Finite Group Databasebeta

← #220183

Subgroups — #220183

Order 512.

1101 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2952C211yesmaximal, characteristic
2#17855C212yesmaximal, complemented
3#17353C211yesmaximal, characteristic, complemented
4#17351C211yesmaximal, characteristic, complemented
5#9667C212yesmaximal, complemented
6#9089C212yesmaximal, complemented
7#9088C212yesmaximal, complemented
8#8601C211yesmaximal, characteristic, complemented
9#8324C211yesmaximal, characteristic, complemented
10#8324C211yesmaximal, characteristic, complemented
11#8156C211yesmaximal, characteristic, complemented
12C22×#135C411yescharacteristic
13C2×#520C2211yescharacteristic
14C2×#520C2211yescharacteristic
15#2982C2212yes—
16#2972C2212yes—
17#2952—24no—
18#2952—24no—
19C2×#442C411yescharacteristic
20C2×#442C2211yescharacteristic
21C2×#441C411yescharacteristic
22C2×#441C2211yescharacteristic
23C2×C4×SD16C411yescharacteristic
24C2×C4×SD16C411yescharacteristic
25C2×C4×SD16C2211yescharacteristic
26C2×C4×SD16C2211yescharacteristic
27C22×#106C411yescharacteristic
28C2×#417C411yescharacteristic
29C2×#417C411yescharacteristic
30C2×#405C2211yescharacteristic
31SD16⋊C8—24no—
32C8×SD16—24no—
33#1564C2212yes—
34#1479C2212yes—
35#1479C2212yes—
36#1478C2212yes—
37#1478C2212yes—
38#1461C2212yes—
39#1460—22no—
40#1460C2212yes—
41#1459—22no—
42#1458—22no—
43#1458C2211yescharacteristic
44#1458C2211yescharacteristic
45#1458C2211yescharacteristic
46#1458C2211yescharacteristic
47C2×#324—22no—
48#1450C2212yes—
49#1281C2212yes—
50#1275C2212yes—
51C22×#41C2×C411yescharacteristic
52C22×#41C2×C411yescharacteristic
53C2×#135—22no—
54C2×#135—22no—
55C2×#135—24no—
56C2×#135C818yescomplemented
57C22×Q16C2×C411yescharacteristic
58C22×SD16C2×C411yescharacteristic
59C22×SD16C2×C411yescharacteristic
60C22×D8C2×C411yescharacteristic
61C22×M16C2×C411yescharacteristic
62C23×C8C2×C411yescharacteristic
63#520—24no—
64#520—24no—
65#520—42no—
66#520—42no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×#117—21no—
80C2×#117—21no—
81C2×#117C2311yescharacteristic
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C8×Q8—24no—
87C8×Q8—42no—
88#442—24no—
89#442—24no—
90#441—24no—
91#441—24no—
92C4×SD16—22no—
93C4×SD16—22no—
94C4×SD16—22no—
95C4×SD16—22no—
96C4×SD16—22no—
97C4×SD16—22no—
98C4×SD16D414yes—
99C4×SD16D414yes—
100C8×D4—24no—
101C8×D4—42no—
102#431—24no—
103#431C2312yes—
104#431C2312yes—
105#431C2312yes—
106#431C2312yes—
107C2×#106—22no—
108C2×#106—22no—
109C2×#106C2×C411yescharacteristic
110C2×#106C2×C411yescharacteristic
111C2×#106C2×C411yescharacteristic
112C2×#106C2×C411yescharacteristic
113#417—24no—
114#417—24no—
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×C22⋊C8—24no—
124#405—22no—
125#405—22no—
126C2×#97C2×C411yescharacteristic
127C2×#97C2311yescharacteristic
128C2×C4×C8C2×C411yescharacteristic
129C2×C4×C8C2311yescharacteristic
130#335—22no—
131#334—22no—
132Q8⋊C8—24no—
133Q8⋊C8—42no—
134D4⋊C8—24no—
135D4⋊C8—42no—
136#324—21no—
137#324—21no—
138#324—21no—
139#324—24no—
140#324D412yes—
141#323—21no—
142#323—21no—
143#323—21no—
144#323—24no—
145#323D412yes—
146#322—22no—
147C82—22no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41—41no—
155C2×#41—41no—
156C2×#41—41no—
157C2×#41—41no—
158C2×#41C2×C814yes—
159C2×#41C2×C814yescomplemented
160C22×Q8C22×C411yescharacteristic
161C22×Q8C22×C411yescharacteristic
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22×C411yescharacteristic
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4C22×C411yescharacteristic
175#135—216no—
176#135—216no—
177C2×Q16—22no—
178C2×Q16—22no—
179C2×Q16C2×C814yescomplemented
180C2×SD16—21no—
181C2×SD16—21no—
182C2×SD16—22no—
183C2×SD16C22⋊C412yes—
184C2×SD16C22⋊C412yes—
185C2×SD16C2×C814yes—
186C2×SD16C2×C814yescomplemented
187C2×SD16M1614yes—
188C2×D8—22no—
189C2×D8—22no—
190C2×D8C2×C814yes—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—41no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—44no—
198C22×C8C2×C814yescomplemented
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8C2×D414yes—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—44no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—44no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4C2×D414yes—
223#117—22no—
224#117—22no—
225#117—44no—
226#117—44no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4#4111yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4C2×D411yescharacteristic
250C2×C22⋊C4#4111yescharacteristic
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42C22×C411yescharacteristic
256C2×C42C2411yescharacteristic
257#106—21no—
258#106—21no—
259#106—22no—
260#106—22no—
261#106C22⋊C412yes—
262#106C22⋊C412yes—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—44no—
268#103—22no—
269#103—22no—
270#103—22no—
271#103C22⋊C414yes—
272#102—22no—
273#102—22no—
274#102—22no—
275#102C22⋊C414yes—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—44no—
280C22⋊C8C2×D414yes—
281#97—22no—
282#97—22no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—22no—
286C4×C8—22no—
287C4×C8—22no—
288C4×C8—22no—
289C4×C8—44no—
290C4×C8C2×D412yes—
291C4×C8C2×D412yes—
292C24—21no—
293C24—21no—
294C24C4×D411yescharacteristic
295#41—28no—
296#41—28no—
297#41—44no—
298#41—44no—
299#41—44no—
300#41—44no—
301#41—44no—
302#41—44no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8C2×C22⋊C412yes—
312C2×Q8C22×C812yes—
313C2×Q8C22×C812yes—
314C2×Q8C2×M1612yes—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4C2×C22⋊C412yes—
330C2×D4C22×C812yes—
331C2×D4C22×C812yes—
332C2×D4C2×M1612yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—161no—
356C22×C4—161no—
357C22×C4—161no—
358C22×C4—161no—
359C22×C4—161no—
360C22×C4—161no—
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4C22×C812yes—
365C22×C4#13112yes—
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369Q16—28no—
370SD16—24no—
371SD16—24no—
372SD16C22⋊C818yes—
373SD16C22⋊C818yes—
374D8—28no—
375M16—44no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—21no—
383C2×C8—24no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—164no—
390C2×C8C2×C22⋊C412yes—
391C2×C8C2×C22⋊C411yescharacteristic
392C2×C8C2×C22⋊C411yescharacteristic
393C2×C8C22×C812yes—
394C2×C8C22×C812yes—
395C2×C8C2×M1612yes—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4C2×C22⋊C412yes—
411C4⋊C4C2×C22⋊C412yes—
412C22⋊C4—22no—
413C22⋊C4—24no—
414C22⋊C4—22no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C42—21no—
419C42—21no—
420C42—24no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42C22×D412yes—
425C42H(5,𝔽2)12yes—
426C23—21no—
427C23—22no—
428C23—21no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23C8×D412yes—
441C23C2×C4×D411yescharacteristic
442Q8—22no—
443Q8—22no—
444Q8—22no—
445Q8—22no—
446Q8—22no—
447Q8—41no—
448Q8—41no—
449Q8C2×C22⋊C814yes—
450D4—22no—
451D4—22no—
452D4—22no—
453D4—22no—
454D4—22no—
455D4—48no—
456D4—48no—
457D4—41no—
458D4—41no—
459D4C2×C22⋊C814yes—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—84no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—162no—
490C2×C4—162no—
491C2×C4—162no—
492C2×C4—162no—
493C2×C4—162no—
494C2×C4C8×D412yes—
495C2×C4C22×C22⋊C411yescharacteristic
496C2×C4#51811yescharacteristic
497C2×C4D4212yes—
498C2×C4#54612yes—
499C2×C4C23×C811yescharacteristic
500C2×C4C22×M1611yescharacteristic
501C2×C4C2×#13111yescharacteristic
502C2×C4#56811yescharacteristic
503C8—22no—
504C8—22no—
505C8—44no—
506C8—41no—
507C8—82no—
508C8—82no—
509C8—82no—
510C8C2×C22⋊C812yes—
511C8C2×C22⋊C812yes—
512C22—21no—
513C22—21no—
514C22—22no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—41no—
520C22—41no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—81no—
528C22—81no—
529C22D4×C22⋊C411yescharacteristic
530C22C2×C8×D411yescharacteristic
531C22D4×M1611yescharacteristic
532C4—22no—
533C4—24no—
534C4—22no—
535C4—22no—
536C4—42no—
537C4—41no—
538C4—42no—
539C4—41no—
540C4—42no—
541C4—81no—
542C4—81no—
543C4C22×C22⋊C811yescharacteristic
544C4#288011yescharacteristic
545C4#297211yescharacteristic
546C4#297411yescharacteristic
547C2—21no—
548C2—21no—
549C2—42no—
550C2—44no—
551C2—82no—
552C2D4×C22⋊C811yescharacteristic
553C2#2245611yescharacteristic
554C2#2245911yescharacteristic
subgroupG/H
center, socleC22D4×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC22×#135C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17855C2semidirect
3. #17353C2semidirect
4. #17351C2semidirect
5. #9667C2semidirect
6. #9089C2semidirect
7. #9088C2semidirect
8. #8601C2semidirect
9. #8324C2semidirect
10. #8324C2semidirect
11. #8156C2semidirect
56. C2×#135C8semidirect
159. C2×#41C2×C8semidirect
179. C2×Q16C2×C8semidirect
186. C2×SD16C2×C8semidirect
198. C22×C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#29522
#178552
#173532
#173512
#96672
#90892
#90882
#86012
#83242
#81562