Finite Group Databasebeta

← #161913

Subgroups — #161913

Order 512.

960 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3C2×#1839C211yesmaximal, characteristic, complemented
4#7703C212yesmaximal, complemented
5#7699C212yesmaximal, complemented
6#7654C214yesmaximal, complemented
7#7594C214yesmaximal, complemented
8C2×#529—21no—
9#2861—21no—
10#2860—21no—
11#2860C412yes—
12#2859C2211yescharacteristic
13#2858C2211yescharacteristic
14#2857C411yescharacteristic
15#2856C411yescharacteristic
16#2854—21no—
17#2854C2212yes—
18#2850—21no—
19C4×C4⋊Q8C411yescharacteristic
20C4×#127C411yescharacteristic
21C4×#124—21no—
22C4×#121—21no—
23C4×#121C412yes—
24#2293—21no—
25#2293C2212yes—
26C2×C43C2211yescharacteristic
27#1839—21no—
28#1839—21no—
29#1839—21no—
30#1839C2212yes—
31#1839C2212yes—
32#1831C2212yes—
33#1817C2212yes—
34#1797—21no—
35#1767—21no—
36#1749—21no—
37#1745C2212yes—
38C2×#336—21no—
39C2×#336—21no—
40C2×#336C2212yes—
41#1271C2218yes—
42#1270C2218yes—
43C23×C8—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#127C2×C411yescharacteristic
46#529—48no—
47C2×#126—22no—
48C2×#125—21no—
49C2×#124—21no—
50C2×#124—21no—
51C2×#123—22no—
52C2×C22⋊Q8—21no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121C2×C412yes—
56C2×C4×Q8—21no—
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4C2×C412yes—
61C2×#117—22no—
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×#97—22no—
72#391—21no—
73#391—21no—
74#390—21no—
75#390—21no—
76#390—21no—
77#390—21no—
78#390—21no—
79#390—21no—
80#390C2×C412yes—
81#390C2×C412yes—
82#390C2×C412yes—
83#390C2312yes—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2312yes—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#386—22no—
101#386—22no—
102#384C2×C411yescharacteristic
103#384C2×C411yescharacteristic
104#382—21no—
105#380—21no—
106#380C2×C412yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C412yes—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C43C2×C411yescharacteristic
112C43C2×C411yescharacteristic
113C43D411yescharacteristic
114C43D411yescharacteristic
115C43D411yescharacteristic
116C43D411yescharacteristic
117C43C2311yescharacteristic
118C43C2311yescharacteristic
119#336—22no—
120#336—22no—
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336C2314yes—
128#336C2314yes—
129C22×Q8—21no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4C22×C412yes—
134C23×C4—22no—
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C22×C8—24no—
143C22×C8—24no—
144C22×C8—44no—
145C22×C8—44no—
146C22×C8—44no—
147C4⋊Q8—42no—
148#127—42no—
149#126—48no—
150#125—44no—
151#124—44no—
152#124—44no—
153#123—48no—
154C22⋊Q8—44no—
155#121—44no—
156#121—44no—
157#121—44no—
158C4×Q8—44no—
159C4×D4—28no—
160C4×D4—44no—
161C4×D4—44no—
162C4×D4—48no—
163#117—48no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2411yescharacteristic
223C4⋊C8—44no—
224C22⋊C8—44no—
225#97—44no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232C24—22no—
233C24C2×C22⋊C411yescharacteristic
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×D4—24no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—42no—
248C2×D4—42no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4AGL(1,ℤ/8)14yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—88no—
297C4⋊C4—24no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42—44no—
348C42#13114yes—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23#41012yes—
361C23#41812yes—
362C23#42112yes—
363C23C22×C22⋊C411yescharacteristic
364Q8—84no—
365D4—84no—
366D4—84no—
367D4—84no—
368D4—88no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—28no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—88no—
420C2×C4#40812yes—
421C2×C4#40812yes—
422C2×C4#41312yes—
423C2×C4#41812yes—
424C2×C4#42112yes—
425C2×C4C2×#13112yes—
426C2×C4C2×AGL(1,ℤ/8)12yes—
427C2×C4#57512yes—
428C8—84no—
429C8—84no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—82no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22#151414yes—
444C22C2×#40811yescharacteristic
445C22C2×#41011yescharacteristic
446C22C2×#41311yescharacteristic
447C22C2×#41811yescharacteristic
448C22#289611yescharacteristic
449C22C2×#42111yescharacteristic
450C22C2×#42111yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—41no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—82no—
466C4—82no—
467C4—82no—
468C2—24no—
469C2—162no—
470C2#1737611yescharacteristic
471C2#1746311yescharacteristic
472C2#1748211yescharacteristic
473C2C2×#151411yescharacteristic
474C2#1759111yescharacteristic
475C2#1766811yescharacteristic
476C2#1773111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. C2×#1839C2semidirect
4. #7703C2semidirect
5. #7699C2semidirect
6. #7654C2semidirect
7. #7594C2semidirect

Maximal subgroups

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

subgroupindex
#366312
#207782
C2×#18392
#77032
#76992
#76542
#75942