Finite Group Databasebeta

← #164841

Subgroups — #164841

Order 512.

761 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#36290C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic
3#22012C211yesmaximal, characteristic
4#7731C212yesmaximal, complemented
5#7731C212yesmaximal, complemented
6#7683C211yesmaximal, characteristic, complemented
7#7680C211yesmaximal, characteristic, complemented
8#7641C212yesmaximal, complemented
9#7636C211yesmaximal, characteristic, complemented
10#7593C212yesmaximal, complemented
11#7568C211yesmaximal, characteristic, complemented
12#2859C2211yescharacteristic
13#2851C2212yes—
14#2571C411yescharacteristic
15#2552C411yescharacteristic
16#2509—21no—
17#2507—21no—
18#2505—22no—
19#2473—22no—
20#2442C412yes—
21#2442C412yes—
22#2349—21no—
23D4×C4⋊C4—21no—
24D4×C4⋊C4C411yescharacteristic
25D4×C4⋊C4C411yescharacteristic
26C4×#125C2211yescharacteristic
27C2×#389C2212yes—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1939—21no—
32#1939—21no—
33#1924—22no—
34#1924—22no—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39#1767C2211yescharacteristic
40#1767C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×#125—21no—
54C2×#125C2×C411yescharacteristic
55C2×#125C2×C412yes—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C412yes—
64C22×C42C2312yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—21no—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#400—22no—
78#400—22no—
79#400—22no—
80#398—22no—
81#398—22no—
82#398—22no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—22no—
90#389—22no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2311yescharacteristic
95#389C2311yescharacteristic
96#389C2312yes—
97#387—21no—
98#387—21no—
99#387—22no—
100#387—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#384—21no—
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#382—21no—
111#382—21no—
112#382C2×C411yescharacteristic
113#382C2×C412yes—
114#380—22no—
115#380—22no—
116#380C2×C412yes—
117#380C2×C412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4C2×C412yes—
126C4×C22⋊C4D412yes—
127C4×C22⋊C4D412yes—
128C2×#95—22no—
129C43C2311yescharacteristic
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336C2311yescharacteristic
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×D4C22⋊C411yescharacteristic
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C22×C412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152#125—22no—
153#125—41no—
154#125—81no—
155#125—81no—
156#125—82no—
157C4×D4—24no—
158C4×D4—42no—
159C4×D4—42no—
160C4×D4—44no—
161C4×D4—44no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C42—21no—
198C2×C42—21no—
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—42no—
211C4⋊C8—42no—
212C4⋊C8—82no—
213C4⋊C8—82no—
214C22⋊C8—82no—
215C22⋊C8—82no—
216C22⋊C8—82no—
217C22⋊C8—82no—
218#97—81no—
219#97—81no—
220C4×C8—42no—
221C4×C8—42no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—21no—
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#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239C24C2×C22⋊C412yes—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4#10114yes—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—81no—
324C4⋊C4—81no—
325C4⋊C4#13112yes—
326C22⋊C4—24no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—42no—
341C42—42no—
342C42—41no—
343C42—42no—
344C42—42no—
345C42—42no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23C2×#10114yes—
353C23C22×C22⋊C411yescharacteristic
354D4—28no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—88no—
404C2×C4—164no—
405C2×C4C2×#10112yes—
406C2×C4#42112yes—
407C2×C4#42112yes—
408C2×C4C2×#13111yescharacteristic
409C2×C4#56811yescharacteristic
410C2×C4#57611yescharacteristic
411C2×C4#57711yescharacteristic
412C2×C4#57711yescharacteristic
413C2×C4#57811yescharacteristic
414C8—81no—
415C8—82no—
416C8—81no—
417C8—81no—
418C8—82no—
419C8—81no—
420C22—24no—
421C22—24no—
422C22—24no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22#288011yescharacteristic
428C22#288311yescharacteristic
429C22C22×#10111yescharacteristic
430C22#289711yescharacteristic
431C22#289811yescharacteristic
432C22C2×#42111yescharacteristic
433C22C2×#42111yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—42no—
441C4—41no—
442C4—41no—
443C4—42no—
444C4—41no—
445C4—42no—
446C4—42no—
447C4—82no—
448C4—81no—
449C4—82no—
450C4—81no—
451C4—81no—
452C4#153414yes—
453C2—44no—
454C2#1742611yescharacteristic
455C2#1752211yescharacteristic
456C2#1752511yescharacteristic
457C2#1759911yescharacteristic
458C2#1760011yescharacteristic
459C2C2×#153411yescharacteristic
460C2#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. #7731C2semidirect
5. #7731C2semidirect
6. #7683C2semidirect
7. #7680C2semidirect
8. #7641C2semidirect
9. #7636C2semidirect
10. #7593C2semidirect
11. #7568C2semidirect

Maximal subgroups

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

subgroupindex
#362902
#220122
#77312
#76832
#76802
#76412
#76362
#75932
#75682