Finite Group Databasebeta

← #213679

Subgroups — #213679

Order 512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4#9140C214yesmaximal, complemented
5#9137C214yesmaximal, complemented
6#8723C212yesmaximal, complemented
7#8340C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×C22⋊Q8—21no—
10C23×C42C2211yescharacteristic
11C2×#432C2212yes—
12C2×#423C2212yes—
13C2×#407—21no—
14#2857—21no—
15#2857—21no—
16#2857—21no—
17#2857—21no—
18#2857C412yes—
19#2857C412yes—
20#2857C2212yes—
21#2857C2212yes—
22C2×#391—21no—
23C2×#389—21no—
24C2×#389C412yes—
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27C2×#380C412yes—
28#1817—21no—
29#1797—21no—
30#1797—21no—
31#1797C2212yes—
32#1797C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1779—21no—
36#1749—21no—
37#1749—21no—
38#1728—21no—
39C2×#332C2218yescomplemented
40#1296C2218yescomplemented
41C23×Q8—21no—
42C24×C4C2×C411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C22×M16—22no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—44no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2312yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—48no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C22⋊C4—22no—
63C22×C42—44no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#432—44no—
77#423—44no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8C2314yes—
80C2×C4⋊C8C2314yes—
81#407—44no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×#97—22no—
88#391—22no—
89#391—22no—
90#391—22no—
91#391—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389C2×C414yes—
99#389C2×C414yes—
100#384—21no—
101#384—21no—
102#384C2×C412yes—
103#384C2312yes—
104#380—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380C2×C414yes—
111#380C2×C414yes—
112C2×#95—22no—
113C2×#95—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#332—48no—
119C25C22⋊C411yescharacteristic
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—44no—
123C22×Q8—44no—
124C22×Q8—44no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
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
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×M16—44no—
156C2×M16—44no—
157C2×M16—44no—
158C22×C8—24no—
159C22×C8—24no—
160C4⋊Q8—28no—
161C4⋊Q8—44no—
162C4⋊Q8—44no—
163C4⋊Q8—44no—
164C22⋊Q8—48no—
165C22⋊Q8—48no—
166C22⋊Q8—48no—
167C22⋊Q8—48no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C22×C414yes—
186C2×C4⋊C4C22×C414yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—48no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2411yescharacteristic
218C4⋊C8—28no—
219C4⋊C8—28no—
220C4⋊C8—44no—
221C22⋊C8—44no—
222C22⋊C8—44no—
223C22⋊C8—44no—
224C22⋊C8—44no—
225C22⋊C8—44no—
226#97—44no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—41no—
237C24—41no—
238C24C2×C22⋊C411yescharacteristic
239C24C2×C22⋊C411yescharacteristic
240C24C2×C22⋊C411yescharacteristic
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—816no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—416no—
305C22×C4—416no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4#10114yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4AGL(1,ℤ/8)12yes—
323C22×C4AGL(1,ℤ/8)14yes—
324C22×C4#13712yes—
325C22×C4#13714yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329M16—88no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C4⋊C4—28no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—42no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42H(5,𝔽2)14yes—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—28no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23C2×#10112yes—
391C23#41312yes—
392C23#41712yes—
393C23#41812yes—
394C23#41912yes—
395C23C22×C22⋊C411yescharacteristic
396Q8—88no—
397Q8—88no—
398C2×C4—28no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—88no—
450C2×C4—88no—
451C2×C4C2×#10112yes—
452C2×C4C2×#10112yes—
453C2×C4#41712yes—
454C2×C4#41812yes—
455C2×C4#41912yes—
456C2×C4#51812yes—
457C2×C4#53412yes—
458C2×C4#53412yes—
459C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
460C2×C4C2×AGL(1,ℤ/8)12yes—
461C2×C4C2×#13712yes—
462C2×C4C2×#13711yescharacteristic
463C2×C4#57512yes—
464C8—84no—
465C8—84no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—416no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22#155714yes—
488C22#230811yescharacteristic
489C22C22×#10111yescharacteristic
490C22C2×#41311yescharacteristic
491C22C2×#41811yescharacteristic
492C22C2×#41911yescharacteristic
493C22C2×#41711yescharacteristic
494C22#289611yescharacteristic
495C4—22no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—84no—
506C4—84no—
507C4#319312yes—
508C4#319414yes—
509C4#319512yes—
510C2—24no—
511C2—42no—
512C2—42no—
513C2C2×#155711yescharacteristic
514C2#1774211yescharacteristic
515C2#1774811yescharacteristic
516C2#1775111yescharacteristic
517C2#2175311yescharacteristic
518C2#2176511yescharacteristic
519C2#2176711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9140C2semidirect
5. #9137C2semidirect
6. #8723C2semidirect
7. #8340C2semidirect
39. C2×#332C22semidirect
40. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#207152
#206262
#91402
#91372
#87232
#83402