Finite Group Databasebeta

← #205971

Subgroups — #205971

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3207C211yesmaximal, characteristic
2#21424C211yesmaximal, characteristic, complemented
3#21424C211yesmaximal, characteristic, complemented
4#17815C211yesmaximal, characteristic, complemented
5#17717C211yesmaximal, characteristic, complemented
6#17714C211yesmaximal, characteristic, complemented
7#17319C211yesmaximal, characteristic, complemented
8#9812C212yesmaximal, complemented
9#9811C212yesmaximal, complemented
10#9805C212yesmaximal, complemented
11#9803C212yesmaximal, 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×#431C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29#1997C2212yes—
30#1997C2212yes—
31#1965C2212yes—
32#1965C2212yes—
33#1900—22no—
34#1900—22no—
35C4×#100—22no—
36C4×#100—22no—
37#1572—22no—
38#1572—22no—
39#1567C2212yes—
40#1566C2212yes—
41#1553—22no—
42#1551—22no—
43#1549C2212yes—
44#1548C2212yes—
45#1545C2212yes—
46#1545C2212yes—
47#1543—22no—
48#1540—22no—
49#1452—22no—
50#1451—22no—
51#1451C2212yes—
52#1451C2212yes—
53C22×#41—21no—
54C22×#41C2×C412yes—
55C22×SD16C2×C414yes—
56C22×D8C2×C414yes—
57C22×M16D412yes—
58#532—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—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×#127C2311yescharacteristic
77C2×#127C2311yescharacteristic
78C2×#124—21no—
79C2×#124C2×C412yes—
80C2×#121—21no—
81C2×#121—21no—
82C2×#121—21no—
83C2×#121C2×C412yes—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4C2×C412yes—
88C22×C42C2311yescharacteristic
89#493—22no—
90#493—22no—
91#492—22no—
92#492—22no—
93#487—22no—
94#487—22no—
95#486—22no—
96#486—22no—
97#463—28no—
98#463—28no—
99#449—28no—
100#449—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#431D412yes—
115#431D412yes—
116#431C2312yes—
117#423C2312yes—
118#423C2312yes—
119C2×C4⋊C8C2×C412yes—
120C2×C4⋊C8D412yes—
121C2×#103C2×C414yes—
122C2×#102C2×C414yes—
123C2×#100—24no—
124C2×#100—24no—
125#407—21no—
126#407—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8C2×C412yes—
129C4×M16—21no—
130C4×M16—21no—
131C4×M16—21no—
132C4×M16—21no—
133C4×M16D412yes—
134C4×M16C2312yes—
135C2×#97—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140#343—24no—
141#343—24no—
142#335—24no—
143#334—24no—
144#333—24no—
145#332—24no—
146#331—24no—
147#331—24no—
148#322—24no—
149#322—24no—
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×#41—84no—
159C2×#41—84no—
160C22×Q8—21no—
161C22×Q8C22×C412yes—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4C22×C412yes—
169C22×D4C22×C412yes—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4#4111yescharacteristic
175C2×SD16—28no—
176C2×SD16—84no—
177C2×D8—28no—
178C2×D8—84no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—44no—
183C2×M16C2×D414yes—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8C22×C412yes—
187C22×C8C22×C412yes—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196#127—21no—
197#127—21no—
198#127—21no—
199#127—21no—
200#127—21no—
201#127—21no—
202#127—21no—
203#127—21no—
204#124—22no—
205#124—22no—
206#124—22no—
207#124—22no—
208#124—41no—
209#124—41no—
210#124—41no—
211#124—41no—
212#121—22no—
213#121—22no—
214#121—22no—
215#121—22no—
216#121—41no—
217#121—41no—
218#121—41no—
219#121—41no—
220#121—41no—
221#121—41no—
222#121—41no—
223#121—41no—
224#121—41no—
225#121—41no—
226#121—41no—
227#121—41no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—41no—
242C4×D4—41no—
243C4×D4—41no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22×C412yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42C2411yescharacteristic
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8C22⋊C414yes—
270C4⋊C8C22⋊C414yes—
271C4⋊C8C2×D414yes—
272#103—28no—
273#102—28no—
274#100—48no—
275#100—48no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—41no—
279C22⋊C8—41no—
280C22⋊C8—41no—
281C22⋊C8—41no—
282C22⋊C8C22⋊C414yes—
283C22⋊C8C22⋊C414yes—
284#97—22no—
285#97—22no—
286#97—41no—
287#97—41no—
288C4×C8—22no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—22no—
292C4×C8—41no—
293C4×C8—41no—
294C4×C8C2×D412yes—
295C4×C8C2×D412yes—
296C4×C8C2×D412yes—
297C4×C8C2×D412yes—
298C24—22no—
299C24—22no—
300C24C4×D411yescharacteristic
301#41—84no—
302#41—84no—
303#41—84no—
304#41—84no—
305#41—84no—
306#41—84no—
307#41—84no—
308#41—84no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—44no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—82no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—82no—
336C2×D4—82no—
337C2×D4—84no—
338C2×D4—84no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4C4×D411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4#12411yescharacteristic
362C22×C4#12411yescharacteristic
363C22×C4#12711yescharacteristic
364C22×C4#12711yescharacteristic
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4111yescharacteristic
368SD16—88no—
369D8—88no—
370M16—24no—
371M16—24no—
372M16—44no—
373M16—44no—
374M16—44no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—44no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—82no—
383C2×C8C2×C22⋊C414yes—
384C2×C8C2×C22⋊C414yes—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C42C2×C22⋊C411yescharacteristic
402C42C2×C22⋊C411yescharacteristic
403C42C4×D411yescharacteristic
404C42C4×D411yescharacteristic
405C42#12111yescharacteristic
406C42#12111yescharacteristic
407C42#12111yescharacteristic
408C42#12111yescharacteristic
409C42#12411yescharacteristic
410C42#12411yescharacteristic
411C42#12711yescharacteristic
412C42#12711yescharacteristic
413C42C22×D411yescharacteristic
414C42C22×D411yescharacteristic
415C42C22×D411yescharacteristic
416C42C2×#4111yescharacteristic
417C23—22no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23#39011yescharacteristic
429C23#39011yescharacteristic
430C23C2×C4×D411yescharacteristic
431Q8—44no—
432Q8—82no—
433Q8—82no—
434Q8—82no—
435D4—44no—
436D4—44no—
437D4—82no—
438D4—82no—
439D4—82no—
440D4—82no—
441D4—84no—
442D4—84no—
443D4—84no—
444D4—84no—
445D4—82no—
446D4—82no—
447D4—82no—
448D4—82no—
449D4—82no—
450D4—82no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4#39011yescharacteristic
477C2×C4#39011yescharacteristic
478C2×C4#39011yescharacteristic
479C2×C4#39011yescharacteristic
480C2×C4#39011yescharacteristic
481C2×C4#39011yescharacteristic
482C2×C4C22×C22⋊C411yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12411yescharacteristic
487C2×C4C2×#12711yescharacteristic
488C2×C4#57514yes—
489C8—24no—
490C8—24no—
491C8—44no—
492C8—44no—
493C8—82no—
494C8—82no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—164no—
511C22—164no—
512C22C2×#39011yescharacteristic
513C22#294312yes—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—82no—
523C4—82no—
524C4#288812yes—
525C4#305412yes—
526C4#306812yes—
527C4#315012yes—
528C2—22no—
529C2—41no—
530C2—41no—
531C2—162no—
532C2—162no—
533C2#2121911yescharacteristic
534C2#2122811yescharacteristic
535C2#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. #21424C2semidirect
3. #21424C2semidirect
4. #17815C2semidirect
5. #17717C2semidirect
6. #17714C2semidirect
7. #17319C2semidirect
8. #9812C2semidirect
9. #9811C2semidirect
10. #9805C2semidirect
11. #9803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32072
#214242
#178152
#177172
#177142
#173192
#98122
#98112
#98052
#98032