Finite Group Databasebeta

← #164422

Subgroups — #164422

Order 512.

783 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36300C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic, complemented
3#20573C211yesmaximal, characteristic, complemented
4#7706C212yesmaximal, complemented
5#7679C211yesmaximal, characteristic
6#7679C211yesmaximal, characteristic
7#7672C212yesmaximal, complemented
8#7636C211yesmaximal, characteristic, complemented
9#7608C212yesmaximal, complemented
10#7574C212yesmaximal, complemented
11#7568C211yesmaximal, characteristic, complemented
12C2×#446—21no—
13#2860C2212yes—
14#2858C2211yescharacteristic
15#2567C411yescharacteristic
16#2548C411yescharacteristic
17#2521—22no—
18#2513C412yes—
19#2512—21no—
20#2474—22no—
21#2454C412yes—
22#2448—21no—
23#2369C411yescharacteristic
24#2367—21no—
25#2366C411yescharacteristic
26#2354—21no—
27C4×#125C2211yescharacteristic
28#2293C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1839C2212yes—
34#1831C2212yes—
35#1771—22no—
36C8×C4⋊C4—21no—
37#1767C2211yescharacteristic
38#1767C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40#1745C2212yes—
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#126—22no—
56C2×#125C2×C411yescharacteristic
57C2×#124—21no—
58C2×#124—21no—
59C2×#124—21no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×#117—22no—
66#446—28no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×#97—22no—
72C2×#97—21no—
73C2×#97—21no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—22no—
77#399—22no—
78#397—22no—
79#397—22no—
80#396—22no—
81#395—22no—
82#394—22no—
83#391—21no—
84#391—22no—
85#390—21no—
86#390—21no—
87#390—21no—
88#390—21no—
89#390C2×C412yes—
90#390C2×C412yes—
91#390C2×C412yes—
92#390C2×C412yes—
93#390C2312yes—
94#389—21no—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2311yescharacteristic
99#389C2311yescharacteristic
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#387—21no—
108#387—21no—
109#386—22no—
110#385—22no—
111#384—21no—
112#382C2×C411yescharacteristic
113#380—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C22⋊C4—22no—
127C43C2311yescharacteristic
128C43C2312yes—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336—22no—
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141C22×Q8—21no—
142C22×Q8C22⋊C411yescharacteristic
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C22×C412yes—
146C23×C4—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C4⋊Q8—42no—
154#127—42no—
155#126—82no—
156#126—82no—
157#125—22no—
158#125—41no—
159#124—44no—
160#124—81no—
161#124—81no—
162#124—81no—
163#124—81no—
164C4×Q8—24no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×D4—24no—
168C4×D4—42no—
169C4×D4—44no—
170#117—82no—
171#117—82no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
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×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42C22×C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2411yescharacteristic
218C4⋊C8—24no—
219C4⋊C8—28no—
220C4⋊C8—42no—
221C4⋊C8—82no—
222#97—28no—
223#97—81no—
224#97—81no—
225C4×C8—24no—
226C4×C8—42no—
227C4×C8—82no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241C24—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—42no—
246C2×Q8—81no—
247C2×Q8—81no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—81no—
257C2×D4—81no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—82no—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
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—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4#10414yes—
317C4⋊C4#13112yes—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—44no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C42—22no—
332C42—21no—
333C42—22no—
334C42—21no—
335C42—22no—
336C42—21no—
337C42—21no—
338C42—22no—
339C42—22no—
340C42—21no—
341C42—21no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—24no—
346C42—22no—
347C42—42no—
348C42—42no—
349C42—41no—
350C42—42no—
351C42—42no—
352C42—82no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23C22×C22⋊C411yescharacteristic
361Q8—44no—
362Q8—162no—
363D4—82no—
364D4—82no—
365D4—84no—
366D4—162no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
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—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—88no—
409C2×C4#41312yes—
410C2×C4C2×#10414yes—
411C2×C4C2×#10412yes—
412C2×C4#42112yes—
413C2×C4C2×#13111yescharacteristic
414C2×C4#56811yescharacteristic
415C2×C4#57611yescharacteristic
416C2×C4#57611yescharacteristic
417C2×C4#57711yescharacteristic
418C2×C4#57711yescharacteristic
419C8—42no—
420C8—44no—
421C8—42no—
422C8—81no—
423C8—82no—
424C8—81no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22#288011yescharacteristic
433C22#288411yescharacteristic
434C22C2×#41311yescharacteristic
435C22#289711yescharacteristic
436C22#289711yescharacteristic
437C22C22×#10411yescharacteristic
438C22C2×#42111yescharacteristic
439C4—21no—
440C4—21no—
441C4—24no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—41no—
447C4—42no—
448C4—42no—
449C4—41no—
450C4—42no—
451C4—41no—
452C4—42no—
453C4—82no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4#152914yes—
458C2—162no—
459C2#1742811yescharacteristic
460C2#1751711yescharacteristic
461C2#1752611yescharacteristic
462C2#1755011yescharacteristic
463C2#1760211yescharacteristic
464C2C2×#152911yescharacteristic
465C2#1777611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22022C2semidirect
3. #20573C2semidirect
4. #7706C2semidirect
7. #7672C2semidirect
8. #7636C2semidirect
9. #7608C2semidirect
10. #7574C2semidirect
11. #7568C2semidirect

Maximal subgroups

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

subgroupindex
#363002
#220222
#205732
#77062
#76792
#76722
#76362
#76082
#75742
#75682