Finite Group Databasebeta

← #164506

Subgroups — #164506

Order 512.

795 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#36454C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic
3#20558C211yesmaximal, characteristic
4#7701C212yesmaximal, complemented
5#7682C211yesmaximal, characteristic, complemented
6#7682C211yesmaximal, characteristic, complemented
7#7655C212yesmaximal, complemented
8#7636C211yesmaximal, characteristic, complemented
9#7623C212yesmaximal, complemented
10#7589C212yesmaximal, complemented
11#7568C211yesmaximal, characteristic, complemented
12C2×C8×D4—21no—
13#2862C2211yescharacteristic
14#2855C2212yes—
15#2575—22no—
16#2547C411yescharacteristic
17#2542C411yescharacteristic
18#2517—22no—
19#2511—21no—
20#2489C412yes—
21#2447—21no—
22#2421C412yes—
23#2367—21no—
24#2365C411yescharacteristic
25#2365C411yescharacteristic
26#2355—21no—
27C4×#125C2211yescharacteristic
28C4×C22⋊Q8C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1924—22no—
33#1817C2212yes—
34#1797C2212yes—
35#1770—21no—
36#1767C2211yescharacteristic
37#1767C2211yescharacteristic
38#1752—22no—
39#1749C2212yes—
40#1744C2211yescharacteristic
41#1744C2211yescharacteristic
42C2×#336C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C23×C8—22no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C411yescharacteristic
57C2×#124—21no—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C412yes—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C22×C42C2312yes—
66C8×D4—28no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—21no—
74C2×#97—22no—
75C2×#97—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#400—22no—
79#400—22no—
80#399—22no—
81#397—22no—
82#397—22no—
83#397—22no—
84#391—21no—
85#391—21no—
86#391—21no—
87#391—21no—
88#391C2×C412yes—
89#391C2×C412yes—
90#391C2312yes—
91#390—21no—
92#389—21no—
93#389C2×C412yes—
94#389C2311yescharacteristic
95#389C2311yescharacteristic
96#388—22no—
97#388—22no—
98#387—21no—
99#387—21no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#385—22no—
107#385—22no—
108#384—21no—
109#382—21no—
110#382—21no—
111#382—22no—
112#382C2×C411yescharacteristic
113#381—22no—
114#380C2×C412yes—
115#380C2×C412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—22no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130C43C2311yescharacteristic
131#336—22no—
132#336—21no—
133#336—22no—
134#336—21no—
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141C22×Q8—21no—
142C22×Q8C22×C412yes—
143C22×D4C22⋊C411yescharacteristic
144C23×C4C22×C412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—28no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—82no—
155C22×C8—82no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158#125—22no—
159#125—41no—
160#124—44no—
161C22⋊Q8—24no—
162C22⋊Q8—42no—
163C22⋊Q8—82no—
164C4×Q8—44no—
165C4×Q8—82no—
166C4×D4—24no—
167C4×D4—42no—
168C4×D4—42no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—82no—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2411yescharacteristic
210C4⋊C8—24no—
211C4⋊C8—42no—
212C4⋊C8—82no—
213C22⋊C8—28no—
214C22⋊C8—82no—
215C22⋊C8—82no—
216#97—81no—
217#97—81no—
218#97—82no—
219C4×C8—24no—
220C4×C8—42no—
221#95—22no—
222#95—22no—
223#95—21no—
224#95—21no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95—21no—
236#95—22no—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240C24C2×C22⋊C412yes—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4#10414yes—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—88no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4#13112yes—
321C22⋊C4—24no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—82no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—42no—
338C42—42no—
339C42—41no—
340C42—42no—
341C42—44no—
342C42—42no—
343C42—42no—
344C42—82no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23C2×#10414yes—
352C23C22×C22⋊C411yescharacteristic
353Q8—82no—
354Q8—82no—
355Q8—84no—
356D4—28no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4#41312yes—
406C2×C4C2×#10412yes—
407C2×C4#42112yes—
408C2×C4C2×#13111yescharacteristic
409C2×C4#56811yescharacteristic
410C2×C4#57711yescharacteristic
411C2×C4#57711yescharacteristic
412C2×C4#57811yescharacteristic
413C2×C4#57811yescharacteristic
414C8—42no—
415C8—44no—
416C8—42no—
417C8—81no—
418C8—82no—
419C8—81no—
420C22—24no—
421C22—24no—
422C22—24no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22#288011yescharacteristic
428C22#288311yescharacteristic
429C22C2×#41311yescharacteristic
430C22#289811yescharacteristic
431C22#289811yescharacteristic
432C22C22×#10411yescharacteristic
433C22C2×#42111yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—42no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—82no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—82no—
452C4#153214yes—
453C2—44no—
454C2#1742911yescharacteristic
455C2#1751911yescharacteristic
456C2#1752311yescharacteristic
457C2#1754811yescharacteristic
458C2#1760011yescharacteristic
459C2C2×#153211yescharacteristic
460C2#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. #7701C2semidirect
5. #7682C2semidirect
6. #7682C2semidirect
7. #7655C2semidirect
8. #7636C2semidirect
9. #7623C2semidirect
10. #7589C2semidirect
11. #7568C2semidirect

Maximal subgroups

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

subgroupindex
#364542
#220122
#205582
#77012
#76822
#76552
#76362
#76232
#75892
#75682