Finite Group Databasebeta

← #222051

Subgroups — #222051

Order 512.

1167 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic
2#22750C212yesmaximal
3C2×#1558C211yesmaximal, characteristic, complemented
4C22×#333C211yesmaximal, characteristic, complemented
5#9702C212yesmaximal, complemented
6#9136C212yesmaximal, complemented
7#9127C212yesmaximal, complemented
8#8594C211yesmaximal, characteristic, complemented
9#8342C211yesmaximal, characteristic, complemented
10#8342C211yesmaximal, characteristic, complemented
11#8232C211yesmaximal, characteristic, complemented
12C2×D4×Q8C412yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C2×#432C2212yes—
16C22×C4⋊C8C2212yes—
17#2868C2211yescharacteristic
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#2551—22no—
21#2546C412yes—
22#2440—22no—
23#2440—22no—
24#2434C412yes—
25#2434C412yes—
26D4×C4⋊C4—22no—
27C4×C4⋊Q8C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1924—22no—
30#1917—22no—
31#1869—22no—
32#1831C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1767C2212yes—
36#1752—22no—
37#1558—24no—
38C2×#333—24no—
39C2×#333C2211yescharacteristic
40C2×#333C2212yes—
41C2×#333C2211yescharacteristic
42C2×#333C2211yescharacteristic
43C2×#333C2212yes—
44C2×#333C2212yes—
45C2×#333C2212yes—
46C2×#333C2211yescharacteristic
47#1296C2212yes—
48#1296C2212yes—
49C23×Q8C2×C412yes—
50C23×Q8C2×C412yes—
51C23×C8—22no—
52D4×Q8—48no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8—42no—
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8C2×C412yes—
64C2×C4⋊Q8C2×C412yes—
65C2×C4⋊Q8C2×C412yes—
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8C2×C412yes—
72C2×C22⋊Q8C2×C412yes—
73C2×C4×Q8C2×C412yes—
74C2×C4×D4—22no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4D412yes—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#432—24no—
82#432—42no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—42no—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×#97—22no—
94C2×#97—22no—
95#398—24no—
96#395—24no—
97#395—24no—
98#391C2×C412yes—
99#391C2×C412yes—
100#391C2×C412yes—
101#389—24no—
102#389—22no—
103#389—22no—
104#389—22no—
105#385—24no—
106#385—24no—
107#384—22no—
108#384—22no—
109#384C2311yescharacteristic
110#380—22no—
111#380—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4D412yes—
115C4×C22⋊C4C2×C411yescharacteristic
116C4×C22⋊C4C2×C411yescharacteristic
117C4×C22⋊C4D412yes—
118C4×C22⋊C4D412yes—
119C43C2311yescharacteristic
120#336—22no—
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#333—22no—
126#333—22no—
127#333—22no—
128#333—24no—
129#333—24no—
130#333—42no—
131#333—42no—
132#333D414yescomplemented
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—24no—
136C22×Q8—24no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—44no—
140C22×Q8—44no—
141C22×Q8—82no—
142C22×Q8—82no—
143C22×Q8C22×C412yes—
144C22×Q8C22×C412yes—
145C22×Q8C22×C412yes—
146C22×D4C22⋊C411yescharacteristic
147C23×C4—22no—
148C23×C4—22no—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—44no—
158C22×C8—44no—
159C22×C8—82no—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—44no—
170C4⋊Q8—44no—
171C4⋊Q8—44no—
172C4⋊Q8—44no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—44no—
176C4⋊Q8—48no—
177C4⋊Q8—82no—
178C4⋊Q8C2×D412yes—
179C4⋊Q8#4112yes—
180C22⋊Q8—48no—
181C22⋊Q8—48no—
182C22⋊Q8—44no—
183C22⋊Q8—44no—
184C4×Q8—44no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—48no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
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—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4C22⋊C411yescharacteristic
211C2×C22⋊C4C22⋊C411yescharacteristic
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C412yes—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2411yescharacteristic
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—44no—
238C4⋊C8C2×D414yes—
239C22⋊C8—44no—
240C22⋊C8—44no—
241C22⋊C8—82no—
242#97—44no—
243#97—82no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250C24C2×C22⋊C411yescharacteristic
251C24C2×C22⋊C411yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8—88no—
270C2×Q8—88no—
271C2×Q8—88no—
272C2×Q8—88no—
273C2×Q8C4×D414yes—
274C2×D4—24no—
275C2×D4#10014yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C412yes—
311C22×C4#13714yes—
312C22×C4#13714yes—
313C22×C4C23×C411yescharacteristic
314C22×C4C22×D412yes—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—88no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—48no—
333C4⋊C4—48no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—84no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—41no—
367C42—44no—
368C42C22×D411yescharacteristic
369C42C2×#4111yescharacteristic
370C42H(5,𝔽2)12yes—
371C23—21no—
372C23—21no—
373C23—22no—
374C23—21no—
375C23—21no—
376C23—22no—
377C23—22no—
378C23C2×#10012yes—
379C23C2×#10012yes—
380C23#41914yes—
381C23C22×C22⋊C411yescharacteristic
382Q8—44no—
383Q8—44no—
384Q8—82no—
385Q8—82no—
386Q8—82no—
387Q8—82no—
388Q8—88no—
389Q8—88no—
390D4—28no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—24no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—88no—
434C2×C4—88no—
435C2×C4C2×#10012yes—
436C2×C4#41914yes—
437C2×C4C2×C4×D411yescharacteristic
438C2×C4#51811yescharacteristic
439C2×C4D4212yes—
440C2×C4#54612yes—
441C2×C4C2×#13712yes—
442C2×C4C2×#13712yes—
443C2×C4#57712yes—
444C2×C4#57812yes—
445C8—44no—
446C8—82no—
447C8—82no—
448C8—82no—
449C22—24no—
450C22—22no—
451C22—22no—
452C22—24no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—42no—
457C22—42no—
458C22#155814yescomplemented
459C22D4×C22⋊C411yescharacteristic
460C22C22×#10011yescharacteristic
461C22#288811yescharacteristic
462C22C2×#41912yes—
463C22#289812yes—
464C4—22no—
465C4—21no—
466C4—21no—
467C4—24no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—44no—
473C4—42no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4#326612yes—
478C4#327012yes—
479C4#331212yes—
480C4#332012yes—
481C2—24no—
482C2—42no—
483C2C2×#155811yescharacteristic
484C2#1776012yes—
485C2#1777911yescharacteristic
486C2D4×#10011yescharacteristic
487C2#2242512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1558C2semidirect
4. C22×#333C2semidirect
5. #9702C2semidirect
6. #9136C2semidirect
7. #9127C2semidirect
8. #8594C2semidirect
9. #8342C2semidirect
10. #8342C2semidirect
11. #8232C2semidirect
132. #333D4semidirect
458. C22#1558semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#227502
C2×#15582
C22×#3332
#97022
#91362
#91272
#85942
#83422
#82322