Finite Group Databasebeta

← #6379827

Subgroups — #6379827

Order 512.

1928 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2924C212yesmaximal
2C2×#2911C211yesmaximal, characteristic
3#22608C218yesmaximal, complemented
4#22607C214yesmaximal, complemented
5#22606C214yesmaximal, complemented
6#21164C212yesmaximal, complemented
7#21041C214yesmaximal, complemented
8#20705C212yesmaximal
9#20667C211yesmaximal, characteristic
10#20444C212yesmaximal, complemented
11#20389C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2212yes—
14#2924—22no—
15#2924—22no—
16#2924—24no—
17#2924C2214yes—
18#2924C2214yes—
19#2924C2218yes—
20C2×#433—24no—
21C2×#432—22no—
22C2×#431—22no—
23C2×#430C2212yes—
24#2911C414yescomplemented
25#2911C414yescomplemented
26#2911C2212yes—
27#2911C2214yes—
28#2911C2211yescharacteristic
29#2911C2211yescharacteristic
30C2×#428C414yescomplemented
31C2×#428C414yescomplemented
32C2×#427C2214yes—
33C2×#424C2214yes—
34C2×#407C2212yes—
35C22×C22⋊C8C2211yescharacteristic
36C2×#405C2214yes—
37C2×#405C2212yes—
38#2280C2211yescharacteristic
39#1946C2218yescomplemented
40#1945C2218yescomplemented
41#1922C22116yescomplemented
42#1922C22116yescomplemented
43#1826C2218yes—
44#1826C2218yes—
45#1811C2214yes—
46#1810C2214yes—
47#1773C2218yescomplemented
48#1772C2218yescomplemented
49#1753C2214yes—
50#1752C2212yes—
51C8×C22⋊C4C2212yes—
52#1738C2212yes—
53#1733C2214yes—
54#1733C2214yes—
55#1729C2212yes—
56C24×C4C2311yescharacteristic
57C22×M16—28no—
58C22×M16C2×C414yes—
59C22×M16C2×C414yes—
60C22×M16C2314yes—
61C22×M16C2312yes—
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C23×C8C2312yes—
65C23×C8C2312yes—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117C2×C414yes—
73C2×#117C2×C412yes—
74C2×#117C2×C412yes—
75C2×#117C2×C414yes—
76C2×#117C2314yes—
77C2×#117C2314yes—
78C2×#117C2312yes—
79C2×#117C2312yes—
80C2×#117C2312yes—
81C22×C4⋊C4C2×C412yes—
82C22×C4⋊C4C2×C412yes—
83C22×C22⋊C4—22no—
84C22×C42—22no—
85#433—28no—
86#433—28no—
87#433—28no—
88#433—28no—
89#432—24no—
90#432—24no—
91#432—28no—
92#431—24no—
93#431—24no—
94#431—28no—
95#430—28no—
96C2×#108C2318yes—
97#428C2×C414yes—
98#428C2×C414yes—
99#428C2×C414yes—
100#428C2×C414yes—
101#428C2×C418yes—
102#428C2×C418yes—
103#428C2×C418yes—
104#428C2×C418yes—
105#428C2×C414yes—
106#428C2×C414yes—
107#428C2×C414yes—
108#428C2×C414yes—
109#427C2×C414yes—
110#427C2×C414yes—
111#427C2×C418yes—
112#427C2318yes—
113#427C2314yes—
114#427C2314yes—
115C2×#107C2×C414yes—
116C2×#107C2×C414yes—
117C2×#106C2×C418yes—
118#424—24no—
119#424—24no—
120#424C2318yes—
121#424C2318yes—
122C2×C4⋊C8—24no—
123C2×C4⋊C8—24no—
124#407—22no—
125#407—22no—
126#407C2314yes—
127#407C2314yes—
128C2×C22⋊C8—22no—
129C2×C22⋊C8—22no—
130C2×C22⋊C8D412yes—
131C2×C22⋊C8D412yes—
132C2×C22⋊C8D414yes—
133C2×C22⋊C8C2311yescharacteristic
134C2×C22⋊C8C2311yescharacteristic
135C2×C22⋊C8C2312yes—
136#405—28no—
137#405—24no—
138#405—24no—
139#405C2318yes—
140#405C2318yes—
141C2×#97—24no—
142C2×#97C2314yes—
143C2×C4×C8—24no—
144C2×C4×C8C2314yes—
145#381—21no—
146#380—21no—
147#380—21no—
148#379—21no—
149C4×C22⋊C4C2312yes—
150C4×C22⋊C4C2312yes—
151#341C2318yes—
152#339C23116yes—
153#339C23116yes—
154#336C2314yes—
155#336C2314yes—
156C25#4111yescharacteristic
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4C22×C412yes—
162C23×C4C22×C412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4#4111yescharacteristic
166C23×C4#4111yescharacteristic
167C23×C4#4111yescharacteristic
168C23×C4C2411yescharacteristic
169C23×C4C2411yescharacteristic
170C23×C4C2411yescharacteristic
171C2×M16—24no—
172C2×M16—216no—
173C2×M16—216no—
174C2×M16—24no—
175C2×M16—28no—
176C2×M16—28no—
177C2×M16—28no—
178C2×M16C22×C414yes—
179C2×M16C22×C418yes—
180C2×M16C22×C414yes—
181C2×M16#4118yes—
182C2×M16C2418yes—
183C22×C8—24no—
184C22×C8—24no—
185C22×C8—28no—
186C22×C8—24no—
187C22×C8C22×C414yes—
188C22×C8C22×C414yes—
189C22×C8C2×D414yes—
190C22×C8C2×D414yes—
191C22×C8C2412yes—
192C22×C8C2412yes—
193C22×C8C2412yes—
194C22×C8C2412yes—
195#117—22no—
196#117—22no—
197#117—24no—
198#117—24no—
199#117—24no—
200#117—24no—
201#117—22no—
202#117—22no—
203#117—22no—
204#117—22no—
205#117—24no—
206#117—24no—
207#117—24no—
208#117—24no—
209#117—22no—
210#117—22no—
211#117—24no—
212#117—24no—
213#117C22×C414yes—
214#117C22×C414yes—
215#117C22×C414yes—
216#117C22×C414yes—
217#117C22×C414yes—
218#117C22×C414yes—
219#117C2414yes—
220#117C2414yes—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C414yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42C2414yes—
256#108—216no—
257#107C22×C418yes—
258#107C22×C418yes—
259#107C22×C414yes—
260#107C22×C414yes—
261#107C22×C414yes—
262#107C22×C414yes—
263#106C22×C418yes—
264#106C22×C418yes—
265#106C22×C418yes—
266#106C22×C418yes—
267C4⋊C8—28no—
268C4⋊C8—28no—
269C4⋊C8—28no—
270C4⋊C8—28no—
271C22⋊C8—24no—
272C22⋊C8—24no—
273C22⋊C8—24no—
274C22⋊C8—24no—
275C22⋊C8C2×D411yescharacteristic
276C22⋊C8C2×D412yes—
277C22⋊C8C2×D414yes—
278C22⋊C8C2×D411yescharacteristic
279C22⋊C8C2×D414yes—
280C22⋊C8C2×D412yes—
281C22⋊C8C2×D412yes—
282#97—28no—
283#97—28no—
284#97—28no—
285C4×C8—28no—
286C4×C8—28no—
287C4×C8—28no—
288#95—22no—
289#95—22no—
290C24—24no—
291C24—42no—
292C24—81no—
293C24C4×D412yes—
294C24C4×D412yes—
295C24C2×#4111yescharacteristic
296C24C2×#4111yescharacteristic
297C24C2×#4111yescharacteristic
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—48no—
321C22×C4—84no—
322C22×C4C4×D412yes—
323C22×C4C4×D412yes—
324C22×C4C4×D412yes—
325C22×C4C4×D412yes—
326C22×C4C4×D412yes—
327C22×C4C4×D412yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C23×C412yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C23×C412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D412yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D412yes—
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4114yes—
339C22×C4C2×#4111yescharacteristic
340C22×C42−1+414yes—
341C22×C4C2511yescharacteristic
342M16—28no—
343M16—216no—
344M16—28no—
345M16—28no—
346M16—48no—
347M16#117116yes—
348C2×C8—28no—
349C2×C8—28no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—28no—
356C2×C8—28no—
357C2×C8—28no—
358C2×C8—48no—
359C2×C8C23×C414yes—
360C2×C8C23×C414yes—
361C2×C8C22×D412yes—
362C2×C8C22×D414yes—
363C2×C8C22×D412yes—
364C2×C8C2×#4118yes—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—44no—
374C4⋊C4C23×C414yes—
375C4⋊C4C23×C414yes—
376C4⋊C4C23×C414yes—
377C4⋊C4C23×C414yes—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—82no—
389C42—24no—
390C42—24no—
391C42—24no—
392C42—24no—
393C42—24no—
394C42—24no—
395C42—24no—
396C42—24no—
397C42—24no—
398C42—24no—
399C23—24no—
400C23—22no—
401C23—24no—
402C23—22no—
403C23—24no—
404C23—22no—
405C23—42no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—41no—
412C23—42no—
413C23—88no—
414C23C2×C4×D411yescharacteristic
415C23C2×C4×D412yes—
416C23C2×C4×D412yes—
417C23C2×C4×D411yescharacteristic
418C23#55214yes—
419C23C22×#4111yescharacteristic
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—48no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4C2×#11714yes—
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×C4×D412yes—
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×C4×D412yes—
457C2×C4#51914yes—
458C2×C4#54718yes—
459C2×C4#55214yes—
460C2×C4C24×C411yescharacteristic
461C2×C4C23×D411yescharacteristic
462C2×C4C22×#4112yes—
463C2×C4C2×2−1+412yes—
464C8—24no—
465C8—22no—
466C8—22no—
467C8—24no—
468C8—48no—
469C8C2×#11718yes—
470C22—22no—
471C22—21no—
472C22—22no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—21no—
478C22—22no—
479C22—21no—
480C22—21no—
481C22—22no—
482C22—44no—
483C22—44no—
484C22—48no—
485C22—82no—
486C22—84no—
487C22#234414yes—
488C22#295512yes—
489C22C22×C4×D411yescharacteristic
490C22C2×#55212yes—
491C4—21no—
492C4—21no—
493C4—22no—
494C4—44no—
495C4—44no—
496C4—44no—
497C4—44no—
498C4—84no—
499C4#234414yes—
500C4C22×#11711yescharacteristic
501C4C2×#51911yescharacteristic
502C4C2×#54712yes—
503C2—22no—
504C2—22no—
505C2—24no—
506C2—42no—
507C2—41no—
508C2C2×#234411yescharacteristic
509C2C2×#295511yescharacteristic
510C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radicalC25#41
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22608C2semidirect
4. #22607C2semidirect
5. #22606C2semidirect
6. #21164C2semidirect
7. #21041C2semidirect
10. #20444C2semidirect
24. #2911C4semidirect
25. #2911C4semidirect
30. C2×#428C4semidirect
31. C2×#428C4semidirect
39. #1946C22semidirect
40. #1945C22semidirect
41. #1922C22semidirect
42. #1922C22semidirect
47. #1773C22semidirect
48. #1772C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29242
C2×#29112
#226082
#226072
#226062
#211642
#210412
#207052
#206672
#204442
#203892