Finite Group Databasebeta

← #232395

Subgroups — #232395

Order 512.

2138 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 2138 conjugacy classes of subgroups — too many to draw. Only its 258 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
1C2×#3186C212yesmaximal, complemented
2#35001C211yesmaximal, characteristic, complemented
3C22×#327C212yesmaximal, complemented
4#14152C211yesmaximal, characteristic, complemented
5#9970C214yesmaximal, complemented
6#9962C214yesmaximal, complemented
7#7907C211yesmaximal, characteristic, complemented
8C2×#534—22no—
9C22×#121C2212yes—
10C22×#121C2211yescharacteristic
11#3186—28no—
12#3186—28no—
13C2×#480—22no—
14C2×#480C2214yescomplemented
15C2×#466—24no—
16C2×#447—22no—
17C2×#447C2214yescomplemented
18C22×C22⋊C8C2212yes—
19#2856—22no—
20#2656C2214yes—
21#2539—24no—
22#2501—22no—
23#2433—22no—
24C2×#386—22no—
25C2×C4×C22⋊C4C2211yescharacteristic
26C22×#95C2211yescharacteristic
27C2×#327—28no—
28C2×#327C2218yescomplemented
29#1445C2212yescomplemented
30#1438C2212yescomplemented
31#1437—22no—
32#1280C2214yescomplemented
33C23×D4—22no—
34C23×D4—21no—
35C23×D4D412yes—
36C23×D4C2311yescharacteristic
37C24×C4—21no—
38C24×C4C2311yescharacteristic
39C22×D8—24no—
40C23×C8—22no—
41#534—48no—
42#534—48no—
43C2×#127—22no—
44C2×#123—28no—
45C2×#121—22no—
46C2×#121—28no—
47C2×#121—22no—
48C2×#121—22no—
49C2×#121—22no—
50C2×#121—24no—
51C2×#121—24no—
52C2×#121—24no—
53C2×#121—24no—
54C2×#121—22no—
55C2×#121—24no—
56C2×#121D414yes—
57C2×#121D414yescomplemented
58C2×#121C2314yes—
59C2×#120—28no—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—24no—
65C22×C22⋊C4D412yes—
66C22×C42D412yes—
67#480—216no—
68#480—28no—
69#480—28no—
70#466—232no—
71#447—216no—
72#447—28no—
73#447—28no—
74C2×#107—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8D418yescomplemented
80C2×C22⋊C8C2314yes—
81#399—28no—
82#392—28no—
83#390—24no—
84#390—28no—
85#386—48no—
86#384—22no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C2×#95—24no—
90C2×#95—21no—
91C2×#95—24no—
92C2×#95C2312yes—
93C2×#95C2312yes—
94#327—216no—
95C25—21no—
96C25—22no—
97C25C2×D411yescharacteristic
98C22×D4—24no—
99C22×D4—24no—
100C22×D4—24no—
101C22×D4—28no—
102C22×D4—24no—
103C22×D4—22no—
104C22×D4—24no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—21no—
108C22×D4—42no—
109C22×D4—44no—
110C22×D4—42no—
111C22×D4—44no—
112C22×D4—48no—
113C22×D4—44no—
114C22×D4—48no—
115C22×D4—44no—
116C22×D4—82no—
117C22×D4—84no—
118C22×D4C2×D414yes—
119C22×D4C2×D412yes—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—44no—
124C23×C4—42no—
125C23×C4—44no—
126C23×C4—82no—
127C23×C4—82no—
128C23×C4C2×D414yes—
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D414yes—
134C23×C4#4112yes—
135C23×C4C2411yescharacteristic
136C2×D8—216no—
137C2×D8—88no—
138C22×C8—24no—
139C22×C8—28no—
140C22×C8—84no—
141#127—84no—
142#123—432no—
143#121—28no—
144#121—28no—
145#121—28no—
146#121—28no—
147#121—24no—
148#121—24no—
149#121—24no—
150#121—24no—
151#121—416no—
152#121—44no—
153#121—44no—
154#121—44no—
155#121—44no—
156#121—44no—
157#121—44no—
158#121—416no—
159#120—432no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2×D412yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—216no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—48no—
181C2×C22⋊C4—48no—
182C2×C22⋊C4—84no—
183C2×C22⋊C4D818yescomplemented
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—82no—
187#107—216no—
188#102—216no—
189#102—216no—
190C22⋊C8—28no—
191C22⋊C8—24no—
192C22⋊C8C2×D418yes—
193#95—24no—
194#95—24no—
195#95—42no—
196#95—48no—
197C24—24no—
198C24—22no—
199C24—42no—
200C24—42no—
201C24—44no—
202C24—44no—
203C24—41no—
204C24—41no—
205C24—42no—
206C24—42no—
207C24—44no—
208C24—48no—
209C24—84no—
210C24—84no—
211C24#12012yes—
212C24#12112yes—
213C24#12712yes—
214C24C2×D814yes—
215C24AGL(1,ℤ/8)14yes—
216C24C22×D411yescharacteristic
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—48no—
221C2×D4—48no—
222C2×D4—48no—
223C2×D4—416no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—48no—
227C2×D4—48no—
228C2×D4—48no—
229C2×D4—48no—
230C2×D4—416no—
231C2×D4—416no—
232C2×D4—42no—
233C2×D4—44no—
234C2×D4—82no—
235C2×D4—88no—
236C2×D4—84no—
237C2×D4—82no—
238C2×D4—88no—
239C2×D4—88no—
240C2×D4—88no—
241C2×D4—88no—
242C2×D4—816no—
243C2×D4—84no—
244C2×D4—88no—
245C2×D4—84no—
246C2×D4—88no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—48no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—82no—
270C22×C4—84no—
271C22×C4—82no—
272C22×C4—816no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4#12014yes—
278C22×C4#12014yes—
279C22×C4#12114yes—
280C22×C4#12114yes—
281C22×C4#12112yes—
282C22×C4#12712yes—
283C22×C4C2×D818yes—
284C22×C4C22×D412yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D412yes—
287C22×C4C2×#4111yescharacteristic
288D8—816no—
289C2×C8—28no—
290C2×C8—48no—
291C2×C8—88no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—416no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C22⋊C4—216no—
299C22⋊C4—432no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—48no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—816no—
306C42—44no—
307C42—82no—
308C23—22no—
309C23—24no—
310C23—22no—
311C23—24no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—48no—
316C23—42no—
317C23—48no—
318C23—44no—
319C23—44no—
320C23—42no—
321C23—44no—
322C23—48no—
323C23—48no—
324C23—48no—
325C23—44no—
326C23—84no—
327C23—82no—
328C23—88no—
329C23—82no—
330C23—84no—
331C23—88no—
332C23—88no—
333C23—81no—
334C23—81no—
335C23—82no—
336C23—82no—
337C23—82no—
338C23—88no—
339C23—1616no—
340C23#39214yes—
341C23#44718yes—
342C23#45918yes—
343C23#49314yes—
344C23#49614yes—
345C23C2×#12011yescharacteristic
346C23C2×#12111yescharacteristic
347C23C2×#12711yescharacteristic
348C23C22×D812yes—
349C23C2×AGL(1,ℤ/8)12yes—
350D4—48no—
351D4—48no—
352D4—84no—
353D4—84no—
354D4—88no—
355D4—84no—
356D4—84no—
357D4—88no—
358D4—816no—
359D4—816no—
360D4—1616no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—28no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—48no—
377C2×C4—48no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—48no—
382C2×C4—44no—
383C2×C4—84no—
384C2×C4—82no—
385C2×C4—84no—
386C2×C4—816no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—88no—
392C2×C4#39212yes—
393C2×C4#39212yes—
394C2×C4C2×#12012yes—
395C2×C4C2×#12112yes—
396C8—48no—
397C22—21no—
398C22—21no—
399C22—24no—
400C22—21no—
401C22—21no—
402C22—24no—
403C22—24no—
404C22—24no—
405C22—216no—
406C22—44no—
407C22—48no—
408C22—44no—
409C22—88no—
410C22—82no—
411C22—84no—
412C22—84no—
413C22—82no—
414C22—82no—
415C22—84no—
416C22—816no—
417C22—162no—
418C22—162no—
419C22—1616no—
420C22—164no—
421C22#201118yes—
422C22C2×#39211yescharacteristic
423C22C2×#44712yes—
424C22#301812yes—
425C22#302612yes—
426C22C2×#45912yes—
427C22C2×#49311yescharacteristic
428C22C2×#49611yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—48no—
434C4—84no—
435C4—84no—
436C4—82no—
437C2—21no—
438C2—22no—
439C2—21no—
440C2—28no—
441C2—162no—
442C2—164no—
443C2#1838014yes—
444C2#2340611yescharacteristic
445C2C2×#201111yescharacteristic
446C2#2344411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3186C2semidirect
2. #35001C2semidirect
3. C22×#327C2semidirect
4. #14152C2semidirect
5. #9970C2semidirect
6. #9962C2semidirect
7. #7907C2semidirect
14. C2×#480C22semidirect
17. C2×#447C22semidirect
28. C2×#327C22semidirect
29. #1445C22semidirect
30. #1438C22semidirect
32. #1280C22semidirect
57. C2×#121D4semidirect
79. C2×C22⋊C8D4semidirect
183. C2×C22⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#31862
#350012
C22×#3272
#141522
#99702
#99622
#79072