Finite Group Databasebeta

← #181736

Subgroups — #181736

Order 512.

1527 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#60277C211yesmaximal, characteristic
2#60125C211yesmaximal, characteristic
3C4⋊C4×#41C211yesmaximal, characteristic
4#8098C214yesmaximal, complemented
5#8095C212yesmaximal, complemented
6#8094C214yesmaximal, complemented
7C2×#1303C212yesmaximal, complemented
8C2×C4×#41—21no—
9C2×C4×#41C2211yescharacteristic
10#2989C418yescomplemented
11#2983—21no—
12#2975—21no—
13#2973—22no—
14#2965—24no—
15#2925—24no—
16#2925C418yescomplemented
17#2921—22no—
18#2920C2214yes—
19#2920C2214yes—
20C2×#431C2212yes—
21C2×#423C2212yes—
22#2879—21no—
23#2873—21no—
24#2351C412yes—
25Q8×C4⋊C4—21no—
26#2349C414yes—
27#2349C412yes—
28D4×C4⋊C4—22no—
29D4×C4⋊C4—21no—
30#2296—21no—
31#2296—22no—
32#2285C2212yes—
33#2285C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36#1303C2214yes—
37#1303C2214yes—
38#1303C2218yes—
39C22×#41D411yescharacteristic
40C22×M16—22no—
41C4×#41—48no—
42C4×#41C2×C414yes—
43C4×#41C2×C414yes—
44C4×#41D414yes—
45C4×#41Q814yes—
46C2×C4×Q8—22no—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×D4—24no—
50C2×C4×D4—22no—
51C2×C4×D4C2×C412yes—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C2×C4×D4D411yescharacteristic
55C2×#117—22no—
56C2×#117—21no—
57C2×#117D411yescharacteristic
58C2×#117D412yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C42—22no—
62C22×C42—21no—
63C22×C42C2311yescharacteristic
64C22×C42C2312yes—
65#446—28no—
66#436—28no—
67#435—216no—
68#433—216no—
69#432—24no—
70#432—24no—
71#432C2×C418yescomplemented
72#432C2×C418yescomplemented
73#431—22no—
74#431—22no—
75#431—24no—
76#431—24no—
77#431C2×C414yescomplemented
78#431C2×C418yescomplemented
79#431C2314yes—
80#423—22no—
81#423—22no—
82#423C2×C414yescomplemented
83#423C2×C418yescomplemented
84#423C2314yes—
85C2×C4⋊C8—22no—
86C2×C4⋊C8C2×C414yescomplemented
87C2×C4⋊C8C2×C414yescomplemented
88#408—28no—
89#407—24no—
90#407—24no—
91C2×C22⋊C8—22no—
92#405—28no—
93C4×M16—24no—
94C4×M16—24no—
95C2×C4×C8—22no—
96#389—24no—
97#389—22no—
98#389—24no—
99#389—22no—
100#385—24no—
101#385—24no—
102#385—24no—
103#384—21no—
104#384—22no—
105#382C2×C414yes—
106#382C2×C412yes—
107#380—21no—
108#380—22no—
109#380C2×C414yes—
110#380C2×C412yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C414yes—
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2312yes—
122C2×#95C2×C412yes—
123C2×#95C2×C414yes—
124C2×#41—24no—
125C2×#41—24no—
126C2×#41C22⋊C414yes—
127C2×#41C4⋊C414yes—
128C22×Q8C22⋊C411yescharacteristic
129C22×D4C22⋊C412yes—
130C22×D4C22⋊C411yescharacteristic
131C23×C4—24no—
132C23×C4—22no—
133C23×C4C22×C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C2×M16—24no—
141C2×M16—28no—
142C2×M16—44no—
143C2×M16—44no—
144C22×C8—24no—
145C22×C8—24no—
146C4×Q8—48no—
147C4×Q8C4214yes—
148C4×Q8C22⋊C414yes—
149C4×Q8C4⋊C414yes—
150C4×Q8C4⋊C414yes—
151C4×D4—416no—
152C4×D4—48no—
153C4×D4C4218yes—
154C4×D4C4214yes—
155C4×D4C22⋊C418yes—
156C4×D4C22⋊C414yes—
157C4×D4C4⋊C418yes—
158C4×D4C4⋊C414yes—
159C4×D4C4⋊C418yes—
160C4×D4C4⋊C414yes—
161#117—48no—
162#117—44no—
163#117C22⋊C414yes—
164#117C22⋊C418yes—
165#117C4⋊C418yes—
166#117C4⋊C414yes—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22⋊C412yes—
187C2×C42—24no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—48no—
193C2×C42—44no—
194C2×C42C22×C412yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C412yes—
197C2×C42C22×C412yes—
198C2×C42C22×C414yes—
199C2×C42C22×C414yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D414yes—
206C2×C42C2×Q814yes—
207C2×C42C2×Q812yes—
208C2×C42C2411yescharacteristic
209C4⋊C8—24no—
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8—28no—
213C4⋊C8—24no—
214C4⋊C8C22×C418yes—
215C4⋊C8C22×C418yes—
216C22⋊C8—24no—
217C22⋊C8—28no—
218C22⋊C8—28no—
219C22⋊C8—28no—
220C22⋊C8—24no—
221#97—28no—
222#97—28no—
223C4×C8—24no—
224C4×C8—28no—
225C4×C8—24no—
226#95—24no—
227#95—22no—
228#95—24no—
229#95—22no—
230#95C22×C412yes—
231#95C22×C412yes—
232#95C22×C414yes—
233C24C2×C22⋊C412yes—
234C24C2×C22⋊C411yescharacteristic
235#41—28no—
236#41—28no—
237#41—216no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8#9514yes—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4#9518yes—
246C2×D4#9514yes—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—48no—
256C22×C4—48no—
257C22×C4—44no—
258C22×C4—81no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—84no—
262C22×C4C2×C4214yes—
263C22×C4C2×C4212yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C414yes—
266C22×C4C2×C22⋊C414yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C4⋊C414yes—
275C22×C4C2×C4⋊C414yes—
276C22×C4C2×C4⋊C412yes—
277C22×C4C2×C4⋊C414yes—
278C22×C4C2×C4⋊C412yes—
279C22×C4C2×C4⋊C412yes—
280C22×C4AGL(1,ℤ/8)14yes—
281C22×C4#13714yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285M16—44no—
286M16—44no—
287M16—48no—
288M16—88no—
289C2×C8—28no—
290C2×C8—28no—
291C2×C8—48no—
292C2×C8—84no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—44no—
300C4⋊C4—84no—
301C4⋊C4—82no—
302C4⋊C4#9518yes—
303C4⋊C4#9514yes—
304C4⋊C4#9518yes—
305C4⋊C4#9514yes—
306C22⋊C4—48no—
307C22⋊C4—44no—
308C22⋊C4#9518yes—
309C22⋊C4#9518yes—
310C22⋊C4#9518yes—
311C42—48no—
312C42—42no—
313C42—44no—
314C42—42no—
315C42—44no—
316C42C2×C4214yes—
317C42C2×C4212yes—
318C42C2×C22⋊C412yes—
319C42C2×C22⋊C414yes—
320C42C2×C22⋊C414yes—
321C42C2×C22⋊C412yes—
322C42C2×C4⋊C412yes—
323C42C2×C4⋊C414yes—
324C42C2×C4⋊C414yes—
325C42C2×C4⋊C412yes—
326C42C2×C4⋊C414yes—
327C42C2×C4⋊C412yes—
328C42C23×C411yescharacteristic
329C42C23×C411yescharacteristic
330C42C22×D411yescharacteristic
331C42C22×Q811yescharacteristic
332C23—22no—
333C23—22no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—42no—
338C23C2×#9514yes—
339C23C2×#9512yes—
340C23#41812yes—
341C23#41912yes—
342C23C22×C22⋊C411yescharacteristic
343Q8—28no—
344D4—216no—
345D4—28no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—28no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—24no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—88no—
374C2×C4—816no—
375C2×C4C2×#9514yes—
376C2×C4C2×#9512yes—
377C2×C4C2×#9514yes—
378C2×C4C2×#9512yes—
379C2×C4C2×#9514yes—
380C2×C4C2×#9512yes—
381C2×C4#41714yes—
382C2×C4#41714yes—
383C2×C4#41812yes—
384C2×C4#41912yes—
385C2×C4#42818yes—
386C2×C4C22×C4211yescharacteristic
387C2×C4C22×C22⋊C411yescharacteristic
388C2×C4C22×C22⋊C411yescharacteristic
389C2×C4C22×C4⋊C411yescharacteristic
390C2×C4C22×C4⋊C411yescharacteristic
391C2×C4C22×C4⋊C411yescharacteristic
392C2×C4C2×AGL(1,ℤ/8)12yes—
393C2×C4C2×#13712yes—
394C8—48no—
395C8—48no—
396C22—21no—
397C22—22no—
398C22—21no—
399C22—22no—
400C22—24no—
401C22—24no—
402C22—24no—
403C22—42no—
404C22—42no—
405C22—44no—
406C22—44no—
407C22#173614yes—
408C22C22×#9511yescharacteristic
409C22C2×#41811yescharacteristic
410C22C2×#41911yescharacteristic
411C4—21no—
412C4—22no—
413C4—24no—
414C4—24no—
415C4—21no—
416C4—21no—
417C4—22no—
418C4—24no—
419C4—44no—
420C4—84no—
421C4—82no—
422C4#173614yes—
423C4C2×#41712yes—
424C4#290112yes—
425C4C2×#42812yes—
426C2—21no—
427C2—21no—
428C2—24no—
429C2—44no—
430C2#2034311yescharacteristic
431C2C2×#173611yescharacteristic
432C2#2036011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC22C22×#95
prime radicalC22×#41D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8098C2semidirect
5. #8095C2semidirect
6. #8094C2semidirect
7. C2×#1303C2semidirect
10. #2989C4semidirect
16. #2925C4semidirect
71. #432C2×C4semidirect
72. #432C2×C4semidirect
77. #431C2×C4semidirect
78. #431C2×C4semidirect
82. #423C2×C4semidirect
83. #423C2×C4semidirect
86. C2×C4⋊C8C2×C4semidirect
87. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602772
#601252
C4⋊C4×#412
#80982
#80952
#80942
C2×#13032