Finite Group Databasebeta

← #205983

Subgroups — #205983

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3207C211yesmaximal, characteristic
2#21878C211yesmaximal, characteristic, complemented
3#21878C211yesmaximal, characteristic, complemented
4#17814C211yesmaximal, characteristic, complemented
5#17750C211yesmaximal, characteristic, complemented
6#17745C211yesmaximal, characteristic, complemented
7#17324C211yesmaximal, characteristic, complemented
8#9812C212yesmaximal, complemented
9#9805C212yesmaximal, complemented
10#9802C212yesmaximal, complemented
11#9795C212yesmaximal, complemented
12C2×#532C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3207—22no—
15#3207—22no—
16#3207—22no—
17#3207—22no—
18C2×#493C2211yescharacteristic
19C2×#492C2211yescharacteristic
20C2×#487C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C2×#463C414yes—
23C2×#449C414yes—
24#2920C2211yescharacteristic
25#2920C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#423C2211yescharacteristic
29#1997C2212yes—
30#1997C2212yes—
31#1965C2212yes—
32#1965C2212yes—
33#1900—22no—
34#1900—22no—
35#1800—22no—
36#1800—22no—
37#1571—22no—
38#1571—22no—
39#1567C2212yes—
40#1567C2212yes—
41#1560—22no—
42#1557—22no—
43#1551—22no—
44#1549C2212yes—
45#1548C2212yes—
46#1547—22no—
47#1545C2212yes—
48#1545C2212yes—
49#1454—22no—
50#1453—22no—
51#1451C2212yes—
52C8×M16C2212yes—
53C22×#41—21no—
54C22×#41C2×C412yes—
55C22×SD16C2×C414yes—
56C22×D8C2×C414yes—
57C22×M16—21no—
58C22×M16—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532—21no—
67#532—21no—
68#532—21no—
69#532—21no—
70#532—21no—
71#532—21no—
72#532—21no—
73#532—21no—
74#532—21no—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×#127C2311yescharacteristic
78C2×#127C2311yescharacteristic
79C2×#124—21no—
80C2×#124C2×C412yes—
81C2×#121—21no—
82C2×#121—21no—
83C2×#121—21no—
84C2×#121C2×C412yes—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4C2×C412yes—
89C22×C42C2311yescharacteristic
90#493—22no—
91#493—22no—
92#492—22no—
93#492—22no—
94#487—22no—
95#487—22no—
96#486—22no—
97#486—22no—
98#463—28no—
99#463—28no—
100#449—28no—
101#449—28no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432D412yes—
106#431—21no—
107#431—21no—
108#431—21no—
109#431D412yes—
110#431D412yes—
111#431D412yes—
112#431D412yes—
113#431C2312yes—
114#423—21no—
115#423—21no—
116#423—21no—
117#423—21no—
118#423—21no—
119#423—21no—
120#423D412yes—
121#423D412yes—
122#423C2312yes—
123#423C2312yes—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8C2×C412yes—
129C2×#103C2×C414yes—
130C2×#102C2×C414yes—
131C2×#100—24no—
132C2×#100—24no—
133#407—21no—
134#407D412yes—
135C2×C22⋊C8C2×C412yes—
136C4×M16—21no—
137C4×M16C2312yes—
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140#343—24no—
141#343—24no—
142#335—24no—
143#335—24no—
144#333—24no—
145#332—24no—
146#331—24no—
147#331—24no—
148#322—22no—
149#322—24no—
150C82—22no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—41no—
156C2×#41—41no—
157C2×#41—41no—
158C2×#41—41no—
159C2×#41—84no—
160C2×#41—84no—
161C22×Q8—21no—
162C22×Q8C22×C412yes—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4C22×C412yes—
170C22×D4C22×C412yes—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4#4111yescharacteristic
176C2×SD16—28no—
177C2×SD16—84no—
178C2×D8—28no—
179C2×D8—84no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—41no—
184C2×M16—41no—
185C2×M16—82no—
186C2×M16—82no—
187C2×M16C2×D414yes—
188C22×C8—41no—
189C22×C8—41no—
190C22×C8C22×C412yes—
191C22×C8C22×C412yes—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200#127—21no—
201#127—21no—
202#127—21no—
203#127—21no—
204#127—21no—
205#127—21no—
206#127—21no—
207#127—21no—
208#124—22no—
209#124—22no—
210#124—22no—
211#124—22no—
212#124—41no—
213#124—41no—
214#124—41no—
215#124—41no—
216#121—22no—
217#121—22no—
218#121—22no—
219#121—22no—
220#121—41no—
221#121—41no—
222#121—41no—
223#121—41no—
224#121—41no—
225#121—41no—
226#121—41no—
227#121—41no—
228#121—41no—
229#121—41no—
230#121—41no—
231#121—41no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—41no—
242C4×D4—41no—
243C4×D4—41no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C22×C412yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C42C22×C411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42C2411yescharacteristic
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8—41no—
278C4⋊C8—41no—
279C4⋊C8C22⋊C414yes—
280C4⋊C8C22⋊C414yes—
281C4⋊C8C2×D414yes—
282#103—28no—
283#102—28no—
284#100—84no—
285#100—84no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8C22⋊C414yes—
291C22⋊C8C22⋊C414yes—
292#97—22no—
293#97—22no—
294C4×C8—22no—
295C4×C8—22no—
296C4×C8C2×D412yes—
297C4×C8C2×D412yes—
298C4×C8C2×D412yes—
299C4×C8C2×D412yes—
300C24—22no—
301C24—22no—
302C24C4×D411yescharacteristic
303#41—84no—
304#41—84no—
305#41—84no—
306#41—84no—
307#41—84no—
308#41—84no—
309#41—84no—
310#41—84no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—44no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—82no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—82no—
340C2×D4—82no—
341C2×D4—84no—
342C2×D4—84no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12411yescharacteristic
367C22×C4#12411yescharacteristic
368C22×C4#12711yescharacteristic
369C22×C4#12711yescharacteristic
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4111yescharacteristic
373SD16—88no—
374D8—88no—
375M16—24no—
376M16—24no—
377M16—44no—
378M16—44no—
379M16—82no—
380M16—82no—
381M16—82no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—82no—
389C2×C8—82no—
390C2×C8C2×C22⋊C414yes—
391C2×C8C2×C22⋊C414yes—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C42C2×C22⋊C411yescharacteristic
409C42C2×C22⋊C411yescharacteristic
410C42C4×D411yescharacteristic
411C42C4×D411yescharacteristic
412C42#12111yescharacteristic
413C42#12111yescharacteristic
414C42#12111yescharacteristic
415C42#12111yescharacteristic
416C42#12411yescharacteristic
417C42#12411yescharacteristic
418C42#12711yescharacteristic
419C42#12711yescharacteristic
420C42C22×D411yescharacteristic
421C42C22×D411yescharacteristic
422C42C22×D411yescharacteristic
423C42C2×#4111yescharacteristic
424C23—22no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23#39011yescharacteristic
436C23#39011yescharacteristic
437C23C2×C4×D411yescharacteristic
438Q8—44no—
439Q8—82no—
440Q8—82no—
441Q8—82no—
442D4—44no—
443D4—44no—
444D4—82no—
445D4—82no—
446D4—82no—
447D4—82no—
448D4—84no—
449D4—84no—
450D4—84no—
451D4—84no—
452D4—81no—
453D4—81no—
454D4—81no—
455D4—81no—
456D4—82no—
457D4—82no—
458D4—82no—
459D4—82no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4—84no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4#39011yescharacteristic
488C2×C4#39011yescharacteristic
489C2×C4#39011yescharacteristic
490C2×C4#39011yescharacteristic
491C2×C4#39011yescharacteristic
492C2×C4#39011yescharacteristic
493C2×C4C22×C22⋊C411yescharacteristic
494C2×C4C2×C4×D411yescharacteristic
495C2×C4C2×#12111yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×#12411yescharacteristic
498C2×C4C2×#12711yescharacteristic
499C2×C4#57612yes—
500C2×C4#57812yes—
501C8—24no—
502C8—24no—
503C8—44no—
504C8—44no—
505C8—82no—
506C8—82no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—164no—
523C22—164no—
524C22C2×#39011yescharacteristic
525C22#295112yes—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—82no—
535C4—82no—
536C4#288812yes—
537C4#306112yes—
538C4#306812yes—
539C4#315511yescharacteristic
540C4#315611yescharacteristic
541C2—22no—
542C2—41no—
543C2—41no—
544C2—162no—
545C2—162no—
546C2#2121911yescharacteristic
547C2#2123011yescharacteristic
548C2#2124911yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3207C2
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. #21878C2semidirect
3. #21878C2semidirect
4. #17814C2semidirect
5. #17750C2semidirect
6. #17745C2semidirect
7. #17324C2semidirect
8. #9812C2semidirect
9. #9805C2semidirect
10. #9802C2semidirect
11. #9795C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32072
#218782
#178142
#177502
#177452
#173242
#98122
#98052
#98022
#97952