Finite Group Databasebeta

← #165027

Subgroups — #165027

Order 512.

821 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36477C211yesmaximal, characteristic
2#22700C211yesmaximal, characteristic
3#22696C211yesmaximal, characteristic
4#7715C212yesmaximal, complemented
5#7681C212yesmaximal, complemented
6#7663C211yesmaximal, characteristic, complemented
7#7660C211yesmaximal, characteristic, complemented
8#7653C211yesmaximal, characteristic, complemented
9#7617C212yesmaximal, complemented
10#7589C212yesmaximal, complemented
11#7571C211yesmaximal, characteristic, complemented
12C2×#436—21no—
13C2×C8×D4—21no—
14#2859C2211yescharacteristic
15#2855C2212yes—
16#2608—22no—
17#2587—22no—
18#2570—21no—
19#2566—21no—
20#2538C411yescharacteristic
21#2538C411yescharacteristic
22#2497C412yes—
23#2429C412yes—
24#2370—21no—
25#2357—21no—
26#2355C411yescharacteristic
27Q8×C4⋊C4C411yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×C22⋊Q8C2212yes—
30C42×D4C2211yescharacteristic
31#1939—21no—
32#1939—21no—
33#1851—22no—
34#1850—22no—
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37#1797C2212yes—
38#1797C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1272C2211yescharacteristic
47#1272C2211yescharacteristic
48#1272C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53C22×M16—22no—
54C23×C8—22no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C412yes—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C22×C42C2312yes—
66#436—44no—
67C8×D4—44no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#400—22no—
79#398—22no—
80#396—22no—
81#396—22no—
82#396—22no—
83#396—22no—
84#391C2×C412yes—
85#391C2311yescharacteristic
86#391C2311yescharacteristic
87#390—21no—
88#390—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—22no—
96#389—22no—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2312yes—
100#388—22no—
101#388—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#385—22no—
108#385—22no—
109#384—21no—
110#384C2×C411yescharacteristic
111#384C2×C411yescharacteristic
112#384C2×C411yescharacteristic
113#382—21no—
114#382—21no—
115#381—22no—
116#380C2×C412yes—
117#380C2×C412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—22no—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
131C43C2311yescharacteristic
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8C22×C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C22×D4C22⋊C411yescharacteristic
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C2×M16—44no—
149C2×M16—44no—
150C2×M16—82no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—44no—
158C22×C8—44no—
159C22×C8—82no—
160C4⋊Q8—22no—
161C4⋊Q8—41no—
162#125—42no—
163#125—42no—
164C22⋊Q8—24no—
165C22⋊Q8—42no—
166C22⋊Q8—82no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—82no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—42no—
173C4×D4SD1614yes—
174C4×D4Q1614yes—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—82no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2411yescharacteristic
220C4⋊C8—42no—
221C4⋊C8—42no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224C4⋊C8—82no—
225C4⋊C8—82no—
226C22⋊C8—44no—
227C22⋊C8—44no—
228C4×C8—42no—
229C4×C8—42no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—22no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245C24C2×C22⋊C412yes—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8#13112yes—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4#10314yes—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×SD1614yes—
281C22×C4C2×Q1614yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285M16—88no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—88no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—81no—
323C4⋊C4—81no—
324C4⋊C4—81no—
325C4⋊C4—81no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4#10314yes—
330C22⋊C4—24no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4#10318yes—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—41no—
341C42—41no—
342C42—42no—
343C42—42no—
344C42—41no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—82no—
349C42—82no—
350C42C2×SD1612yes—
351C42C2×Q1612yes—
352C42AGL(1,ℤ/8)12yes—
353C42#13712yes—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23C2×#10314yes—
361C23C22×C22⋊C411yescharacteristic
362Q8—42no—
363Q8—42no—
364Q8—81no—
365Q8—81no—
366D4—28no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4#41312yes—
413C2×C4C2×#10312yes—
414C2×C4C2×#10314yes—
415C2×C4C2×#10312yes—
416C2×C4#41712yes—
417C2×C4#41712yes—
418C2×C4C2×#13111yescharacteristic
419C2×C4#56811yescharacteristic
420C2×C4C22×SD1611yescharacteristic
421C2×C4C22×Q1611yescharacteristic
422C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
423C2×C4C2×#13711yescharacteristic
424C8—42no—
425C8—42no—
426C8—81no—
427C8—81no—
428C8—82no—
429C8—82no—
430C22—24no—
431C22—24no—
432C22—24no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22#288011yescharacteristic
438C22#288311yescharacteristic
439C22C2×#41311yescharacteristic
440C22C22×#10311yescharacteristic
441C22C2×#41711yescharacteristic
442C22#290111yescharacteristic
443C22#290111yescharacteristic
444C4—21no—
445C4—21no—
446C4—22no—
447C4—22no—
448C4—21no—
449C4—21no—
450C4—24no—
451C4—21no—
452C4—21no—
453C4—42no—
454C4—42no—
455C4—41no—
456C4—42no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—82no—
461C4—82no—
462C4#155014yes—
463C2—44no—
464C2#1742411yescharacteristic
465C2#1750911yescharacteristic
466C2#1751511yescharacteristic
467C2#1761411yescharacteristic
468C2#1763011yescharacteristic
469C2#1768711yescharacteristic
470C2C2×#155011yescharacteristic
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 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7715C2semidirect
5. #7681C2semidirect
6. #7663C2semidirect
7. #7660C2semidirect
8. #7653C2semidirect
9. #7617C2semidirect
10. #7589C2semidirect
11. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#364772
#227002
#226962
#77152
#76812
#76632
#76602
#76532
#76172
#75892
#75712