Finite Group Databasebeta

← #240790

Subgroups — #240790

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#21306C211yesmaximal, characteristic
5#17787C211yesmaximal, characteristic
6#17708C211yesmaximal, characteristic
7#10505C218yesmaximal, complemented
8#8095C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3209—22no—
12#3209—22no—
13#3209—22no—
14#3209C2214yes—
15C2×#494—21no—
16C2×#487—21no—
17C2×#462—22no—
18C2×#451—22no—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2285C2211yescharacteristic
24#1997—22no—
25#1997C2214yes—
26#1986—22no—
27#1986C2214yes—
28#1981C2214yes—
29#1956—22no—
30#1787—22no—
31C2×#343—21no—
32C2×#343—21no—
33#1733—21no—
34#1733—21no—
35#1565—22no—
36#1565C2214yes—
37#1564—22no—
38#1550C2214yes—
39#1549—21no—
40#1548—21no—
41#1542—22no—
42#1303C2214yes—
43C23×Q8D412yes—
44C22×Q16—24no—
45C22×M16—21no—
46C22×M16D411yescharacteristic
47C22×M16D411yescharacteristic
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533D412yes—
55#533C2312yes—
56C2×C4⋊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⋊Q8D412yes—
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8D411yescharacteristic
69C2×C4⋊Q8C2312yes—
70C2×#125—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×#117—21no—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C42C2311yescharacteristic
80#494—24no—
81#494—24no—
82#487—24no—
83#487—24no—
84#462—28no—
85#462—28no—
86#451—28no—
87#451—28no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431D412yes—
95#431C2312yes—
96#423—21no—
97#423—21no—
98#423—21no—
99#423C2312yes—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8D412yes—
103C2×C4⋊C8C2312yes—
104C2×#103—24no—
105C2×#103—28no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C4×M16—21no—
109C4×M16—21no—
110C4×M16D412yes—
111C4×M16C2312yes—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116#384—21no—
117#382—21no—
118#380—21no—
119#380—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2312yes—
122#343—24no—
123#343—24no—
124#339—24no—
125#339—24no—
126#335—24no—
127#334—24no—
128#333—24no—
129#332—24no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—84no—
138C22×Q8C2×D414yes—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4#4111yescharacteristic
143C2×Q16—216no—
144C2×Q16—88no—
145C2×M16—21no—
146C2×M16—22no—
147C2×M16—21no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—41no—
152C2×M16—42no—
153C2×M16—42no—
154C2×M16Q1614yescomplemented
155C2×M16C2×D412yes—
156C2×M16C2×D412yes—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—41no—
162C4⋊Q8—21no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—21no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—41no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C4⋊Q8C2×D412yes—
185C4⋊Q8C2×D412yes—
186#125—22no—
187#125—22no—
188#125—41no—
189#125—41no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—41no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206#117—22no—
207#117—22no—
208#117—41no—
209#117—41no—
210#117—42no—
211#117—42no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
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×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C42—22no—
236C2×C42—41no—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42C2411yescharacteristic
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8C2×D414yes—
257#103—216no—
258#103—88no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—41no—
262C22⋊C8—41no—
263#97—22no—
264#97—22no—
265C4×C8—22no—
266C4×C8—22no—
267C4×C8—22no—
268C4×C8—22no—
269C4×C8—41no—
270C4×C8—41no—
271#95—22no—
272#95—41no—
273C24#12111yescharacteristic
274C2×Q8—24no—
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—22no—
278C2×Q8—48no—
279C2×Q8—44no—
280C2×Q8—48no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—84no—
286C2×Q8—82no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×Q8—84no—
290C22×C4—24no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—84no—
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12311yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4#12311yescharacteristic
312C22×C4#12411yescharacteristic
313C22×C4#12411yescharacteristic
314C22×C4C2×Q1612yes—
315C22×C4#13512yes—
316C22×C4#13712yes—
317C22×C4#13712yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4111yescharacteristic
321Q16—816no—
322M16—22no—
323M16—24no—
324M16—22no—
325M16—22no—
326M16—42no—
327M16—42no—
328M16—42no—
329M16—82no—
330M16C2×Q1614yescomplemented
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—22no—
335C2×C8—44no—
336C2×C8—42no—
337C2×C8—44no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—82no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—44no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—82no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42#12011yescharacteristic
372C42#12011yescharacteristic
373C42#12111yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42#12111yescharacteristic
377C42#12311yescharacteristic
378C42#12311yescharacteristic
379C42#12311yescharacteristic
380C42#12311yescharacteristic
381C42#12411yescharacteristic
382C42#12411yescharacteristic
383C42C22×D411yescharacteristic
384C42C22×D411yescharacteristic
385C42C2×#4111yescharacteristic
386C42C2×#4111yescharacteristic
387C23—41no—
388C23—41no—
389C23#39411yescharacteristic
390C23#39411yescharacteristic
391C23#46212yes—
392C23#46412yes—
393C23C2×#12111yescharacteristic
394Q8—48no—
395Q8—44no—
396Q8—48no—
397Q8—82no—
398Q8—82no—
399Q8—84no—
400Q8—84no—
401Q8—82no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—84no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4#39411yescharacteristic
426C2×C4#39411yescharacteristic
427C2×C4#39411yescharacteristic
428C2×C4#39411yescharacteristic
429C2×C4#39411yescharacteristic
430C2×C4#39411yescharacteristic
431C2×C4#46212yes—
432C2×C4#46212yes—
433C2×C4#46212yes—
434C2×C4#46412yes—
435C2×C4#46412yes—
436C2×C4#46412yes—
437C2×C4#46712yes—
438C2×C4#46712yes—
439C2×C4#47012yes—
440C2×C4#47012yes—
441C2×C4C2×#12011yescharacteristic
442C2×C4C2×#12111yescharacteristic
443C2×C4C2×#12111yescharacteristic
444C2×C4C2×#12311yescharacteristic
445C2×C4C2×#12311yescharacteristic
446C2×C4C2×#12411yescharacteristic
447C2×C4C22×Q1611yescharacteristic
448C2×C4C2×#13511yescharacteristic
449C2×C4C2×#13711yescharacteristic
450C2×C4C2×#13711yescharacteristic
451C8—24no—
452C8—44no—
453C8—42no—
454C8—82no—
455C8—82no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—42no—
463C22—42no—
464C22#205612yes—
465C22#205612yes—
466C22C2×#39411yescharacteristic
467C22C2×#46211yescharacteristic
468C22C2×#46411yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—84no—
476C4—82no—
477C4—82no—
478C4—162no—
479C4#204912yes—
480C4#205612yes—
481C4#205612yes—
482C4#301912yes—
483C4C2×#46211yescharacteristic
484C4C2×#46411yescharacteristic
485C4C2×#46711yescharacteristic
486C4C2×#47011yescharacteristic
487C2—21no—
488C2—22no—
489C2—22no—
490C2—21no—
491C2#2349311yescharacteristic
492C2C2×#204911yescharacteristic
493C2C2×#205611yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10505C2semidirect
154. C2×M16Q16semidirect
330. M16C2×Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#218042
C2×#19972
#213062
#177872
#177082
#105052
#80952