Finite Group Databasebeta

← #188331

Subgroups — #188331

Order 512.

1084 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
1C2×#2347C211yesmaximal, characteristic
2#21953C212yesmaximal, complemented
3C2×#1464C211yesmaximal, characteristic
4C2×#1458C211yesmaximal, characteristic
5#8561C214yesmaximal
6#8560C212yesmaximal, complemented
7#8558C212yesmaximal, complemented
8#7916C212yesmaximal, complemented
9C22×#117C2211yescharacteristic
10C2×#432C2212yes—
11C2×#431C2212yes—
12#2347—24no—
13#2347—22no—
14#2347—22no—
15C2×#389C412yes—
16C2×#388C412yes—
17C2×#388C2212yes—
18C2×#383C411yescharacteristic
19C2×#383C2211yescharacteristic
20C2×#380C411yescharacteristic
21C2×C4×C4⋊C4C411yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C411yescharacteristic
25#1940—22no—
26#1940—22no—
27#1939—24no—
28#1797C2214yes—
29#1797C2214yes—
30#1464—24no—
31#1458—24no—
32C2×#324C2211yescharacteristic
33C2×#324C2212yes—
34C2×#324C2211yescharacteristic
35C2×#323C2212yes—
36C2×#323C2212yes—
37#1280C2214yes—
38#1280C2214yes—
39C24×C4C2×C411yescharacteristic
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—24no—
43C2×#117D414yes—
44C2×#117Q814yes—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2×C412yes—
53C22×C22⋊C4C2311yescharacteristic
54C22×C42C2×C412yes—
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C412yes—
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59#432—24no—
60#432—24no—
61#431—24no—
62#431—24no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—24no—
67C2×C22⋊C8C2314yes—
68C2×C22⋊C8C2312yes—
69C2×C22⋊C8C2312yes—
70C2×#97—22no—
71C2×C4×C8—22no—
72#389—24no—
73#389—24no—
74#388—24no—
75#388—24no—
76#388—24no—
77#388—24no—
78#383—22no—
79#383—22no—
80#383—22no—
81#383—22no—
82#380—22no—
83#380—22no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4D414yes—
90C4×C4⋊C4D412yes—
91C4×C4⋊C4D412yes—
92C4×C22⋊C4—21no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—21no—
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
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—22no—
107#324—22no—
108#323—24no—
109#323—24no—
110C25C22×C411yescharacteristic
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—42no—
116C23×C4C4⋊C414yes—
117C23×C4C22×C412yes—
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C412yes—
121C23×C4C22×C412yes—
122C23×C4C22×C412yes—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—24no—
128C22×C8—22no—
129C22×C8—22no—
130#117—28no—
131#117—28no—
132#117—28no—
133#117—28no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4C22⋊C414yes—
146C2×C4⋊C4C22⋊C414yes—
147C2×C4⋊C4C4⋊C414yes—
148C2×C4⋊C4C2×D412yes—
149C2×C4⋊C4C2×Q812yes—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—44no—
158C2×C22⋊C4—82no—
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—24no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—42no—
171C2×C42—82no—
172C2×C42—82no—
173C2×C42C4⋊C414yes—
174C2×C42C2×D414yes—
175C2×C42C2×D414yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×Q812yes—
180C2×C42#4114yes—
181C4⋊C8—44no—
182C4⋊C8—44no—
183C4⋊C8—84no—
184C4⋊C8—84no—
185C22⋊C8—24no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—24no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191#97—44no—
192C4×C8—44no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
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—22no—
210C24—42no—
211C24—41no—
212C24—41no—
213C24C2×C4⋊C412yes—
214C24C23×C411yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—41no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—48no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4C2×C22⋊C414yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C4⋊C412yes—
262C22×C4C2×C4⋊C412yes—
263C22×C4C2×C4⋊C412yes—
264C22×C4C2×C4⋊C412yes—
265C22×C4C2×C4⋊C412yes—
266C22×C4#11712yes—
267C22×C4#11712yes—
268C22×C4#13114yes—
269C22×C4C22×D412yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×Q811yescharacteristic
272C22×C4C2×#4112yes—
273C2×C8—44no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—44no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—44no—
280C2×C8—42no—
281C2×C8—42no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—28no—
285C4⋊C4—28no—
286C4⋊C4—44no—
287C4⋊C4—48no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—28no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—84no—
296C42—24no—
297C42—24no—
298C42—44no—
299C42—44no—
300C42—44no—
301C42—44no—
302C42—82no—
303C42—82no—
304C42#12114yes—
305C42#12114yes—
306C42C22⋊Q814yes—
307C42C22⋊Q814yes—
308C23—21no—
309C23—21no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23—82no—
324C23—82no—
325C23—82no—
326C23—82no—
327C23C22×C22⋊C411yescharacteristic
328C23C22×C4⋊C411yescharacteristic
329C23C2×#11711yescharacteristic
330C23C2×#13112yes—
331C23#56812yes—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4#38012yes—
383C2×C4#38012yes—
384C2×C4#38014yes—
385C2×C4#41314yes—
386C2×C4#42414yes—
387C2×C4C2×#12112yes—
388C2×C4C2×C22⋊Q812yes—
389C8—84no—
390C8—82no—
391C8—82no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—82no—
405C22—84no—
406C22—82no—
407C22—82no—
408C22C2×#38011yescharacteristic
409C22#288012yes—
410C22#288311yescharacteristic
411C22#288411yescharacteristic
412C22C2×#41312yes—
413C22C2×#42411yescharacteristic
414C22#290511yescharacteristic
415C22#292412yes—
416C4—24no—
417C4—22no—
418C4—22no—
419C4—24no—
420C4—22no—
421C4—22no—
422C4—24no—
423C4—42no—
424C4—44no—
425C4—42no—
426C4—42no—
427C4—82no—
428C4—82no—
429C2—21no—
430C2—21no—
431C2—22no—
432C2—82no—
433C2#1742712yes—
434C2#1742812yes—
435C2#2062811yescharacteristic
436C2#2063411yescharacteristic
437C2#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
6. #8560C2semidirect
7. #8558C2semidirect
8. #7916C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23472
#219532
C2×#14642
C2×#14582
#85612
#85602
#85582
#79162