Finite Group Databasebeta

← #205927

Subgroups — #205927

Order 512.

1139 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3208C211yesmaximal, characteristic
2#21879C211yesmaximal, characteristic, complemented
3#21797C211yesmaximal, characteristic
4#17817C211yesmaximal, characteristic, complemented
5#17718C211yesmaximal, characteristic, complemented
6C2×#1545C211yesmaximal, characteristic
7#17326C211yesmaximal, characteristic, complemented
8#9805C212yesmaximal, complemented
9#9804C212yesmaximal, complemented
10#9796C212yesmaximal, complemented
11#9793C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C22×#127C2211yescharacteristic
14#3208—22no—
15#3208—22no—
16#3208—21no—
17#3208—21no—
18#3208—22no—
19C2×#492C2212yes—
20C2×#486C2212yes—
21C2×#460C414yes—
22C2×#450C414yes—
23#2920C2211yescharacteristic
24#2920C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28#1997C2212yescomplemented
29#1966C2212yescomplemented
30#1965—22no—
31#1965C2212yes—
32#1965C2212yes—
33#1901—22no—
34#1801—22no—
35#1783—22no—
36#1574—22no—
37#1573—22no—
38#1567C2212yescomplemented
39#1566C2212yescomplemented
40#1554—22no—
41#1548C2211yescharacteristic, complemented
42#1548C2211yescharacteristic, complemented
43#1546C2211yescharacteristic, complemented
44#1546C2211yescharacteristic, complemented
45#1545—22no—
46#1545C2212yes—
47#1545C2212yes—
48#1542—22no—
49C2×#331—22no—
50#1454—22no—
51#1452—21no—
52#1451C2212yescomplemented
53#1450—21no—
54C8×M16C2212yescomplemented
55C23×D4—21no—
56C23×D4—21no—
57C23×D4C2×C412yes—
58C22×SD16C2×C418yes—
59C22×M16—21no—
60#533—21no—
61#533—21no—
62#533—21no—
63#533—21no—
64#533—21no—
65#533—21no—
66#533—21no—
67#533—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127—21no—
71C2×#127—21no—
72C2×#127—21no—
73C2×#127—21no—
74C2×#127—21no—
75C2×#127—21no—
76C2×#127—21no—
77C2×#127—21no—
78C2×#127—21no—
79C2×#127—21no—
80C2×#127C2×C412yes—
81C2×#127C2312yes—
82C2×#125—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8C2×C412yes—
85C2×C4×Q8—21no—
86C2×C4×Q8C2×C412yes—
87C2×#117—21no—
88C22×C42C2311yescharacteristic
89#492—22no—
90#492—22no—
91#492—24no—
92#486—22no—
93#486—22no—
94#486—24no—
95#460—28no—
96#460—24no—
97#460—24no—
98#450—24no—
99#450—24no—
100#450—28no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431D412yes—
112#431D412yes—
113#431D412yes—
114#431D411yescharacteristic
115#431D411yescharacteristic
116#431D412yes—
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119#423—21no—
120#423—21no—
121#423—21no—
122#423D412yes—
123#423C2312yes—
124#423C2311yescharacteristic
125#423C2311yescharacteristic
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8C2×C412yes—
129C2×C4⋊C8D412yes—
130C2×#102C2×C418yes—
131C2×#101—24no—
132C2×#100—24no—
133C2×#100—24no—
134#407—21no—
135#407D412yes—
136C2×C22⋊C8—21no—
137C2×C22⋊C8C2×C412yes—
138C4×M16—21no—
139C4×M16C2312yes—
140C2×C4×C8—21no—
141C2×C4×C8C2312yes—
142#343—24no—
143#335—24no—
144#334—24no—
145#333—22no—
146#333—22no—
147#332—22no—
148#332—22no—
149#331—24no—
150#331—24no—
151#322—22no—
152#322—22no—
153#322—22no—
154C82—22no—
155C25—22no—
156C22×Q8—21no—
157C22×Q8C22×C412yes—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4—82no—
176C22×D4—82no—
177C22×D4—84no—
178C22×D4—84no—
179C22×D4C22×C414yes—
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4#4111yescharacteristic
183C2×SD16—28no—
184C2×SD16—28no—
185C2×SD16—88no—
186C2×M16—22no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—41no—
190C2×M16—82no—
191C2×M16C2×D414yes—
192C22×C8—22no—
193C22×C8—41no—
194C22×C8C22×C414yes—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203#127—22no—
204#127—22no—
205#127—22no—
206#127—22no—
207#127—22no—
208#127—22no—
209#127—22no—
210#127—22no—
211#127—41no—
212#127—41no—
213#127—41no—
214#127—41no—
215#127—41no—
216#127—41no—
217#127—41no—
218#127—41no—
219#125—41no—
220#125—41no—
221#125—41no—
222#125—41no—
223C22⋊Q8—22no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227C22⋊Q8—41no—
228C22⋊Q8—41no—
229C22⋊Q8—41no—
230C22⋊Q8—41no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8—41no—
238C4×Q8—41no—
239#117—41no—
240#117—41no—
241#117—41no—
242#117—41no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C22×C412yes—
249C2×C22⋊C4—22no—
250C2×C42—22no—
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42C2411yescharacteristic
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—22no—
271C4⋊C8—21no—
272C4⋊C8—41no—
273C4⋊C8—41no—
274C4⋊C8C22⋊C414yes—
275C4⋊C8C22⋊C412yes—
276C4⋊C8C22⋊C412yes—
277C4⋊C8C2×D412yes—
278C4⋊C8C2×D412yes—
279#102—28no—
280#102—28no—
281#101—84no—
282#100—84no—
283#100—84no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—41no—
290C22⋊C8—41no—
291C22⋊C8C22⋊C412yes—
292C22⋊C8C22⋊C412yes—
293C22⋊C8C22⋊C414yes—
294#97—22no—
295#97—41no—
296C4×C8—22no—
297C4×C8—21no—
298C4×C8—22no—
299C4×C8—21no—
300C4×C8—41no—
301C4×C8C2×D412yes—
302C4×C8C2×D412yes—
303C4×C8C2×D414yes—
304C24—24no—
305C24—42no—
306C24—42no—
307C24—84no—
308C24—84no—
309C24—84no—
310C24C4×D411yescharacteristic
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—44no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—48no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—84no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—84no—
337C2×D4—82no—
338C2×D4—82no—
339C2×D4—82no—
340C2×D4—82no—
341C2×D4—84no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—84no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12411yescharacteristic
369C22×C4#12411yescharacteristic
370C22×C4#12711yescharacteristic
371C22×C4#12711yescharacteristic
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375SD16—88no—
376SD16—88no—
377M16—24no—
378M16—24no—
379M16—42no—
380M16—44no—
381M16—42no—
382M16—82no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—44no—
387C2×C8—44no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—42no—
391C2×C8—82no—
392C2×C8C2×C22⋊C414yes—
393C2×C8C2×C22⋊C414yes—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C42—42no—
409C42—42no—
410C42C2×C22⋊C411yescharacteristic
411C42C2×C22⋊C411yescharacteristic
412C42C4×D411yescharacteristic
413C42C4×D411yescharacteristic
414C42#12111yescharacteristic
415C42#12111yescharacteristic
416C42#12111yescharacteristic
417C42#12111yescharacteristic
418C42#12411yescharacteristic
419C42#12411yescharacteristic
420C42#12711yescharacteristic
421C42#12711yescharacteristic
422C42C22×D411yescharacteristic
423C42C22×D411yescharacteristic
424C42C22×D411yescharacteristic
425C42C2×#4111yescharacteristic
426C23—22no—
427C23—22no—
428C23—41no—
429C23—44no—
430C23—48no—
431C23—84no—
432C23—84no—
433C23—84no—
434C23—84no—
435C23—84no—
436C23—84no—
437C23—84no—
438C23—84no—
439C23—1616no—
440C23#39011yescharacteristic
441C23#39011yescharacteristic
442C23C2×C4×D411yescharacteristic
443Q8—44no—
444Q8—82no—
445Q8—81no—
446Q8—81no—
447Q8—82no—
448D4—48no—
449D4—84no—
450D4—84no—
451D4—84no—
452D4—84no—
453D4—84no—
454D4—84no—
455D4—84no—
456D4—82no—
457D4—82no—
458D4—84no—
459D4—84no—
460D4—84no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4#39011yescharacteristic
484C2×C4#39011yescharacteristic
485C2×C4#39011yescharacteristic
486C2×C4#39011yescharacteristic
487C2×C4#39011yescharacteristic
488C2×C4#39011yescharacteristic
489C2×C4#47114yes—
490C2×C4C22×C22⋊C411yescharacteristic
491C2×C4C2×C4×D411yescharacteristic
492C2×C4C2×#12111yescharacteristic
493C2×C4C2×#12111yescharacteristic
494C2×C4C2×#12411yescharacteristic
495C2×C4C2×#12711yescharacteristic
496C2×C4#57612yes—
497C2×C4#57712yes—
498C8—24no—
499C8—24no—
500C8—44no—
501C8—42no—
502C8—42no—
503C8—81no—
504C8—82no—
505C8—81no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—44no—
514C22—84no—
515C22—84no—
516C22—84no—
517C22—168no—
518C22—168no—
519C22—168no—
520C22#197712yes—
521C22C2×#39011yescharacteristic
522C22#294612yes—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—82no—
531C4—82no—
532C4#197712yes—
533C4#288812yes—
534C4#306212yes—
535C4C2×#47112yes—
536C4#315411yescharacteristic
537C4#315511yescharacteristic
538C2—22no—
539C2—22no—
540C2—41no—
541C2—164no—
542C2#2121911yescharacteristic
543C2#2123511yescharacteristic
544C2C2×#197711yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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
2. #21879C2semidirect
4. #17817C2semidirect
5. #17718C2semidirect
7. #17326C2semidirect
8. #9805C2semidirect
9. #9804C2semidirect
10. #9796C2semidirect
11. #9793C2semidirect
28. #1997C22semidirect
29. #1966C22semidirect
38. #1567C22semidirect
39. #1566C22semidirect
41. #1548C22semidirect
42. #1548C22semidirect
43. #1546C22semidirect
44. #1546C22semidirect
52. #1451C22semidirect
54. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32082
#218792
#217972
#178172
#177182
C2×#15452
#173262
#98052
#98042
#97962
#97932