Finite Group Databasebeta

← #203192

Subgroups — #203192

Order 512.

764 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22008C211yesmaximal, characteristic, complemented
2#20555C211yesmaximal, characteristic, complemented
3C42×D8C211yesmaximal, characteristic
4#9541C212yesmaximal, complemented
5#9536C212yesmaximal, complemented
6#8947C212yesmaximal
7#8843C212yesmaximal, complemented
8C4×#335C211yesmaximal, characteristic, complemented
9#8170C211yesmaximal, characteristic, complemented
10#8170C211yesmaximal, characteristic, complemented
11#8133C211yesmaximal, characteristic, complemented
12C2×C4×D8C411yescharacteristic
13C2×C4×D8C411yescharacteristic
14C2×C4×D8C2211yescharacteristic
15C2×C8×D4C2212yes—
16C42×D4C2211yescharacteristic
17C42×D4C2211yescharacteristic
18#1954C2212yes—
19#1939C2212yes—
20#1923—22no—
21#1817—21no—
22#1797—21no—
23C4×#107C411yescharacteristic
24C4×#107C411yescharacteristic
25C4×#107C2211yescharacteristic
26C8×C4⋊C4C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C4×C4⋊C8C2211yescharacteristic
29C4×#102C411yescharacteristic
30C4×#102C411yescharacteristic
31C4×#102C411yescharacteristic
32C4×#102C411yescharacteristic
33C4×#102C2211yescharacteristic
34C4×#102C2211yescharacteristic
35C8×C22⋊C4—22no—
36C4×C22⋊C8—21no—
37#1744C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#336—21no—
40C42×C8C2211yescharacteristic
41C2×#335C2212yes—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44#1279C2212yes—
45#1279C2212yes—
46#1277C2212yes—
47#1273C2212yes—
48#1273C2212yes—
49C22×D8C2×C411yescharacteristic
50C23×C8—22no—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C22×C42—21no—
58C22×C42—21no—
59C4×D8—24no—
60C4×D8—24no—
61C4×D8C818yescomplemented
62C8×D4—24no—
63C8×D4—24no—
64C2×#107C2×C411yescharacteristic
65C2×#107C2×C411yescharacteristic
66C2×#107C2×C411yescharacteristic
67C2×C4⋊C8—22no—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×#102C2×C411yescharacteristic
71C2×#102C2×C411yescharacteristic
72C2×#102C2×C411yescharacteristic
73C2×#102C2×C411yescharacteristic
74C2×#102C2×C411yescharacteristic
75C2×#102C2×C411yescharacteristic
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8C2×C411yescharacteristic
83C2×C4×C8C2×C411yescharacteristic
84C2×C4×C8C2×C411yescharacteristic
85C2×C4×C8C2×C411yescharacteristic
86C2×C4×C8C2312yes—
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2312yes—
90C4×C4⋊C4C2×C411yescharacteristic
91C4×C4⋊C4C2×C411yescharacteristic
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C4⋊C4C2311yescharacteristic
95C4×C4⋊C4C2311yescharacteristic
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4—21no—
102C43C2311yescharacteristic
103#340C2×C411yescharacteristic
104#340C2×C411yescharacteristic
105#340C2×C411yescharacteristic
106#340C2×C411yescharacteristic
107#340C2×C411yescharacteristic
108#340C2×C411yescharacteristic
109#336—22no—
110#336—22no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336D411yescharacteristic
116#336D411yescharacteristic
117#336D412yes—
118#335—24no—
119D4⋊C8—24no—
120D4⋊C8—24no—
121C22×D4C22×C411yescharacteristic
122C22×D4C22×C411yescharacteristic
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C2×D8—22no—
130C2×D8—22no—
131C2×D8C2×C814yes—
132C22×C8—22no—
133C22×C8—28no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—82no—
137C22×C8—82no—
138C22×C8C22×C411yescharacteristic
139C22×C8C22×C411yescharacteristic
140C22×C8C22×C411yescharacteristic
141C22×C8C22×C411yescharacteristic
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4C2×C814yes—
153C4×D4C2×C814yescomplemented
154C4×D4M1614yes—
155C4×D4#4114yes—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C22×C411yescharacteristic
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C4⋊C4C22×C411yescharacteristic
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—82no—
175C2×C42—82no—
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42C2411yescharacteristic
183#107—22no—
184#107—22no—
185#107C2×C814yescomplemented
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—82no—
190C4⋊C8#4114yes—
191#102—22no—
192#102—22no—
193#102—22no—
194#102—22no—
195#102C2×C814yes—
196#102C2×C814yescomplemented
197C22⋊C8—28no—
198C22⋊C8—82no—
199C22⋊C8—82no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—81no—
210C4×C8—81no—
211C4×C8—82no—
212C4×C8C2×C814yescomplemented
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219C24—21no—
220C24—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4#11712yes—
230C2×D4C22×C812yes—
231C2×D4C22×C812yes—
232C2×D4C2×M1612yes—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4C4×D411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C2×#4111yescharacteristic
266D8—28no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—88no—
293C2×C8C22×C812yes—
294C2×C8C22×C812yes—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4#11712yes—
304C4⋊C4C22×C812yes—
305C4⋊C4C22×C812yes—
306C4⋊C4C2×M1612yes—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C42—21no—
314C42—22no—
315C42—21no—
316C42—41no—
317C42—41no—
318C42—42no—
319C42—42no—
320C42—81no—
321C42—81no—
322C42—81no—
323C42—81no—
324C42C22×C812yes—
325C42C2×M1612yes—
326C42#13112yes—
327C42#13112yes—
328C42C2×#4112yes—
329C42C2×#4112yes—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23C2×C4×D411yescharacteristic
345D4—22no—
346D4—22no—
347D4—22no—
348D4—22no—
349D4—22no—
350D4—41no—
351D4—41no—
352D4#43114yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4C8×D412yes—
390C2×C4C8×D412yes—
391C2×C4#43512yes—
392C2×C4#43512yes—
393C2×C4C2×#11711yescharacteristic
394C2×C4C4×#4111yescharacteristic
395C2×C4#52512yes—
396C2×C4#52912yes—
397C2×C4C23×C811yescharacteristic
398C2×C4C22×M1611yescharacteristic
399C2×C4C2×#13111yescharacteristic
400C2×C4C2×#13111yescharacteristic
401C2×C4#57611yescharacteristic
402C2×C4#57612yes—
403C2×C4#57811yescharacteristic
404C8—22no—
405C8—22no—
406C8—42no—
407C8—44no—
408C8—42no—
409C8—41no—
410C8—41no—
411C8—82no—
412C8—82no—
413C22—24no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22#229011yescharacteristic
428C22C2×C8×D411yescharacteristic
429C22C2×#43511yescharacteristic
430C22#294811yescharacteristic
431C22#295111yescharacteristic
432C22#295711yescharacteristic
433C22#295711yescharacteristic
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—81no—
444C4—81no—
445C4C2×#43111yescharacteristic
446C4#292211yescharacteristic
447C4C8×#4111yescharacteristic
448C4#296511yescharacteristic
449C4#312212yes—
450C4#312412yes—
451C2—44no—
452C2—82no—
453C2#1785411yescharacteristic
454C2#1785711yescharacteristic
455C2#1787711yescharacteristic
456C2#1788111yescharacteristic
457C2#2110111yescharacteristic
458C2#2112911yescharacteristic
459C2#2114011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22008C2semidirect
2. #20555C2semidirect
4. #9541C2semidirect
5. #9536C2semidirect
7. #8843C2semidirect
8. C4×#335C2semidirect
9. #8170C2semidirect
10. #8170C2semidirect
11. #8133C2semidirect
61. C4×D8C8semidirect
153. C4×D4C2×C8semidirect
185. #107C2×C8semidirect
196. #102C2×C8semidirect
212. C4×C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#220082
#205552
C42×D82
#95412
#95362
#89472
#88432
C4×#3352
#81702
#81332