Finite Group Databasebeta

← #382889

Subgroups — #382889

Order 512.

1098 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2331C211yesmaximal, characteristic
2C2×#1832C211yesmaximal, characteristic
3#14350C211yesmaximal, characteristic
4#11384C212yesmaximal
5#11358C212yesmaximal
6#11228C212yesmaximal
7#11227C212yesmaximal
8#10807C212yesmaximal
9#10806C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊C8C2212yes—
12#2331—24no—
13#2331C418yescomplemented
14C2×#391—22no—
15C2×#389C2212yes—
16C2×#387—22no—
17C2×#382C2211yescharacteristic
18C2×C4×C4⋊C4C2212yes—
19C2×#376C2211yescharacteristic
20#1832—24no—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32#1443—22no—
33#1443—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C23×Q8—21no—
39C23×C8C2312yes—
40C23×C8C2312yes—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8C2×C414yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C42—22no—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C2×C4⋊C8—24no—
56C2×C4⋊C8—42no—
57C2×C4⋊C8—42no—
58#391—216no—
59#389—24no—
60#389C2×C418yes—
61#387—216no—
62#382—22no—
63#382C2×C414yes—
64C4×C4⋊C4—24no—
65C4×C4⋊C4C2×C418yes—
66#376C2×C414yes—
67#376D414yes—
68C2×#95—22no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95D412yes—
79C2×#95D412yes—
80#340—24no—
81#340—24no—
82#340—24no—
83#340—24no—
84#340—42no—
85#340—42no—
86#340—42no—
87#340—42no—
88#336—24no—
89#336—22no—
90#336—22no—
91C22×Q8—24no—
92C22×Q8—24no—
93C22×Q8—24no—
94C22×Q8—82no—
95C22×Q8—82no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4C2411yescharacteristic
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C4⋊Q8—216no—
120C4⋊Q8—216no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—82no—
146C2×C4⋊C4—82no—
147C2×C4⋊C4C22×C414yes—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4#4114yes—
150C2×C4⋊C4#4114yes—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—28no—
158C2×C42—82no—
159C2×C42—82no—
160C2×C42C22×C412yes—
161C2×C42C22×C412yes—
162C2×C42C22×C412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C4⋊C8—48no—
167#95—28no—
168#95—24no—
169#95—24no—
170#95—28no—
171#95—82no—
172#95—82no—
173#95—82no—
174#95—82no—
175#95—82no—
176#95—82no—
177#95C22⋊C418yes—
178#95C22⋊C418yes—
179C24C22×D411yescharacteristic
180C2×Q8—216no—
181C2×Q8—88no—
182C2×Q8—88no—
183C2×Q8—84no—
184C2×Q8—84no—
185C22×C4—24no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4C2×C22⋊C414yes—
234C22×C4C2×C22⋊C414yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C23×C411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C22×C4C2×#4112yes—
240C22×C4C2×#4112yes—
241C22×C4H(5,𝔽2)12yes—
242C22×C42−1+412yes—
243C22×C42−1+412yes—
244C22×C42−1+412yes—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C4⋊C4—216no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—88no—
276C4⋊C4#11718yes—
277C42—24no—
278C42—22no—
279C42—22no—
280C42—24no—
281C42—24no—
282C42—22no—
283C42—24no—
284C42—22no—
285C42—88no—
286C23C22×C22⋊C411yescharacteristic
287C23#52511yescharacteristic
288C23#52511yescharacteristic
289C23#53612yes—
290C23#53712yes—
291C23#57612yes—
292C23#57712yes—
293C23#57712yes—
294C23#57812yes—
295Q8—816no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—28no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—44no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4C2×#11712yes—
349C2×C4#51911yescharacteristic
350C2×C4#51911yescharacteristic
351C2×C4#52012yes—
352C2×C4#52512yes—
353C2×C4#53511yescharacteristic
354C2×C4#53612yes—
355C2×C4#53611yescharacteristic
356C2×C4#54112yes—
357C2×C4#54112yes—
358C2×C4#54112yes—
359C2×C4#54112yes—
360C2×C4#54212yes—
361C2×C4#54212yes—
362C2×C4#54212yes—
363C2×C4#54212yes—
364C8—82no—
365C8—82no—
366C22#228011yescharacteristic
367C22#231011yescharacteristic
368C22#231111yescharacteristic
369C22#263712yes—
370C22#263712yes—
371C22#289712yes—
372C22#289812yes—
373C22#311812yes—
374C22#312112yes—
375C22#312211yescharacteristic
376C22#312311yescharacteristic
377C22#312611yescharacteristic
378C22#312711yescharacteristic
379C22#324112yes—
380C22#324212yes—
381C22#324312yes—
382C22#324412yes—
383C22#325712yes—
384C22#325812yes—
385C22#326212yes—
386C22#326312yes—
387C4—42no—
388C4—48no—
389C4—42no—
390C4—44no—
391C4—44no—
392C4—84no—
393C4—162no—
394C4—162no—
395C4#232012yes—
396C4#232012yes—
397C4#263212yes—
398C4#263412yes—
399C2#1434611yescharacteristic
400C2#2074411yescharacteristic
401C2#2074511yescharacteristic
402C2#2184312yes—
403C2#2189612yes—
404C2#3015712yes—
405C2#3015812yes—
406C2#3015912yes—
407C2#3016012yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2331C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23312
C2×#18322
#143502
#113842
#113582
#112282
#112272
#108072
#108062