Finite Group Databasebeta

← #240956

Subgroups — #240956

Order 512.

918 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#21177C211yesmaximal, characteristic
3#17708C211yesmaximal, characteristic
4C2×#1549C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#17486C211yesmaximal, characteristic
7#10555C218yesmaximal, complemented
8#7932C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3165—22no—
12#3165—22no—
13#3165—22no—
14#3165C2214yes—
15C2×#501—22no—
16C2×#500—21no—
17C2×#498—21no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#430—21no—
21C2×#428—21no—
22C2×#428—21no—
23C2×#423C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26#1950—22no—
27#1950C2214yes—
28#1738—21no—
29#1550C2214yes—
30#1549—22no—
31#1549—21no—
32#1549C2214yes—
33#1548—22no—
34#1548—22no—
35#1548—21no—
36#1542—22no—
37C2×#333C2214yes—
38C2×#332—22no—
39#1493—22no—
40#1487—22no—
41#1481C2214yes—
42#1283C2214yes—
43C23×Q8—21no—
44C22×M16—21no—
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⋊Q8D412yes—
66C2×C4⋊Q8D411yescharacteristic
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8D412yes—
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—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C42C2311yescharacteristic
79#501—28no—
80#501—28no—
81#500—24no—
82#500—24no—
83#498—24no—
84#498—24no—
85#431—21no—
86#431—21no—
87#431—21no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431C2312yes—
93#430—44no—
94C2×#108—24no—
95#428—24no—
96#428—24no—
97#428—24no—
98#428—24no—
99C2×#107—22no—
100C2×#107—22no—
101C2×#106—22no—
102C2×#106—22no—
103#423—21no—
104#423—21no—
105#423—21no—
106#423—21no—
107#423—21no—
108#423D412yes—
109#423D412yes—
110#423C2312yes—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2312yes—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C4×M16—21no—
120C4×M16—21no—
121C4×M16D412yes—
122C4×M16C2312yes—
123C2×#97—21no—
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127#341—24no—
128#333—24no—
129#333—28no—
130#332—28no—
131#332—24no—
132Q8⋊C8—28no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—41no—
139C22×Q8—41no—
140C22×Q8—41no—
141C22×Q8—84no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C23×C4#4111yescharacteristic
145C23×C4#4111yescharacteristic
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—42no—
154C2×M16—42no—
155C2×M16—41no—
156C2×M16—42no—
157C2×M16—42no—
158C2×M16SD1614yes—
159C2×M16Q1614yes—
160C2×M16C2×D412yes—
161C2×M16C2×D412yes—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—41no—
167C22×C8—41no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—21no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185C4⋊Q8—41no—
186C4⋊Q8—41no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C4⋊Q8C2×D414yes—
190C4⋊Q8C2×D412yes—
191C4⋊Q8C2×D412yes—
192#125—22no—
193#125—22no—
194#125—41no—
195#125—41no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212#117—22no—
213#117—22no—
214#117—41no—
215#117—41no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
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⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C42—41no—
236C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
248#108—48no—
249#107—28no—
250#107—28no—
251#106—28no—
252#106—28no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—41no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8C2×D414yes—
265C22⋊C8—22no—
266C22⋊C8—41no—
267C22⋊C8—41no—
268C22⋊C8—41no—
269#97—22no—
270#97—22no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—41no—
275C4×C8—41no—
276C4×C8—41no—
277C24#12411yescharacteristic
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—42no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—82no—
287C2×Q8—82no—
288C2×Q8—84no—
289C2×Q8—84no—
290C2×Q8—84no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×Q8—84no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—84no—
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4#12311yescharacteristic
312C22×C4#12311yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4#12411yescharacteristic
315C22×C4#12411yescharacteristic
316C22×C4#12411yescharacteristic
317C22×C4C2×SD1612yes—
318C22×C4C2×Q1612yes—
319C22×C4#13712yes—
320C22×C4#13712yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324M16—22no—
325M16—22no—
326M16—24no—
327M16—24no—
328M16—44no—
329M16—44no—
330M16—42no—
331M16—42no—
332M16—82no—
333M16C2×SD1614yes—
334M16C2×Q1614yes—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—82no—
346C2×C8—82no—
347C2×C8—82no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C42—42no—
371C42—42no—
372C42#12011yescharacteristic
373C42#12011yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42#12111yescharacteristic
377C42#12111yescharacteristic
378C42#12111yescharacteristic
379C42#12111yescharacteristic
380C42#12311yescharacteristic
381C42#12311yescharacteristic
382C42#12311yescharacteristic
383C42#12311yescharacteristic
384C42C22×D411yescharacteristic
385C42C22×D411yescharacteristic
386C42C22×D411yescharacteristic
387C42C2×#4111yescharacteristic
388C23—41no—
389C23—41no—
390C23#39411yescharacteristic
391C23#39411yescharacteristic
392C23#48712yes—
393C23#49112yes—
394C23C2×#12411yescharacteristic
395Q8—44no—
396Q8—82no—
397Q8—84no—
398Q8—82no—
399Q8—84no—
400Q8—84no—
401Q8—82no—
402Q8—84no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4#39411yescharacteristic
424C2×C4#39411yescharacteristic
425C2×C4#39411yescharacteristic
426C2×C4#39411yescharacteristic
427C2×C4#39411yescharacteristic
428C2×C4#39411yescharacteristic
429C2×C4#44812yes—
430C2×C4#44812yes—
431C2×C4#45112yes—
432C2×C4#45112yes—
433C2×C4#46112yes—
434C2×C4#46112yes—
435C2×C4#46212yes—
436C2×C4#46212yes—
437C2×C4#48712yes—
438C2×C4#49112yes—
439C2×C4C2×#12011yescharacteristic
440C2×C4C2×#12111yescharacteristic
441C2×C4C2×#12111yescharacteristic
442C2×C4C2×#12111yescharacteristic
443C2×C4C2×#12311yescharacteristic
444C2×C4C2×#12311yescharacteristic
445C2×C4C22×SD1611yescharacteristic
446C2×C4C22×Q1611yescharacteristic
447C2×C4C2×#13711yescharacteristic
448C2×C4C2×#13711yescharacteristic
449C8—24no—
450C8—24no—
451C8—44no—
452C8—82no—
453C8—82no—
454C8—82no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—42no—
462C22—42no—
463C22#204312yes—
464C22#204312yes—
465C22C2×#39411yescharacteristic
466C22C2×#48711yescharacteristic
467C22C2×#49111yescharacteristic
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—84no—
475C4—82no—
476C4—82no—
477C4#204312yes—
478C4#204312yes—
479C4#204812yes—
480C4C2×#44811yescharacteristic
481C4C2×#45111yescharacteristic
482C4C2×#46111yescharacteristic
483C4C2×#46211yescharacteristic
484C4#306912yes—
485C2—21no—
486C2—21no—
487C2—22no—
488C2—22no—
489C2C2×#204311yescharacteristic
490C2C2×#204811yescharacteristic
491C2#2353211yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#124
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. #10555C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#211772
#177082
C2×#15492
C2×#15482
#174862
#105552
#79322