Finite Group Databasebeta

← #164496

Subgroups — #164496

Order 512.

813 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36407C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic
3#20558C211yesmaximal, characteristic
4#7708C212yesmaximal, complemented
5#7683C211yesmaximal, characteristic, complemented
6#7680C211yesmaximal, characteristic, complemented
7#7665C212yesmaximal, complemented
8#7636C211yesmaximal, characteristic, complemented
9#7613C212yesmaximal, complemented
10#7579C212yesmaximal, complemented
11#7568C211yesmaximal, characteristic, complemented
12C2×C8×D4—21no—
13#2859C2211yescharacteristic
14#2857C2212yes—
15#2574—22no—
16#2556—22no—
17#2552C411yescharacteristic
18#2551C411yescharacteristic
19#2454—21no—
20#2452—21no—
21#2441C412yes—
22#2439C412yes—
23#2370—21no—
24#2369—21no—
25#2354C411yescharacteristic
26#2354C411yescharacteristic
27C4×#125C2211yescharacteristic
28#2296C2212yes—
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#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C23×C8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8C2×C412yes—
56C2×#125—21no—
57C2×#125—21no—
58C2×#125C2×C411yescharacteristic
59C2×C22⋊Q8—22no—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×#117C2×C412yes—
64C2×#117C2×C412yes—
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#399—22no—
79#398—22no—
80#396—22no—
81#396—22no—
82#395—22no—
83#393—22no—
84#391—22no—
85#391—22no—
86#390—21no—
87#390—21no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389—21no—
92#389—21no—
93#389—22no—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2311yescharacteristic
98#389C2311yescharacteristic
99#389C2312yes—
100#388—22no—
101#388—22no—
102#387—21no—
103#387—21no—
104#387—22no—
105#387—22no—
106#385—22no—
107#385—22no—
108#384C2×C412yes—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#382—21no—
112#382C2×C411yescharacteristic
113#380—22no—
114#380C2×C412yes—
115#380C2×C412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
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—22no—
142C22×D4C22⋊C411yescharacteristic
143C23×C4C22×C412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—28no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—82no—
154C22×C8—82no—
155C4⋊Q8—44no—
156C4⋊Q8—82no—
157#125—22no—
158#125—41no—
159#125—44no—
160#125—81no—
161#125—81no—
162C22⋊Q8—82no—
163C22⋊Q8—82no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—42no—
167#117—24no—
168#117—42no—
169#117—82no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
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⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—82no—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—24no—
217C4⋊C8—42no—
218C4⋊C8—82no—
219C22⋊C8—28no—
220C22⋊C8—82no—
221C22⋊C8—82no—
222#97—81no—
223#97—81no—
224#97—82no—
225C4×C8—24no—
226C4×C8—42no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—21no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95—22no—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242C24C2×C22⋊C412yes—
243C2×Q8—44no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4#10114yes—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—88no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—44no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—82no—
320C4⋊C4—81no—
321C4⋊C4—82no—
322C4⋊C4—81no—
323C4⋊C4—82no—
324C4⋊C4#10414yes—
325C4⋊C4#13112yes—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—42no—
343C42—42no—
344C42—41no—
345C42—42no—
346C42—44no—
347C42—42no—
348C42—42no—
349C42—82no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23C2×#10114yes—
357C23C22×C22⋊C411yescharacteristic
358Q8—164no—
359D4—28no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4C2×#10112yes—
408C2×C4C2×#10414yes—
409C2×C4C2×#10412yes—
410C2×C4#42112yes—
411C2×C4C2×#13111yescharacteristic
412C2×C4#56811yescharacteristic
413C2×C4#57611yescharacteristic
414C2×C4#57711yescharacteristic
415C2×C4#57711yescharacteristic
416C2×C4#57811yescharacteristic
417C8—42no—
418C8—44no—
419C8—42no—
420C8—81no—
421C8—82no—
422C8—81no—
423C22—24no—
424C22—24no—
425C22—24no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22#288011yescharacteristic
431C22#288311yescharacteristic
432C22C22×#10111yescharacteristic
433C22#289711yescharacteristic
434C22#289811yescharacteristic
435C22C22×#10411yescharacteristic
436C22C2×#42111yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—24no—
442C4—21no—
443C4—21no—
444C4—42no—
445C4—41no—
446C4—41no—
447C4—41no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—82no—
454C4—81no—
455C4#153114yes—
456C2—44no—
457C2#1742611yescharacteristic
458C2#1752011yescharacteristic
459C2#1752311yescharacteristic
460C2#1754811yescharacteristic
461C2#1759911yescharacteristic
462C2C2×#153111yescharacteristic
463C2#1777411yescharacteristic
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. #7708C2semidirect
5. #7683C2semidirect
6. #7680C2semidirect
7. #7665C2semidirect
8. #7636C2semidirect
9. #7613C2semidirect
10. #7579C2semidirect
11. #7568C2semidirect

Maximal subgroups

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

subgroupindex
#364072
#220122
#205582
#77082
#76832
#76802
#76652
#76362
#76132
#75792
#75682