Finite Group Databasebeta

← #496209

Subgroups — #496209

Order 512.

1744 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C211yesmaximal, characteristic
2C2×#1784C212yesmaximal
3#13094C218yesmaximal, complemented
4#12860C212yesmaximal, complemented
5#12700C212yesmaximal, complemented
6C24×Q8C2211yescharacteristic
7C22×C4⋊Q8C411yescharacteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×C22⋊Q8C411yescharacteristic
10C22×#101—22no—
11C22×#101C2214yes—
12C2×#407—22no—
13#2434—22no—
14#2434—22no—
15#2434C414yes—
16#2434C2214yes—
17C2×#395—22no—
18C2×#391C411yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#385—22no—
21C2×C4×C22⋊C4C411yescharacteristic
22C2×#356C2218yescomplemented
23#1784—22no—
24#1784—22no—
25#1784C2214yes—
26#1784C2214yes—
27#1394C2218yescomplemented
28C23×Q8—41no—
29C23×Q8—42no—
30C23×Q8—41no—
31C23×Q8—44no—
32C23×Q8—44no—
33C23×Q8C2×C411yescharacteristic
34C23×Q8C2×C411yescharacteristic
35C23×Q8C2×C412yes—
36C23×Q8D412yes—
37C23×Q8D412yes—
38C23×Q8D412yes—
39C23×Q8C2312yes—
40C23×Q8C2311yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4D412yes—
43C22×M16—24no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—44no—
48C2×C4⋊Q8—44no—
49C2×C4⋊Q8C2×C414yes—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2312yes—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—42no—
54C2×C22⋊Q8—41no—
55C2×C22⋊Q8—41no—
56C2×C22⋊Q8C2×C414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C2×#101—24no—
66C2×#101—24no—
67C2×#101—24no—
68C2×#101—48no—
69C2×#101—48no—
70C2×#101C2318yes—
71#407—24no—
72#407—24no—
73#407—42no—
74#407—42no—
75C2×C22⋊C8—42no—
76C2×C22⋊C8—42no—
77#395—24no—
78#395—24no—
79#395—24no—
80#395—24no—
81#391—21no—
82#391—21no—
83#391—22no—
84#391C2×C412yes—
85#391C2×C414yes—
86#391C2312yes—
87#385—24no—
88#385—24no—
89#385—24no—
90#385—24no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4C2×C414yes—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96#356—48no—
97C25C22⋊C411yescharacteristic
98C22×Q8—24no—
99C22×Q8—28no—
100C22×Q8—24no—
101C22×Q8—28no—
102C22×Q8—21no—
103C22×Q8—41no—
104C22×Q8—41no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8—41no—
114C22×Q8—41no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8—42no—
118C22×Q8—44no—
119C22×Q8—44no—
120C22×Q8—44no—
121C22×Q8—44no—
122C22×Q8—44no—
123C22×Q8—44no—
124C22×Q8—44no—
125C22×Q8—42no—
126C22×Q8—44no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—41no—
130C22×Q8—41no—
131C22×Q8—416no—
132C22×Q8—432no—
133C22×Q8—416no—
134C22×Q8C22⋊C411yescharacteristic
135C22×Q8C22⋊C412yes—
136C22×Q8C22⋊C412yes—
137C22×Q8C22⋊C411yescharacteristic
138C22×Q8C22×C412yes—
139C22×Q8C22×C412yes—
140C22×Q8C22×C411yescharacteristic
141C22×Q8C2×D412yes—
142C22×Q8C2×D412yes—
143C22×Q8C2×D412yes—
144C22×Q8C2411yescharacteristic
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4C22⋊C412yes—
161C23×C4C22⋊C412yes—
162C23×C4C22⋊C412yes—
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C2×M16—28no—
171C2×M16—44no—
172C2×M16—48no—
173C2×M16—48no—
174C22×C8—44no—
175C4⋊Q8—28no—
176C4⋊Q8—44no—
177C4⋊Q8—48no—
178C4⋊Q8—44no—
179C4⋊Q8—44no—
180C4⋊Q8—48no—
181C22⋊Q8—44no—
182C22⋊Q8—44no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4C22×C412yes—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210#101—216no—
211#101—48no—
212#101—48no—
213#101—48no—
214C22⋊C8—44no—
215C22⋊C8—44no—
216C22⋊C8—44no—
217C22⋊C8—44no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223C24—24no—
224C24—41no—
225C24—41no—
226C24—42no—
227C24#9912yes—
228C24#9911yescharacteristic
229C24#9911yescharacteristic
230C24C2×C22⋊C411yescharacteristic
231C24C2×C22⋊C412yes—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—22no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—416no—
262C2×Q8—416no—
263C2×Q8—416no—
264C2×Q8—416no—
265C2×Q8—48no—
266C2×Q8—48no—
267C2×Q8—48no—
268C2×Q8—48no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—416no—
274C2×Q8—416no—
275C2×Q8—416no—
276C2×Q8—416no—
277C2×Q8—416no—
278C2×Q8—416no—
279C2×Q8—42no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8#9914yes—
283C2×Q8H(5,𝔽2)14yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—44no—
335C22×C4—48no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D412yes—
355M16—48no—
356M16—48no—
357M16—88no—
358C2×C8—48no—
359C2×C8—88no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C22⋊C4—216no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C42—24no—
371C42—42no—
372C42—42no—
373C42—42no—
374C23—21no—
375C23—21no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—24no—
382C23—22no—
383C23—24no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—48no—
391C23—48no—
392C23#35614yescomplemented
393C23C2×#9912yes—
394C23C2×#9912yes—
395C23C2×#9911yescharacteristic
396C23C2×#9911yescharacteristic
397C23C22×C22⋊C411yescharacteristic
398Q8—44no—
399Q8—44no—
400Q8—48no—
401Q8—48no—
402Q8—48no—
403Q8—48no—
404Q8—44no—
405Q8—48no—
406Q8—44no—
407Q8—48no—
408Q8—84no—
409Q8—84no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4C2×#9912yes—
465C2×C4C2×#9914yes—
466C2×C4#51812yes—
467C2×C4#53414yes—
468C8—88no—
469C22—24no—
470C22—24no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—22no—
476C22—21no—
477C22—21no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22—48no—
486C22—44no—
487C22C2×#35612yes—
488C22C2×#35614yes—
489C22#213312yes—
490C22#230811yescharacteristic
491C22C22×#9911yescharacteristic
492C22C22×#9911yescharacteristic
493C4—24no—
494C4—22no—
495C4—24no—
496C4—22no—
497C4—44no—
498C4—44no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—82no—
504C2—21no—
505C2—21no—
506C2—22no—
507C2—22no—
508C2—24no—
509C2—42no—
510C2#2174611yescharacteristic
511C2C22×#35611yescharacteristic
512C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C24×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13094C2semidirect
4. #12860C2semidirect
5. #12700C2semidirect
22. C2×#356C22semidirect
27. #1394C22semidirect
392. C23#356semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
C2×#17842
#130942
#128602
#127002