Finite Group Databasebeta

← #205026

Subgroups — #205026

Order 512.

1156 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35180C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic, complemented
3C2×#1515C211yesmaximal, characteristic, complemented
4C2×#1514C211yesmaximal, characteristic
5C2×#1506C211yesmaximal, characteristic, complemented
6C2×#1505C211yesmaximal, characteristic
7#13806C211yesmaximal, characteristic, complemented
8#9716C214yesmaximal, complemented
9#9713C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14#2850C2214yes—
15#2639—21no—
16#2639—21no—
17#2592—22no—
18#2568C412yescomplemented
19#2563C412yescomplemented
20#2562—22no—
21#2548C412yescomplemented
22#2547C412yescomplemented
23#2311—22no—
24C2×#383C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1940—22no—
28#1940—22no—
29#1797C2214yes—
30#1515—24no—
31#1514—24no—
32#1506—24no—
33#1505—24no—
34C2×#328C2212yes—
35C2×#328C2212yes—
36C2×#328C2214yes—
37C2×#327C2212yes—
38C2×#327C2212yes—
39C2×#327C2214yes—
40#1436—22no—
41#1434—21no—
42#1434C2212yes—
43C4×#95—21no—
44C4×#95C2212yes—
45C22×#41—21no—
46#533—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50#532—22no—
51#532—22no—
52#532—22no—
53#532—22no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#127C2×C411yescharacteristic
58C2×#125—21no—
59C2×#124—21no—
60C2×#124C2×C414yes—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8C2312yes—
63C2×#121—21no—
64C2×#121C2312yes—
65C2×C4×Q8—21no—
66C2×C4×Q8C2×C412yes—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×#117—21no—
70C22×C4⋊C4D412yes—
71C22×C42—21no—
72C22×C42—22no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77#432—22no—
78#432—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—24no—
82C2×C22⋊C8C2312yes—
83C2×C22⋊C8C2312yes—
84C2×#97—21no—
85C2×#97—21no—
86#400—22no—
87#400—22no—
88#397—22no—
89#397—22no—
90#394—22no—
91#394—22no—
92#393—22no—
93#393—22no—
94#391—24no—
95#391C2×C414yes—
96#390—24no—
97#390C2×C414yes—
98#388—24no—
99#388—24no—
100#386—24no—
101#386—24no—
102#383—22no—
103#383D414yes—
104#382C2×C414yes—
105#380—24no—
106#380—24no—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4D412yes—
111#376—22no—
112#376D414yes—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121C2×#95C2311yescharacteristic
122C2×#95C2312yes—
123C2×#95C2311yescharacteristic
124#336—24no—
125#328—24no—
126#328—24no—
127#328—28no—
128#327—24no—
129#327—24no—
130#327—28no—
131C2×#41—44no—
132C2×#41—44no—
133C2×#41—42no—
134C2×#41—42no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8C22×C412yes—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4C22×C412yes—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159#127—22no—
160#127—22no—
161#125—42no—
162#125—42no—
163#124—42no—
164#124—42no—
165#124—84no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172#121—22no—
173#121—22no—
174#121—22no—
175#121—22no—
176#121—42no—
177#121—42no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—42no—
185C4×D4—42no—
186#117—42no—
187#117—42no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4C22⋊C414yes—
197C2×C4⋊C4C22⋊C414yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—44no—
210C2×C42—44no—
211C2×C42—44no—
212C2×C42—82no—
213C2×C42—82no—
214C2×C42C22×C414yes—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D414yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4112yes—
223C2×C42#4112yes—
224C4⋊C8—42no—
225C4⋊C8—42no—
226C4⋊C8—84no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231#97—42no—
232#97—42no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—21no—
241#95—21no—
242#95—22no—
243#95—44no—
244#95—44no—
245#95—82no—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C2×D412yes—
251#95C2×D412yes—
252#95C2×D414yes—
253C24—22no—
254C24C22×D411yescharacteristic
255#41—84no—
256#41—84no—
257#41—88no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—82no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—82no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—48no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4C2×C22⋊C414yes—
314C22×C4C2×C22⋊C414yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4#12112yes—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—44no—
353C22⋊C4—84no—
354C22⋊C4—84no—
355C22⋊C4—84no—
356C22⋊C4—84no—
357C42—44no—
358C42—44no—
359C42—48no—
360C42—44no—
361C42—88no—
362C42C4×D412yes—
363C42C4×D412yes—
364C42C4×D412yes—
365C42C4×D412yes—
366C42#12412yes—
367C42#12412yes—
368C42#12712yes—
369C42#12712yes—
370C23—21no—
371C23—21no—
372C23—22no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—44no—
377C23—42no—
378C23C22×C22⋊C411yescharacteristic
379C23C2×#12111yescharacteristic
380C23C2×#12111yescharacteristic
381C23#57512yes—
382C23#57512yes—
383Q8—82no—
384Q8—82no—
385Q8—84no—
386Q8—84no—
387D4—82no—
388D4—82no—
389D4—84no—
390D4—84no—
391D4—82no—
392D4—84no—
393D4—82no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—88no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4#39012yes—
447C2×C4#39012yes—
448C2×C4#39012yes—
449C2×C4#39012yes—
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×#12411yescharacteristic
453C2×C4C2×#12711yescharacteristic
454C8—82no—
455C8—82no—
456C22—22no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—22no—
462C22—21no—
463C22—21no—
464C22—82no—
465C22—84no—
466C22—82no—
467C22—84no—
468C22—82no—
469C22C2×#39011yescharacteristic
470C22#288311yescharacteristic
471C22#288411yescharacteristic
472C22#289612yes—
473C22#290112yes—
474C22#304011yescharacteristic
475C22#304012yes—
476C22#304011yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—22no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4—82no—
489C4—84no—
490C4—82no—
491C4—82no—
492C4—82no—
493C2—21no—
494C2—22no—
495C2—21no—
496C2—162no—
497C2#1762412yes—
498C2#1762912yes—
499C2#2121011yescharacteristic
500C2#2121411yescharacteristic
501C2#2126111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #21955C2semidirect
3. C2×#1515C2semidirect
5. C2×#1506C2semidirect
7. #13806C2semidirect
8. #9716C2semidirect
9. #9713C2semidirect
18. #2568C4semidirect
19. #2563C4semidirect
21. #2548C4semidirect
22. #2547C4semidirect

Maximal subgroups

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

subgroupindex
#351802
#219552
C2×#15152
C2×#15142
C2×#15062
C2×#15052
#138062
#97162
#97132