Finite Group Databasebeta

← #223160

Subgroups — #223160

Order 512.

1049 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic, complemented
3D4×C4⋊C8C211yesmaximal, characteristic, complemented
4#9898C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9134C212yesmaximal, complemented
7#9133C212yesmaximal, complemented
8#8726C212yesmaximal, complemented
9#8585C211yesmaximal, characteristic, complemented
10C4×#331C211yesmaximal, characteristic, complemented
11#8039C212yesmaximal, complemented
12C2×#555—21no—
13C2×C8×D4—21no—
14C2×#431C2212yes—
15C22×C4⋊C8C2212yes—
16#2868C2211yescharacteristic
17#2856C2212yes—
18#2563—21no—
19#2537—21no—
20#2537C412yes—
21#2500—22no—
22#2491—22no—
23#2491C414yes—
24#2354—21no—
25#2354C412yes—
26C4×#127C2211yescharacteristic
27#2285C2212yes—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1939—21no—
31#1923—22no—
32#1850—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1797C2212yes—
36#1797C2212yes—
37C8×C4⋊C4—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—44no—
52#555—44no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8C2×C412yes—
56C2×#127C2311yescharacteristic
57C2×#124—24no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121C2×C414yes—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×#117—22no—
69C2×#117—22no—
70C22×C42C2312yes—
71C8×D4—44no—
72#431—24no—
73#431—42no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—42no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93#394—24no—
94#394—24no—
95#393—24no—
96#390—21no—
97#390—21no—
98#390—22no—
99#390—22no—
100#390C2×C412yes—
101#390C2×C414yes—
102#388—24no—
103#388—24no—
104#384—21no—
105#384—21no—
106#384C2×C412yes—
107#384C2311yescharacteristic
108#384C2312yes—
109#382—22no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380C2×C414yes—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C414yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C22⋊C4C2×C412yes—
120C4×C22⋊C4D412yes—
121C4×C22⋊C4D412yes—
122C43C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—21no—
126#336—21no—
127#331—22no—
128#331—22no—
129#331—42no—
130C22×Q8—22no—
131C22×D4—24no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4C22⋊C411yescharacteristic
135C22×D4C22×C412yes—
136C23×C4—22no—
137C23×C4C22×C412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—44no—
147C22×C8—44no—
148C22×C8—82no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8—81no—
152C4⋊Q8—81no—
153C4⋊Q8—82no—
154#127—21no—
155#127—21no—
156#127—21no—
157#127—21no—
158#124—416no—
159#121—48no—
160#121—44no—
161#121—44no—
162#121—82no—
163#121—82no—
164#121—84no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—42no—
168C4×D4—44no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4D814yes—
174C4×D4SD1614yes—
175#117—44no—
176#117—44no—
177#117—48no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C414yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—24no—
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C412yes—
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2411yescharacteristic
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—44no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8—81no—
229C4⋊C8—81no—
230C4⋊C8#4112yes—
231C4⋊C8#4112yes—
232C22⋊C8—44no—
233C22⋊C8—44no—
234C22⋊C8—82no—
235C4×C8—44no—
236C4×C8—42no—
237C4×C8—42no—
238#95—24no—
239#95—24no—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243C24—22no—
244C24C2×C22⋊C412yes—
245C2×Q8—44no—
246C2×Q8—82no—
247C2×Q8—82no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—48no—
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—41no—
266C2×D4—41no—
267C2×D4—84no—
268C2×D4—84no—
269C2×D4#10214yes—
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—24no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×D814yes—
297C22×C4C2×SD1614yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—88no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4#10214yes—
336C22⋊C4—24no—
337C22⋊C4—42no—
338C22⋊C4—44no—
339C22⋊C4—48no—
340C22⋊C4—48no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—84no—
344C22⋊C4—84no—
345C22⋊C4#10218yes—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—22no—
357C42—41no—
358C42—48no—
359C42C2×D812yes—
360C42C2×SD1612yes—
361C42#13512yes—
362C42#13512yes—
363C42C2×#4111yescharacteristic
364C42C2×#4111yescharacteristic
365C42H(5,𝔽2)11yescharacteristic
366C422−1+411yescharacteristic
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23C2×#10214yes—
381C23C22×C22⋊C411yescharacteristic
382Q8—164no—
383D4—28no—
384D4—82no—
385D4—82no—
386D4—81no—
387D4—81no—
388D4—81no—
389D4—81no—
390D4—81no—
391D4—81no—
392D4—81no—
393D4—81no—
394D4—168no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—88no—
429C2×C4#41312yes—
430C2×C4C2×#10214yes—
431C2×C4C2×#10212yes—
432C2×C4C2×#10212yes—
433C2×C4#41612yes—
434C2×C4#41612yes—
435C2×C4C4×#4111yescharacteristic
436C2×C4#52011yescharacteristic
437C2×C4#54611yescharacteristic
438C2×C4#54611yescharacteristic
439C2×C4#54711yescharacteristic
440C2×C4#54711yescharacteristic
441C2×C4C22×D811yescharacteristic
442C2×C4C22×SD1611yescharacteristic
443C2×C4C2×#13511yescharacteristic
444C2×C4C2×#13511yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57811yescharacteristic
449C8—42no—
450C8—42no—
451C8—81no—
452C8—81no—
453C8—82no—
454C8—82no—
455C22—24no—
456C22—24no—
457C22—24no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22#233911yescharacteristic
470C22C2×#41311yescharacteristic
471C22#288811yescharacteristic
472C22C22×#10211yescharacteristic
473C22C2×#41611yescharacteristic
474C22#289711yescharacteristic
475C22#289811yescharacteristic
476C4—22no—
477C4—22no—
478C4—21no—
479C4—21no—
480C4—24no—
481C4—41no—
482C4—42no—
483C4—42no—
484C4—82no—
485C4—84no—
486C4#154014yes—
487C4#329411yescharacteristic
488C4#329811yescharacteristic
489C4#330311yescharacteristic
490C4#330511yescharacteristic
491C4#333411yescharacteristic
492C4#333511yescharacteristic
493C4#334011yescharacteristic
494C4#334111yescharacteristic
495C2—44no—
496C2—162no—
497C2C2×#154011yescharacteristic
498C2#1772011yescharacteristic
499C2#1772611yescharacteristic
500C2#1778111yescharacteristic
501C2#2248311yescharacteristic
502C2#2248511yescharacteristic
503C2#2248911yescharacteristic
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. #22795C2semidirect
3. D4×C4⋊C8C2semidirect
4. #9898C2semidirect
5. #9896C2semidirect
6. #9134C2semidirect
7. #9133C2semidirect
8. #8726C2semidirect
9. #8585C2semidirect
10. C4×#331C2semidirect
11. #8039C2semidirect

Maximal subgroups

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

subgroupindex
#361872
#227952
D4×C4⋊C82
#98982
#98962
#91342
#91332
#87262
#85852
C4×#3312
#80392