Finite Group Databasebeta

← #197982

Subgroups — #197982

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3246C211yesmaximal, characteristic
2C2×#2979C211yesmaximal, characteristic
3#21869C211yesmaximal, characteristic
4#21835C211yesmaximal, characteristic, complemented
5C2×#1554C211yesmaximal, characteristic, complemented
6#9225C212yesmaximal, complemented
7#9223C212yesmaximal, complemented
8#9217C212yesmaximal, complemented
9#9215C212yesmaximal, complemented
10#8039C211yesmaximal, characteristic, complemented
11#7879C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3246—24no—
15#3246C418yescomplemented
16C2×#475—21no—
17C2×#476C2212yes—
18C2×#465—22no—
19C2×#460—21no—
20C2×#461C2212yes—
21#2979—24no—
22#2979—24no—
23C2×#436—21no—
24C2×#436C2212yes—
25C2×C8×D4—21no—
26C2×C8×D4C2212yes—
27C2×#432C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C2×#408—22no—
32#2285C2211yescharacteristic
33#1993—21no—
34#1990C2212yes—
35#1990C2212yes—
36#1989—21no—
37#1989—21no—
38#1961—21no—
39#1875—22no—
40#1825—22no—
41#1802C2212yes—
42#1802C2212yes—
43#1554—24no—
44C2×#333C2211yescharacteristic
45C2×#333C2211yescharacteristic
46C2×#332C2211yescharacteristic
47C2×#332C2211yescharacteristic
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51#1277C2212yes—
52C22×#41—21no—
53C22×SD16—22no—
54C22×M16—22no—
55C23×C8—21no—
56C23×C8—22no—
57#533—22no—
58#533—22no—
59#533—22no—
60#533C2×C414yes—
61#532—22no—
62#532—22no—
63#532C2×C414yes—
64#532D414yes—
65C2×C4⋊Q8D411yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127D411yescharacteristic
69C2×#125—21no—
70C2×#124—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×#121—21no—
74C2×#121D412yes—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2312yes—
79C2×#117—21no—
80C2×#117—21no—
81C22×C42C2311yescharacteristic
82#476—24no—
83#476C2×C418yes—
84#475—28no—
85#465—216no—
86#461—24no—
87#461C2×C418yes—
88#460—28no—
89#436—28no—
90#436—24no—
91#436—24no—
92C8×D4—28no—
93C8×D4—24no—
94C8×D4—24no—
95#432—22no—
96#432—22no—
97#431—22no—
98#431—22no—
99#431—22no—
100#431D414yes—
101C2×#106—22no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8C2×C414yes—
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×#103—22no—
113C2×#102—22no—
114C2×#102—22no—
115C2×#102—22no—
116#408—216no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—22no—
119C2×C22⋊C8—22no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8D412yes—
122C2×#97—21no—
123C2×C4×C8—21no—
124C2×C4×C8C2312yes—
125#384C2312yes—
126#382—21no—
127#380—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#333—22no—
138#333—22no—
139#332—22no—
140#332—22no—
141C2×#41—24no—
142C2×#41—24no—
143C2×#41—82no—
144C2×#41—82no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C2×D412yes—
152C23×C4—22no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4#4111yescharacteristic
156C2×SD16—28no—
157C2×SD16—82no—
158C2×SD16—82no—
159C2×M16—28no—
160C2×M16—84no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—28no—
165C22×C8—24no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—84no—
169C22×C8—81no—
170C22×C8—81no—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C4⋊Q8—22no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8C22⋊C414yes—
181C4⋊Q8C22×C412yes—
182C4⋊Q8C22×C412yes—
183#127—22no—
184#127C22⋊C414yes—
185#125—24no—
186#125—42no—
187#124—24no—
188#124—24no—
189C22⋊Q8—24no—
190C22⋊Q8—24no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193#121—24no—
194#121—24no—
195#121—24no—
196#121C22⋊C418yes—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4C22×C414yes—
206C4×D4C2×D414yes—
207#117—24no—
208#117—42no—
209#117—81no—
210#117—81no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4C2×D412yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42C22×C412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42#4111yescharacteristic
234C2×C42#4112yes—
235C2×C42C2411yescharacteristic
236#106—28no—
237C4⋊C8—24no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8C22×C414yes—
248C4⋊C8C2×D412yes—
249C4⋊C8C2×D412yes—
250#103—28no—
251#102—28no—
252#102—82no—
253#102—82no—
254C22⋊C8—28no—
255C22⋊C8—28no—
256C22⋊C8—24no—
257C22⋊C8—24no—
258C22⋊C8—42no—
259#97—42no—
260C4×C8—24no—
261C4×C8—22no—
262C4×C8—22no—
263#95—41no—
264#95—41no—
265C24—22no—
266C24#12111yescharacteristic
267#41—88no—
268#41—84no—
269#41—84no—
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×Q8—82no—
281C2×D4—28no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—44no—
290C2×D4—82no—
291C2×D4—84no—
292C2×D4C2×C22⋊C414yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4#12012yes—
312C22×C4#12112yes—
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12312yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320SD16—88no—
321M16—88no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—84no—
340C2×C8—88no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—82no—
358C4⋊C4C2×C22⋊C414yes—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—28no—
362C22⋊C4—44no—
363C22⋊C4—82no—
364C42—44no—
365C42—82no—
366C42C2×C22⋊C412yes—
367C42C4×D412yes—
368C42C4×D412yes—
369C42#12012yes—
370C42#12112yes—
371C42#12312yes—
372C42C23×C411yescharacteristic
373C42C22×D411yescharacteristic
374C42C22×D411yescharacteristic
375C42C2×#4111yescharacteristic
376C23—22no—
377C23—21no—
378C23—21no—
379C23—22no—
380C23—24no—
381C23—44no—
382C23—42no—
383C23—42no—
384C23—44no—
385C23#38612yes—
386C23C2×#12111yescharacteristic
387Q8—42no—
388Q8—42no—
389Q8—84no—
390Q8—82no—
391Q8—82no—
392Q8—81no—
393Q8—81no—
394D4—24no—
395D4—24no—
396D4—48no—
397D4—42no—
398D4—42no—
399D4—84no—
400D4—88no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4#38612yes—
429C2×C4#38612yes—
430C2×C4#38612yes—
431C2×C4C22×C22⋊C411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×C4×D411yescharacteristic
434C2×C4C2×#12011yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C2×C4C2×#12311yescharacteristic
437C2×C4#57611yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57712yes—
441C2×C4#57712yes—
442C2×C4#57811yescharacteristic
443C8—44no—
444C8—42no—
445C8—42no—
446C8—42no—
447C8—42no—
448C8—81no—
449C8—81no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—42no—
454C22—42no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22—84no—
459C22—84no—
460C22C2×#38611yescharacteristic
461C22#304212yes—
462C22#306211yescharacteristic
463C22#306311yescharacteristic
464C22#306712yes—
465C4—21no—
466C4—21no—
467C4—22no—
468C4—22no—
469C4—22no—
470C4—44no—
471C4—82no—
472C4—82no—
473C4—162no—
474C4#289711yescharacteristic
475C4#289811yescharacteristic
476C4#294912yes—
477C4#300911yescharacteristic
478C4#300911yescharacteristic
479C4#306011yescharacteristic
480C4#306111yescharacteristic
481C2—42no—
482C2—84no—
483C2#1956212yes—
484C2#1956312yes—
485C2#2093311yescharacteristic
486C2#2097611yescharacteristic
487C2#2100311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21835C2semidirect
5. C2×#1554C2semidirect
6. #9225C2semidirect
7. #9223C2semidirect
8. #9217C2semidirect
9. #9215C2semidirect
10. #8039C2semidirect
11. #7879C2semidirect
15. #3246C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32462
C2×#29792
#218692
#218352
C2×#15542
#92252
#92232
#92172
#92152
#80392
#78792