Finite Group Databasebeta

← #223947

Subgroups — #223947

Order 512.

965 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#22750C212yesmaximal
3#9905C212yesmaximal, complemented
4#9136C212yesmaximal, complemented
5#9127C212yesmaximal, complemented
6#8725C211yesmaximal, characteristic, complemented
7#8723C211yesmaximal, characteristic, complemented
8#8594C211yesmaximal, characteristic, complemented
9#8232C211yesmaximal, characteristic, complemented
10#8040C211yesmaximal, characteristic, complemented
11C2×#1296C211yesmaximal, characteristic, complemented
12C2×#554C412yes—
13C2×#432C2212yes—
14C22×C4⋊C8C2212yes—
15#2868C2211yescharacteristic
16#2857C2211yescharacteristic
17#2857C2211yescharacteristic
18#2567C412yes—
19#2552—22no—
20#2502C412yes—
21#2502C412yes—
22#2440—22no—
23#2440—22no—
24D4×C4⋊C4—22no—
25C4×C4⋊Q8C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1924—22no—
30#1917—22no—
31#1869—22no—
32#1831C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1767C2212yes—
36#1752—22no—
37C2×#333C2212yes—
38C2×#333C2212yes—
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2212yes—
42#1296C2211yescharacteristic
43#1296C2212yes—
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1296C2212yes—
47C23×C8—22no—
48#554—48no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2311yescharacteristic
52C2×#125—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8C2×C412yes—
56C2×C22⋊Q8C2×C412yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×D4—22no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D412yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67#432—24no—
68#432—42no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×#97—22no—
80C2×#97—22no—
81#398—24no—
82#398—24no—
83#395—24no—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2×C412yes—
87#389—24no—
88#389—22no—
89#389—22no—
90#389—22no—
91#385—24no—
92#385—24no—
93#384—22no—
94#384—22no—
95#384—22no—
96#384C2×C412yes—
97#384C2×C412yes—
98#384C2×C412yes—
99#384C2311yescharacteristic
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#380—22no—
103#380—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4D412yes—
107C4×C22⋊C4C2×C411yescharacteristic
108C4×C22⋊C4C2×C411yescharacteristic
109C4×C22⋊C4D412yes—
110C4×C22⋊C4D412yes—
111C43C2311yescharacteristic
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#333—24no—
118#333—42no—
119C22×Q8—22no—
120C22×Q8C22×C412yes—
121C22×D4C22⋊C411yescharacteristic
122C23×C4—22no—
123C23×C4—22no—
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—44no—
133C22×C8—44no—
134C22×C8—82no—
135C4⋊Q8—22no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—44no—
139C4⋊Q8—82no—
140C4⋊Q8—82no—
141#125—48no—
142C22⋊Q8—48no—
143C22⋊Q8—48no—
144C22⋊Q8—44no—
145C22⋊Q8—44no—
146C4×Q8—44no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—48no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4C22⋊C411yescharacteristic
178C2×C22⋊C4C22⋊C411yescharacteristic
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C22⋊C412yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2411yescharacteristic
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—44no—
205C4⋊C8#4114yes—
206C22⋊C8—44no—
207C22⋊C8—44no—
208C22⋊C8—82no—
209#97—44no—
210#97—82no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95C22⋊C412yes—
215#95C22⋊C412yes—
216#95C22⋊C412yes—
217C24C2×C22⋊C411yescharacteristic
218C24C2×C22⋊C411yescharacteristic
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—44no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—84no—
225C2×D4—24no—
226C2×D4#10014yes—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—84no—
252C22×C4—84no—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4#13714yes—
261C22×C4#13714yes—
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D412yes—
264C2×C8—24no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—88no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C4⋊C4—84no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—41no—
319C42—44no—
320C42C2×#4112yes—
321C42H(5,𝔽2)11yescharacteristic
322C422−1+411yescharacteristic
323C23—21no—
324C23—21no—
325C23—22no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23C2×#10012yes—
331C23C2×#10012yes—
332C23#41914yes—
333C23C22×C22⋊C411yescharacteristic
334Q8—82no—
335Q8—82no—
336Q8—164no—
337D4—28no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—164no—
379C2×C4—164no—
380C2×C4C2×#10012yes—
381C2×C4#41914yes—
382C2×C4C4×#4111yescharacteristic
383C2×C4#52011yescharacteristic
384C2×C4#54612yes—
385C2×C4#54712yes—
386C2×C4C2×#13712yes—
387C2×C4C2×#13712yes—
388C2×C4#57612yes—
389C2×C4#57712yes—
390C8—44no—
391C8—82no—
392C8—82no—
393C8—82no—
394C22—24no—
395C22—22no—
396C22—22no—
397C22—24no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—42no—
402C22—42no—
403C22#155814yes—
404C22#233911yescharacteristic
405C22C22×#10011yescharacteristic
406C22#288911yescharacteristic
407C22C2×#41912yes—
408C22#289712yes—
409C4—22no—
410C4—21no—
411C4—21no—
412C4—24no—
413C4—42no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—42no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4#331412yes—
423C4#331612yes—
424C4#334412yes—
425C4#334612yes—
426C2—24no—
427C2—42no—
428C2C2×#155811yescharacteristic
429C2#1776112yes—
430C2#1777011yescharacteristic
431C2#2247911yescharacteristic
432C2#2249712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9905C2semidirect
4. #9136C2semidirect
5. #9127C2semidirect
6. #8725C2semidirect
7. #8723C2semidirect
8. #8594C2semidirect
9. #8232C2semidirect
10. #8040C2semidirect
11. C2×#1296C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#227502
#99052
#91362
#91272
#87252
#87232
#85942
#82322
#80402
C2×#12962