Finite Group Databasebeta

← #208454

Subgroups — #208454

Order 512.

930 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35082C211yesmaximal, characteristic
2#21914C211yesmaximal, characteristic, complemented
3#13781C211yesmaximal, characteristic, complemented
4#9913C212yesmaximal, complemented
5#9715C212yesmaximal
6#9157C212yesmaximal, complemented
7#9156C212yesmaximal, 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×#383C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1917—22no—
26#1916—22no—
27#1817C2212yes—
28#1797C2212yes—
29#1749C2211yescharacteristic
30C4×C22⋊C8C2211yescharacteristic
31C2×#327C2212yes—
32C2×#327C2212yes—
33C2×#327C2212yes—
34C2×#327C2212yes—
35#1436—22no—
36#1436—22no—
37#1434C2212yes—
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—22no—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×C22⋊C8C2312yes—
59C2×C22⋊C8C2312yes—
60C2×#97—21no—
61C2×C4×C8—21no—
62#399—22no—
63#398—22no—
64#398—22no—
65#397—22no—
66#397—22no—
67#396—22no—
68#394—22no—
69#394—22no—
70#390—22no—
71#390C2×C412yes—
72#390C2×C412yes—
73#390C2×C412yes—
74#390C2×C412yes—
75#389—22no—
76#389C2×C412yes—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#388—22no—
85#387—22no—
86#387—22no—
87#386—22no—
88#386—22no—
89#386—22no—
90#385—22no—
91#385—22no—
92#384C2×C411yescharacteristic
93#383—24no—
94#382—24no—
95#382C2×C411yescharacteristic
96#380—22no—
97#380C2312yes—
98C4×C4⋊C4C2×C412yes—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C22⋊C4—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D412yes—
111C2×#95C2312yes—
112C2×#95C2312yes—
113#336—22no—
114#336—24no—
115#336—21no—
116#336—21no—
117#327—24no—
118#327—24no—
119#327—24no—
120#327—24no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4C22×C412yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134#127—81no—
135#125—81no—
136#123—48no—
137#123—82no—
138#123—82no—
139#121—22no—
140#121—22no—
141#121—22no—
142#121—22no—
143#121—82no—
144#121—82no—
145C4×D4—82no—
146C4×D4—82no—
147C4×D4—82no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4C22⋊C412yes—
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×C22⋊C4—22no—
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×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—44no—
179C2×C42—81no—
180C2×C42—81no—
181C2×C42—82no—
182C2×C42—82no—
183C2×C42C22×C412yes—
184C2×C42C22×C412yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C2×D412yes—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D411yescharacteristic
191C4⋊C8—82no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196#97—81no—
197C4×C8—81no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—44no—
211#95—82no—
212#95—82no—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215#95C22⋊C412yes—
216#95C22⋊C412yes—
217#95#4114yes—
218#95#4114yes—
219C24—22no—
220C24C22×D411yescharacteristic
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—44no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—82no—
233C2×D4—82no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C2×#4112yes—
280C22×C4C2×#4112yes—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—84no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—44no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C42—24no—
316C42—24no—
317C42—41no—
318C42—41no—
319C42—42no—
320C42—42no—
321C42—41no—
322C42—41no—
323C42—82no—
324C42—88no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23C22×C22⋊C411yescharacteristic
336C23#52511yescharacteristic
337C23#52511yescharacteristic
338C23#57612yes—
339C23#57712yes—
340D4—82no—
341D4—82no—
342D4—82no—
343D4—82no—
344D4—82no—
345D4—82no—
346D4—164no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—164no—
403C2×C4#42114yes—
404C2×C4C4×#4112yes—
405C2×C4#53212yes—
406C8—82no—
407C8—82no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22#229711yescharacteristic
424C22#288311yescharacteristic
425C22#288411yescharacteristic
426C22#289712yes—
427C22C2×#42112yes—
428C22#311912yes—
429C22#312211yescharacteristic
430C22#312611yescharacteristic
431C4—22no—
432C4—22no—
433C4—24no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—42no—
439C4—84no—
440C4—84no—
441C4—82no—
442C4—82no—
443C4—82no—
444C2—21no—
445C2—22no—
446C2—21no—
447C2—162no—
448C2#1759712yes—
449C2#1760212yes—
450C2#2139911yescharacteristic
451C2#2141611yescharacteristic
452C2#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. #13781C2semidirect
4. #9913C2semidirect
6. #9157C2semidirect
7. #9156C2semidirect
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
#137812
#99132
#97152
#91572
#91562
#82182
C4×#3272