Finite Group Databasebeta

← #201258

Subgroups — #201258

Order 512.

1169 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3C2×#1556C212yesmaximal, complemented
4#17363C211yesmaximal, characteristic
5#9456C214yesmaximal, complemented
6#9455C214yesmaximal, complemented
7#7952C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2990—24no—
11#2983C2212yes—
12#2973—21no—
13#2973C2212yes—
14#2972—21no—
15#2970—24no—
16#2920—21no—
17C2×#432—21no—
18C2×#432C2212yes—
19C2×#432C2212yes—
20#2857—21no—
21#2857—21no—
22#2628—21no—
23#2628—21no—
24#2568C2212yes—
25#2567C412yes—
26#2563C412yes—
27#2542C2212yes—
28#2487—21no—
29#2487—21no—
30#2409—21no—
31#2409—21no—
32#2317C412yes—
33#2311C412yes—
34#2285C2211yescharacteristic
35#1556—24no—
36#1556C2218yes—
37C2×#332—24no—
38#1467C2212yes—
39#1466C2212yes—
40#1464—21no—
41#1464—21no—
42#1459—21no—
43#1459—21no—
44#1285C2218yes—
45C22×#41C2×C411yescharacteristic
46C23×Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C412yes—
54C2×C4⋊Q8C2312yes—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C412yes—
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C412yes—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78#446—28no—
79#435—28no—
80#435—28no—
81C8×D4—28no—
82#432—22no—
83#432—22no—
84#432—22no—
85#432—22no—
86#432—22no—
87#432—22no—
88#432—22no—
89#432—22no—
90#432—22no—
91#432D414yes—
92#432C2314yes—
93#432C2314yes—
94#431—22no—
95#431—22no—
96#423—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8D414yes—
100#407—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C4×M16—22no—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97—22no—
108#400—22no—
109#400—22no—
110#399—22no—
111#399—22no—
112#395—22no—
113#395—22no—
114#393—22no—
115#393—22no—
116#391—22no—
117#391—22no—
118#391C2×C414yes—
119#390C2×C414yes—
120#389—22no—
121#389—22no—
122#389—22no—
123#389—22no—
124#386—22no—
125#386—22no—
126#385—22no—
127#385—22no—
128#384—21no—
129#384C2×C414yes—
130#382—21no—
131#382C2×C414yes—
132#380—22no—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—21no—
137#380—21no—
138C4×C4⋊C4C2×C412yes—
139C4×C4⋊C4C2312yes—
140#332—216no—
141#324—22no—
142#324—22no—
143#323—22no—
144#323—22no—
145C2×#41—24no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—42no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—84no—
154C22×Q8C22×C411yescharacteristic
155C22×D4C22⋊C411yescharacteristic
156C22×D4C22⋊C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C23×C4—22no—
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C2×M16—24no—
169C22×C8—24no—
170C22×C8—24no—
171C22×C8—42no—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—24no—
176C4⋊Q8—41no—
177C4⋊Q8—41no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182C22⋊Q8—84no—
183C22⋊Q8—84no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200#117—22no—
201#117—22no—
202#117—22no—
203#117—22no—
204#117—22no—
205#117—22no—
206#117—44no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22×C414yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4C22⋊C411yescharacteristic
230C2×C22⋊C4C22⋊C411yescharacteristic
231C2×C22⋊C4C22⋊C411yescharacteristic
232C2×C22⋊C4C22⋊C411yescharacteristic
233C2×C22⋊C4C2×D411yescharacteristic
234C2×C22⋊C4C2×D411yescharacteristic
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22×C414yes—
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42#4112yes—
256C2×C42#4112yes—
257C2×C42C2411yescharacteristic
258C4⋊C8—24no—
259C4⋊C8—24no—
260C4⋊C8—24no—
261C4⋊C8—24no—
262C4⋊C8—42no—
263C4⋊C8C2×D418yes—
264C22⋊C8—24no—
265C22⋊C8—24no—
266C22⋊C8—24no—
267C22⋊C8—24no—
268C22⋊C8—24no—
269C22⋊C8—42no—
270#97—24no—
271#97—24no—
272#97—24no—
273#97—24no—
274#97—42no—
275C4×C8—24no—
276#95—22no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—22no—
282C24C2×C22⋊C411yescharacteristic
283C24C2×C22⋊C411yescharacteristic
284C24C2×C22⋊C411yescharacteristic
285#41—48no—
286#41—48no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—44no—
295C2×Q8—88no—
296C2×Q8—84no—
297C2×Q8—88no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—41no—
306C2×D4#10114yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—84no—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C4×D412yes—
345C22×C4C4×D412yes—
346C22×C4#12012yes—
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4#12312yes—
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353M16—84no—
354C2×C8—28no—
355C2×C8—44no—
356C2×C8—84no—
357C2×C8—84no—
358C2×C8—84no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—28no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—44no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—44no—
398C42C4×D412yes—
399C42C4×D412yes—
400C42#12011yescharacteristic
401C42#12011yescharacteristic
402C42#12112yes—
403C42#12112yes—
404C42#12311yescharacteristic
405C42#12311yescharacteristic
406C42C22×D412yes—
407C42C2×#4112yes—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23#38612yes—
420C23C2×#10112yes—
421C23C2×#10112yes—
422C23#41312yes—
423C23C22×C22⋊C411yescharacteristic
424Q8—44no—
425Q8—84no—
426Q8—88no—
427Q8—84no—
428D4—28no—
429D4—44no—
430D4—44no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—88no—
477C2×C4#38611yescharacteristic
478C2×C4#38611yescharacteristic
479C2×C4#38611yescharacteristic
480C2×C4#38612yes—
481C2×C4#38611yescharacteristic
482C2×C4C2×#10112yes—
483C2×C4C2×C4×D412yes—
484C2×C4C2×#12011yescharacteristic
485C2×C4C2×#12112yes—
486C2×C4C2×#12311yescharacteristic
487C8—48no—
488C8—84no—
489C22—21no—
490C22—22no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—81no—
504C22—81no—
505C22#190114yes—
506C22C2×#38611yescharacteristic
507C22C22×#10111yescharacteristic
508C22C2×#41311yescharacteristic
509C4—22no—
510C4—21no—
511C4—21no—
512C4—22no—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—22no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—84no—
523C4—84no—
524C4#190912yes—
525C4#302112yes—
526C2—21no—
527C2—21no—
528C2—24no—
529C2—42no—
530C2—81no—
531C2#2095611yescharacteristic
532C2C2×#190111yescharacteristic
533C2C2×#190911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
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
3. C2×#1556C2semidirect
5. #9456C2semidirect
6. #9455C2semidirect
7. #7952C2semidirect

Maximal subgroups

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

subgroupindex
#602822
#352792
C2×#15562
#173632
#94562
#94552
#79522