Finite Group Databasebeta

← #236046

Subgroups — #236046

Order 512.

1428 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2C2×#3078C211yesmaximal, characteristic
3C2×#2567C211yesmaximal, characteristic
4#21385C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6C2×#1479C211yesmaximal, characteristic
7#10073C214yesmaximal, complemented
8#10057C214yesmaximal, complemented
9#7901C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#3209—24no—
14#3209—24no—
15C2×#494C2212yes—
16C2×#487C2212yes—
17#3078—24no—
18#3078—24no—
19C2×#479—22no—
20C2×#477C2212yes—
21C2×#475C2212yes—
22C2×#462—22no—
23C2×#451—22no—
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#428—21no—
27C2×#423C2211yescharacteristic
28C2×#419—21no—
29#2567—22no—
30#2567—22no—
31#2567—22no—
32#2567C2214yes—
33C2×#395—22no—
34C2×#391—21no—
35C2×#389—22no—
36C2×#384—21no—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1987C2212yes—
39#1981C2212yes—
40#1872—22no—
41C4×#103—22no—
42#1548—24no—
43C2×#333C2214yes—
44#1479—24no—
45C2×Q8⋊C8C2214yes—
46#1279C2214yes—
47C23×Q8D411yescharacteristic
48C23×Q8D412yes—
49C22×Q16—24no—
50C22×M16—21no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533D414yes—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8D412yes—
62C2×C4⋊Q8C2312yes—
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—22no—
76C2×C4×Q8D412yes—
77C2×C4×Q8C2312yes—
78C2×#117—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4D412yes—
84C22×C4⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#494—24no—
89#494—24no—
90#487—24no—
91#487—24no—
92#479—216no—
93#477—24no—
94#477—24no—
95#475—24no—
96#475—24no—
97#462—216no—
98#451—216no—
99#431—22no—
100#431—22no—
101#431—22no—
102#431D414yes—
103#428—28no—
104C2×#107—22no—
105C2×#106—22no—
106#423—22no—
107#423D414yes—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2312yes—
112#419—28no—
113C2×#103—22no—
114C2×#103—22no—
115C2×#103—24no—
116C2×#103—24no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8C2312yes—
121#395—28no—
122#395—28no—
123#391—24no—
124#391—24no—
125#389—28no—
126#389—28no—
127#384—24no—
128#384—24no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D412yes—
132C4×C4⋊C4C2312yes—
133C2×#95—24no—
134C2×#95—21no—
135#340—24no—
136#333—28no—
137Q8⋊C8—28no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—24no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—84no—
150C22×Q8Q1614yescomplemented
151C22×Q8C2×D412yes—
152C22×Q8C2×D414yes—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×Q16—216no—
161C2×Q16—88no—
162C2×M16—24no—
163C2×M16—42no—
164C2×M16—42no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—22no—
179C4⋊Q8—21no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184C4⋊Q8C2×D412yes—
185C4⋊Q8C2×D412yes—
186#125—24no—
187#125—42no—
188C22⋊Q8—24no—
189C22⋊Q8—24no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206#117—24no—
207#117—42no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
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—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—216no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—88no—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C22⋊C4—41no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—41no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42#4112yes—
252C2×C42C2411yescharacteristic
253#107—28no—
254#106—28no—
255C4⋊C8—24no—
256C4⋊C8—24no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8C2×D414yes—
261#103—28no—
262#103—28no—
263#103—216no—
264#103—48no—
265C22⋊C8—24no—
266C22⋊C8—42no—
267C4×C8—22no—
268C4×C8—42no—
269C4×C8C2×D414yes—
270#95—216no—
271#95—22no—
272#95—22no—
273C24#12011yescharacteristic
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—24no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—28no—
285C2×Q8—28no—
286C2×Q8—48no—
287C2×Q8—44no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—44no—
292C2×Q8—42no—
293C2×Q8—44no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—84no—
297C2×Q8—88no—
298C2×Q8—88no—
299C2×Q8C2×Q1614yes—
300C22×C4—24no—
301C22×C4—28no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—48no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12012yes—
321C22×C4#12012yes—
322C22×C4#12011yescharacteristic
323C22×C4#12112yes—
324C22×C4#12112yes—
325C22×C4#12112yes—
326C22×C4#12712yes—
327C22×C4C2×Q1612yes—
328C22×C4#13512yes—
329C22×C4#13712yes—
330C22×C4#13712yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334Q16—816no—
335M16—48no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—44no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—44no—
343C4⋊C4—28no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
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—42no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—816no—
368C4⋊C4#13718yes—
369C22⋊C4—44no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—44no—
375C42—44no—
376C42—44no—
377C42#12012yes—
378C42#12012yes—
379C42#12112yes—
380C42#12112yes—
381C42#12112yes—
382C42#12712yes—
383C42C22×D411yescharacteristic
384C42C22×D411yescharacteristic
385C42C22×D411yescharacteristic
386C42C2×#4111yescharacteristic
387C23—22no—
388C23—22no—
389C23#39212yes—
390C23#45112yes—
391C23#45212yes—
392C23C2×#12011yescharacteristic
393Q8—24no—
394Q8—24no—
395Q8—44no—
396Q8—48no—
397Q8—42no—
398Q8—48no—
399Q8—42no—
400Q8—44no—
401Q8—84no—
402Q8—82no—
403Q8—88no—
404Q8—82no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—28no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—416no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—48no—
426C2×C4—84no—
427C2×C4—88no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4#39212yes—
431C2×C4#39212yes—
432C2×C4#39212yes—
433C2×C4#45114yes—
434C2×C4#45112yes—
435C2×C4#45214yes—
436C2×C4#45212yes—
437C2×C4#46714yes—
438C2×C4#47014yes—
439C2×C4C2×#12011yescharacteristic
440C2×C4C2×#12011yescharacteristic
441C2×C4C2×#12111yescharacteristic
442C2×C4C2×#12111yescharacteristic
443C2×C4C2×#12111yescharacteristic
444C2×C4C2×#12711yescharacteristic
445C2×C4C22×Q1611yescharacteristic
446C2×C4C2×#13511yescharacteristic
447C2×C4C2×#13714yes—
448C2×C4C2×#13711yescharacteristic
449C2×C4C2×#13711yescharacteristic
450C8—44no—
451C8—44no—
452C8—82no—
453C22—24no—
454C22—24no—
455C22—22no—
456C22—22no—
457C22#200114yes—
458C22C2×#39211yescharacteristic
459C22C2×#45111yescharacteristic
460C22C2×#45211yescharacteristic
461C22#300412yes—
462C22#301912yes—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—22no—
467C4—48no—
468C4—44no—
469C4—82no—
470C4—84no—
471C4—82no—
472C4—82no—
473C4#200114yes—
474C4C2×#45111yescharacteristic
475C4C2×#45211yescharacteristic
476C4#303912yes—
477C4C2×#46711yescharacteristic
478C4C2×#47011yescharacteristic
479C4#315212yes—
480C2—24no—
481C2#1875312yes—
482C2#1876912yes—
483C2C2×#200111yescharacteristic
484C2#2340211yescharacteristic
485C2#2343211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10073C2semidirect
8. #10057C2semidirect
150. C22×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C2×#30782
C2×#25672
#213852
C2×#15482
C2×#14792
#100732
#100572
#79012