Finite Group Databasebeta

← #223209

Subgroups — #223209

Order 512.

1033 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic, complemented
3D4×C4⋊C8C211yesmaximal, characteristic, complemented
4#9896C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9142C212yesmaximal, complemented
7#9133C212yesmaximal, complemented
8#8729C212yesmaximal, complemented
9#8585C211yesmaximal, characteristic, complemented
10C4×#331C211yesmaximal, characteristic, complemented
11#8044C212yesmaximal, complemented
12C2×#555C412yes—
13C2×C8×D4—21no—
14C2×#432C2212yes—
15C22×C4⋊C8C2212yes—
16#2868C2211yescharacteristic
17#2856C2212yes—
18#2563C412yes—
19#2537—21no—
20#2537—21no—
21#2500C414yes—
22#2491—22no—
23#2491—22no—
24#2354—21no—
25#2354—21no—
26C4×#127C2211yescharacteristic
27#2285C2212yes—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1939—21no—
31#1924—22no—
32#1850—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1797C2212yes—
36#1797C2212yes—
37#1770—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C2×#331C2211yescharacteristic
41C2×#331C2211yescharacteristic
42C2×#331C2212yes—
43#1296C2211yescharacteristic
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2211yescharacteristic
48#1296C2212yes—
49#1296C2212yes—
50C23×C8—22no—
51#555—48no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C412yes—
55C2×#127C2311yescharacteristic
56C2×#124—24no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121C2×C414yes—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×#117—22no—
68C2×#117C2×C414yes—
69C22×C42C2312yes—
70C8×D4—44no—
71#432—24no—
72#432—42no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×#97—22no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92#394—24no—
93#394—24no—
94#393—24no—
95#390—21no—
96#390—21no—
97#390—22no—
98#390—22no—
99#390C2×C412yes—
100#390C2×C414yes—
101#388—24no—
102#388—24no—
103#384—21no—
104#384—21no—
105#384C2×C412yes—
106#384C2311yescharacteristic
107#384C2312yes—
108#382C2×C414yes—
109#380—22no—
110#380—22no—
111#380—22no—
112#380—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4D412yes—
120C4×C22⋊C4D412yes—
121C43C2311yescharacteristic
122#336—22no—
123#336—22no—
124#336—21no—
125#336—21no—
126#331—22no—
127#331—22no—
128#331—42no—
129C22×Q8—22no—
130C22×D4—24no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4C22⋊C411yescharacteristic
134C22×D4C22×C412yes—
135C23×C4—22no—
136C23×C4C22×C412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—44no—
146C22×C8—44no—
147C22×C8—82no—
148C4⋊Q8—44no—
149C4⋊Q8—81no—
150C4⋊Q8—81no—
151C4⋊Q8—81no—
152C4⋊Q8—81no—
153#127—21no—
154#127—21no—
155#127—21no—
156#127—21no—
157#124—416no—
158#121—48no—
159#121—48no—
160#121—82no—
161#121—82no—
162#121—82no—
163#121—82no—
164C4×D4—24no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—44no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174#117—48no—
175#117—48no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C414yes—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C412yes—
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—44no—
224C4⋊C8—42no—
225C4⋊C8—42no—
226C4⋊C8—81no—
227C4⋊C8—81no—
228C4⋊C8#4112yes—
229C4⋊C8#4112yes—
230C22⋊C8—44no—
231C22⋊C8—44no—
232C22⋊C8—82no—
233#97—44no—
234C4×C8—42no—
235C4×C8—42no—
236#95—24no—
237#95—24no—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241C24—22no—
242C24C2×C22⋊C412yes—
243C2×Q8—44no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—48no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—84no—
266C2×D4—84no—
267C2×D4#10014yes—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—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—24no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—41no—
317C2×C8—88no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—48no—
338C22⋊C4—44no—
339C22⋊C4—48no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—22no—
355C42—41no—
356C42—48no—
357C42#13512yes—
358C42#13512yes—
359C42AGL(1,ℤ/8)12yes—
360C42#13712yes—
361C42C2×#4111yescharacteristic
362C42C2×#4111yescharacteristic
363C42H(5,𝔽2)11yescharacteristic
364C422−1+411yescharacteristic
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23C2×#10014yes—
379C23C22×C22⋊C411yescharacteristic
380Q8—164no—
381D4—28no—
382D4—82no—
383D4—82no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—81no—
388D4—81no—
389D4—81no—
390D4—81no—
391D4—81no—
392D4—168no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—88no—
428C2×C4C2×#10012yes—
429C2×C4#41612yes—
430C2×C4#41612yes—
431C2×C4#41712yes—
432C2×C4#41712yes—
433C2×C4C4×#4111yescharacteristic
434C2×C4#52011yescharacteristic
435C2×C4#54611yescharacteristic
436C2×C4#54611yescharacteristic
437C2×C4#54711yescharacteristic
438C2×C4#54711yescharacteristic
439C2×C4C2×#13511yescharacteristic
440C2×C4C2×#13511yescharacteristic
441C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4#57611yescharacteristic
444C2×C4#57611yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57711yescharacteristic
447C8—42no—
448C8—42no—
449C8—81no—
450C8—81no—
451C8—82no—
452C8—82no—
453C22—24no—
454C22—24no—
455C22—24no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22#233911yescharacteristic
468C22C22×#10011yescharacteristic
469C22#288911yescharacteristic
470C22C2×#41611yescharacteristic
471C22C2×#41711yescharacteristic
472C22#289711yescharacteristic
473C22#289711yescharacteristic
474C4—22no—
475C4—22no—
476C4—21no—
477C4—21no—
478C4—41no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—82no—
483C4—84no—
484C4#155114yes—
485C4#330311yescharacteristic
486C4#330511yescharacteristic
487C4#332511yescharacteristic
488C4#332911yescharacteristic
489C4#334011yescharacteristic
490C4#334111yescharacteristic
491C4#335411yescharacteristic
492C4#335711yescharacteristic
493C2—44no—
494C2—162no—
495C2C2×#155111yescharacteristic
496C2#1772711yescharacteristic
497C2#1776511yescharacteristic
498C2#1777011yescharacteristic
499C2#2247911yescharacteristic
500C2#2248911yescharacteristic
501C2#2250311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36187C2
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
2. #22799C2semidirect
3. D4×C4⋊C8C2semidirect
4. #9896C2semidirect
5. #9896C2semidirect
6. #9142C2semidirect
7. #9133C2semidirect
8. #8729C2semidirect
9. #8585C2semidirect
10. C4×#331C2semidirect
11. #8044C2semidirect

Maximal subgroups

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

subgroupindex
#361872
#227992
D4×C4⋊C82
#98962
#91422
#91332
#87292
#85852
C4×#3312
#80442