Finite Group Databasebeta

← #101756

Subgroups — #101756

Order 512.

649 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29328C211yesmaximal, characteristic
2#13825C211yesmaximal, characteristic, complemented
3#9712C211yesmaximal, characteristic
4#9712C211yesmaximal, characteristic
5#7641C211yesmaximal, characteristic, complemented
6#7028C211yesmaximal, characteristic, complemented
7#7028C211yesmaximal, characteristic, complemented
8#2856C2211yescharacteristic
9C2×#389C2211yescharacteristic
10C2×#387—21no—
11C2×C4×C4⋊C4—21no—
12#2099C412yescomplemented
13#2041C412yescomplemented
14#1934—22no—
15#1874—21no—
16#1874—21no—
17#1797C2211yescharacteristic
18#1797C2211yescharacteristic
19C4×C22⋊C8C2211yescharacteristic
20C2×#327—21no—
21C2×#327—21no—
22C2×#327—21no—
23C2×#327—21no—
24#1436—21no—
25#1436—21no—
26#1436—21no—
27C4×#95—21no—
28C4×#95C2211yescharacteristic
29C4×#95C2211yescharacteristic
30#1271—22no—
31C2×#127C2×C411yescharacteristic
32C2×#121—21no—
33C2×#121—21no—
34C2×#121—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C42—21no—
40C22×C42—21no—
41C22×C42—21no—
42C22×C42—21no—
43C22×C42C2311yescharacteristic
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×#102—22no—
47C2×#102—22no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C4×C8—21no—
52#390—21no—
53#390—21no—
54#390—21no—
55#389—21no—
56#389—21no—
57#389C2×C412yes—
58#389D412yes—
59#387—28no—
60#384C2×C411yescharacteristic
61C4×C4⋊C4—42no—
62C4×C4⋊C4—42no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95D411yescharacteristic
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74C2×#95D411yescharacteristic
75#340—22no—
76#340—22no—
77#336—21no—
78#336—21no—
79#336C2×C412yes—
80#327—42no—
81#327—42no—
82#327—42no—
83#327—42no—
84C22×D4—21no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—41no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C22×C8—22no—
98C22×C8—41no—
99C22×C8—41no—
100#127—81no—
101#121—42no—
102#121—42no—
103#121—41no—
104#121—41no—
105#121—41no—
106#121—41no—
107#121—41no—
108#121—41no—
109#121—41no—
110#121—41no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—81no—
127C2×C4⋊C4C22⋊C412yes—
128C2×C4⋊C4C22⋊C412yes—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C42—24no—
133C2×C42—42no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42—42no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42—41no—
142C2×C42—41no—
143C2×C42—81no—
144C2×C42—81no—
145C2×C42—81no—
146C2×C42—81no—
147C2×C42C22×C411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151C4⋊C8—81no—
152C4⋊C8—81no—
153#102—44no—
154#102—82no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C4×C8—81no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—22no—
167#95—22no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95—21no—
173#95—21no—
174#95—42no—
175#95—42no—
176#95—81no—
177#95D812yes—
178#95SD1612yes—
179C24—41no—
180C24#12011yescharacteristic
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—81no—
189C2×D4—81no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—81no—
194C2×D4—81no—
195C2×D4—81no—
196C2×D4—81no—
197C2×D4—81no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×SD1611yescharacteristic
257C22×C4#13511yescharacteristic
258C22×C4#13511yescharacteristic
259C22×C4AGL(1,ℤ/8)11yescharacteristic
260C22×C4AGL(1,ℤ/8)11yescharacteristic
261C22×C4AGL(1,ℤ/8)11yescharacteristic
262C22×C4#13711yescharacteristic
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—81no—
267C2×C8—81no—
268C2×C8—81no—
269C2×C8—81no—
270C2×C8—81no—
271C2×C8—81no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—48no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—81no—
286C4⋊C4#10414yes—
287C22⋊C4—82no—
288C22⋊C4—81no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C42—41no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42—48no—
297C42—82no—
298C42—82no—
299C42—82no—
300C42—82no—
301C42—84no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23#38611yescharacteristic
314C23#44711yescharacteristic
315C23#44811yescharacteristic
316C23#44911yescharacteristic
317C23#44911yescharacteristic
318C23#45411yescharacteristic
319C23#45411yescharacteristic
320D4—84no—
321D4—161no—
322D4—161no—
323D4—161no—
324D4—161no—
325D4—161no—
326D4—161no—
327D4—161no—
328D4—161no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4C2×#10412yes—
398C2×C4C4×D811yescharacteristic
399C2×C4C4×SD1611yescharacteristic
400C2×C4#44011yescharacteristic
401C2×C4#44211yescharacteristic
402C2×C4#45911yescharacteristic
403C2×C4#46011yescharacteristic
404C2×C4#46311yescharacteristic
405C2×C4#46411yescharacteristic
406C8—82no—
407C8—161no—
408C8—161no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22#162511yescharacteristic
429C22#162611yescharacteristic
430C22#162711yescharacteristic
431C22#162811yescharacteristic
432C22#187911yescharacteristic
433C22#188011yescharacteristic
434C22#188811yescharacteristic
435C22#189611yescharacteristic
436C22#190611yescharacteristic
437C22#190611yescharacteristic
438C22#191311yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—22no—
448C4—44no—
449C4—44no—
450C4—44no—
451C4—41no—
452C4—82no—
453C4—82no—
454C4—82no—
455C4—161no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—321no—
461C2#906211yescharacteristic
462C2#923311yescharacteristic
463C2#923411yescharacteristic
464C2#927711yescharacteristic
465C2#927811yescharacteristic
466C2#941011yescharacteristic
467C2#944211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13825C2semidirect
5. #7641C2semidirect
6. #7028C2semidirect
7. #7028C2semidirect
12. #2099C4semidirect
13. #2041C4semidirect

Maximal subgroups

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

subgroupindex
#293282
#138252
#97122
#76412
#70282