Finite Group Databasebeta

← #196934

Subgroups — #196934

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35240C211yesmaximal, characteristic
2C2×#2319C211yesmaximal, characteristic
3#21788C211yesmaximal, characteristic
4C2×#1506C211yesmaximal, characteristic, complemented
5C2×#1503C211yesmaximal, characteristic, complemented
6#9154C214yesmaximal, complemented
7#9151C214yesmaximal, complemented
8#8021C211yesmaximal, characteristic, complemented
9#8013C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2848C2214yes—
15#2633—22no—
16#2547C412yescomplemented
17#2542C412yescomplemented
18#2536C414yescomplemented
19#2502—21no—
20#2482—22no—
21#2435—21no—
22#2326—22no—
23#2319—24no—
24#2319—24no—
25C2×#390—21no—
26C2×#389—21no—
27C2×#388—22no—
28#2283C2211yescharacteristic
29C2×#381C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1955—22no—
34#1954—22no—
35#1779C2214yes—
36#1506—24no—
37#1503—24no—
38C2×#328C2214yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C23×D4D411yescharacteristic
45C24×C4C2312yes—
46#533—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#125—21no—
53C2×#124—22no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2×C418yes—
56C2×C22⋊Q8C2312yes—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C412yes—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C412yes—
65C2×C4×D4D412yes—
66C2×C4×D4D412yes—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D411yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4D412yes—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#432—22no—
78#432—22no—
79#431—22no—
80#431—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C22⋊C8—28no—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86C2×#97—21no—
87C2×C4×C8—21no—
88#399—24no—
89#398—24no—
90#397—24no—
91#394—24no—
92#391—22no—
93#390—28no—
94#390—22no—
95#389—28no—
96#388—216no—
97#388—28no—
98#385—28no—
99#385—24no—
100#384—22no—
101#383—21no—
102#382C2×C412yes—
103#381—21no—
104#381—24no—
105#381—24no—
106#380—24no—
107#380—24no—
108#380C2×C418yes—
109#380C2312yes—
110#379—24no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4D414yes—
114C4×C22⋊C4—21no—
115#376—22no—
116#376—21no—
117#376D414yes—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95C2311yescharacteristic
121C2×#95C2311yescharacteristic
122C2×#95C2312yes—
123#328—28no—
124#328—24no—
125#328—24no—
126C25C2×D412yes—
127C22×Q8—21no—
128C22×Q8C22×C412yes—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—44no—
132C22×D4C22⋊C412yes—
133C22×D4C22⋊C412yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—84no—
142C23×C4C22×C414yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D414yes—
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154#125—42no—
155#125—42no—
156#124—48no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C22⋊Q8—88no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—24no—
176#117—42no—
177#117—42no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—84no—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C414yes—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—84no—
203C2×C22⋊C4—84no—
204C2×C22⋊C4C22⋊C414yes—
205C2×C22⋊C4C22⋊C414yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—44no—
211C2×C42—82no—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4112yes—
219C2×C42#4112yes—
220C4⋊C8—42no—
221C4⋊C8—42no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—88no—
227#97—42no—
228C4×C8—42no—
229#95—28no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—28no—
234#95—24no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95C2×D412yes—
240#95C2×D412yes—
241#95C2×D414yes—
242C24—22no—
243C24—22no—
244C24—24no—
245C24—24no—
246C24—44no—
247C24—44no—
248C24C2×C22⋊C414yes—
249C24C22×D411yescharacteristic
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—44no—
253C2×Q8—42no—
254C2×Q8—44no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—44no—
260C2×D4—48no—
261C2×D4—48no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4C2×C22⋊C414yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4#12012yes—
300C22×C4#12012yes—
301C22×C4#12312yes—
302C22×C4#12312yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—42no—
312C2×C8—42no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—44no—
318C4⋊C4—48no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—88no—
326C4⋊C4—88no—
327C22⋊C4—28no—
328C22⋊C4—28no—
329C22⋊C4—416no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—88no—
335C22⋊C4—88no—
336C22⋊C4—88no—
337C42—48no—
338C42—44no—
339C42—44no—
340C42—84no—
341C42C4×D412yes—
342C42C4×D412yes—
343C42C4×D412yes—
344C42C4×D412yes—
345C42#12112yes—
346C42#12112yes—
347C42#12112yes—
348C42#12112yes—
349C23—22no—
350C23—22no—
351C23—28no—
352C23—28no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—44no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23—42no—
362C23—42no—
363C23—88no—
364C23—88no—
365C23C22×C22⋊C411yescharacteristic
366C23C2×#12011yescharacteristic
367C23C2×#12311yescharacteristic
368C23#57712yes—
369C23#57812yes—
370Q8—82no—
371Q8—82no—
372Q8—84no—
373D4—48no—
374D4—48no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—24no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—88no—
415C2×C4—84no—
416C2×C4#38612yes—
417C2×C4#38612yes—
418C2×C4#38612yes—
419C2×C4#38612yes—
420C2×C4#42114yes—
421C2×C4C2×C4×D411yescharacteristic
422C2×C4C2×C4×D411yescharacteristic
423C2×C4C2×#12111yescharacteristic
424C2×C4C2×#12111yescharacteristic
425C8—82no—
426C8—82no—
427C22—24no—
428C22—24no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—44no—
434C22—48no—
435C22—48no—
436C22—44no—
437C22—44no—
438C22—84no—
439C22—84no—
440C22—88no—
441C22C2×#38611yescharacteristic
442C22#288312yes—
443C22#289812yes—
444C22C2×#42112yes—
445C22#300911yescharacteristic
446C22#301111yescharacteristic
447C22#309712yes—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—44no—
456C4—82no—
457C4—84no—
458C4—82no—
459C4—84no—
460C4—82no—
461C4—82no—
462C4#189714yes—
463C2—21no—
464C2—21no—
465C2—22no—
466C2—84no—
467C2#1760014yes—
468C2#2091311yescharacteristic
469C2C2×#189711yescharacteristic
470C2#2098511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1506C2semidirect
5. C2×#1503C2semidirect
6. #9154C2semidirect
7. #9151C2semidirect
8. #8021C2semidirect
9. #8013C2semidirect
16. #2547C4semidirect
17. #2542C4semidirect
18. #2536C4semidirect

Maximal subgroups

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

subgroupindex
#352402
C2×#23192
#217882
C2×#15062
C2×#15032
#91542
#91512
#80212
#80132