Finite Group Databasebeta

← #190774

Subgroups — #190774

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34210C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20786C211yesmaximal, characteristic, complemented
4#8726C212yesmaximal, complemented
5#8592C212yesmaximal, complemented
6#8588C212yesmaximal, complemented
7#8339C212yesmaximal
8#8234C212yesmaximal, complemented
9#8231C212yesmaximal, complemented
10C22×#127C2211yescharacteristic
11C2×#518—21no—
12#2366—22no—
13#2366C414yes—
14D4×C22⋊C4—22no—
15D4×C22⋊C4C414yes—
16#2328—21no—
17#2328—21no—
18#2309—21no—
19C4×#127C2212yes—
20C4×#127C2212yes—
21#2285C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1924—22no—
24#1917—22no—
25#1842C2211yescharacteristic, complemented
26#1840—21no—
27#1831C2212yescomplemented
28#1831C2211yescharacteristic, complemented
29#1818—21no—
30#1797C2212yes—
31#1797C2212yes—
32#1767C2211yescharacteristic, complemented
33#1767C2212yescomplemented
34C4×C4⋊C8C2211yescharacteristic, complemented
35#1750—21no—
36#1746—21no—
37C2×#331C2212yes—
38C2×#331C2212yes—
39C2×#331C2212yes—
40C2×#331C2212yes—
41#1296C2212yescomplemented
42#1296C2212yescomplemented
43#1296C2212yescomplemented
44#1296C2212yescomplemented
45C23×D4—21no—
46C23×D4—21no—
47C23×D4C2×C412yes—
48C2×#127—24no—
49C2×#127—24no—
50C2×#127—21no—
51C2×#127—21no—
52C2×#127C2×C412yes—
53C2×#127C2312yes—
54#518—48no—
55C2×C4×D4—24no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4C2×C414yes—
61C2×C4×D4C2×C414yes—
62C2×#117—21no—
63C2×#117D411yescharacteristic
64C2×#117D411yescharacteristic
65C22×C22⋊C4—24no—
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—22no—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×#97—21no—
75C2×#97—22no—
76C2×C4×C8—21no—
77#390—24no—
78#390—22no—
79#390—22no—
80#390—22no—
81#390—22no—
82#390C2×C414yes—
83#390C2×C414yes—
84#386—24no—
85#386—24no—
86#386—24no—
87#386—24no—
88#384C2312yes—
89#382—21no—
90#380—21no—
91#380C2×C412yes—
92#379—24no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4D412yes—
97C4×C22⋊C4C2×C412yes—
98C4×C22⋊C4D412yes—
99#376D411yescharacteristic
100#376D411yescharacteristic
101C43C2311yescharacteristic
102C43C2311yescharacteristic
103#336—22no—
104#336—22no—
105#336—22no—
106#336—21no—
107#336—21no—
108#336—22no—
109#336—22no—
110#336—22no—
111#331—24no—
112#331—24no—
113#331—24no—
114#331—24no—
115C25—22no—
116C22×D4—28no—
117C22×D4—24no—
118C22×D4—24no—
119C22×D4—28no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—42no—
126C22×D4—42no—
127C22×D4—44no—
128C22×D4—82no—
129C22×D4—82no—
130C22×D4—84no—
131C22×D4C22×C414yes—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141#127—24no—
142#127—24no—
143#127—28no—
144#127—28no—
145#127—22no—
146#127—22no—
147#127#4114yes—
148#127#4114yes—
149C4×D4—44no—
150C4×D4—44no—
151C4×D4—44no—
152C4×D4—44no—
153C4×D4—416no—
154C4×D4—82no—
155C4×D4—82no—
156C4×D4—82no—
157C4×D4—82no—
158C4×D4—84no—
159C4×D4—84no—
160#117—24no—
161#117—41no—
162#117—41no—
163#117—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22×C414yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—416no—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22⋊C412yes—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C412yes—
184C2×C42C22×C412yes—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2411yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—82no—
203C22⋊C8—44no—
204C22⋊C8—82no—
205#97—42no—
206#97—82no—
207C4×C8—42no—
208#95—22no—
209#95—22no—
210#95C22⋊C412yes—
211#95C22⋊C412yes—
212#95C22⋊C412yes—
213#95C22⋊C412yes—
214C24—28no—
215C24—24no—
216C24—44no—
217C24—44no—
218C24—84no—
219C24—84no—
220C24C2×C22⋊C411yescharacteristic
221C2×D4—28no—
222C2×D4—28no—
223C2×D4—28no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—416no—
233C2×D4—416no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—44no—
243C2×D4—84no—
244C2×D4—84no—
245C2×D4—816no—
246C2×D4—88no—
247C2×D4—84no—
248C2×D4—84no—
249C2×D4#11718yes—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—42no—
260C22×C4—48no—
261C22×C4—48no—
262C22×C4—84no—
263C22×C4—84no—
264C22×C4—84no—
265C22×C4—84no—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4AGL(1,ℤ/8)14yes—
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C4⋊C4—24no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C22⋊C4—24no—
297C22⋊C4—42no—
298C22⋊C4—48no—
299C22⋊C4—48no—
300C22⋊C4—44no—
301C22⋊C4—84no—
302C22⋊C4—84no—
303C22⋊C4—84no—
304C22⋊C4—84no—
305C42—22no—
306C42—21no—
307C42—21no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42—42no—
321C42—44no—
322C42C2×#4112yes—
323C42C2×#4112yes—
324C23—22no—
325C23—21no—
326C23—22no—
327C23—21no—
328C23—28no—
329C23—416no—
330C23—44no—
331C23—44no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23—88no—
337C23—88no—
338C23—168no—
339C23—168no—
340C23#41812yes—
341C23C22×C22⋊C411yescharacteristic
342D4—48no—
343D4—416no—
344D4—48no—
345D4—44no—
346D4—44no—
347D4—44no—
348D4—44no—
349D4—44no—
350D4—44no—
351D4—44no—
352D4—44no—
353D4—88no—
354D4—88no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
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—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—168no—
384C2×C4—168no—
385C2×C4#41812yes—
386C2×C4C2×#11712yes—
387C2×C4#52512yes—
388C2×C4#52511yescharacteristic
389C2×C4#52511yescharacteristic
390C2×C4C2×AGL(1,ℤ/8)12yes—
391C2×C4#57511yescharacteristic
392C2×C4#57511yescharacteristic
393C2×C4#57612yes—
394C2×C4#57712yes—
395C8—82no—
396C8—82no—
397C8—82no—
398C8—82no—
399C22—22no—
400C22—24no—
401C22—41no—
402C22—42no—
403C22—41no—
404C22—48no—
405C22—48no—
406C22—48no—
407C22—84no—
408C22—84no—
409C22—84no—
410C22—84no—
411C22—1616no—
412C22#228011yescharacteristic
413C22#288812yes—
414C22C2×#41811yescharacteristic
415C22#289611yescharacteristic
416C22#289712yes—
417C4—22no—
418C4—22no—
419C4—41no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—41no—
425C4—84no—
426C4#311412yes—
427C4#311412yes—
428C4#311812yes—
429C4#312211yescharacteristic
430C4#312311yescharacteristic
431C2—42no—
432C2—88no—
433C2#1775812yes—
434C2#1776612yes—
435C2#2072911yescharacteristic
436C2#2073911yescharacteristic
437C2#2074411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21926C2semidirect
3. #20786C2semidirect
4. #8726C2semidirect
5. #8592C2semidirect
6. #8588C2semidirect
8. #8234C2semidirect
9. #8231C2semidirect
25. #1842C22semidirect
27. #1831C22semidirect
28. #1831C22semidirect
32. #1767C22semidirect
33. #1767C22semidirect
34. C4×C4⋊C8C22semidirect
41. #1296C22semidirect
42. #1296C22semidirect
43. #1296C22semidirect
44. #1296C22semidirect

Maximal subgroups

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

subgroupindex
#342102
#219262
#207862
#87262
#85922
#85882
#83392
#82342
#82312