Finite Group Databasebeta

← #217004

Subgroups — #217004

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3#20752C211yesmaximal, characteristic, complemented
4#9907C212yesmaximal, complemented
5#9710C212yesmaximal, complemented
6#9141C212yesmaximal
7#9139C212yesmaximal, complemented
8#8593C212yesmaximal, complemented
9#8592C212yesmaximal, complemented
10C2×#529—21no—
11C2×#423C2212yes—
12#2861—21no—
13#2860—21no—
14#2860C412yes—
15#2859C2211yescharacteristic
16#2858C2211yescharacteristic
17#2857C2211yescharacteristic
18#2856C2211yescharacteristic
19#2854—21no—
20#2854C412yes—
21#2850—21no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×#127C2211yescharacteristic
24C4×#124—21no—
25C4×#121—21no—
26C4×#121C412yes—
27#2293—21no—
28#2293C412yes—
29C2×C43C2211yescharacteristic
30#1839—21no—
31#1831—21no—
32#1831—21no—
33#1831C2212yes—
34#1831C2212yes—
35#1831C2212yes—
36#1817—21no—
37#1797C2212yes—
38#1767C2212yes—
39#1749—21no—
40#1745—21no—
41#1728—21no—
42#1728—21no—
43C2×#332C2214yes—
44C2×#331C2214yes—
45#1296C2214yes—
46#1296C2214yes—
47C22×M16—22no—
48C2×C4⋊Q8C2311yescharacteristic
49C2×#127C2311yescharacteristic
50#529—48no—
51C2×#126—22no—
52C2×#125—21no—
53C2×#124—21no—
54C2×#124—21no—
55C2×#123—22no—
56C2×C22⋊Q8—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×#117—22no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73#423—44no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8C2314yes—
76C2×C4⋊C8C2314yes—
77C2×C22⋊C8—22no—
78C2×#97—22no—
79#391—21no—
80#391—21no—
81#390—21no—
82#390—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390C2×C412yes—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2×C412yes—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#386—22no—
108#386—22no—
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#382—21no—
112#380—21no—
113#380C2×C412yes—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43C2311yescharacteristic
125C43C2311yescharacteristic
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#332—44no—
135#331—44no—
136C22×Q8—21no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C22×C412yes—
141C23×C4—22no—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—44no—
150C2×M16—44no—
151C2×M16—44no—
152C22×C8—24no—
153C22×C8—24no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156#127—22no—
157#127—22no—
158#126—48no—
159#125—44no—
160#124—44no—
161#124—44no—
162#123—48no—
163C22⋊Q8—44no—
164#121—44no—
165#121—44no—
166#121—44no—
167C4×Q8—44no—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4—44no—
171C4×D4—48no—
172#117—48no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2411yescharacteristic
233C4⋊C8—24no—
234C4⋊C8—24no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237C4⋊C8—44no—
238C22⋊C8—44no—
239#97—44no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246C24—22no—
247C24C2×C22⋊C411yescharacteristic
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×D4—24no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
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—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4AGL(1,ℤ/8)14yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307M16—88no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—44no—
313C2×C8—44no—
314C4⋊C4—24no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C42—22no—
338C42—21no—
339C42—21no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—44no—
366C42H(5,𝔽2)12yes—
367C422−1+412yes—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23#41812yes—
382C23C22×C22⋊C411yescharacteristic
383Q8—82no—
384Q8—82no—
385D4—84no—
386D4—88no—
387D4—82no—
388D4—82no—
389D4—84no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—88no—
440C2×C4#41312yes—
441C2×C4#41312yes—
442C2×C4#41812yes—
443C2×C4#52012yes—
444C2×C4#53511yescharacteristic
445C2×C4#53611yescharacteristic
446C2×C4#53712yes—
447C2×C4C2×AGL(1,ℤ/8)12yes—
448C2×C4#57512yes—
449C2×C4#57512yes—
450C2×C4#57512yes—
451C8—84no—
452C8—84no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22#231111yescharacteristic
468C22C2×#41311yescharacteristic
469C22C2×#41311yescharacteristic
470C22C2×#41811yescharacteristic
471C22#289611yescharacteristic
472C22#289611yescharacteristic
473C22#289611yescharacteristic
474C4—22no—
475C4—22no—
476C4—22no—
477C4—22no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4#321312yes—
490C4#323412yes—
491C4#325614yes—
492C2—42no—
493C2—162no—
494C2#1774011yescharacteristic
495C2#1774311yescharacteristic
496C2#1774411yescharacteristic
497C2#1775711yescharacteristic
498C2#2186511yescharacteristic
499C2#2188011yescharacteristic
500C2#2189511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. #20752C2semidirect
4. #9907C2semidirect
5. #9710C2semidirect
7. #9139C2semidirect
8. #8593C2semidirect
9. #8592C2semidirect

Maximal subgroups

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

subgroupindex
#366312
#207782
#207522
#99072
#97102
#91412
#91392
#85932
#85922