Finite Group Databasebeta

← #482065

Subgroups — #482065

Order 512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#20637C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#11883C214yesmaximal, complemented
5#11881C214yesmaximal, complemented
6#11700C214yesmaximal, complemented
7C2×#536—21no—
8C22×C4⋊Q8C411yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C2×#431—21no—
11C2×#423—21no—
12C2×#413—21no—
13#2857—21no—
14#2857—21no—
15#2621C412yes—
16#2621C2212yes—
17#2547—21no—
18#2547C412yes—
19#2546—21no—
20#2546C2212yes—
21#2486—21no—
22#2486—21no—
23#2486C412yes—
24#2486C2212yes—
25#2310—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#1346C22116yes—
39C22×#41D411yescharacteristic
40C23×Q8—21no—
41C22×M16—22no—
42#536—44no—
43#536—44no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—41no—
51C2×C4⋊Q8C2×C412yes—
52C2×C4⋊Q8C2×C412yes—
53C2×#124—22no—
54C2×#123—22no—
55C2×#123—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C4×#41—41no—
59C4×#41—41no—
60C4×#41—41no—
61C4×#41—41no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117C2×C411yescharacteristic
72C2×#117D411yescharacteristic
73C2×#117C2311yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2×C412yes—
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79#431—22no—
80#431—22no—
81#431—41no—
82#431—41no—
83#423—22no—
84#423—22no—
85#423—41no—
86#423—41no—
87C2×C4⋊C8—41no—
88C2×C4⋊C8—41no—
89C2×C4⋊C8—41no—
90#413—44no—
91C2×#101—22no—
92C2×#100—22no—
93C2×C22⋊C8—22no—
94C2×C4×C8—41no—
95#398—22no—
96#398—22no—
97#398C2×C414yes—
98#398C2314yes—
99#397—22no—
100#397—22no—
101#397C2×C414yes—
102#397C2×C414yes—
103#391—22no—
104#391—22no—
105#391—22no—
106#391—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#384—21no—
112#382C2×C412yes—
113#381—22no—
114#381—22no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380—21no—
119#380—22no—
120#380C2×C412yes—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C412yes—
123#343—22no—
124#343C2314yes—
125#324—22no—
126#324C2314yes—
127#323—22no—
128#323—22no—
129C2×#41—41no—
130C2×#41—41no—
131C2×#41—42no—
132C2×#41D814yes—
133C2×#41SD1614yes—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—44no—
139C22×Q8—44no—
140C22×Q8C2×D411yescharacteristic
141C22×D4C22⋊C411yescharacteristic
142C22×D4C22⋊C411yescharacteristic
143C22×D4C2×D411yescharacteristic
144C23×C4—22no—
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C2×M16—24no—
155C2×M16—42no—
156C2×M16—44no—
157C2×M16—44no—
158C22×C8—42no—
159C22×C8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172#124—48no—
173#123—48no—
174#123—44no—
175#123—44no—
176C22⋊Q8—48no—
177C22⋊Q8—44no—
178C22⋊Q8—44no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195#117—41no—
196#117—41no—
197#117—41no—
198#117—41no—
199#117—41no—
200#117—41no—
201#117—44no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22×C414yes—
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2411yescharacteristic
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4C22⋊C411yescharacteristic
225C2×C22⋊C4C22⋊C411yescharacteristic
226C2×C22⋊C4C22×C411yescharacteristic
227C2×C22⋊C4C22×C411yescharacteristic
228C2×C22⋊C4C2×D411yescharacteristic
229C2×C22⋊C4C2×D411yescharacteristic
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258C4⋊C8—42no—
259#101—44no—
260#100—44no—
261C22⋊C8—24no—
262C22⋊C8—42no—
263C4×C8—42no—
264C4×C8—42no—
265#95—22no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95C22×C414yes—
271C24C2×C22⋊C411yescharacteristic
272C24C2×C22⋊C411yescharacteristic
273C24C2×C22⋊C411yescharacteristic
274#41—44no—
275#41—44no—
276#41—44no—
277#41—44no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—88no—
288C2×Q8—88no—
289C2×Q8C2×D812yes—
290C2×Q8C2×SD1612yes—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4#10214yes—
298C2×D4#10214yes—
299C2×D4C2×D812yes—
300C2×D4C2×SD1612yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4#10214yes—
334C22×C4#10214yes—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×D812yes—
345C22×C4C2×SD1612yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349M16—44no—
350M16—44no—
351M16—84no—
352C2×C8—44no—
353C2×C8—44no—
354C2×C8—84no—
355C2×C8—84no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—48no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—44no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—44no—
402C42#9911yescharacteristic
403C42#9911yescharacteristic
404C42#9911yescharacteristic
405C42#9911yescharacteristic
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23C2×#10212yes—
419C23C2×#10212yes—
420C23C2×#10212yes—
421C23C22×C22⋊C411yescharacteristic
422Q8—42no—
423Q8—42no—
424Q8—88no—
425Q8—84no—
426Q8—84no—
427D4—44no—
428D4—44no—
429D4—42no—
430D4—42no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—88no—
478C2×C4C2×#9911yescharacteristic
479C2×C4C2×#9911yescharacteristic
480C2×C4C2×#9911yescharacteristic
481C2×C4C2×#9911yescharacteristic
482C2×C4C2×#9911yescharacteristic
483C2×C4C2×#9911yescharacteristic
484C2×C4C2×#10212yes—
485C2×C4C2×#10212yes—
486C2×C4C2×#10212yes—
487C2×C4#42112yes—
488C2×C4C22×D811yescharacteristic
489C2×C4C22×SD1611yescharacteristic
490C8—84no—
491C8—84no—
492C22—21no—
493C22—21no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—81no—
508C22—81no—
509C22C22×#9911yescharacteristic
510C22C22×#10211yescharacteristic
511C22C2×#42111yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—22no—
517C4—22no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—84no—
525C4—84no—
526C4#151912yes—
527C4#151912yes—
528C4#212312yes—
529C4#213512yes—
530C2—21no—
531C2—21no—
532C2—42no—
533C2—42no—
534C2—42no—
535C2C2×#151911yescharacteristic
536C2C2×#212311yescharacteristic
537C2C2×#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. #11883C2semidirect
5. #11881C2semidirect
6. #11700C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#206372
#173492
#118832
#118812
#117002