Finite Group Databasebeta

← #190270

Subgroups — #190270

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#8765C214yesmaximal, complemented
5#8764C214yesmaximal, complemented
6#8098C212yesmaximal, complemented
7#7959C212yesmaximal, complemented
8C22×C4×D4—21no—
9C22×C4×D4C2211yescharacteristic
10#2920C2212yes—
11#2920C2212yes—
12#2349—22no—
13#2349—21no—
14#2349—21no—
15#2349C414yes—
16#2349C2212yes—
17#2349C2212yes—
18C2×#389—21no—
19C2×#385—22no—
20C2×#382C2211yescharacteristic
21C2×#380C412yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23C22×#95C412yes—
24#1821—21no—
25#1800C2212yes—
26#1800C2212yes—
27#1783—21no—
28C4×#100—21no—
29C4×#100—21no—
30#1466—22no—
31#1462—22no—
32#1460C2212yes—
33#1460C2212yes—
34#1303C2218yescomplemented
35#1286C2218yescomplemented
36C23×D4D411yescharacteristic
37C24×C4—22no—
38C24×C4C2×C412yes—
39C2×C4×D4—24no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—48no—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D411yescharacteristic
49C2×C4×D4D411yescharacteristic
50C2×C4×D4D412yes—
51C2×C4×D4C2311yescharacteristic
52C2×C4×D4C2311yescharacteristic
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4D411yescharacteristic
57C22×C22⋊C4—22no—
58C22×C22⋊C4D412yes—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#432—22no—
62#432—22no—
63#432—22no—
64#432—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68#431C2314yes—
69#423—22no—
70#423C2314yes—
71C2×#100—22no—
72C2×#100—22no—
73#407—22no—
74#407—22no—
75C4×M16—22no—
76C4×M16—22no—
77#389—24no—
78#389—22no—
79#389—22no—
80#385—24no—
81#385—24no—
82#385—24no—
83#385—24no—
84#382—21no—
85#382—21no—
86#382C2×C412yes—
87#382C2312yes—
88#380—22no—
89#380—22no—
90#380C2×C414yes—
91#380C2×C414yes—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4C2312yes—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95C2×C414yes—
101C2×#95C2×C414yes—
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C412yes—
104C2×#95C2×C411yescharacteristic
105#343—22no—
106#343—22no—
107#323—24no—
108#323—24no—
109C25C22⋊C412yes—
110C22×D4—24no—
111C22×D4—24no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4C22⋊C412yes—
115C22×D4C2×D411yescharacteristic
116C22×D4C2×D411yescharacteristic
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—48no—
127C23×C4—48no—
128C23×C4—84no—
129C23×C4—84no—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22×C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—24no—
139C2×M16—42no—
140C4×D4—24no—
141C4×D4—24no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—24no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—24no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—416no—
153C4×D4—48no—
154C4×D4—48no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—48no—
174C2×C22⋊C4C22⋊C412yes—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C2×D412yes—
177C2×C22⋊C4C2×D412yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—44no—
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42#4112yes—
185C2×C42#4112yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188C2×C42C2411yescharacteristic
189C4⋊C8—24no—
190C4⋊C8—24no—
191C4⋊C8—24no—
192C4⋊C8—42no—
193#100—44no—
194#100—44no—
195C22⋊C8—24no—
196C22⋊C8—42no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199#97—42no—
200#97—42no—
201C4×C8—24no—
202C4×C8—42no—
203#95—21no—
204#95—22no—
205#95—24no—
206#95—24no—
207#95—22no—
208#95—21no—
209#95—22no—
210#95C22×C412yes—
211#95C22×C414yes—
212#95C22×C412yes—
213C24—22no—
214C24—22no—
215C24—24no—
216C24—24no—
217C24—44no—
218C24—44no—
219C24C2×C22⋊C411yescharacteristic
220C24C2×C22⋊C412yes—
221C24C2×C22⋊C412yes—
222C2×D4—216no—
223C2×D4—216no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—24no—
231C2×D4AGL(1,ℤ/8)14yes—
232C2×D4#13714yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—48no—
262C22×C4—44no—
263C22×C4—48no—
264C22×C4—84no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—832no—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4#11711yescharacteristic
281C22×C4#11711yescharacteristic
282C22×C4#11711yescharacteristic
283C22×C4#11711yescharacteristic
284C22×C4#12312yes—
285C22×C4#12312yes—
286C22×C4#12312yes—
287C22×C4#12312yes—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291M16—44no—
292M16—44no—
293M16—84no—
294C2×C8—44no—
295C2×C8—44no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—48no—
310C22⋊C4—48no—
311C42—44no—
312C42—44no—
313C42#11712yes—
314C42#11712yes—
315C42#12312yes—
316C42#12312yes—
317C42#12312yes—
318C42#12312yes—
319C42C2×#4112yes—
320C42C2×#4112yes—
321C23—22no—
322C23—22no—
323C23—21no—
324C23—21no—
325C23—28no—
326C23—28no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—416no—
336C23—44no—
337C23—416no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23#38111yescharacteristic
342C23#38111yescharacteristic
343C23#41814yes—
344C23#41914yes—
345C23C22×C22⋊C411yescharacteristic
346D4—216no—
347D4—28no—
348D4—28no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—48no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—816no—
388C2×C4—816no—
389C2×C4—84no—
390C2×C4—88no—
391C2×C4#38112yes—
392C2×C4#38112yes—
393C2×C4#38111yescharacteristic
394C2×C4#38111yescharacteristic
395C2×C4C2×#11711yescharacteristic
396C2×C4C2×#11711yescharacteristic
397C2×C4C2×#12312yes—
398C2×C4C2×#12312yes—
399C2×C4C2×AGL(1,ℤ/8)12yes—
400C2×C4C2×#13712yes—
401C8—84no—
402C8—84no—
403C22—24no—
404C22—24no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—24no—
412C22—24no—
413C22—44no—
414C22—44no—
415C22—44no—
416C22—416no—
417C22—416no—
418C22—44no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22#182712yes—
424C22#182912yes—
425C22C2×#38111yescharacteristic
426C22C2×#41811yescharacteristic
427C22C2×#41911yescharacteristic
428C4—22no—
429C4—22no—
430C4—22no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—44no—
436C4—42no—
437C4—42no—
438C4—82no—
439C4—84no—
440C4#309214yes—
441C2—21no—
442C2—22no—
443C2—22no—
444C2—21no—
445C2—44no—
446C2—44no—
447C2#2070011yescharacteristic
448C2C2×#182711yescharacteristic
449C2C2×#182911yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8765C2semidirect
5. #8764C2semidirect
6. #8098C2semidirect
7. #7959C2semidirect
34. #1303C22semidirect
35. #1286C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
#207262
#173642
#87652
#87642
#80982
#79592