Finite Group Databasebeta

← #181626

Subgroups — #181626

Order 512.

1621 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2#34126C211yesmaximal, characteristic
3#34058C211yesmaximal, characteristic
4#8052C212yesmaximal, complemented
5#8051C214yesmaximal, complemented
6#8049C214yesmaximal, complemented
7#8048C212yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9#2975—21no—
10#2974—24no—
11#2973—22no—
12#2972—21no—
13D4×M16C418yescomplemented
14#2920C2212yes—
15C2×#407C2214yes—
16#2357C412yescomplemented
17#2353C414yescomplemented
18#2349C412yescomplemented
19#2345—21no—
20#2342—22no—
21D4×C22⋊C4—21no—
22#2319C418yescomplemented
23#2316—21no—
24#2312—22no—
25#2309—21no—
26#2293—24no—
27#2293—21no—
28#2290—22no—
29#2283C2211yescharacteristic
30#2280C2212yes—
31#2280C2214yes—
32C42×D4—21no—
33C4×#117C2211yescharacteristic
34C4×#117C2212yes—
35C2×C4×C22⋊C4C2212yes—
36#1297C2214yes—
37#1297C2214yes—
38#1297C2218yes—
39C23×D4D411yescharacteristic
40C24×C4C2312yes—
41C22×M16C2×C418yescomplemented
42C2×C4×D4—22no—
43C2×C4×D4C2×C414yes—
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4C2×C412yes—
46C2×C4×D4C2×C411yescharacteristic
47C2×C4×D4C2×C414yes—
48C2×C4×D4D411yescharacteristic
49C2×C4×D4D412yes—
50C2×C4×D4D414yes—
51C2×C4×D4D411yescharacteristic
52C2×C4×D4Q814yes—
53C2×#117—22no—
54C2×#117—21no—
55C2×#117—22no—
56C2×#117—24no—
57C22×C4⋊C4D411yescharacteristic
58C22×C22⋊C4—22no—
59C22×C22⋊C4D412yes—
60C22×C42—22no—
61C22×C42C2311yescharacteristic
62#436—28no—
63#435—216no—
64C8×D4—28no—
65#432—22no—
66#432—22no—
67#431—22no—
68#431—22no—
69#423C2×C414yescomplemented
70#407—24no—
71#407—24no—
72#407C2×C418yescomplemented
73#407C2314yes—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C4×M16C2×C414yescomplemented
77#390—28no—
78#390—22no—
79#389—28no—
80#389—22no—
81#388—216no—
82#388—24no—
83#388—24no—
84#387—24no—
85#386—24no—
86#385—24no—
87#383C2×C412yes—
88#382C2×C412yes—
89#381—21no—
90#381C2×C414yes—
91#381C2×C418yescomplemented
92#380—24no—
93#380—21no—
94#380—22no—
95#380C2×C414yes—
96#380C2×C418yescomplemented
97#379—24no—
98#379—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4C2×C414yescomplemented
105C4×C22⋊C4—22no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—24no—
111C4×C22⋊C4—24no—
112C4×C22⋊C4C2×C414yes—
113C4×C22⋊C4C2311yescharacteristic
114C4×C22⋊C4C2312yes—
115C4×C22⋊C4C2311yescharacteristic
116C4×C22⋊C4C2314yes—
117#376—21no—
118#376C2×C412yes—
119#376C2×C414yescomplemented
120C2×#95C2×C414yes—
121C43—21no—
122C43—22no—
123C25C2×D412yes—
124C22×D4—44no—
125C22×D4C22⋊C411yescharacteristic
126C22×D4C22⋊C411yescharacteristic
127C22×D4C22⋊C412yes—
128C22×D4C22⋊C414yes—
129C22×D4C4⋊C414yes—
130C23×C4—24no—
131C23×C4—84no—
132C23×C4C22×C414yes—
133C23×C4C22×C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C414yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D414yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×Q814yes—
143C23×C4C2411yescharacteristic
144C2×M16—28no—
145C2×M16—88no—
146C2×M16C22×C418yes—
147C22×C8—28no—
148C4×D4—48no—
149C4×D4C4214yes—
150C4×D4C4214yes—
151C4×D4C4218yes—
152C4×D4C22⋊C414yes—
153C4×D4C22⋊C418yes—
154C4×D4C22⋊C414yes—
155C4×D4C4⋊C414yes—
156C4×D4C4⋊C414yes—
157C4×D4C4⋊C414yes—
158C4×D4C4⋊C414yes—
159C4×D4C4⋊C418yes—
160C4×D4C4⋊C418yes—
161#117—44no—
162#117—48no—
163#117—84no—
164#117—88no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C414yes—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C4⋊C414yes—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—48no—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C22⋊C418yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C4⋊C418yes—
191C2×C42—22no—
192C2×C42—24no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—24no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—84no—
201C2×C42C22×C412yes—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×Q812yes—
209C4⋊C8—24no—
210C4⋊C8—24no—
211C22⋊C8—28no—
212C22⋊C8—28no—
213C22⋊C8—24no—
214C22⋊C8—24no—
215#97—24no—
216C4×C8—24no—
217#95—24no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—24no—
222#95—24no—
223#95—22no—
224#95—28no—
225#95C22×C412yes—
226#95C22×C414yes—
227#95C22×C412yes—
228#95C22×C418yes—
229C24—44no—
230C24—44no—
231C24—82no—
232C24C2×C22⋊C412yes—
233C24C2×C22⋊C414yes—
234C24C2×C22⋊C412yes—
235C24C2×C4⋊C414yes—
236C24C22×D411yescharacteristic
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—48no—
240C2×D4—48no—
241C2×D4—44no—
242C2×D4#9514yes—
243C2×D4#9514yes—
244C2×D4#9518yes—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—28no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—48no—
253C22×C4—48no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—48no—
257C22×C4—48no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—84no—
261C22×C4—82no—
262C22×C4—84no—
263C22×C4—84no—
264C22×C4C2×C4214yes—
265C22×C4C2×C4214yes—
266C22×C4C2×C22⋊C414yes—
267C22×C4C2×C22⋊C414yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C414yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C4⋊C412yes—
278C22×C4C2×C4⋊C414yes—
279C22×C4C2×C4⋊C414yes—
280C22×C4C2×C4⋊C414yes—
281C22×C4C2×C4⋊C414yes—
282C22×C4C2×C4⋊C412yes—
283C22×C4C2×C4⋊C414yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C23×C411yescharacteristic
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×Q811yescharacteristic
290M16—48no—
291M16—88no—
292C2×C8—28no—
293C2×C8—88no—
294C4⋊C4—48no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—84no—
298C4⋊C4—88no—
299C4⋊C4#9518yes—
300C4⋊C4#9514yes—
301C4⋊C4#9514yes—
302C22⋊C4—24no—
303C22⋊C4—48no—
304C22⋊C4—44no—
305C22⋊C4—48no—
306C22⋊C4—42no—
307C22⋊C4—48no—
308C22⋊C4—88no—
309C22⋊C4—84no—
310C22⋊C4—88no—
311C22⋊C4#9518yes—
312C22⋊C4#9518yes—
313C22⋊C4#9518yes—
314C22⋊C4#9518yes—
315C42—42no—
316C42—44no—
317C42—48no—
318C42—44no—
319C42—42no—
320C42—48no—
321C42—44no—
322C42—84no—
323C42—84no—
324C42C2×C4212yes—
325C42C2×C4212yes—
326C42C2×C22⋊C412yes—
327C42C2×C22⋊C412yes—
328C42C2×C4⋊C412yes—
329C42C2×C4⋊C412yes—
330C42C2×C4⋊C412yes—
331C42C2×C4⋊C412yes—
332C23—22no—
333C23—24no—
334C23—22no—
335C23—24no—
336C23—28no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—42no—
341C23—42no—
342C23—44no—
343C23—44no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—816no—
348C23C2×#9514yes—
349C23C2×#9514yes—
350C23C22×C22⋊C411yescharacteristic
351C23C22×C22⋊C411yescharacteristic
352C23C22×C4⋊C411yescharacteristic
353D4—28no—
354D4—28no—
355D4—48no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—28no—
361C2×C4—22no—
362C2×C4—28no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—48no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—816no—
394C2×C4C2×#9512yes—
395C2×C4C2×#9512yes—
396C2×C4C2×#9512yes—
397C2×C4C2×#9514yes—
398C2×C4C2×#9514yes—
399C2×C4C2×#9512yes—
400C2×C4#42114yes—
401C2×C4C22×C4211yescharacteristic
402C2×C4C22×C22⋊C411yescharacteristic
403C2×C4C22×C4⋊C411yescharacteristic
404C2×C4C22×C4⋊C411yescharacteristic
405C8—48no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—24no—
412C22—24no—
413C22—21no—
414C22—24no—
415C22—24no—
416C22—21no—
417C22—24no—
418C22—24no—
419C22—44no—
420C22—48no—
421C22—84no—
422C22—88no—
423C22—88no—
424C22C22×#9511yescharacteristic
425C22C2×#42112yes—
426C22#290112yes—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—24no—
434C4—24no—
435C4—44no—
436C4—82no—
437C4—88no—
438C4—84no—
439C4#173314yes—
440C4#288312yes—
441C2—22no—
442C2—22no—
443C2—44no—
444C2—41no—
445C2—44no—
446C2C2×#173311yescharacteristic
447C2#2034311yescharacteristic
448C2#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. #8049C2semidirect
7. #8048C2semidirect
13. D4×M16C4semidirect
16. #2357C4semidirect
17. #2353C4semidirect
18. #2349C4semidirect
22. #2319C4semidirect
41. C22×M16C2×C4semidirect
69. #423C2×C4semidirect
72. #407C2×C4semidirect
76. C4×M16C2×C4semidirect
91. #381C2×C4semidirect
96. #380C2×C4semidirect
104. C4×C4⋊C4C2×C4semidirect
119. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602302
#341262
#340582
#80522
#80512
#80492
#80482