Finite Group Databasebeta

← #100374

Subgroups — #100374

Order 512.

789 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 789 conjugacy classes of subgroups — too many to draw. Only its 112 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#32903C211yesmaximal, characteristic
2#13817C211yesmaximal, characteristic, complemented
3#9712C211yesmaximal, characteristic
4#9680C211yesmaximal, characteristic
5#8311C211yesmaximal, characteristic, complemented
6#7039C211yesmaximal, characteristic, complemented
7#7028C211yesmaximal, characteristic, complemented
8C22×C4×Q8C2211yescharacteristic
9C2×#460C412yescomplemented
10C2×#448—21no—
11#2856C2211yescharacteristic
12C2×#391—21no—
13C2×C4×C4⋊C4—21no—
14#1880—22no—
15#1874—21no—
16#1803C412yescomplemented
17#1797C2211yescharacteristic
18#1797C2211yescharacteristic
19C4×C22⋊C8C2211yescharacteristic
20C2×#327—22no—
21C2×#327—21no—
22C2×#327—21no—
23C2×Q8⋊C8—22no—
24#1436—22no—
25C4⋊#95C2211yescharacteristic
26#1434—21no—
27C4×#95—21no—
28C4×#95C2211yescharacteristic
29C23×Q8D411yescharacteristic
30C22×SD16—22no—
31C2×#127C2×C411yescharacteristic
32C2×#121—21no—
33C2×#121—21no—
34C2×#121—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—44no—
38C2×C4×Q8C2×C412yes—
39C2×C4×Q8D412yes—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4D411yescharacteristic
43C22×C42—21no—
44C22×C42—21no—
45C22×C42—22no—
46C22×C42C2311yescharacteristic
47#460—44no—
48#448—42no—
49#448—42no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8C2×C412yes—
52C2×#103—22no—
53C2×#102—22no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C4×C8—21no—
58#391—28no—
59#390—21no—
60#390—21no—
61#390—21no—
62#384C2×C411yescharacteristic
63C4×C4⋊C4—44no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95D411yescharacteristic
70C2×#95D411yescharacteristic
71C2×#95D412yes—
72#340—22no—
73#336—21no—
74#336—21no—
75#336—21no—
76#327—42no—
77#327—44no—
78#327—42no—
79Q8⋊C8—44no—
80C22×Q8—24no—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—21no—
84C22×Q8—21no—
85C22×Q8—21no—
86C22×Q8—42no—
87C22×Q8C22⋊C412yes—
88C22×D4—21no—
89C22×D4—21no—
90C22×D4—21no—
91C22×D4—41no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C2×SD16—44no—
101C2×SD16—44no—
102C2×SD16—44no—
103C22×C8—22no—
104C22×C8—41no—
105C22×C8—41no—
106#127—42no—
107#121—42no—
108#121—42no—
109#121—41no—
110#121—41no—
111#121—41no—
112#121—41no—
113#121—41no—
114#121—41no—
115#121—41no—
116#121—41no—
117C4×Q8—24no—
118C4×Q8—24no—
119C4×Q8—42no—
120C4×Q8—42no—
121C4×Q8—44no—
122C4×Q8—44no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4C22⋊C412yes—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C42—24no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—42no—
143C2×C42—44no—
144C2×C42—42no—
145C2×C42—41no—
146C2×C42—41no—
147C2×C42—44no—
148C2×C42—44no—
149C2×C42—82no—
150C2×C42C22×C411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42#4111yescharacteristic
154C4⋊C8—42no—
155C4⋊C8—42no—
156#103—44no—
157#102—44no—
158C22⋊C8—22no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C4×C8—42no—
165#95—28no—
166#95—28no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—42no—
171#95—42no—
172#95—44no—
173#95—41no—
174#95—41no—
175#95SD1612yescomplemented
176C24—41no—
177C24#12011yescharacteristic
178C2×Q8—24no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—44no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—42no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8#10414yes—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—42no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—81no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C2×D4—81no—
205C2×D4—81no—
206C2×D4—81no—
207C2×D4—81no—
208C2×D4—81no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
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—42no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
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×SD1611yescharacteristic
256C22×C4#13511yescharacteristic
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
263SD16—88no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—81no—
269C2×C8—81no—
270C2×C8—81no—
271C2×C8—81no—
272C2×C8—81no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4#9914yes—
285C22⋊C4—82no—
286C22⋊C4—81no—
287C22⋊C4—81no—
288C22⋊C4—81no—
289C22⋊C4—81no—
290C42—22no—
291C42—22no—
292C42—41no—
293C42—41no—
294C42—44no—
295C42—44no—
296C42—44no—
297C42—44no—
298C42—88no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23#38611yescharacteristic
311C23#44811yescharacteristic
312C23#44911yescharacteristic
313C23#44911yescharacteristic
314C23#44911yescharacteristic
315C23#45311yescharacteristic
316C23#45411yescharacteristic
317Q8—24no—
318Q8—24no—
319Q8—44no—
320Q8—44no—
321Q8—42no—
322Q8—42no—
323D4—82no—
324D4—82no—
325D4—161no—
326D4—161no—
327D4—162no—
328D4—161no—
329D4—161no—
330D4—161no—
331D4—161no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—24no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4C2×#9912yes—
389C2×C4#41611yescharacteristic
390C2×C4#41811yescharacteristic
391C2×C4C2×#10412yes—
392C2×C4C4×SD1611yescharacteristic
393C2×C4#44011yescharacteristic
394C2×C4#46011yescharacteristic
395C2×C4#46411yescharacteristic
396C2×C4#48211yescharacteristic
397C2×C4#48211yescharacteristic
398C8—82no—
399C8—82no—
400C8—161no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22#162011yescharacteristic
421C22#162211yescharacteristic
422C22#162611yescharacteristic
423C22#162811yescharacteristic
424C22#187511yescharacteristic
425C22#188011yescharacteristic
426C22#188311yescharacteristic
427C22#189611yescharacteristic
428C22#189811yescharacteristic
429C22#190511yescharacteristic
430C22#190611yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—44no—
438C4—41no—
439C4—41no—
440C4—44no—
441C4—44no—
442C4—84no—
443C4—82no—
444C4—161no—
445C4#150112yes—
446C4#150712yes—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—321no—
452C2#897811yescharacteristic
453C2#915011yescharacteristic
454C2#918211yescharacteristic
455C2#923411yescharacteristic
456C2#926211yescharacteristic
457C2#932611yescharacteristic
458C2#936511yescharacteristic
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. #13817C2semidirect
5. #8311C2semidirect
6. #7039C2semidirect
7. #7028C2semidirect
9. C2×#460C4semidirect
16. #1803C4semidirect
175. #95SD16semidirect

Maximal subgroups

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

subgroupindex
#329032
#138172
#97122
#96802
#83112
#70392
#70282