Finite Group Databasebeta

← #100363

Subgroups — #100363

Order 512.

1118 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 128 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#32956C211yesmaximal, characteristic
2D4×#95C211yesmaximal, characteristic
3#9908C211yesmaximal, characteristic, complemented
4#9679C211yesmaximal, characteristic, complemented
5#8310C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic, complemented
7#7028C211yesmaximal, characteristic, complemented
8C22×C4×D4—21no—
9C22×C4×D4C2211yescharacteristic
10C2×#476C412yescomplemented
11#2857C2211yescharacteristic
12C22×#95—21no—
13#1864—22no—
14#1853—21no—
15#1802C412yescomplemented
16#1797C2211yescharacteristic
17#1797C2211yescharacteristic
18#1786—21no—
19C4×C22⋊C8C2211yescharacteristic
20C2×#328—22no—
21C2×D4⋊C8—22no—
22C4⋊#95—21no—
23C4⋊#95C2211yescharacteristic
24C22⋊#95—21no—
25C22⋊#95—22no—
26C4×#95C2211yescharacteristic
27#1294—21no—
28#1294—21no—
29C23×D4D411yescharacteristic
30C24×C4—22no—
31C24×C4—21no—
32C2×C4⋊Q8C2×C411yescharacteristic, complemented
33C2×C22⋊Q8—21no—
34C2×C4×D4—28no—
35C2×C4×D4—21no—
36C2×C4×D4—21no—
37C2×C4×D4—44no—
38C2×C4×D4—44no—
39C2×C4×D4—42no—
40C2×C4×D4—42no—
41C2×C4×D4C2×C412yes—
42C2×C4×D4D412yes—
43C22×C4⋊C4—21no—
44C22×C4⋊C4D411yescharacteristic
45C22×C22⋊C4—22no—
46C22×C22⋊C4—21no—
47C22×C42—21no—
48C22×C42C2311yescharacteristic
49#476—44no—
50C2×#106—22no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8C2×C412yescomplemented
53C2×#102—22no—
54C2×#102—22no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C4×C8—21no—
59#391—21no—
60#389—21no—
61#389—21no—
62#384C2×C411yescharacteristic, complemented
63#380—21no—
64#380—21no—
65C2×#95—22no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—22no—
69C2×#95—42no—
70C2×#95—42no—
71C2×#95D411yescharacteristic
72C2×#95D411yescharacteristic
73C2×#95D412yes—
74#340—22no—
75#336—21no—
76#336—21no—
77#336—21no—
78#328—44no—
79D4⋊C8—44no—
80C25—21no—
81C22×Q8—21no—
82C22×D4—24no—
83C22×D4—24no—
84C22×D4—21no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4C22⋊C412yes—
88C22×D4SD1612yes—
89C22×D4Q1612yes—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—22no—
93C23×C4—21no—
94C23×C4—28no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—44no—
98C23×C4—44no—
99C23×C4—44no—
100C23×C4—44no—
101C23×C4—44no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—82no—
107C23×C4—82no—
108C23×C4—82no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C22×C8—22no—
113C22×C8—41no—
114C22×C8—41no—
115C4⋊Q8—42no—
116C22⋊Q8—42no—
117C22⋊Q8—42no—
118C4×D4—24no—
119C4×D4—24no—
120C4×D4—42no—
121C4×D4—42no—
122C4×D4—44no—
123C4×D4—44no—
124C4×D4—416no—
125C4×D4—416no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4C22⋊C412yes—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4—41no—
146C2×C42—24no—
147C2×C42—42no—
148C2×C42—42no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155#106—44no—
156C4⋊C8—42no—
157C4⋊C8—42no—
158#102—44no—
159#102—44no—
160C22⋊C8—22no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C22⋊C8—41no—
165C22⋊C8—41no—
166C4×C8—42no—
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#95—41no—
176#95—41no—
177#95—82no—
178#95SD1612yes—
179C24—24no—
180C24—24no—
181C24—22no—
182C24—22no—
183C24—22no—
184C24—42no—
185C24—42no—
186C24—41no—
187C24—41no—
188C24—41no—
189C24—41no—
190C24#12011yescharacteristic
191C2×Q8—42no—
192C2×Q8—81no—
193C2×Q8—81no—
194C2×D4—24no—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—28no—
198C2×D4—28no—
199C2×D4—28no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4#9914yes—
206C2×D4#10314yes—
207C2×D4#10414yes—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—416no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—416no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—88no—
255C22×C4—84no—
256C22×C4—84no—
257C22×C4—84no—
258C22×C4—84no—
259C22×C4—88no—
260C22×C4—88no—
261C22×C4—88no—
262C22×C4—88no—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4C4×D411yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12311yescharacteristic
269C22×C4C2×SD1611yescharacteristic
270C22×C4C2×SD1611yescharacteristic
271C22×C4C2×Q1611yescharacteristic
272C22×C4#13511yescharacteristic
273C22×C4AGL(1,ℤ/8)11yescharacteristic
274C22×C4AGL(1,ℤ/8)11yescharacteristic
275C22×C4#13711yescharacteristic
276C22×C4#13711yescharacteristic
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—81no—
282C2×C8—81no—
283C2×C8—81no—
284C2×C8—81no—
285C2×C8—81no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4—82no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—416no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42—41no—
306C42—44no—
307C42—44no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—28no—
313C23—28no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23#38611yescharacteristic
336C23#45011yescharacteristic
337C23#45011yescharacteristic
338C23#45111yescharacteristic
339C23#45211yescharacteristic
340C23#45311yescharacteristic
341C23#45411yescharacteristic
342Q8—162no—
343D4—28no—
344D4—28no—
345D4—24no—
346D4—24no—
347D4—24no—
348D4#32818yescomplemented
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—832no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4C2×#9912yes—
409C2×C4C2×#10311yescharacteristic
410C2×C4#41711yescharacteristic
411C2×C4C2×#10412yes—
412C2×C4C4×SD1611yescharacteristic
413C2×C4#44011yescharacteristic
414C2×C4#46111yescharacteristic
415C2×C4#46311yescharacteristic
416C2×C4#48311yescharacteristic
417C2×C4#48411yescharacteristic
418C8—82no—
419C8—82no—
420C8—161no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—28no—
434C22—48no—
435C22—48no—
436C22—48no—
437C22—48no—
438C22—44no—
439C22—44no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—44no—
444C22#162911yescharacteristic
445C22#163211yescharacteristic
446C22#163511yescharacteristic
447C22#163711yescharacteristic
448C22#187211yescharacteristic
449C22#187711yescharacteristic
450C22#188311yescharacteristic
451C22#189611yescharacteristic
452C22#189811yescharacteristic
453C22#190411yescharacteristic
454C22#190611yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—44no—
462C4—41no—
463C4—41no—
464C4—82no—
465C4—84no—
466C4—82no—
467C4—161no—
468C4—161no—
469C4C2×#32812yes—
470C4#150412yes—
471C2—21no—
472C2—21no—
473C2—21no—
474C2—21no—
475C2—48no—
476C2#897311yescharacteristic
477C2#914311yescharacteristic
478C2#917511yescharacteristic
479C2#923111yescharacteristic
480C2#925511yescharacteristic
481C2#932511yescharacteristic
482C2#936511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
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
3. #9908C2semidirect
4. #9679C2semidirect
6. #7039C2semidirect
7. #7028C2semidirect
10. C2×#476C4semidirect
15. #1802C4semidirect
32. C2×C4⋊Q8C2×C4semidirect
52. C2×C4⋊C8C2×C4semidirect
62. #384C2×C4semidirect
348. D4#328semidirect

Maximal subgroups

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

subgroupindex
#329562
D4×#952
#99082
#96792
#83102
#70392
#70282