103—44no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8D412yes—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C2×#95—21no—
118#340—24no—
119#340—24no—
120#332—28no—
121Q8⋊C8—28no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8—41no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—42no—
134C23×C4—22no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4#4111yescharacteristic
139C2×M16—41no—
140C2×M16—41no—
141C2×M16—42no—
142C2×M16—42no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—41no—
147C22×C8—41no—
148C22×C8—41no—
149C22×C8—42no—
150C22×C8—42no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—22no—
160C4⋊Q8—21no—
161C4⋊Q8—22no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—22no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8C22⋊C414yes—
173C4⋊Q8C22⋊C412yes—
174C4⋊Q8C22⋊C412yes—
175#125—22no—
176#125—22no—
177#125—41no—
178#125—41no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195#117—22no—
196#117—22no—
197#117—41no—
198#117—41no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
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⋊C4SD1614yes—
223C2×C22⋊C4—41no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42—82no—
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42#4111yescharacteristic
240#106—216no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—41no—
249C4⋊C8—41no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252C4⋊C8—41no—
253#103—216no—
254#103—416no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—41no—
258C22⋊C8C22⋊C414yes—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—22no—
262C4×C8—41no—
263#95—41no—
264#95—41no—
265C24#12111yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—44no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—41no—
274C2×Q8—41no—
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—81no—
288C2×Q8—81no—
289C2×Q8—81no—
290C2×Q8—81no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4#12011yescharacteristic
312C22×C4#12011yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12311yescharacteristic
319C22×C4#12311yescharacteristic
320C22×C4#12411yescharacteristic
321C22×C4#12411yescharacteristic
322C22×C4#12711yescharacteristic
323C22×C4#12711yescharacteristic
324C22×C4C2×SD1612yes—
325C22×C4#13512yes—
326C22×C4AGL(1,ℤ/8)12yes—
327C22×C4#13712yes—
328M16—82no—
329M16—81no—
330M16—81no—
331C2×C8—24no—
332C2×C8—44no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—82no—
339C2×C8—82no—
340C2×C8—81no—
341C2×C8—81no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C42—42no—
369C42—42no—
370C42—82no—
371C42—82no—
372C42C2×C22⋊C411yescharacteristic
373C42C2×C22⋊C411yescharacteristic
374C42C4×D411yescharacteristic
375C42C4×D411yescharacteristic
376C42#12011yescharacteristic
377C42#12011yescharacteristic
378C42#12011yescharacteristic
379C42#12011yescharacteristic
380C42#12111yescharacteristic
381C42#12111yescharacteristic
382C42#12111yescharacteristic
383C42#12111yescharacteristic
384C42#12111yescharacteristic
385C42#12111yescharacteristic
386C42#12311yescharacteristic
387C42#12311yescharacteristic
388C23—22no—
389C23—21no—
390C23—21no—
391C23#39011yescharacteristic
392C23#39211yescharacteristic
393C23#39411yescharacteristic
394C23#46512yes—
395C23#46812yes—
396Q8—44no—
397Q8—48no—
398Q8—44no—
399Q8—82no—
400Q8—82no—
401Q8—84no—
402Q8—82no—
403Q8—82no—
404Q8—82no—
405Q8—82no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—84no—
423C2×C4—84no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—84no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—84no—
433C2×C4#37911yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#38611yescharacteristic
437C2×C4#38611yescharacteristic
438C2×C4#39011yescharacteristic
439C2×C4#39211yescharacteristic
440C2×C4#39211yescharacteristic
441C2×C4#39211yescharacteristic
442C2×C4#39411yescharacteristic
443C2×C4#39411yescharacteristic
444C2×C4#39411yescharacteristic
445C2×C4C4×SD1612yes—
446C2×C4#44012yes—
447C2×C4#44812yes—
448C2×C4#44912yes—
449C2×C4#45012yes—
450C2×C4#45212yes—
451C2×C4#46012yes—
452C2×C4#46112yes—
453C2×C4#46312yes—
454C2×C4#46412yes—
455C2×C4#47112yes—
456C2×C4#47312yes—
457C2×C4#49211yescharacteristic
458C2×C4#49511yescharacteristic
459C2×C4#49611yescharacteristic
460C2×C4#49711yescharacteristic
461C8—44no—
462C8—82no—
463C8—82no—
464C8—82no—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—21no—
472C22—21no—
473C22#143811yescharacteristic
474C22#196811yescharacteristic
475C22#197311yescharacteristic
476C22#200012yes—
477C22#200912yes—
478C22#201911yescharacteristic
479C22#202011yescharacteristic
480C22#205512yes—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—82no—
488C4—84no—
489C4—84no—
490C4—82no—
491C4#187712yes—
492C4#188012yes—
493C4#190812yes—
494C4#201211yescharacteristic
495C4#201311yescharacteristic
496C4#201411yescharacteristic
497C4#201511yescharacteristic
498C4#205212yes—
499C2—22no—
500C2—21no—
501C2—21no—
502C2#1018612yes—
503C2#1018712yes—
504C2#1100511yescharacteristic
505C2#1102111yescharacteristic
506C2#1102311yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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. #3229C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#213122
C2×#19812
C2×#15462
C2×#14812
C2×#13012
#79082