Finite Group Databasebeta

← #232356

Subgroups — #232356

Order 512.

1566 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1566 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×#3189C212yesmaximal
2C2×#2659C211yesmaximal, characteristic
3C2×#1520C212yesmaximal
4#14281C211yesmaximal, characteristic
5#9972C214yesmaximal, complemented
6#9967C214yesmaximal, complemented
7C2×#1280C211yesmaximal, characteristic
8C22×C22⋊Q8C2212yes—
9C22×C22⋊Q8C2211yescharacteristic
10#3189—28no—
11#3189—28no—
12C2×#484—22no—
13C2×#484C2214yes—
14C2×#467—24no—
15C2×#451—22no—
16C2×#451C2214yes—
17C22×C22⋊C8C2212yes—
18#2659—22no—
19#2659—24no—
20#2659C2214yes—
21C2×#398—22no—
22C2×#395—22no—
23C2×#380—22no—
24C2×#380C2211yescharacteristic
25C22×#95C2211yescharacteristic
26#1520—28no—
27C2×#328C2218yes—
28#1446C2212yes—
29#1445C2212yes—
30#1441—22no—
31C22⋊#95—22no—
32#1280C2214yes—
33C23×Q8D412yes—
34C23×Q8C2311yescharacteristic
35C24×C4D412yes—
36C24×C4C2311yescharacteristic
37C22×Q16—24no—
38C23×C8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8D414yes—
50C2×C22⋊Q8D414yes—
51C2×C22⋊Q8C2314yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62#484—216no—
63#484—28no—
64#484—28no—
65#467—232no—
66#451—216no—
67#451—28no—
68#451—28no—
69C2×#107—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8D418yes—
75C2×C22⋊C8C2314yes—
76#398—28no—
77#398—28no—
78#395—28no—
79#395—28no—
80#380—28no—
81#380—28no—
82#380—24no—
83C2×#95—24no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—24no—
87C2×#95—24no—
88C2×#95D412yes—
89C2×#95C2312yes—
90C2×#95C2312yes—
91#328—216no—
92C25C2×D411yescharacteristic
93C22×Q8—22no—
94C22×Q8—21no—
95C22×Q8—24no—
96C22×Q8—24no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—82no—
100C22×Q8—84no—
101C22×Q8C2×D414yes—
102C22×Q8C2×D412yes—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—24no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—82no—
112C23×C4—82no—
113C23×C4—82no—
114C23×C4Q1618yescomplemented
115C23×C4C2×D414yes—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2×D414yes—
121C23×C4#4112yes—
122C23×C4C2411yescharacteristic
123C2×Q16—216no—
124C2×Q16—88no—
125C22×C8—24no—
126C22×C8—28no—
127C22×C8—84no—
128C22⋊Q8—28no—
129C22⋊Q8—28no—
130C22⋊Q8—28no—
131C22⋊Q8—28no—
132C22⋊Q8—24no—
133C22⋊Q8—24no—
134C22⋊Q8—24no—
135C22⋊Q8—24no—
136C22⋊Q8—44no—
137C22⋊Q8—44no—
138C22⋊Q8—44no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—84no—
159C2×C4⋊C4—84no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4C2×D414yes—
162C2×C4⋊C4C2×D412yes—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—44no—
173#107—216no—
174#103—216no—
175#103—216no—
176C22⋊C8—28no—
177C22⋊C8—24no—
178C22⋊C8C2×D418yes—
179#95—216no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—84no—
184#95—84no—
185C24—41no—
186C24—42no—
187C24—41no—
188C24#12011yescharacteristic
189C24#12011yescharacteristic
190C24#12112yes—
191C24#12712yes—
192C24C2×Q1614yes—
193C24#13714yes—
194C24C22×D411yescharacteristic
195C2×Q8—24no—
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—48no—
199C2×Q8—48no—
200C2×Q8—42no—
201C2×Q8—44no—
202C2×Q8—84no—
203C2×Q8—82no—
204C2×Q8—82no—
205C2×Q8—88no—
206C2×Q8—84no—
207C2×Q8—88no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—48no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—84no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—88no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—88no—
248C22×C4#12012yes—
249C22×C4#12014yes—
250C22×C4#12014yes—
251C22×C4#12114yes—
252C22×C4#12114yes—
253C22×C4#12112yes—
254C22×C4#12712yes—
255C22×C4C2×Q1618yes—
256C22×C4C22×D412yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D412yes—
259C22×C4C2×#4111yescharacteristic
260Q16—816no—
261C2×C8—28no—
262C2×C8—48no—
263C2×C8—88no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—48no—
273C4⋊C4—88no—
274C4⋊C4—88no—
275C4⋊C4—84no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—48no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—84no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—42no—
292C23—42no—
293C23—44no—
294C23—44no—
295C23—42no—
296C23—42no—
297C23—44no—
298C23—44no—
299C23#39212yes—
300C23#39212yes—
301C23#45114yes—
302C23#45114yes—
303C23#46218yes—
304C23#49414yes—
305C23#49714yes—
306C23C2×#12011yescharacteristic
307C23C2×#12111yescharacteristic
308C23C2×#12711yescharacteristic
309C23C22×Q1612yes—
310C23C2×#13712yes—
311Q8—48no—
312Q8—48no—
313Q8—84no—
314Q8—84no—
315Q8—88no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—28no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—416no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—48no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—48no—
341C2×C4—84no—
342C2×C4—88no—
343C2×C4—82no—
344C2×C4—88no—
345C2×C4—82no—
346C2×C4—88no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—84no—
352C2×C4—82no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4#39214yes—
356C2×C4C2×#12012yes—
357C2×C4C2×#12112yes—
358C8—48no—
359C22—22no—
360C22—24no—
361C22—22no—
362C22—24no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—24no—
370C22—24no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—48no—
376C22—44no—
377C22—44no—
378C22#201614yes—
379C22#201614yes—
380C22C2×#39211yescharacteristic
381C22C2×#45112yes—
382C22#301911yescharacteristic
383C22#302011yescharacteristic
384C22#302612yes—
385C22C2×#46212yes—
386C22C2×#49411yescharacteristic
387C22C2×#49711yescharacteristic
388C4—22no—
389C4—22no—
390C4—24no—
391C4—48no—
392C4—82no—
393C4—84no—
394C4—82no—
395C4—84no—
396C4—162no—
397C2—21no—
398C2—21no—
399C2—22no—
400C2—24no—
401C2—24no—
402C2#1828712yes—
403C2#1829512yes—
404C2#2340911yescharacteristic
405C2C2×#201611yescharacteristic
406C2#2344411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
5. #9972C2semidirect
6. #9967C2semidirect
114. C23×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C2×#26592
C2×#15202
#142812
#99722
#99672
C2×#12802