Finite Group Databasebeta

← #208448

Subgroups — #208448

Order 512.

1098 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35082C211yesmaximal, characteristic
2#21914C211yesmaximal, characteristic, complemented
3#13750C211yesmaximal, characteristic, complemented
4#9910C212yesmaximal, complemented
5#9712C212yesmaximal
6#9149C212yesmaximal, complemented
7#9148C212yesmaximal, complemented
8#8218C212yesmaximal, complemented
9C4×#327C212yesmaximal, complemented
10#2856C2211yescharacteristic
11#2854C2212yes—
12#2851C2211yescharacteristic
13#2651—22no—
14#2513C412yescomplemented
15#2507C412yescomplemented
16#2454C412yescomplemented
17#2445C412yescomplemented
18#2443—22no—
19#2425—22no—
20#2335—22no—
21C4×#121C2212yes—
22C2×C4×C4⋊C4—22no—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#1917—22no—
27#1916—22no—
28#1817C2212yes—
29#1797C2212yes—
30#1749C2211yescharacteristic
31C4×C22⋊C8C2211yescharacteristic
32C2×#327C2212yes—
33C2×#327C2212yes—
34C2×#327C2212yes—
35C2×#327C2212yes—
36C4×#95—22no—
37C4×#95C2212yes—
38C4×#95C2212yes—
39#1294C2212yes—
40#1294C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43C2×#127C2×C411yescharacteristic
44C2×#125C2×C411yescharacteristic
45C2×#123—22no—
46C2×#123—22no—
47C2×#121—22no—
48C2×#121C2312yes—
49C2×C4×D4—22no—
50C2×C4×D4C2×C412yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D412yes—
53C22×C42—21no—
54C22×C42—21no—
55C22×C42—22no—
56C22×C42—22no—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—22no—
61C2×C22⋊C8C2312yes—
62C2×C22⋊C8C2312yes—
63C2×#97—21no—
64C2×C4×C8—21no—
65#399—22no—
66#398—22no—
67#398—22no—
68#397—22no—
69#397—22no—
70#396—22no—
71#394—22no—
72#394—22no—
73#390—22no—
74#390C2×C412yes—
75#390C2×C412yes—
76#390C2×C412yes—
77#390C2×C412yes—
78#389—22no—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2×C412yes—
82#389C2×C412yes—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#387—22no—
89#387—22no—
90#386—22no—
91#386—22no—
92#386—22no—
93#385—22no—
94#385—22no—
95#384C2×C411yescharacteristic
96#382C2×C411yescharacteristic
97#380—22no—
98#380C2312yes—
99C4×C4⋊C4—216no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C22⋊C4—22no—
107#376—24no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D412yes—
112C2×#95C2312yes—
113C2×#95C2312yes—
114#336—22no—
115#336—24no—
116#336—21no—
117#336—21no—
118#327—24no—
119#327—24no—
120#327—24no—
121#327—24no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4C22×C412yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135#127—81no—
136#125—81no—
137#123—48no—
138#123—82no—
139#123—82no—
140#121—22no—
141#121—22no—
142#121—22no—
143#121—22no—
144#121—82no—
145#121—82no—
146C4×D4—82no—
147C4×D4—82no—
148C4×D4—82no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
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⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C42—24no—
177C2×C42—28no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—28no—
181C2×C42—28no—
182C2×C42—24no—
183C2×C42—22no—
184C2×C42—81no—
185C2×C42—81no—
186C2×C42—82no—
187C2×C42—82no—
188C2×C42—82no—
189C2×C42—82no—
190C2×C42—82no—
191C2×C42C22×C412yes—
192C2×C42C22×C412yes—
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C4⋊C8—82no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204#97—81no—
205C4×C8—81no—
206#95—24no—
207#95—28no—
208#95—28no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95C22⋊C412yes—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222#95#4114yes—
223#95#4114yes—
224C24—22no—
225C24C22×D411yescharacteristic
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—44no—
237C2×D4—82no—
238C2×D4—82no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C2×#4112yes—
285C22×C4C2×#4112yes—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C4⋊C4—216no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4#10418yes—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C42—216no—
321C42—24no—
322C42—24no—
323C42—41no—
324C42—41no—
325C42—42no—
326C42—42no—
327C42—41no—
328C42—41no—
329C42—82no—
330C42—88no—
331C42—88no—
332C42—84no—
333C23—21no—
334C23—21no—
335C23—22no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23C22×C22⋊C411yescharacteristic
344C23#52511yescharacteristic
345C23#52511yescharacteristic
346C23#57612yes—
347C23#57712yes—
348D4—82no—
349D4—82no—
350D4—82no—
351D4—82no—
352D4—82no—
353D4—82no—
354D4—164no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—41no—
397C2×C4—44no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—164no—
409C2×C4C2×#10414yes—
410C2×C4C2×#10414yes—
411C2×C4C2×#10414yes—
412C2×C4C4×#4112yes—
413C2×C4#53212yes—
414C8—82no—
415C8—82no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22#229711yescharacteristic
432C22#288311yescharacteristic
433C22#288411yescharacteristic
434C22#289712yes—
435C22C22×#10412yes—
436C22#311912yes—
437C22#312211yescharacteristic
438C22#312611yescharacteristic
439C4—24no—
440C4—24no—
441C4—24no—
442C4—22no—
443C4—22no—
444C4—48no—
445C4—48no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—82no—
450C4—82no—
451C4—82no—
452C2—21no—
453C2—22no—
454C2—21no—
455C2—162no—
456C2#1754712yes—
457C2#1755012yes—
458C2#2139911yescharacteristic
459C2#2141511yescharacteristic
460C2#2142311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21914C2semidirect
3. #13750C2semidirect
4. #9910C2semidirect
6. #9149C2semidirect
7. #9148C2semidirect
8. #8218C2semidirect
9. C4×#327C2semidirect
14. #2513C4semidirect
15. #2507C4semidirect
16. #2454C4semidirect
17. #2445C4semidirect

Maximal subgroups

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

subgroupindex
#350822
#219142
#137502
#99102
#97122
#91492
#91482
#82182
C4×#3272