Finite Group Databasebeta

← #482134

Subgroups — #482134

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#20637C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#11908C214yesmaximal, complemented
5#11904C214yesmaximal, complemented
6#11724C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C2×#431—21no—
10C2×#423—21no—
11C2×#413—21no—
12#2857—21no—
13#2857—21no—
14#2628C412yes—
15#2628C2212yes—
16#2568—21no—
17#2567C2212yes—
18#2563C412yes—
19#2542—21no—
20#2487—21no—
21#2487—21no—
22#2409C412yes—
23#2409C2212yes—
24#2317—21no—
25#2311—21no—
26#2285C411yescharacteristic
27#1801—21no—
28#1801C2212yes—
29#1800—21no—
30#1800C2212yes—
31#1742C2212yes—
32#1461C2212yes—
33#1460—21no—
34#1459—21no—
35#1459C2212yes—
36#1458—21no—
37#1458C2212yes—
38#1354C22116yes—
39C22×#41D411yescharacteristic
40C23×Q8C2×C412yes—
41C22×M16—22no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C412yes—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C4×#41—41no—
53C4×#41—41no—
54C4×#41—41no—
55C4×#41—41no—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4C2311yescharacteristic
64C2×#117—21no—
65C2×#117C2×C411yescharacteristic
66C2×#117D411yescharacteristic
67C2×#117C2311yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C42C2×C411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73#431—22no—
74#431—22no—
75#431—41no—
76#431—41no—
77#423—22no—
78#423—22no—
79#423—41no—
80#423—41no—
81C2×C4⋊C8—41no—
82C2×C4⋊C8—41no—
83C2×C4⋊C8—41no—
84#413—44no—
85C2×#101—22no—
86C2×#100—22no—
87C2×C22⋊C8—22no—
88C2×C4×C8—41no—
89#400—22no—
90#400—22no—
91#399—22no—
92#399—22no—
93#395C2×C414yes—
94#395C2314yes—
95#393C2×C414yes—
96#393C2×C414yes—
97#391—22no—
98#391—22no—
99#391—22no—
100#390—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#386—22no—
106#386—22no—
107#385—22no—
108#385—22no—
109#384—22no—
110#384—21no—
111#382—22no—
112#382C2×C412yes—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—21no—
117#380—22no—
118#380C2×C412yes—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121#343—22no—
122#343C2314yes—
123#324—22no—
124#324C2314yes—
125#323—22no—
126#323—22no—
127C2×#41—41no—
128C2×#41—41no—
129C2×#41—42no—
130C2×#41SD1614yes—
131C2×#41Q1614yes—
132C22×Q8—22no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—84no—
136C22×Q8C22×C414yes—
137C22×Q8C2×D411yescharacteristic
138C22×D4C22⋊C411yescharacteristic
139C22×D4C22⋊C411yescharacteristic
140C22×D4C2×D411yescharacteristic
141C23×C4—22no—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—24no—
152C2×M16—42no—
153C2×M16—44no—
154C2×M16—44no—
155C22×C8—42no—
156C22×C8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C22⋊Q8—84no—
168C22⋊Q8—84no—
169C4×Q8—41no—
170C4×Q8—41no—
171C4×Q8—41no—
172C4×Q8—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—41no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—41no—
185#117—41no—
186#117—41no—
187#117—41no—
188#117—41no—
189#117—41no—
190#117—41no—
191#117—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2411yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4C22⋊C411yescharacteristic
215C2×C22⋊C4C22⋊C411yescharacteristic
216C2×C22⋊C4C22×C411yescharacteristic
217C2×C22⋊C4C22×C411yescharacteristic
218C2×C22⋊C4C2×D411yescharacteristic
219C2×C22⋊C4C2×D411yescharacteristic
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249#101—44no—
250#100—44no—
251C22⋊C8—24no—
252C22⋊C8—42no—
253C4×C8—42no—
254C4×C8—42no—
255#95—22no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95C22×C414yes—
261C24C2×C22⋊C411yescharacteristic
262C24C2×C22⋊C411yescharacteristic
263C24C2×C22⋊C411yescharacteristic
264#41—44no—
265#41—44no—
266#41—44no—
267#41—44no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—84no—
276C2×Q8—88no—
277C2×Q8—88no—
278C2×Q8C2×SD1612yes—
279C2×Q8C2×Q1612yes—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4#10314yes—
287C2×D4#10314yes—
288C2×D4C2×SD1612yes—
289C2×D4C2×Q1612yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—84no—
321C22×C4#10314yes—
322C22×C4#10314yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×SD1612yes—
333C22×C4C2×Q1612yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337M16—44no—
338M16—44no—
339M16—84no—
340C2×C8—44no—
341C2×C8—44no—
342C2×C8—84no—
343C2×C8—84no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—84no—
360C4⋊C4—84no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—44no—
368C22⋊C4—84no—
369C22⋊C4—84no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—44no—
389C42#9911yescharacteristic
390C42#9911yescharacteristic
391C42#9911yescharacteristic
392C42#9911yescharacteristic
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23C2×#10312yes—
406C23C2×#10312yes—
407C23C2×#10312yes—
408C23C22×C22⋊C411yescharacteristic
409Q8—42no—
410Q8—42no—
411Q8—84no—
412Q8—88no—
413Q8—84no—
414D4—44no—
415D4—44no—
416D4—42no—
417D4—42no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—88no—
465C2×C4C2×#9911yescharacteristic
466C2×C4C2×#9911yescharacteristic
467C2×C4C2×#9911yescharacteristic
468C2×C4C2×#9911yescharacteristic
469C2×C4C2×#9911yescharacteristic
470C2×C4C2×#9911yescharacteristic
471C2×C4C2×#10312yes—
472C2×C4C2×#10312yes—
473C2×C4C2×#10312yes—
474C2×C4#42112yes—
475C2×C4C22×SD1611yescharacteristic
476C2×C4C22×Q1611yescharacteristic
477C8—84no—
478C8—84no—
479C22—21no—
480C22—21no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—81no—
495C22—81no—
496C22C22×#9911yescharacteristic
497C22C22×#10311yescharacteristic
498C22C2×#42111yescharacteristic
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—22no—
504C4—22no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—84no—
512C4—84no—
513C4#152012yes—
514C4#152012yes—
515C4#212212yes—
516C4#213512yes—
517C2—21no—
518C2—21no—
519C2—42no—
520C2—42no—
521C2—42no—
522C2C2×#152011yescharacteristic
523C2C2×#212211yescharacteristic
524C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11908C2semidirect
5. #11904C2semidirect
6. #11724C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#206372
#173492
#119082
#119042
#117242