Finite Group Databasebeta

← #206376

Subgroups — #206376

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60944C211yesmaximal, characteristic
2#21415C212yesmaximal, complemented
3#20463C211yesmaximal, characteristic, complemented
4#17815C211yesmaximal, characteristic, complemented
5#17489C212yesmaximal, complemented
6#9854C212yesmaximal, complemented
7#9853C211yesmaximal, characteristic, complemented
8#9852C211yesmaximal, characteristic, complemented
9#9851C212yesmaximal, complemented
10#9850C211yesmaximal, characteristic, complemented
11#9849C211yesmaximal, characteristic, complemented
12#3444C2212yes—
13#3250—22no—
14#3231C412yescomplemented
15#3228C412yescomplemented
16#3210C412yescomplemented
17#3207C412yescomplemented
18#3191—22no—
19#3170C2211yescharacteristic
20#3169C2211yescharacteristic
21#3167C2212yes—
22#3102—24no—
23#2920C2212yes—
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×C43C2211yescharacteristic
27#1994C2212yes—
28#1994C2212yes—
29#1962C2212yes—
30#1962C2212yes—
31#1896—24no—
32#1767—22no—
33C4×#104—24no—
34#1749—22no—
35#1744C2211yescharacteristic
36#1744C2211yescharacteristic
37C4×#97C2211yescharacteristic
38C4×#97C2211yescharacteristic
39#1572—24no—
40#1567C2212yes—
41#1566C2211yescharacteristic
42#1566C2211yescharacteristic
43#1493C2212yes—
44#1493C2212yes—
45#1492C2212yes—
46#1492C2212yes—
47#1488—24no—
48#1482—24no—
49#533C2×C412yes—
50#533C2×C412yes—
51#533C2312yes—
52#532C2×C412yes—
53#532C2×C412yes—
54#532C2312yes—
55#529—22no—
56#529—22no—
57#529—22no—
58#529—22no—
59#529—22no—
60#529—22no—
61#525—22no—
62#525—22no—
63#525—22no—
64C4×#41—22no—
65C4×#41—22no—
66C4×#41—22no—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42D412yes—
70C22×C42C2311yescharacteristic
71#500C2×C411yescharacteristic, complemented
72#500C2×C411yescharacteristic, complemented
73#498C2×C412yescomplemented
74#494C2×C411yescharacteristic, complemented
75#493C2×C411yescharacteristic, complemented
76#492C2×C412yescomplemented
77#487C2×C412yescomplemented
78#487C2×C412yescomplemented
79#486C2×C412yescomplemented
80#486C2×C412yescomplemented
81#485—24no—
82#482—24no—
83#479—24no—
84#478—24no—
85#467—22no—
86#466—22no—
87#465—24no—
88#464—24no—
89#463—24no—
90#452—24no—
91#449—24no—
92C4×Q16—22no—
93C4×SD16—24no—
94C4×D8—22no—
95#432—22no—
96#432—22no—
97#431—22no—
98#431—22no—
99#431—22no—
100#431C2×C412yes—
101#431C2×C412yes—
102#431D414yes—
103#431D414yes—
104#423C2312yes—
105#423C2312yes—
106C2×C4⋊C8—21no—
107C2×C4⋊C8D412yes—
108C2×#104—24no—
109C2×#104—24no—
110#407—22no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8D412yes—
113C4×M16—22no—
114C2×#97—22no—
115C2×#97—22no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C43—21no—
121C43—21no—
122C43D412yes—
123C43C2311yescharacteristic
124C43C2311yescharacteristic
125#339—24no—
126#339—24no—
127#336—22no—
128#336—22no—
129#335—24no—
130#334—22no—
131#334—22no—
132Q8⋊C8—24no—
133Q8⋊C8—24no—
134D4⋊C8—24no—
135D4⋊C8—24no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—22no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4#4111yescharacteristic
143C2×Q16—22no—
144C2×SD16—24no—
145C2×D8—22no—
146C2×M16—22no—
147C2×M16—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C4⋊Q8C22×C412yes—
153C4⋊Q8C22×C412yes—
154C4⋊Q8C22×C412yes—
155#127C22×C412yes—
156#126—42no—
157#126—42no—
158#125—22no—
159#125—22no—
160#125—22no—
161#124—22no—
162#124—22no—
163#124—22no—
164#123—42no—
165#123—42no—
166#123—42no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173#121—22no—
174#121—22no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—22no—
179#120—42no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4—42no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—42no—
202#117—42no—
203#117—42no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—44no—
212C2×C42—44no—
213C2×C42—44no—
214C2×C42—42no—
215C2×C42C22×C412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42#4111yescharacteristic
223C2×C42#4112yes—
224C2×C42C2411yescharacteristic
225#107—22no—
226#107—22no—
227#106—24no—
228C4⋊C8—24no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8C22⋊C414yes—
234C4⋊C8C22⋊C414yes—
235#104—48no—
236#104—48no—
237#103—24no—
238#103—24no—
239#102—24no—
240#102—24no—
241C22⋊C8—24no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C22⋊C8C22⋊C414yes—
245C22⋊C8C22⋊C414yes—
246#97—44no—
247#97—44no—
248#97—42no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—42no—
252C4×C8C22×C411yescharacteristic
253C4×C8C22×C412yes—
254C4×C8C22×C411yescharacteristic
255C4×C8C2×D412yes—
256C4×C8C2×D412yes—
257C4×C8C2×D412yes—
258C4×C8C2×D412yes—
259C24#12111yescharacteristic
260#41—44no—
261#41—44no—
262#41—84no—
263#41—84no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4C2×C22⋊C412yes—
285C22×C4C4×D412yes—
286C22×C4C4×D412yes—
287C22×C4#12111yescharacteristic
288C22×C4#12112yes—
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12412yes—
292C22×C4#12712yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4111yescharacteristic
296Q16—82no—
297SD16—84no—
298D8—82no—
299M16—44no—
300M16—44no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8C2×C22⋊C412yes—
310C2×C8C2×C22⋊C414yes—
311C2×C8C2×C22⋊C412yes—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C42—24no—
334C42—24no—
335C42—24no—
336C42—24no—
337C42—44no—
338C42—42no—
339C42—42no—
340C42—44no—
341C42—44no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42C2×C22⋊C412yes—
346C42C4×D412yes—
347C42C4×D412yes—
348C42#12112yes—
349C42#12412yes—
350C42#12712yes—
351C42C23×C411yescharacteristic
352C42C22×D411yescharacteristic
353C42C22×D411yescharacteristic
354C42C2×#4111yescharacteristic
355C23—22no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23#39012yes—
360C23C2×#12111yescharacteristic
361Q8—44no—
362Q8—44no—
363Q8—44no—
364D4—44no—
365D4—44no—
366D4—44no—
367D4—84no—
368D4—84no—
369D4—84no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—416no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4#39012yes—
396C2×C4#39012yes—
397C2×C4#39012yes—
398C2×C4C22×C22⋊C411yescharacteristic
399C2×C4C2×C4×D411yescharacteristic
400C2×C4C2×C4×D411yescharacteristic
401C2×C4C2×#12111yescharacteristic
402C2×C4C2×#12411yescharacteristic
403C2×C4C2×#12711yescharacteristic
404C2×C4#57514yes—
405C8—22no—
406C8—24no—
407C8—22no—
408C8—42no—
409C8—42no—
410C8—82no—
411C8—82no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—44no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22C2×#39011yescharacteristic
422C22#305412yes—
423C4—21no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—21no—
428C4—44no—
429C4—44no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4#290112yes—
434C4#294312yes—
435C4#295612yes—
436C4#315012yes—
437C2—22no—
438C2—42no—
439C2—162no—
440C2#2121011yescharacteristic
441C2#2122811yescharacteristic
442C2#2124111yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60944C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21415C2semidirect
3. #20463C2semidirect
4. #17815C2semidirect
5. #17489C2semidirect
6. #9854C2semidirect
7. #9853C2semidirect
8. #9852C2semidirect
9. #9851C2semidirect
10. #9850C2semidirect
11. #9849C2semidirect
14. #3231C4semidirect
15. #3228C4semidirect
16. #3210C4semidirect
17. #3207C4semidirect
71. #500C2×C4semidirect
72. #500C2×C4semidirect
73. #498C2×C4semidirect
74. #494C2×C4semidirect
75. #493C2×C4semidirect
76. #492C2×C4semidirect
77. #487C2×C4semidirect
78. #487C2×C4semidirect
79. #486C2×C4semidirect
80. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#609442
#214152
#204632
#178152
#174892
#98542
#98532
#98522
#98512
#98502
#98492