Finite Group Databasebeta

← #164352

Subgroups — #164352

Order 512.

855 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36454C211yesmaximal, characteristic
2#20567C211yesmaximal, characteristic
3#20558C211yesmaximal, characteristic
4#7682C211yesmaximal, characteristic, complemented
5#7682C211yesmaximal, characteristic, complemented
6#7681C212yesmaximal, complemented
7#7675C212yesmaximal, complemented
8#7636C211yesmaximal, characteristic, complemented
9#7634C212yesmaximal, complemented
10#7568C211yesmaximal, characteristic, complemented
11#7566C212yesmaximal, complemented
12C2×#436—21no—
13C2×C8×D4—21no—
14#2862C2211yescharacteristic
15#2855C2212yes—
16#2575—22no—
17#2547C411yescharacteristic
18#2542C411yescharacteristic
19#2517—22no—
20#2511—21no—
21#2489C412yes—
22#2447—21no—
23#2421C412yes—
24#2367—21no—
25#2365C411yescharacteristic
26#2365C411yescharacteristic
27#2355—21no—
28C4×#125C2211yescharacteristic
29C4×C22⋊Q8C2212yes—
30C42×D4C2211yescharacteristic
31#1770—21no—
32C8×C4⋊C4—21no—
33#1767C2211yescharacteristic
34#1767C2211yescharacteristic
35#1753—22no—
36#1752—22no—
37#1749C2212yes—
38C4×C22⋊C8C2212yes—
39#1744C2211yescharacteristic
40#1744C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2211yescharacteristic
53C22×M16—22no—
54C23×C8—22no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125C2×C411yescharacteristic
58C2×#124—21no—
59C2×C22⋊Q8C2×C412yes—
60C2×C22⋊Q8C2×C412yes—
61C2×C4×Q8—21no—
62C2×C4×Q8C2×C412yes—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C22×C42C2312yes—
67#436—28no—
68C8×D4—28no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—22no—
74C2×#97—21no—
75C2×#97—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—22no—
79#400—22no—
80#400—22no—
81#399—22no—
82#397—22no—
83#397—22no—
84#397—22no—
85#391—21no—
86#391—21no—
87#391—21no—
88#391—21no—
89#391C2×C412yes—
90#391C2×C412yes—
91#391C2312yes—
92#390—21no—
93#389—21no—
94#389C2×C412yes—
95#389C2311yescharacteristic
96#389C2311yescharacteristic
97#388—22no—
98#388—22no—
99#387—21no—
100#387—21no—
101#387—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#382—21no—
111#382—21no—
112#382—22no—
113#382C2×C411yescharacteristic
114#381—22no—
115#380C2×C412yes—
116#380C2×C412yes—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
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—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×Q8C22×C412yes—
144C22×D4C22⋊C411yescharacteristic
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C2×M16—28no—
149C2×M16—82no—
150C2×M16—82no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—28no—
158C22×C8—82no—
159C22×C8—82no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162#125—22no—
163#125—41no—
164#124—44no—
165C22⋊Q8—24no—
166C22⋊Q8—42no—
167C22⋊Q8—82no—
168C4×Q8—44no—
169C4×Q8—82no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—42no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C412yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—82no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2411yescharacteristic
214C4⋊C8—24no—
215C4⋊C8—24no—
216C22⋊C8—28no—
217C22⋊C8—28no—
218#97—81no—
219#97—81no—
220#97—82no—
221C4×C8—24no—
222C4×C8—24no—
223C4×C8—82no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243C24C2×C22⋊C412yes—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4#10414yes—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282M16—88no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—88no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4#10414yes—
324C4⋊C4#13112yes—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—82no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—42no—
342C42—42no—
343C42—41no—
344C42—42no—
345C42—42no—
346C42—44no—
347C42—42no—
348C42—82no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23C2×#10414yes—
356C23C22×C22⋊C411yescharacteristic
357Q8—82no—
358Q8—82no—
359Q8—84no—
360D4—28no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4#41312yes—
409C2×C4C2×#10414yes—
410C2×C4C2×#10412yes—
411C2×C4C2×#10412yes—
412C2×C4C2×#13111yescharacteristic
413C2×C4#56811yescharacteristic
414C2×C4#57711yescharacteristic
415C2×C4#57711yescharacteristic
416C2×C4#57811yescharacteristic
417C2×C4#57811yescharacteristic
418C8—42no—
419C8—42no—
420C8—44no—
421C8—42no—
422C8—44no—
423C8—42no—
424C22—24no—
425C22—24no—
426C22—24no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22#288011yescharacteristic
432C22#288311yescharacteristic
433C22C2×#41311yescharacteristic
434C22#289811yescharacteristic
435C22#289811yescharacteristic
436C22C22×#10411yescharacteristic
437C22C22×#10411yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—24no—
445C4—42no—
446C4—41no—
447C4—42no—
448C4—41no—
449C4—41no—
450C4—42no—
451C4—82no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4—82no—
456C4#152414yes—
457C2—44no—
458C2#1742911yescharacteristic
459C2#1751811yescharacteristic
460C2#1751911yescharacteristic
461C2#1754811yescharacteristic
462C2#1754811yescharacteristic
463C2C2×#152411yescharacteristic
464C2#1777811yescharacteristic
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. #7682C2semidirect
5. #7682C2semidirect
6. #7681C2semidirect
7. #7675C2semidirect
8. #7636C2semidirect
9. #7634C2semidirect
10. #7568C2semidirect
11. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#364542
#205672
#205582
#76822
#76812
#76752
#76362
#76342
#75682
#75662