Finite Group Databasebeta

← #205786

Subgroups — #205786

Order 512.

1143 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3191C211yesmaximal, characteristic
2#21940C211yesmaximal, characteristic, complemented
3#21874C212yesmaximal, complemented
4#20527C211yesmaximal, characteristic, complemented
5#17592C212yesmaximal, complemented
6#9786C214yesmaximal, complemented
7#9783C212yesmaximal, complemented
8#9782C212yesmaximal, complemented
9C2×#525C2212yes—
10#3191—24no—
11#3191—22no—
12#3191—22no—
13C2×#485C412yescomplemented
14C2×#482C412yescomplemented
15C2×#467C2211yescharacteristic
16C2×#466C2211yescharacteristic
17C2×#465C2212yes—
18C2×#452C412yescomplemented
19C2×#449C412yescomplemented
20C2×#407C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2280C2211yescharacteristic
23C2×#380C2212yes—
24#1996C2214yes—
25#1964C2214yes—
26#1898—24no—
27#1852—24no—
28#1808C2212yes—
29#1808C2212yes—
30#1799—24no—
31#1753—22no—
32#1752C2212yes—
33C8×C22⋊C4C2212yes—
34#1750—22no—
35#1520C2214yes—
36#1519C2214yes—
37#1513—24no—
38#1504—24no—
39C22×#41C2×C412yes—
40C24×C4C2311yescharacteristic
41C22×Q16C2×C411yescharacteristic, complemented
42C22×SD16C2×C412yescomplemented
43C22×D8C2×C411yescharacteristic, complemented
44C22×M16—21no—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47#525—22no—
48#525—22no—
49#525—22no—
50#525—22no—
51#525—22no—
52#525—22no—
53#525—22no—
54#525—22no—
55C2×#123—22no—
56C2×C22⋊Q8C2×C412yes—
57C2×C22⋊Q8C2312yes—
58C2×#121C2×C412yes—
59C2×#121C2312yes—
60C2×#120—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×#117—21no—
64C2×#117C2×C412yes—
65C22×C4⋊C4—22no—
66C22×C22⋊C4—22no—
67#485—24no—
68#485—24no—
69#482—24no—
70#482—24no—
71#467—22no—
72#467—22no—
73#466—22no—
74#466—22no—
75#465—24no—
76#465—24no—
77#452—24no—
78#452—24no—
79#449—24no—
80#449—24no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107C2×C411yescharacteristic, complemented
84C2×#107C2×C411yescharacteristic, complemented
85C2×#106C2×C412yescomplemented
86C2×#103C2×C412yescomplemented
87C2×#103C2×C412yescomplemented
88C2×#102C2×C412yescomplemented
89C2×#102C2×C412yescomplemented
90C2×#99—28no—
91#407—22no—
92#407D414yes—
93C2×C22⋊C8—21no—
94C2×C22⋊C8C2×C412yes—
95C2×C22⋊C8C2×C412yes—
96C2×C22⋊C8D414yes—
97C2×C22⋊C8D412yes—
98C2×C22⋊C8D412yes—
99C2×C22⋊C8C2312yes—
100C2×#97—22no—
101C2×C4×C8—22no—
102#381—21no—
103#380—21no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380D412yes—
108#380C2314yes—
109#379D412yes—
110C4×C22⋊C4—21no—
111C4×C22⋊C4C2312yes—
112C2×#95—22no—
113#342—28no—
114#340—22no—
115#340—22no—
116#336—22no—
117#336—22no—
118#328—28no—
119#327—28no—
120C25C2×D411yescharacteristic
121C2×#41—22no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41—82no—
126C2×#41—82no—
127C22×Q8C22×C412yes—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4C22×C412yes—
132C23×C4—22no—
133C23×C4—42no—
134C23×C4—81no—
135C23×C4—81no—
136C23×C4C22×C412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4#4111yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4#4112yes—
145C23×C4C2411yescharacteristic
146C2×Q16—22no—
147C2×Q16—81no—
148C2×SD16—24no—
149C2×SD16—82no—
150C2×D8—22no—
151C2×D8—81no—
152C2×M16—24no—
153C2×M16—81no—
154C2×M16—81no—
155C22×C8—22no—
156C22×C8—81no—
157C22×C8—81no—
158C22×C8C22×C411yescharacteristic
159C22×C8C22×C412yes—
160C22×C8C22×C411yescharacteristic
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C22×C8C2×D412yes—
165#123—42no—
166#123—42no—
167#123—42no—
168#123—42no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177#121—22no—
178#121—22no—
179#121—22no—
180#121—22no—
181#121—22no—
182#121—22no—
183#121—22no—
184#121—22no—
185#120—42no—
186#120—42no—
187#120—42no—
188#120—42no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197#117—22no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—81no—
202#117—81no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—84no—
222C2×C22⋊C4C2×D414yes—
223C2×C42—22no—
224C2×C42—22no—
225#107—22no—
226#107—22no—
227#107—82no—
228#107—82no—
229#106—24no—
230#103—24no—
231#103—24no—
232#102—24no—
233#102—24no—
234#99—88no—
235C22⋊C8—24no—
236C22⋊C8—22no—
237C22⋊C8C22⋊C414yes—
238C22⋊C8C22⋊C414yes—
239C22⋊C8C22⋊C414yes—
240C22⋊C8C22⋊C414yes—
241C22⋊C8C2×D414yes—
242#97—82no—
243C4×C8—82no—
244#95—22no—
245#95—22no—
246#95—24no—
247#95—42no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—81no—
252C24—81no—
253C24C4×D412yes—
254C24#12111yescharacteristic
255C24#12111yescharacteristic
256C24#12712yes—
257C24C22×D411yescharacteristic
258#41—82no—
259#41—82no—
260#41—82no—
261#41—82no—
262#41—84no—
263#41—84no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—84no—
281C2×D4—84no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—81no—
300C22×C4—82no—
301C22×C4—81no—
302C22×C4—82no—
303C22×C4—81no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4—84no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C4×D412yes—
311C22×C4C4×D412yes—
312C22×C4C4×D412yes—
313C22×C4#12112yes—
314C22×C4#12112yes—
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12412yes—
318C22×C4#12412yes—
319C22×C4#12712yes—
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326Q16—82no—
327SD16—84no—
328D8—82no—
329M16—84no—
330C2×C8—24no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8C2×C22⋊C412yes—
341C2×C8C2×C22⋊C414yes—
342C2×C8C2×C22⋊C412yes—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—82no—
354C4⋊C4—84no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—48no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C22⋊C4—84no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C23—22no—
376C23—22no—
377C23—24no—
378C23—22no—
379C23—22no—
380C23—42no—
381C23—41no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—41no—
388C23—42no—
389C23—41no—
390C23—42no—
391C23—44no—
392C23—82no—
393C23—82no—
394C23—84no—
395C23—84no—
396C23#39012yes—
397C23#39012yes—
398C23C2×C4×D411yescharacteristic
399C23C2×#12111yescharacteristic
400C23C2×#12711yescharacteristic
401C23#57612yes—
402C23#57812yes—
403Q8—42no—
404Q8—42no—
405Q8—82no—
406D4—42no—
407D4—42no—
408D4—82no—
409D4—84no—
410D4—84no—
411D4—84no—
412D4—84no—
413D4—82no—
414D4—82no—
415D4—84no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—168no—
454C2×C4#39012yes—
455C2×C4#39012yes—
456C2×C4C22×C22⋊C411yescharacteristic
457C2×C4C2×C4×D411yescharacteristic
458C2×C4C2×#12111yescharacteristic
459C2×C4C2×#12411yescharacteristic
460C8—22no—
461C8—24no—
462C8—22no—
463C8—44no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—42no—
477C22—42no—
478C22—44no—
479C22—44no—
480C22—82no—
481C22—82no—
482C22—84no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—164no—
487C22C2×#39011yescharacteristic
488C22#295112yes—
489C22#306112yes—
490C22#315511yescharacteristic
491C22#315611yescharacteristic
492C4—21no—
493C4—22no—
494C4—22no—
495C4—21no—
496C4—21no—
497C4—84no—
498C4—82no—
499C4—82no—
500C4—82no—
501C4—164no—
502C4#288311yescharacteristic
503C4#288311yescharacteristic
504C2—22no—
505C2—21no—
506C2—21no—
507C2—22no—
508C2—42no—
509C2—42no—
510C2—162no—
511C2#2121311yescharacteristic
512C2#2123011yescharacteristic
513C2#2124411yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3191C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21940C2semidirect
3. #21874C2semidirect
4. #20527C2semidirect
5. #17592C2semidirect
6. #9786C2semidirect
7. #9783C2semidirect
8. #9782C2semidirect
13. C2×#485C4semidirect
14. C2×#482C4semidirect
18. C2×#452C4semidirect
19. C2×#449C4semidirect
41. C22×Q16C2×C4semidirect
42. C22×SD16C2×C4semidirect
43. C22×D8C2×C4semidirect
83. C2×#107C2×C4semidirect
84. C2×#107C2×C4semidirect
85. C2×#106C2×C4semidirect
86. C2×#103C2×C4semidirect
87. C2×#103C2×C4semidirect
88. C2×#102C2×C4semidirect
89. C2×#102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31912
#219402
#218742
#205272
#175922
#97862
#97832
#97822