Finite Group Databasebeta

← #200499

Subgroups — #200499

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60276C211yesmaximal, characteristic
2#34979C211yesmaximal, characteristic
3#20728C211yesmaximal, characteristic
4#17715C212yesmaximal, complemented
5#9396C212yesmaximal, complemented
6#9395C212yesmaximal, complemented
7#9394C212yesmaximal, complemented
8#9393C212yesmaximal, complemented
9#8092C212yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#2988—24no—
13#2982C2212yes—
14#2972—22no—
15#2972C2212yes—
16C8×#41—24no—
17#2920C2212yes—
18C2×#431—22no—
19C2×#431C2212yes—
20#2611C414yes—
21#2568C2211yescharacteristic
22#2547C2211yescharacteristic
23#2546C2211yescharacteristic
24#2542C2211yescharacteristic
25#2499—21no—
26#2432—21no—
27#2415—22no—
28#2395C414yes—
29C4×C4⋊Q8—21no—
30C4×#124—21no—
31C4×C22⋊Q8—22no—
32C4×#117C2211yescharacteristic
33#1822—21no—
34#1801C2212yes—
35#1800C2212yes—
36#1785—21no—
37C4×#101—21no—
38C4×#100—21no—
39#1552—24no—
40#1549C2214yes—
41#1546C2214yes—
42#1544—24no—
43#1303C2214yes—
44#1303C2214yes—
45C22×#41D411yescharacteristic
46#533—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×#125C2×C412yes—
54C2×#124—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—21no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41—22no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4C2×C412yes—
67C2×C4×D4D412yes—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117C2×C412yes—
72C2×#117C2×C412yes—
73C2×#117D411yescharacteristic
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76C8×Q8—28no—
77C8×D4—216no—
78C8×D4—28no—
79#432—22no—
80#432—22no—
81#431—24no—
82#431—24no—
83#431—22no—
84#431—24no—
85#431—22no—
86#431—22no—
87#431—22no—
88#431D414yes—
89#431C2314yes—
90#423C2314yes—
91C2×C4⋊C8—24no—
92C2×C4⋊C8D414yes—
93C2×#101—22no—
94C2×#100—22no—
95#407—22no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—22no—
98C4×M16—22no—
99C2×C4×C8—24no—
100C2×C4×C8—22no—
101#400C2×C414yes—
102#399C2×C414yes—
103#398C2×C414yes—
104#397—24no—
105#397C2×C414yes—
106#394—24no—
107#391—21no—
108#391—21no—
109#391—21no—
110#391—22no—
111#390—21no—
112#389—24no—
113#389—22no—
114#388—24no—
115#387—24no—
116#385—24no—
117#384—21no—
118#382—21no—
119#380—22no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—24no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4C2×C412yes—
127#376C2×C412yes—
128C43—21no—
129#343—22no—
130#343—22no—
131#332—28no—
132#332—28no—
133C2×#41—24no—
134C2×#41—24no—
135C2×#41—22no—
136C2×#41—22no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8C2×D411yescharacteristic
140C22×D4C22⋊C412yes—
141C22×D4C2×D411yescharacteristic
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C2×M16—24no—
149C22×C8—28no—
150C22×C8—24no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—44no—
156#125—24no—
157#125—24no—
158#124—44no—
159C22⋊Q8—48no—
160C22⋊Q8—42no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—41no—
169C4×Q8—41no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—41no—
173C4×Q8—41no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182#117—22no—
183#117—22no—
184#117—24no—
185#117—24no—
186#117—24no—
187#117—24no—
188#117—44no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22×C412yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42C22⋊C412yes—
220C2×C42C22⋊C412yes—
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4112yes—
228C2×C42#4112yes—
229C2×C42C2411yescharacteristic
230C4⋊C8—28no—
231C4⋊C8—24no—
232C4⋊C8—24no—
233C4⋊C8—24no—
234C4⋊C8C2×D418yes—
235#101—44no—
236#100—44no—
237C22⋊C8—28no—
238C22⋊C8—24no—
239C22⋊C8—24no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242#97—42no—
243C4×C8—28no—
244C4×C8—24no—
245C4×C8—24no—
246C4×C8—42no—
247#95—22no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95C22×C412yes—
253#95C22×C412yes—
254C24C2×C22⋊C411yescharacteristic
255C24C2×C22⋊C412yes—
256#41—216no—
257#41—216no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8#13514yes—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4#13514yes—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
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—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C4×D412yes—
298C22×C4C4×D412yes—
299C22×C4#12012yes—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4#12312yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306M16—44no—
307M16—44no—
308C2×C8—28no—
309C2×C8—44no—
310C2×C8—84no—
311C2×C8—88no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—42no—
320C4⋊C4—48no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—48no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—44no—
332C22⋊C4—48no—
333C22⋊C4—44no—
334C22⋊C4—48no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—44no—
342C42—44no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42C4×D412yes—
348C42C4×D412yes—
349C42#12011yescharacteristic
350C42#12011yescharacteristic
351C42#12112yes—
352C42#12112yes—
353C42#12311yescharacteristic
354C42#12311yescharacteristic
355C42C22×D412yes—
356C42C2×#4112yes—
357C23—22no—
358C23—21no—
359C23—21no—
360C23—22no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23#38612yes—
366C23#41614yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—28no—
369Q8—82no—
370Q8—82no—
371Q8—84no—
372D4—216no—
373D4—28no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4#38611yescharacteristic
410C2×C4#38611yescharacteristic
411C2×C4#38611yescharacteristic
412C2×C4#38612yes—
413C2×C4#38611yescharacteristic
414C2×C4#41614yes—
415C2×C4C2×C4×D412yes—
416C2×C4C2×#12011yescharacteristic
417C2×C4C2×#12112yes—
418C2×C4C2×#12311yescharacteristic
419C2×C4C2×#13512yes—
420C2×C4C2×#13512yes—
421C8—48no—
422C8—84no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—24no—
429C22—44no—
430C22—42no—
431C22—42no—
432C22—81no—
433C22—82no—
434C22—81no—
435C22C2×#38611yescharacteristic
436C22C2×#41611yescharacteristic
437C22C2×#41611yescharacteristic
438C4—21no—
439C4—22no—
440C4—21no—
441C4—22no—
442C4—21no—
443C4—21no—
444C4—24no—
445C4—44no—
446C4—42no—
447C4—42no—
448C4—82no—
449C4—82no—
450C4—82no—
451C4#191512yes—
452C4#301812yes—
453C4#305014yes—
454C2—21no—
455C2—21no—
456C2—42no—
457C2—44no—
458C2#2095211yescharacteristic
459C2#2096711yescharacteristic
460C2C2×#191511yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17715C2semidirect
5. #9396C2semidirect
6. #9395C2semidirect
7. #9394C2semidirect
8. #9393C2semidirect
9. #8092C2semidirect

Maximal subgroups

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

subgroupindex
#602762
#349792
#207282
#177152
#93962
#93952
#93942
#93932
#80922