Finite Group Databasebeta

← #181623

Subgroups — #181623

Order 512.

1953 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2#34126C211yesmaximal, characteristic
3C4×#518C211yesmaximal, characteristic
4#8052C212yesmaximal, complemented
5#8051C214yesmaximal, complemented
6#8048C212yesmaximal, complemented
7C2×#1297C214yesmaximal, complemented
8C2×#518—21no—
9C22×C4×D4C2211yescharacteristic
10#2975—21no—
11#2974—24no—
12#2973—22no—
13#2972—21no—
14D4×M16C418yescomplemented
15#2920C2212yes—
16C2×#407C2214yes—
17#2357C412yescomplemented
18#2353C414yescomplemented
19#2349C412yescomplemented
20#2345—21no—
21#2342—22no—
22D4×C22⋊C4—21no—
23#2293—21no—
24#2290—22no—
25#2283C2211yescharacteristic
26#2280C2212yes—
27#2275C418yescomplemented
28#2274—21no—
29#2273—22no—
30C42×D4—21no—
31C42×D4—24no—
32C4×#117C2211yescharacteristic
33C4×#117C2212yes—
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2214yes—
36#1297C2214yes—
37#1297C2218yes—
38#1297C2214yes—
39C23×D4D411yescharacteristic
40C24×C4C2312yes—
41C22×M16C2×C418yescomplemented
42#518—216no—
43C2×C4×D4—28no—
44C2×C4×D4—22no—
45C2×C4×D4C2×C411yescharacteristic
46C2×C4×D4C2×C412yes—
47C2×C4×D4C2×C411yescharacteristic
48C2×C4×D4C2×C414yes—
49C2×C4×D4C2×C414yes—
50C2×C4×D4D411yescharacteristic
51C2×C4×D4D412yes—
52C2×C4×D4D414yes—
53C2×C4×D4D411yescharacteristic
54C2×C4×D4Q814yes—
55C2×#117—22no—
56C2×#117—21no—
57C2×#117—22no—
58C22×C4⋊C4D411yescharacteristic
59C22×C22⋊C4—24no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D412yes—
62C22×C42—24no—
63C22×C42—22no—
64C22×C42C2311yescharacteristic
65#436—28no—
66#435—216no—
67C8×D4—28no—
68#432—22no—
69#432—22no—
70#431—22no—
71#431—22no—
72#423C2×C414yescomplemented
73#407—24no—
74#407—24no—
75#407C2×C418yescomplemented
76#407C2314yes—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C4×M16C2×C414yescomplemented
80#390—22no—
81#389—22no—
82#388—24no—
83#388—24no—
84#387—24no—
85#386—24no—
86#385—24no—
87#383C2×C412yescomplemented
88#382C2×C412yescomplemented
89#381—21no—
90#381C2×C414yescomplemented
91#380—21no—
92#380—22no—
93#380C2×C414yescomplemented
94#379—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—28no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4C2×C412yescomplemented
101C4×C4⋊C4C2×C414yescomplemented
102C4×C22⋊C4—21no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—216no—
111C4×C22⋊C4C2×C414yescomplemented
112C4×C22⋊C4C2×C418yescomplemented
113C4×C22⋊C4C2311yescharacteristic
114C4×C22⋊C4C2314yes—
115C4×C22⋊C4C2312yes—
116C4×C22⋊C4C2311yescharacteristic
117#376—21no—
118#376C2×C412yescomplemented
119#376C2×C414yescomplemented
120C2×#95C2×C414yescomplemented
121C2×#95C2×C418yescomplemented
122C43—22no—
123C43—21no—
124C25C2×D412yes—
125C22×D4—28no—
126C22×D4C22⋊C414yes—
127C22×D4C22⋊C412yes—
128C22×D4C22⋊C411yescharacteristic
129C22×D4C22⋊C411yescharacteristic
130C22×D4C4⋊C414yes—
131C23×C4—28no—
132C23×C4—24no—
133C23×C4—84no—
134C23×C4C22×C414yes—
135C23×C4C22×C414yes—
136C23×C4C22×C412yes—
137C23×C4C22×C412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D414yes—
143C23×C4C2×D412yes—
144C23×C4C2×Q814yes—
145C23×C4C2411yescharacteristic
146C2×M16—28no—
147C2×M16—88no—
148C2×M16C22×C418yes—
149C22×C8—28no—
150C4×D4—264no—
151C4×D4—48no—
152C4×D4C4218yes—
153C4×D4C4214yes—
154C4×D4C4214yes—
155C4×D4C22⋊C414yes—
156C4×D4C22⋊C418yes—
157C4×D4C22⋊C414yes—
158C4×D4C4⋊C418yes—
159C4×D4C4⋊C414yes—
160C4×D4C4⋊C418yes—
161C4×D4C4⋊C414yes—
162C4×D4C4⋊C414yes—
163C4×D4C4⋊C414yes—
164#117—216no—
165#117—44no—
166#117—84no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4C22⋊C414yes—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C4⋊C414yes—
176C2×C22⋊C4—232no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—48no—
186C2×C22⋊C4C22⋊C418yes—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C4⋊C418yes—
192C2×C42—24no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—21no—
199C2×C42—24no—
200C2×C42—28no—
201C2×C42—24no—
202C2×C42—88no—
203C2×C42—84no—
204C2×C42C22×C412yes—
205C2×C42C22×C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×Q812yes—
212C4⋊C8—24no—
213C4⋊C8—24no—
214C22⋊C8—28no—
215C22⋊C8—28no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218#97—24no—
219C4×C8—24no—
220#95—28no—
221#95—24no—
222#95—22no—
223#95—22no—
224#95—24no—
225#95—22no—
226#95—24no—
227#95—22no—
228#95C22×C412yes—
229#95C22×C418yes—
230#95C22×C414yes—
231#95C22×C412yes—
232C24—28no—
233C24—44no—
234C24—82no—
235C24C2×C22⋊C414yes—
236C24C2×C22⋊C412yes—
237C24C2×C22⋊C412yes—
238C24C2×C4⋊C414yes—
239C24C22×D411yescharacteristic
240C2×D4—28no—
241C2×D4—216no—
242C2×D4—216no—
243C2×D4—24no—
244C2×D4#9518yes—
245C2×D4#9514yes—
246C2×D4#9514yes—
247C2×D4#10418yes—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—22no—
251C22×C4—232no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—44no—
255C22×C4—48no—
256C22×C4—44no—
257C22×C4—48no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—48no—
261C22×C4—48no—
262C22×C4—84no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—84no—
267C22×C4—88no—
268C22×C4—88no—
269C22×C4—84no—
270C22×C4C2×C4214yes—
271C22×C4C2×C4214yes—
272C22×C4C2×C22⋊C414yes—
273C22×C4C2×C22⋊C414yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C414yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C4⋊C412yes—
284C22×C4C2×C4⋊C414yes—
285C22×C4C2×C4⋊C414yes—
286C22×C4C2×C4⋊C414yes—
287C22×C4C2×C4⋊C414yes—
288C22×C4C2×C4⋊C412yes—
289C22×C4C2×C4⋊C414yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C23×C411yescharacteristic
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×Q811yescharacteristic
296M16—48no—
297M16—88no—
298C2×C8—28no—
299C2×C8—88no—
300C4⋊C4—216no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—84no—
304C4⋊C4#9518yes—
305C4⋊C4#9514yes—
306C4⋊C4#9514yes—
307C22⋊C4—216no—
308C22⋊C4—232no—
309C22⋊C4—24no—
310C22⋊C4—48no—
311C22⋊C4—44no—
312C22⋊C4—42no—
313C22⋊C4—48no—
314C22⋊C4—84no—
315C22⋊C4#9518yes—
316C22⋊C4#9518yes—
317C22⋊C4#9518yes—
318C22⋊C4#9518yes—
319C42—216no—
320C42—48no—
321C42—42no—
322C42—44no—
323C42—44no—
324C42—42no—
325C42—44no—
326C42—84no—
327C42—88no—
328C42—84no—
329C42C2×C4212yes—
330C42C2×C4212yes—
331C42C2×C22⋊C412yes—
332C42C2×C22⋊C412yes—
333C42C2×C4⋊C412yes—
334C42C2×C4⋊C412yes—
335C42C2×C4⋊C412yes—
336C42C2×C4⋊C412yes—
337C23—22no—
338C23—22no—
339C23—28no—
340C23—28no—
341C23—24no—
342C23—28no—
343C23—42no—
344C23—42no—
345C23—44no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23—44no—
350C23—44no—
351C23—816no—
352C23C2×#9514yes—
353C23C2×#9514yes—
354C23C2×#10418yes—
355C23C22×C22⋊C411yescharacteristic
356C23C22×C22⋊C411yescharacteristic
357C23C22×C4⋊C411yescharacteristic
358D4—216no—
359D4—28no—
360D4#339116yes—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—28no—
366C2×C4—22no—
367C2×C4—28no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—216no—
371C2×C4—24no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—48no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—48no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—832no—
399C2×C4—816no—
400C2×C4C2×#9512yes—
401C2×C4C2×#9512yes—
402C2×C4C2×#9512yes—
403C2×C4C2×#9514yes—
404C2×C4C2×#9514yes—
405C2×C4C2×#9512yes—
406C2×C4C2×#10414yes—
407C2×C4C22×C4211yescharacteristic
408C2×C4C22×C22⋊C411yescharacteristic
409C2×C4C22×C4⋊C411yescharacteristic
410C2×C4C22×C4⋊C411yescharacteristic
411C8—48no—
412C22—21no—
413C22—21no—
414C22—24no—
415C22—21no—
416C22—21no—
417C22—24no—
418C22—21no—
419C22—24no—
420C22—21no—
421C22—24no—
422C22—21no—
423C22—24no—
424C22—48no—
425C22—416no—
426C22—44no—
427C22—84no—
428C22—88no—
429C22C2×#33918yes—
430C22C22×#9511yescharacteristic
431C22C22×#10412yes—
432C22#290112yes—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—24no—
440C4—24no—
441C4—416no—
442C4—44no—
443C4—82no—
444C4—84no—
445C4C2×#33914yes—
446C4#288312yes—
447C2—22no—
448C2—22no—
449C2—28no—
450C2—44no—
451C2—41no—
452C2C22×#33911yescharacteristic
453C2#2034311yescharacteristic
454C2#2035411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC22C22×#95
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8052C2semidirect
5. #8051C2semidirect
6. #8048C2semidirect
7. C2×#1297C2semidirect
14. D4×M16C4semidirect
17. #2357C4semidirect
18. #2353C4semidirect
19. #2349C4semidirect
27. #2275C4semidirect
41. C22×M16C2×C4semidirect
72. #423C2×C4semidirect
75. #407C2×C4semidirect
79. C4×M16C2×C4semidirect
87. #383C2×C4semidirect
88. #382C2×C4semidirect
90. #381C2×C4semidirect
93. #380C2×C4semidirect
100. C4×C4⋊C4C2×C4semidirect
101. C4×C4⋊C4C2×C4semidirect
111. C4×C22⋊C4C2×C4semidirect
112. C4×C22⋊C4C2×C4semidirect
118. #376C2×C4semidirect
119. #376C2×C4semidirect
120. C2×#95C2×C4semidirect
121. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602302
#341262
C4×#5182
#80522
#80512
#80482
C2×#12972