Finite Group Databasebeta

← #146469

Subgroups — #146469

Order 512.

1930 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1930 conjugacy classes of subgroups — too many to draw. Only its 226 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
1C22×#391C211yesmaximal, characteristic
2C2×#1784C212yesmaximal
3C2×#1394C214yesmaximal, complemented
4C24×Q8C2211yescharacteristic
5C23×C4⋊C4C411yescharacteristic
6C23×C42C411yescharacteristic
7C22×#101—22no—
8C22×#101C2214yes—
9C2×#407—22no—
10C2×#391—28no—
11C2×#391—22no—
12C2×#391C412yes—
13C2×#391C2212yes—
14C22×#95—22no—
15#1784—24no—
16#1784C418yescomplemented
17#1394—28no—
18C23×Q8—24no—
19C23×Q8—24no—
20C23×Q8—42no—
21C23×Q8—42no—
22C23×Q8—42no—
23C23×Q8—42no—
24C23×Q8C2×C414yes—
25C23×Q8C2×C411yescharacteristic
26C23×Q8D412yes—
27C23×Q8D412yes—
28C23×Q8Q812yes—
29C23×Q8C2311yescharacteristic
30C24×C4—21no—
31C24×C4—21no—
32C24×C4C2×C411yescharacteristic
33C24×C4D412yes—
34C22×M16—24no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—24no—
37C22×C4⋊C4—42no—
38C22×C4⋊C4—42no—
39C22×C4⋊C4C2×C412yes—
40C22×C42—21no—
41C22×C42—24no—
42C22×C42—42no—
43C22×C42—42no—
44C22×C42C2×C412yes—
45C2×#101—28no—
46C2×#101—28no—
47C2×#101—28no—
48C2×#101—24no—
49C2×#101—44no—
50C2×#101—44no—
51C2×#101C2×C418yescomplemented
52#407—28no—
53#407—44no—
54C2×C22⋊C8—44no—
55#391—216no—
56#391—24no—
57#391—24no—
58#391—24no—
59#391D418yes—
60C2×#95—216no—
61C2×#95—24no—
62C2×#95—24no—
63C25C22⋊C411yescharacteristic
64C22×Q8—216no—
65C22×Q8—28no—
66C22×Q8—28no—
67C22×Q8—24no—
68C22×Q8—24no—
69C22×Q8—22no—
70C22×Q8—22no—
71C22×Q8—24no—
72C22×Q8—24no—
73C22×Q8—48no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8—41no—
77C22×Q8—41no—
78C22×Q8—44no—
79C22×Q8—42no—
80C22×Q8—42no—
81C22×Q8—44no—
82C22×Q8—41no—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8—41no—
86C22×Q8—48no—
87C22×Q8—44no—
88C22×Q8—44no—
89C22×Q8—416no—
90C22×Q8—416no—
91C22×Q8—416no—
92C22×Q8—416no—
93C22×Q8C4212yes—
94C22×Q8C22⋊C414yes—
95C22×Q8C4⋊C414yes—
96C22×Q8C4⋊C412yes—
97C22×Q8C22×C412yes—
98C22×Q8C2×D411yescharacteristic
99C22×Q8C2×D414yes—
100C22×Q8C2×Q811yescharacteristic
101C22×Q8#4114yes—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—28no—
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—22no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—24no—
116C23×C4—22no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—44no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—48no—
124C23×C4—42no—
125C23×C4C4212yes—
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C414yes—
128C23×C4C22⋊C411yescharacteristic
129C23×C4C4⋊C412yes—
130C23×C4C4⋊C414yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D412yes—
133C2×M16—28no—
134C2×M16—44no—
135C2×M16—44no—
136C2×M16—44no—
137C2×M16—44no—
138C2×M16—44no—
139C22×C8—44no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—416no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4C22⋊C414yes—
148C2×C42—24no—
149C2×C42—22no—
150C2×C42—42no—
151C2×C42—42no—
152C2×C42—416no—
153C2×C42—42no—
154C2×C42—42no—
155C2×C42C22⋊C414yes—
156#101—216no—
157#101—416no—
158#101—48no—
159C22⋊C8—416no—
160#95—216no—
161#95—216no—
162C24—24no—
163C24—24no—
164C24—21no—
165C24—21no—
166C24—21no—
167C24—21no—
168C24#9512yes—
169C24#9912yes—
170C24#9912yes—
171C24C2×C22⋊C411yescharacteristic
172C2×Q8—28no—
173C2×Q8—28no—
174C2×Q8—24no—
175C2×Q8—24no—
176C2×Q8—28no—
177C2×Q8—28no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—48no—
195C2×Q8—48no—
196C2×Q8—48no—
197C2×Q8—48no—
198C2×Q8—464no—
199C2×Q8—432no—
200C2×Q8—432no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8C4×D418yes—
206C2×Q8#12114yes—
207C2×Q8C22⋊Q814yes—
208C22×C4—28no—
209C22×C4—28no—
210C22×C4—28no—
211C22×C4—28no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—216no—
231C22×C4—28no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—28no—
242C22×C4—28no—
243C22×C4—24no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—48no—
251C22×C4—48no—
252C22×C4—416no—
253C22×C4—44no—
254C22×C4—416no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—48no—
258C22×C4—416no—
259C22×C4—48no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—88no—
263C22×C4—84no—
264C22×C4—84no—
265C22×C4#9514yes—
266C22×C4#9512yes—
267C22×C4#9912yes—
268C22×C4#9912yes—
269C22×C4C2×C4211yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C4⋊C411yescharacteristic
273C22×C4C2×C4⋊C412yes—
274C22×C4#11712yes—
275C22×C4#12014yes—
276C22×C4#12314yes—
277M16—48no—
278M16—48no—
279M16—84no—
280M16—84no—
281C2×C8—48no—
282C2×C8—84no—
283C2×C8—84no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—44no—
287C42—44no—
288C42—48no—
289C42—44no—
290C23—24no—
291C23—24no—
292C23—28no—
293C23—24no—
294C23—28no—
295C23—24no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—28no—
303C23—28no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23#34214yes—
309C23#35514yes—
310C23#35614yes—
311C23C2×#9511yescharacteristic
312C23#37911yescharacteristic
313C23#38111yescharacteristic
314C23C2×#9911yescharacteristic
315C23C2×#9911yescharacteristic
316C23C2×#9911yescharacteristic
317C23C2×#9911yescharacteristic
318Q8—48no—
319Q8—44no—
320Q8—44no—
321Q8—416no—
322Q8—416no—
323Q8—48no—
324Q8—48no—
325C2×C4—216no—
326C2×C4—216no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—28no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—416no—
354C2×C4—416no—
355C2×C4—432no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—816no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4#34214yes—
370C2×C4C4×C22⋊C412yes—
371C2×C4#38012yes—
372C2×C4#38514yes—
373C2×C4#38614yes—
374C8—88no—
375C22—28no—
376C22—28no—
377C22—28no—
378C22—28no—
379C22—22no—
380C22—24no—
381C22—22no—
382C22—22no—
383C22—24no—
384C22—24no—
385C22—22no—
386C22—21no—
387C22—24no—
388C22—21no—
389C22—21no—
390C22—24no—
391C22—21no—
392C22—24no—
393C22#139418yescomplemented
394C22C22⋊#9511yescharacteristic
395C22C2×#34211yescharacteristic
396C22C2×#34211yescharacteristic
397C22#178111yescharacteristic
398C22#178111yescharacteristic
399C22#181911yescharacteristic
400C22#181911yescharacteristic
401C22C2×#35512yes—
402C22C2×#35612yes—
403C4—24no—
404C4—24no—
405C4—22no—
406C4—22no—
407C4—24no—
408C4—48no—
409C4—44no—
410C4—84no—
411C2—21no—
412C2—21no—
413C2—28no—
414C2—21no—
415C2—21no—
416C2#1072511yescharacteristic
417C2C2×#139412yes—
418C2#1269911yescharacteristic
419C2#1270011yescharacteristic
420C2#1286012yes—
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC23×Q8C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C24×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1394C2semidirect
16. #1784C4semidirect
51. C2×#101C2×C4semidirect
393. C22#1394semidirect

Maximal subgroups

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

subgroupindex
C22×#3912
C2×#17842
C2×#13942