Finite Group Databasebeta

← #192228

Subgroups — #192228

Order 512.

917 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2#20717C211yesmaximal, characteristic
3#17803C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5#8863C214yesmaximal, complemented
6#8862C212yesmaximal, complemented
7#8861C212yesmaximal, complemented
8#8080C212yesmaximal, complemented
9C22×#117C2211yescharacteristic
10#2924—24no—
11#2924—24no—
12#2921—21no—
13#2921C2212yes—
14#2920—21no—
15#2920C2212yes—
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2212yes—
19#2905C2212yes—
20C2×#407—21no—
21#2873—21no—
22#1817—21no—
23#1798C2212yes—
24#1798C2212yes—
25#1779—21no—
26#1750—21no—
27#1750—21no—
28#1572—24no—
29#1571C2214yescomplemented
30#1570C2214yescomplemented
31#1564—24no—
32#1458C412yes—
33#1458C412yes—
34#1458C2211yescharacteristic
35#1458C2211yescharacteristic
36#1458C2212yes—
37C2×#324C412yes—
38C2×#323C412yes—
39#1301C2214yescomplemented
40#1301C2214yescomplemented
41C24×C4D411yescharacteristic
42C22×M16—22no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117D411yescharacteristic
50C2×#117D411yescharacteristic
51C2×#117Q811yescharacteristic
52C2×#117Q811yescharacteristic
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4Q811yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4Q811yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42C2311yescharacteristic
61#433—28no—
62#433—28no—
63#432—22no—
64#432—22no—
65#432—22no—
66#432—22no—
67#432—22no—
68#432D414yes—
69#431—22no—
70#431—22no—
71#431—22no—
72#431—22no—
73#431—22no—
74#431—22no—
75#431—22no—
76#431C2×C414yes—
77#431D414yes—
78#431C2312yes—
79#431C2312yes—
80#424—28no—
81#423—22no—
82#423C2314yes—
83C2×C4⋊C8—22no—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86#407—22no—
87#407—22no—
88#407—22no—
89#407—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8C2×C414yes—
97#405—28no—
98C4×M16—22no—
99C4×M16—22no—
100C2×#97—21no—
101C2×C4×C8—22no—
102C2×C4×C8—21no—
103#336—22no—
104#336—22no—
105#335—28no—
106#334—28no—
107#324C2×C412yes—
108#324C2×C412yes—
109#324C2×C414yes—
110#323C2×C412yes—
111#323C2×C412yes—
112#323C2×C414yes—
113C25C4⋊C411yescharacteristic
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—24no—
119C23×C4—42no—
120C23×C4C4⋊C411yescharacteristic
121C23×C4C4⋊C411yescharacteristic
122C23×C4C4⋊C411yescharacteristic
123C23×C4C4⋊C411yescharacteristic
124C23×C4C4⋊C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×Q811yescharacteristic
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—44no—
133C2×M16—44no—
134C22×C8—24no—
135C22×C8—24no—
136C22×C8—22no—
137C22×C8—22no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—22no—
149#117—22no—
150#117—22no—
151#117—22no—
152#117—22no—
153#117—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C4⋊C411yescharacteristic
159C2×C4⋊C4C4⋊C411yescharacteristic
160C2×C4⋊C4C4⋊C411yescharacteristic
161C2×C4⋊C4C4⋊C411yescharacteristic
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C2×Q811yescharacteristic
165C2×C4⋊C4C2×Q811yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4C4⋊C411yescharacteristic
171C2×C22⋊C4C4⋊C411yescharacteristic
172C2×C22⋊C4C4⋊C411yescharacteristic
173C2×C22⋊C4C4⋊C411yescharacteristic
174C2×C22⋊C4C22×C411yescharacteristic
175C2×C22⋊C4C22×C411yescharacteristic
176C2×C22⋊C4C2×Q811yescharacteristic
177C2×C22⋊C4C2×Q811yescharacteristic
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42C4⋊C411yescharacteristic
183C2×C42C4⋊C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×Q811yescharacteristic
191C2×C42#4112yes—
192C2×C42#4112yes—
193C2×C42C2411yescharacteristic
194C4⋊C8—24no—
195C4⋊C8—24no—
196C4⋊C8—24no—
197C4⋊C8—24no—
198C4⋊C8—24no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—24no—
202C4⋊C8C2×D418yes—
203C22⋊C8—24no—
204C22⋊C8—24no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208C22⋊C8—44no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213C22⋊C8—44no—
214C22⋊C8C22×C414yes—
215C22⋊C8C22×C414yes—
216#97—24no—
217#97—24no—
218#97—42no—
219C4×C8—24no—
220C4×C8—24no—
221C4×C8—24no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—42no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—41no—
229C24—42no—
230C24—81no—
231C24C2×C4⋊C411yescharacteristic
232C24C2×C4⋊C411yescharacteristic
233C24C2×C4⋊C411yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4C2×C4⋊C411yescharacteristic
268C22×C4C2×C4⋊C411yescharacteristic
269C22×C4C2×C4⋊C411yescharacteristic
270C22×C4C2×C4⋊C411yescharacteristic
271C22×C4C2×C4⋊C411yescharacteristic
272C22×C4C2×C4⋊C411yescharacteristic
273C22×C4C2×C4⋊C411yescharacteristic
274C22×C4C2×C4⋊C411yescharacteristic
275C22×C4C2×C4⋊C411yescharacteristic
276C22×C4C4×D412yes—
277C22×C4C4×D412yes—
278C22×C4#12011yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4C22⋊Q812yes—
281C22×C4C22⋊Q812yes—
282C22×C4#12311yescharacteristic
283C22×C4#12311yescharacteristic
284C22×C4#13514yes—
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×Q811yescharacteristic
288M16—48no—
289M16—44no—
290M16—44no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—28no—
294C2×C8—48no—
295C2×C8—48no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42C4×D412yes—
327C42C4×D412yes—
328C42#12012yes—
329C42C22⋊Q812yes—
330C42C22⋊Q812yes—
331C42#12312yes—
332C42C22×D412yes—
333C42C2×#4112yes—
334C23—22no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—81no—
352C23—81no—
353C23—84no—
354C23—84no—
355C23#38511yescharacteristic
356C23#38511yescharacteristic
357C23#42712yes—
358C23C22×C4⋊C411yescharacteristic
359C2×C4—24no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4#38511yescharacteristic
401C2×C4#38511yescharacteristic
402C2×C4#38512yes—
403C2×C4#38512yes—
404C2×C4#42712yes—
405C2×C4C2×C4×D412yes—
406C2×C4C2×#12011yescharacteristic
407C2×C4C2×C22⋊Q812yes—
408C2×C4C2×#12311yescharacteristic
409C2×C4C2×#13512yes—
410C2×C4#57512yes—
411C8—44no—
412C8—44no—
413C8—48no—
414C8—82no—
415C8—82no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—21no—
423C22—21no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—81no—
434C22—84no—
435C22—81no—
436C22—84no—
437C22#186812yes—
438C22C2×#38511yescharacteristic
439C22C2×#42711yescharacteristic
440C22#291111yescharacteristic
441C4—21no—
442C4—21no—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—41no—
449C4—41no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4#300212yes—
454C4#309112yes—
455C2—21no—
456C2—21no—
457C2—22no—
458C2—41no—
459C2—42no—
460C2—81no—
461C2#2084311yescharacteristic
462C2#2086011yescharacteristic
463C2C2×#186811yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17803C2semidirect
5. #8863C2semidirect
6. #8862C2semidirect
7. #8861C2semidirect
8. #8080C2semidirect
29. #1571C22semidirect
30. #1570C22semidirect
39. #1301C22semidirect
40. #1301C22semidirect

Maximal subgroups

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

subgroupindex
#601242
#207172
#178032
C2×#14582
#88632
#88622
#88612
#80802