Finite Group Databasebeta

← #114209

Subgroups — #114209

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4C2×#1548C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#10857C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3165—22no—
11#3165—22no—
12#3165—22no—
13#3165C414yescomplemented
14C2×#501—22no—
15C2×#500—21no—
16C2×#498—21no—
17C2×#431C2211yescharacteristic
18C2×#428—22no—
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#1990—22no—
24#1990—22no—
25#1990—21no—
26#1956—21no—
27#1872—22no—
28#1803—22no—
29C2×#343—22no—
30C2×#340—22no—
31#1548—22no—
32#1548—22no—
33C2×#333—22no—
34#1493—22no—
35#1493—22no—
36C2×Q8⋊C8—22no—
37#1278—22no—
38C23×Q8—21no—
39C22×M16—21no—
40C22×M16D412yes—
41C23×C8—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533D412yes—
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⋊Q8—21no—
59C2×C4⋊Q8C2×C412yes—
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8D412yes—
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D412yes—
75C22×C42C2311yescharacteristic
76#501—28no—
77#501—28no—
78#500—24no—
79#500—24no—
80#498—24no—
81#498—24no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#428—28no—
87#428—28no—
88C2×#107—24no—
89C2×#106—24no—
90#423—21no—
91#423—21no—
92#423—21no—
93#423D412yes—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8D412yes—
97C2×#103—24no—
98C2×C22⋊C8—21no—
99C4×M16—21no—
100C4×M16—21no—
101C4×M16C2×C412yes—
102C4×M16D412yes—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8D412yes—
112#384—21no—
113#384—21no—
114#384—21no—
115#384—21no—
116#343—44no—
117#340—24no—
118#340—28no—
119#333—28no—
120Q8⋊C8—28no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8—82no—
130C22×Q8—82no—
131C23×C4—22no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4#4111yescharacteristic
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—41no—
141C2×M16—42no—
142C2×M16—42no—
143C2×M16C22⋊C414yes—
144C2×M16SD1614yes—
145C2×M16Q1614yes—
146C22×C8—22no—
147C22×C8—24no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—42no—
151C22×C8—41no—
152C22×C8—41no—
153C22×C8—41no—
154C4⋊Q8—22no—
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—22no—
165C4⋊Q8—21no—
166C4⋊Q8—22no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175C4⋊Q8C22⋊C414yes—
176C4⋊Q8C22⋊C412yes—
177C4⋊Q8C22⋊C412yes—
178#125—22no—
179#125—22no—
180#125—41no—
181#125—41no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—41no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189C22⋊Q8—41no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198#117—22no—
199#117—22no—
200#117—41no—
201#117—41no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
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⋊C4—44no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—82no—
230C2×C22⋊C4—41no—
231C2×C42—41no—
232C2×C42C22×C411yescharacteristic
233C2×C42C2×D411yescharacteristic
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
244#107—216no—
245#106—216no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252#103—84no—
253C22⋊C8—41no—
254C22⋊C8—41no—
255#97—22no—
256#97—22no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—41no—
262C4×C8—41no—
263C4×C8—41no—
264C4×C8—41no—
265C24#12111yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—42no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8—82no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—82no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4#12411yescharacteristic
316C22×C4#12411yescharacteristic
317C22×C4#12711yescharacteristic
318C22×C4#12711yescharacteristic
319C22×C4C2×SD1612yes—
320C22×C4C2×Q1612yes—
321C22×C4#13712yes—
322C22×C4#13712yes—
323M16—24no—
324M16—24no—
325M16—24no—
326M16—44no—
327M16—82no—
328M16#10318yescomplemented
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—42no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—81no—
342C2×C8—81no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C42—42no—
373C42—42no—
374C42C2×C22⋊C411yescharacteristic
375C42C2×C22⋊C411yescharacteristic
376C42C4×D411yescharacteristic
377C42C4×D411yescharacteristic
378C42#12011yescharacteristic
379C42#12011yescharacteristic
380C42#12011yescharacteristic
381C42#12011yescharacteristic
382C42#12111yescharacteristic
383C42#12111yescharacteristic
384C42#12111yescharacteristic
385C42#12111yescharacteristic
386C42#12111yescharacteristic
387C42#12111yescharacteristic
388C42#12311yescharacteristic
389C42#12311yescharacteristic
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23#39011yescharacteristic
395C23#39211yescharacteristic
396C23#39411yescharacteristic
397C23#46112yes—
398C23#46212yes—
399Q8—44no—
400Q8—44no—
401Q8—84no—
402Q8—84no—
403Q8—82no—
404Q8—84no—
405Q8—84no—
406Q8—82no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—84no—
423C2×C4—84no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—164no—
434C2×C4#37911yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#38611yescharacteristic
437C2×C4#38611yescharacteristic
438C2×C4#38611yescharacteristic
439C2×C4#39011yescharacteristic
440C2×C4#39211yescharacteristic
441C2×C4#39211yescharacteristic
442C2×C4#39211yescharacteristic
443C2×C4#39411yescharacteristic
444C2×C4#39411yescharacteristic
445C2×C4#39411yescharacteristic
446C2×C4C2×#10312yes—
447C2×C4#41912yes—
448C2×C4#44812yes—
449C2×C4#44812yes—
450C2×C4#45112yes—
451C2×C4#45112yes—
452C2×C4#45512yes—
453C2×C4#45612yes—
454C2×C4#46112yes—
455C2×C4#46212yes—
456C2×C4#47112yes—
457C2×C4#47212yes—
458C2×C4#48712yes—
459C2×C4#49112yes—
460C2×C4#49211yescharacteristic
461C2×C4#49411yescharacteristic
462C2×C4#49711yescharacteristic
463C2×C4#49711yescharacteristic
464C8—28no—
465C8—44no—
466C8—82no—
467C8—82no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22#143811yescharacteristic
477C22#198112yes—
478C22#200411yescharacteristic
479C22#200511yescharacteristic
480C22#201311yescharacteristic
481C22#201611yescharacteristic
482C22#201811yescharacteristic
483C22#201911yescharacteristic
484C22#204312yes—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—82no—
490C4—84no—
491C4—82no—
492C4—162no—
493C4#178712yes—
494C4#188212yes—
495C4#190712yes—
496C4#195611yescharacteristic
497C4#196011yescharacteristic
498C4#200712yes—
499C4#201311yescharacteristic
500C4#201611yescharacteristic
501C4#204312yes—
502C4#205112yes—
503C2—22no—
504C2—22no—
505C2#1016111yescharacteristic
506C2#1016311yescharacteristic
507C2#1016511yescharacteristic
508C2#1016711yescharacteristic
509C2#1101011yescharacteristic
510C2#1101711yescharacteristic
511C2#1102011yescharacteristic
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. #3165C4semidirect
328. M16#103semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#218602
C2×#19902
C2×#15482
C2×#14932
#108572
C2×#12782