Finite Group Databasebeta

← #496192

Subgroups — #496192

Order 512.

1838 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1838 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2432C211yesmaximal, characteristic
2#20584C212yesmaximal
3#13091C218yesmaximal, complemented
4#12856C212yesmaximal, complemented
5#12694C212yesmaximal, complemented
6C23×#41C2211yescharacteristic
7C22×C4⋊Q8C411yescharacteristic, complemented
8C22×#124C2211yescharacteristic
9C22×C22⋊Q8C411yescharacteristic, complemented
10C2×#410—22no—
11C2×#410C2214yes—
12#2432—22no—
13#2432—22no—
14#2432C414yescomplemented
15#2432C2214yes—
16C2×#394—22no—
17C2×#391C411yescharacteristic, complemented
18C2×#390C2211yescharacteristic
19C2×#385—22no—
20#2280—22no—
21C2×C4×C22⋊C4C411yescharacteristic, complemented
22C2×#354C2218yescomplemented
23#1782—22no—
24#1782C2214yes—
25#1781—22no—
26#1781C2214yes—
27#1393C2218yescomplemented
28C22×#41—41no—
29C22×#41—42no—
30C22×#41—41no—
31C22×#41—44no—
32C22×#41—44no—
33C22×#41C2×C412yes—
34C22×#41D412yes—
35C22×#41D412yes—
36C22×#41C2312yes—
37C23×Q8C2×C411yescharacteristic
38C23×Q8C2×C411yescharacteristic
39C23×D4D412yes—
40C23×D4C2311yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4D412yes—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—44no—
46C2×C4⋊Q8C2×C414yes—
47C2×#124—22no—
48C2×#124—44no—
49C2×#124C2×C412yes—
50C2×#124C2312yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—42no—
53C2×C22⋊Q8—41no—
54C2×C22⋊Q8—41no—
55C2×C22⋊Q8C2×C414yes—
56C2×#117—24no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66#410—48no—
67#410—48no—
68C2×#99—24no—
69C2×#99—24no—
70C2×#99—24no—
71C2×#99C2318yes—
72#394—24no—
73#394—24no—
74#394—24no—
75#394—24no—
76#391—21no—
77#391—21no—
78#391C2×C414yes—
79#390—22no—
80#390C2×C412yes—
81#390C2312yes—
82#385—24no—
83#385—24no—
84#385—24no—
85#385—24no—
86#381—24no—
87#380—42no—
88#380—42no—
89#379—24no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—42no—
92C4×C22⋊C4—42no—
93C4×C22⋊C4C2×C414yes—
94C2×#95—22no—
95C2×#95—21no—
96#354—48no—
97C25C22⋊C411yescharacteristic
98C25C2×D412yes—
99C2×#41—28no—
100C2×#41—28no—
101C2×#41—42no—
102C2×#41—42no—
103C2×#41—42no—
104C2×#41—42no—
105C2×#41—42no—
106C2×#41—42no—
107C2×#41—42no—
108C2×#41—42no—
109C2×#41—44no—
110C2×#41—44no—
111C2×#41—44no—
112C2×#41—44no—
113C2×#41—44no—
114C2×#41—44no—
115C2×#41—416no—
116C2×#41—432no—
117C2×#41—416no—
118C22×Q8—24no—
119C22×Q8—21no—
120C22×Q8—41no—
121C22×Q8—41no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C22⋊C411yescharacteristic
129C22×Q8C22×C412yes—
130C22×Q8C22×C412yes—
131C22×D4—24no—
132C22×D4—44no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—41no—
137C22×D4—41no—
138C22×D4—44no—
139C22×D4C22⋊C412yes—
140C22×D4C22⋊C412yes—
141C22×D4C22×C411yescharacteristic
142C22×D4C2×D412yes—
143C22×D4C2×D412yes—
144C22×D4C2×D412yes—
145C22×D4C2411yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4C22⋊C412yes—
162C23×C4C22⋊C412yes—
163C23×C4C22⋊C412yes—
164C23×C4C22⋊C412yes—
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C4⋊Q8—44no—
171C4⋊Q8—44no—
172C4⋊Q8—48no—
173#124—28no—
174#124—44no—
175#124—48no—
176C22⋊Q8—44no—
177C22⋊Q8—44no—
178C22⋊Q8—44no—
179C22⋊Q8—44no—
180#117—48no—
181#117—48no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4C22×C412yes—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—48no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—44no—
213C2×C42C22×C412yes—
214C2×C42C22×C412yes—
215#99—216no—
216#99—48no—
217#99—48no—
218#99—48no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—44no—
223#95—44no—
224C24—24no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—22no—
229C24—41no—
230C24—41no—
231C24—42no—
232C24—44no—
233C24—44no—
234C24—42no—
235C24—42no—
236C24C2×C22⋊C411yescharacteristic
237C24C2×C22⋊C412yes—
238C24C2×C22⋊C412yes—
239C24C22×D412yes—
240#41—416no—
241#41—416no—
242#41—416no—
243#41—416no—
244#41—416no—
245#41—416no—
246#41—416no—
247#41—416no—
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—48no—
258C2×Q8—48no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—84no—
265C2×Q8—84no—
266C2×Q8#9914yes—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—48no—
288C2×D4—48no—
289C2×D4—416no—
290C2×D4—416no—
291C2×D4H(5,𝔽2)14yes—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—48no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—48no—
351C22×C4#9911yescharacteristic
352C22×C4#9911yescharacteristic
353C22×C4#9912yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C23×C411yescharacteristic
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—48no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—48no—
365C4⋊C4—88no—
366C22⋊C4—48no—
367C22⋊C4—48no—
368C22⋊C4—48no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—88no—
375C22⋊C4—88no—
376C42—24no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—88no—
381C23—21no—
382C23—21no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—44no—
394C23—44no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—416no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—42no—
408C23—44no—
409C23—42no—
410C23—82no—
411C23—82no—
412C23—88no—
413C23—84no—
414C23—84no—
415C23C2×#9911yescharacteristic
416C23C2×#9911yescharacteristic
417C23C22×C22⋊C411yescharacteristic
418C23#53414yes—
419Q8—44no—
420Q8—48no—
421Q8—44no—
422Q8—84no—
423Q8—84no—
424D4—44no—
425D4—48no—
426D4—48no—
427D4—44no—
428D4—48no—
429D4—48no—
430D4—48no—
431C2×C4—28no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—48no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—84no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—88no—
479C2×C4—88no—
480C2×C4—88no—
481C2×C4C2×#9912yes—
482C2×C4C2×#9914yes—
483C2×C4C2×#9912yes—
484C2×C4C2×#9912yes—
485C2×C4#51812yes—
486C22—24no—
487C22—24no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—24no—
495C22—44no—
496C22—44no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—88no—
507C22—88no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—84no—
512C22—84no—
513C22#212612yes—
514C22#212912yes—
515C22#230811yescharacteristic
516C22C22×#9911yescharacteristic
517C22C22×#9911yescharacteristic
518C4—22no—
519C4—22no—
520C4—22no—
521C4—24no—
522C4—22no—
523C4—44no—
524C4—42no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—168no—
529C2—21no—
530C2—21no—
531C2—22no—
532C2—22no—
533C2—44no—
534C2—82no—
535C2—82no—
536C2#2174611yescharacteristic
537C2C2×#212611yescharacteristic
538C2C2×#212911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13091C2semidirect
4. #12856C2semidirect
5. #12694C2semidirect
7. C22×C4⋊Q8C4semidirect
9. C22×C22⋊Q8C4semidirect
14. #2432C4semidirect
17. C2×#391C4semidirect
21. C2×C4×C22⋊C4C4semidirect
22. C2×#354C22semidirect
27. #1393C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24322
#205842
#130912
#128562
#126942