Finite Group Databasebeta

← #205776

Subgroups — #205776

Order 512.

1175 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3191C211yesmaximal, characteristic
2#21939C211yesmaximal, characteristic, complemented
3#21420C212yesmaximal, complemented
4#20523C211yesmaximal, characteristic, complemented
5#17572C212yesmaximal, complemented
6#9783C212yesmaximal, complemented
7#9782C212yesmaximal, complemented
8#9781C214yesmaximal, 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
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2280C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1996C2214yes—
25#1964C2214yes—
26#1898—24no—
27#1854—24no—
28#1808C2212yes—
29#1807C2212yes—
30C4×#99—24no—
31C8×C22⋊C4—22no—
32C8×C22⋊C4C2212yes—
33C8×C22⋊C4C2212yes—
34C4×C22⋊C8—22no—
35#1520C2214yes—
36#1519C2214yes—
37#1516—24no—
38#1501—24no—
39C22×#41C2×C412yes—
40C24×C4C2311yescharacteristic
41C22×Q16C2×C411yescharacteristic, complemented
42C22×SD16C2×C412yescomplemented
43C22×D8C2×C411yescharacteristic, complemented
44C23×C8—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—22no—
64C2×#117C2×C412yes—
65C22×C22⋊C4D412yes—
66C22×C42—21no—
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×#107C2×C411yescharacteristic, complemented
83C2×#107C2×C411yescharacteristic, complemented
84C2×#106—22no—
85C2×#106C2×C412yescomplemented
86C2×#103C2×C412yescomplemented
87C2×#103C2×C412yescomplemented
88C2×#102C2×C412yescomplemented
89C2×#102C2×C412yescomplemented
90C2×#99—28no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8C2×C412yes—
94C2×C22⋊C8C2×C412yes—
95C2×C22⋊C8D414yes—
96C2×C22⋊C8D414yes—
97C2×C22⋊C8D412yes—
98C2×C22⋊C8D412yes—
99C2×C22⋊C8C2312yes—
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102#381—22no—
103#380—22no—
104#380C2314yes—
105#379—22no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4D412yes—
111C4×C22⋊C4C2312yes—
112C2×#95—21no—
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—22no—
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—
152C22×C8—22no—
153C22×C8—24no—
154C22×C8—81no—
155C22×C8—81no—
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—82no—
202#117—82no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4C2×D414yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—42no—
223C2×C42—81no—
224C2×C42—81no—
225#107—22no—
226#107—22no—
227#107—82no—
228#106—24no—
229#106—82no—
230#103—24no—
231#103—24no—
232#102—24no—
233#102—24no—
234#99—416no—
235C22⋊C8—24no—
236C22⋊C8—22no—
237C22⋊C8C22⋊C414yes—
238C22⋊C8C22⋊C414yes—
239C22⋊C8C22⋊C414yes—
240C22⋊C8C22⋊C414yes—
241C22⋊C8C2×D414yes—
242C4×C8—82no—
243C4×C8—82no—
244#95—22no—
245#95—22no—
246#95—24no—
247#95—42no—
248C24—24no—
249C24—22no—
250C24—81no—
251C24—81no—
252C24C4×D412yes—
253C24#12111yescharacteristic
254C24#12111yescharacteristic
255C24#12712yes—
256C24C22×D411yescharacteristic
257#41—82no—
258#41—82no—
259#41—82no—
260#41—82no—
261#41—84no—
262#41—84no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—48no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—84no—
279C2×D4—84no—
280C22×C4—24no—
281C22×C4—28no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—81no—
299C22×C4—82no—
300C22×C4—81no—
301C22×C4—82no—
302C22×C4—81no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—84no—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C4×D412yes—
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12412yes—
316C22×C4#12412yes—
317C22×C4#12712yes—
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324Q16—82no—
325SD16—84no—
326D8—82no—
327C2×C8—24no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—84no—
338C2×C8C2×C22⋊C412yes—
339C2×C8C2×C22⋊C414yes—
340C2×C8C2×C22⋊C412yes—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C22⋊C4—28no—
352C22⋊C4—24no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—48no—
364C22⋊C4—84no—
365C22⋊C4—84no—
366C22⋊C4#13518yes—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—82no—
372C42—84no—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—24no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—44no—
389C23—82no—
390C23—82no—
391C23—88no—
392C23#39012yes—
393C23#39012yes—
394C23C2×C4×D411yescharacteristic
395C23C2×#12111yescharacteristic
396C23C2×#12711yescharacteristic
397C23C2×#13514yes—
398Q8—42no—
399Q8—42no—
400Q8—82no—
401D4—42no—
402D4—42no—
403D4—82no—
404D4—84no—
405D4—84no—
406D4—84no—
407D4—84no—
408D4—84no—
409D4—84no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—28no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—88no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4#39012yes—
449C2×C4#39012yes—
450C2×C4C22×C22⋊C411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×#12111yescharacteristic
453C2×C4C2×#12411yescharacteristic
454C8—22no—
455C8—24no—
456C8—22no—
457C8—44no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—44no—
471C22—44no—
472C22—44no—
473C22—84no—
474C22—84no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—164no—
479C22C2×#39011yescharacteristic
480C22C4×#13512yes—
481C22#305012yes—
482C22#314712yes—
483C4—21no—
484C4—22no—
485C4—22no—
486C4—21no—
487C4—21no—
488C4—48no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—164no—
493C4#288311yescharacteristic
494C4#288311yescharacteristic
495C2—22no—
496C2—22no—
497C2—22no—
498C2—42no—
499C2—42no—
500C2—162no—
501C2#2121311yescharacteristic
502C2#2122411yescharacteristic
503C2#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. #21939C2semidirect
3. #21420C2semidirect
4. #20523C2semidirect
5. #17572C2semidirect
6. #9783C2semidirect
7. #9782C2semidirect
8. #9781C2semidirect
13. C2×#485C4semidirect
14. C2×#482C4semidirect
18. C2×#452C4semidirect
19. C2×#449C4semidirect
41. C22×Q16C2×C4semidirect
42. C22×SD16C2×C4semidirect
43. C22×D8C2×C4semidirect
82. C2×#107C2×C4semidirect
83. 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
#219392
#214202
#205232
#175722
#97832
#97822
#97812