Finite Group Databasebeta

← #220114

Subgroups — #220114

Order 512.

1023 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#22650C211yesmaximal, characteristic
3#22650C211yesmaximal, characteristic
4C2×#1561C212yesmaximal, complemented
5#9694C211yesmaximal, characteristic, complemented
6#9692C211yesmaximal, characteristic, complemented
7#9139C212yesmaximal, complemented
8#9137C212yesmaximal, complemented
9#8586C211yesmaximal, characteristic, complemented
10#8345C212yesmaximal, complemented
11C4×#332C211yesmaximal, characteristic, complemented
12C2×#533C2212yes—
13C2×#436—21no—
14C2×#436—21no—
15C2×#423C2212yes—
16C2×#423C2212yes—
17#2868C2211yescharacteristic
18#2857C2212yes—
19#2552C412yes—
20#2542—21no—
21#2538—21no—
22#2441C414yes—
23#2429—22no—
24#2423—22no—
25#2365—21no—
26#2354C412yes—
27Q8×C4⋊C4—21no—
28C4×C4⋊Q8C2211yescharacteristic
29C42×D4C2211yescharacteristic
30#1939—21no—
31#1939—21no—
32#1851—22no—
33#1851—22no—
34#1842C2211yescharacteristic
35#1842C2211yescharacteristic
36#1797C2212yes—
37#1797C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1561—28no—
41C2×#332C2211yescharacteristic
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#332C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2211yescharacteristic
50#1296C2212yes—
51C22×M16—22no—
52C22×M16—22no—
53#533—24no—
54#533—24no—
55#533—42no—
56#533—42no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2312yes—
60C2×C4⋊Q8C2311yescharacteristic
61C2×#125C2×C414yes—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C4×Q8—21no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×#117C2×C414yes—
72C22×C42C2312yes—
73#436—44no—
74#436—44no—
75#423—24no—
76#423—24no—
77#423—42no—
78#423—42no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95#399—24no—
96#398—24no—
97#396—24no—
98#391—21no—
99#391—22no—
100#390—22no—
101#389—22no—
102#389—21no—
103#389—22no—
104#389—24no—
105#389C2×C412yes—
106#389C2×C414yes—
107#388—24no—
108#387—24no—
109#384—22no—
110#384—21no—
111#384—21no—
112#384C2×C412yes—
113#384C2×C412yes—
114#384C2311yescharacteristic
115#380—22no—
116#380C2×C414yes—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4C2×C412yes—
124C4×C22⋊C4D412yes—
125C4×C22⋊C4D412yes—
126C43C2311yescharacteristic
127#336—22no—
128#336—22no—
129#336—21no—
130#336—21no—
131#332—24no—
132#332—22no—
133#332—24no—
134#332—24no—
135#332—42no—
136#332—42no—
137#332D414yescomplemented
138C22×Q8—21no—
139C22×Q8—22no—
140C22×D4C22⋊C411yescharacteristic
141C23×C4C22×C412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C2×M16—44no—
145C2×M16—44no—
146C2×M16—44no—
147C2×M16—44no—
148C2×M16—82no—
149C2×M16—82no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—81no—
165C4⋊Q8—81no—
166C4⋊Q8C2×D412yes—
167C4⋊Q8#4112yes—
168#125—28no—
169#125—44no—
170C22⋊Q8—44no—
171C22⋊Q8—44no—
172C22⋊Q8—44no—
173C22⋊Q8—44no—
174C22⋊Q8—82no—
175C22⋊Q8—82no—
176C4×Q8—44no—
177C4×Q8—44no—
178C4×Q8—44no—
179C4×Q8—44no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×D4—24no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187#117—28no—
188#117—44no—
189C2×C4⋊C4—21no—
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—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22×C414yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C412yes—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C412yes—
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2411yescharacteristic
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—44no—
239C4⋊C8—44no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8—81no—
243C4⋊C8—81no—
244C4⋊C8C2×D412yes—
245C4⋊C8C2×D412yes—
246C22⋊C8—44no—
247C22⋊C8—44no—
248C4×C8—42no—
249C4×C8—42no—
250#95—24no—
251#95—22no—
252#95—22no—
253#95—24no—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257C24C2×C22⋊C412yes—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—42no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
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—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—42no—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302M16—88no—
303M16—88no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—41no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—84no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4#10114yes—
345C4⋊C4C4×D414yes—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—48no—
350C22⋊C4—84no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—22no—
362C42—41no—
363C42—48no—
364C42—44no—
365C42AGL(1,ℤ/8)12yes—
366C42AGL(1,ℤ/8)12yes—
367C42#13712yes—
368C42#13712yes—
369C42C22×D411yescharacteristic
370C42C2×#4111yescharacteristic
371C42H(5,𝔽2)11yescharacteristic
372C42H(5,𝔽2)11yescharacteristic
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23C22×C22⋊C411yescharacteristic
381Q8—42no—
382Q8—42no—
383Q8—42no—
384Q8—42no—
385Q8—82no—
386Q8—82no—
387Q8—82no—
388Q8—82no—
389D4—44no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4C2×#10114yes—
427C2×C4C2×#10112yes—
428C2×C4#41712yes—
429C2×C4#41712yes—
430C2×C4#41712yes—
431C2×C4#41712yes—
432C2×C4C2×C4×D411yescharacteristic
433C2×C4#51811yescharacteristic
434C2×C4D4211yescharacteristic
435C2×C4D4211yescharacteristic
436C2×C4#54611yescharacteristic
437C2×C4#54611yescharacteristic
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
440C2×C4C2×#13711yescharacteristic
441C2×C4C2×#13711yescharacteristic
442C2×C4#57611yescharacteristic
443C2×C4#57711yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57811yescharacteristic
446C8—42no—
447C8—42no—
448C8—81no—
449C8—81no—
450C8—82no—
451C8—82no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22D4×C22⋊C411yescharacteristic
460C22C22×#10111yescharacteristic
461C22#288911yescharacteristic
462C22C2×#41711yescharacteristic
463C22C2×#41711yescharacteristic
464C22#289711yescharacteristic
465C22#289811yescharacteristic
466C4—22no—
467C4—22no—
468C4—21no—
469C4—21no—
470C4—24no—
471C4—41no—
472C4—42no—
473C4—44no—
474C4—42no—
475C4—82no—
476C4—84no—
477C4#156114yescomplemented
478C4#327211yescharacteristic
479C4#327411yescharacteristic
480C4#327611yescharacteristic
481C4#327811yescharacteristic
482C4#333011yescharacteristic
483C4#333111yescharacteristic
484C4#333211yescharacteristic
485C4#333311yescharacteristic
486C2—82no—
487C2C2×#156111yescharacteristic
488C2#1776311yescharacteristic
489C2#1776511yescharacteristic
490C2#1777711yescharacteristic
491C2D4×#10111yescharacteristic
492C2#2243111yescharacteristic
493C2#2243211yescharacteristic
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
4. C2×#1561C2semidirect
5. #9694C2semidirect
6. #9692C2semidirect
7. #9139C2semidirect
8. #9137C2semidirect
9. #8586C2semidirect
10. #8345C2semidirect
11. C4×#332C2semidirect
137. #332D4semidirect
477. C4#1561semidirect

Maximal subgroups

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

subgroupindex
#361342
#226502
C2×#15612
#96942
#96922
#91392
#91372
#85862
#83452
C4×#3322