Finite Group Databasebeta

← #188330

Subgroups — #188330

Order 512.

1138 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2343C211yesmaximal, characteristic
2#21953C212yesmaximal, complemented
3C2×#1464C211yesmaximal, characteristic
4C2×#1458C211yesmaximal, characteristic
5#8562C214yesmaximal, complemented
6#8559C212yesmaximal
7#8557C212yesmaximal
8#7916C212yesmaximal, complemented
9C22×#117C2211yescharacteristic
10C2×#432C2212yes—
11C2×#431C2212yes—
12#2343—24no—
13#2343—22no—
14#2343—22no—
15C2×#387C412yes—
16C2×#387C2212yes—
17C2×#385C412yes—
18C2×#383C2211yescharacteristic
19C2×#382C411yescharacteristic
20C2×#381C411yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23C2×#376C411yescharacteristic
24C22×#95C411yescharacteristic
25#1940—22no—
26#1940—22no—
27#1939—24no—
28#1797C2214yes—
29#1797C2214yes—
30#1464—24no—
31#1458—24no—
32C2×#324C2212yes—
33C2×#324C2212yes—
34C2×#323C2211yescharacteristic
35C2×#323C2212yes—
36C2×#323C2211yescharacteristic
37#1280C2214yes—
38#1280C2214yes—
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C412yes—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—24no—
44C2×#117D414yes—
45C2×#117Q814yes—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C22⋊C4C2×C412yes—
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2311yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C412yes—
55C22×C42C2312yes—
56C22×C42C2311yescharacteristic
57#432—24no—
58#432—24no—
59#431—24no—
60#431—24no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8—24no—
65C2×C22⋊C8C2314yes—
66C2×C22⋊C8C2312yes—
67C2×C22⋊C8C2312yes—
68C2×#97—22no—
69C2×C4×C8—22no—
70#387—24no—
71#387—24no—
72#387—24no—
73#387—24no—
74#385—24no—
75#385—24no—
76#383—22no—
77#383—22no—
78#382—22no—
79#382—22no—
80#381—22no—
81#381—22no—
82C4×C4⋊C4D414yes—
83C4×C4⋊C4D412yes—
84C4×C4⋊C4D412yes—
85C4×C22⋊C4—24no—
86C4×C22⋊C4—24no—
87#376—21no—
88#376—22no—
89#376—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C412yes—
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C412yes—
97C2×#95C2×C412yes—
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2312yes—
103#336—24no—
104#336—24no—
105#324—24no—
106#324—24no—
107#323—24no—
108#323—22no—
109#323—22no—
110C25C22×C411yescharacteristic
111C23×C4—24no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—44no—
121C23×C4—82no—
122C23×C4C4⋊C414yes—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—24no—
134C22×C8—22no—
135C22×C8—22no—
136#117—28no—
137#117—28no—
138#117—28no—
139#117—28no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—82no—
147C2×C4⋊C4C22⋊C414yes—
148C2×C4⋊C4C22⋊C414yes—
149C2×C4⋊C4C4⋊C414yes—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×Q812yes—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4C4⋊C414yes—
160C2×C22⋊C4C2×D412yes—
161C2×C22⋊C4C2×Q812yes—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—44no—
168C2×C42—82no—
169C2×C42C4⋊C414yes—
170C2×C42C2×D414yes—
171C2×C42C2×D414yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×Q812yes—
176C2×C42#4114yes—
177C4⋊C8—44no—
178C4⋊C8—44no—
179C4⋊C8—84no—
180C4⋊C8—84no—
181C22⋊C8—24no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—24no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187#97—44no—
188C4×C8—44no—
189#95—21no—
190#95—21no—
191#95—22no—
192#95—22no—
193#95—21no—
194#95—21no—
195#95—22no—
196#95—22no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95C22⋊C411yescharacteristic
202#95C22⋊C412yes—
203#95C22⋊C411yescharacteristic
204#95C22⋊C412yes—
205#95C22⋊C412yes—
206C24—21no—
207C24—21no—
208C24—22no—
209C24—42no—
210C24—41no—
211C24—41no—
212C24C2×C4⋊C412yes—
213C24#13114yes—
214C24C23×C411yescharacteristic
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—28no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—48no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—88no—
263C22×C4—88no—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C414yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C4⋊C412yes—
271C22×C4C2×C4⋊C412yes—
272C22×C4C2×C4⋊C412yes—
273C22×C4C2×C4⋊C412yes—
274C22×C4C2×C4⋊C412yes—
275C22×C4#11712yes—
276C22×C4#11712yes—
277C22×C4C22×D412yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×Q811yescharacteristic
280C22×C4C2×#4112yes—
281C2×C8—44no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—44no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—44no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—44no—
295C4⋊C4—84no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—28no—
299C22⋊C4—48no—
300C42—24no—
301C42—24no—
302C42—44no—
303C42—44no—
304C42—44no—
305C42—44no—
306C42#12114yes—
307C42#12114yes—
308C42C22⋊Q814yes—
309C42C22⋊Q814yes—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—24no—
314C23—22no—
315C23—24no—
316C23—22no—
317C23—44no—
318C23—42no—
319C23—42no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23—42no—
324C23—42no—
325C23—41no—
326C23—41no—
327C23—42no—
328C23#42414yes—
329C23C22×C22⋊C411yescharacteristic
330C23C22×C4⋊C411yescharacteristic
331C23C2×#11711yescharacteristic
332C23C2×#13112yes—
333C23#56812yes—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—88no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—88no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4#38012yes—
388C2×C4#38012yes—
389C2×C4#38014yes—
390C2×C4#41314yes—
391C2×C4C2×#12112yes—
392C2×C4C2×C22⋊Q812yes—
393C8—84no—
394C8—82no—
395C8—82no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—24no—
405C22—44no—
406C22—44no—
407C22—48no—
408C22—44no—
409C22—44no—
410C22—42no—
411C22—42no—
412C22C2×#38011yescharacteristic
413C22#288012yes—
414C22#288311yescharacteristic
415C22#288411yescharacteristic
416C22C2×#41312yes—
417C22C2×#42411yescharacteristic
418C22#290511yescharacteristic
419C22#292412yes—
420C4—24no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—24no—
426C4—42no—
427C4—44no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—82no—
432C4—82no—
433C2—21no—
434C2—21no—
435C2—22no—
436C2—44no—
437C2#1742212yes—
438C2#1742312yes—
439C2#2062811yescharacteristic
440C2#2063411yescharacteristic
441C2#2063711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21953C2semidirect
5. #8562C2semidirect
8. #7916C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23432
#219532
C2×#14642
C2×#14582
#85622
#85592
#85572
#79162