Finite Group Databasebeta

← #195373

Subgroups — #195373

Order 512.

1171 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3103C211yesmaximal, characteristic
2#60237C211yesmaximal, characteristic
3#21820C211yesmaximal, characteristic, complemented
4#17717C211yesmaximal, characteristic
5#17449C211yesmaximal, characteristic, complemented
6#9047C218yesmaximal, complemented
7#8076C211yesmaximal, characteristic, complemented
8#7922C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#3103—22no—
12#3103—22no—
13#3103C414yes—
14#3103C2214yes—
15C2×#481C412yes—
16C2×#475—21no—
17C2×#460—21no—
18C2×#448C412yes—
19#2981—22no—
20#2980—22no—
21#2974—22no—
22#2973—22no—
23C2×C4×SD16—22no—
24#2935—22no—
25#2932C2214yes—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2880—22no—
32#1985C2214yescomplemented
33#1984—22no—
34#1817—22no—
35#1553—22no—
36#1549—21no—
37#1545—21no—
38#1540C2214yes—
39#1491—22no—
40#1487C2214yescomplemented
41#1479—22no—
42#1301C2214yescomplemented
43#1281C2214yescomplemented
44C22×#41—21no—
45C23×Q8C2×C411yescharacteristic
46C22×SD16—22no—
47C2×#131—24no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532—21no—
52#532C2×C412yes—
53#532D412yes—
54#532D412yes—
55#532C2312yes—
56C2×C4⋊Q8D411yescharacteristic
57C2×#127D411yescharacteristic
58C2×#124—21no—
59C2×#121—21no—
60C2×#121C2×C412yes—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8C2×C412yes—
72C2×C4×Q8C2312yes—
73C2×C4×D4—21no—
74C2×C4×D4D412yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79#481—24no—
80#481C2×C418yes—
81#475—24no—
82#475—24no—
83#460—24no—
84#460—24no—
85#448—24no—
86#448C2×C418yes—
87C4×SD16—28no—
88C4×SD16—28no—
89#436—24no—
90#435—28no—
91#435—28no—
92#435—28no—
93C8×D4—24no—
94#432—21no—
95#432—21no—
96#432—21no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432—21no—
101#432D412yes—
102#431—21no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431C2×C412yes—
111#431D412yes—
112#431C2312yes—
113#431C2312yes—
114C2×#106—22no—
115#423—21no—
116#423C2312yes—
117C2×C4⋊C8—21no—
118C2×C4⋊C8D412yes—
119C2×C4⋊C8C2312yes—
120C2×#103—22no—
121C2×#102—22no—
122#408—28no—
123#408—24no—
124#407—24no—
125#407—21no—
126#407—21no—
127C2×C22⋊C8—24no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8C2×C412yes—
130C2×C22⋊C8D412yes—
131C4×M16—21no—
132C4×M16—21no—
133C2×#97C2312yes—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136#336—28no—
137#332—24no—
138#331—24no—
139Q8⋊C8—28no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—22no—
144C2×#41—84no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—44no—
152C22×Q8C22×C412yes—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4C22⋊C412yes—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C2×SD16—28no—
164C2×SD16—48no—
165#131—88no—
166C2×M16—22no—
167C2×M16—28no—
168C2×M16—41no—
169C22×C8—22no—
170C22×C8—28no—
171C22×C8—22no—
172C22×C8C2×D414yes—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8C22⋊C412yes—
176C4⋊Q8C2×D412yes—
177#127—21no—
178#127—21no—
179#127C22⋊C412yes—
180#127C2×D412yes—
181#124—22no—
182#124—22no—
183#124—22no—
184#124—22no—
185#121—22no—
186#121—22no—
187#121—22no—
188#121—22no—
189#121—22no—
190#121—22no—
191#121C22⋊C414yes—
192#121C22×C414yes—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4C22⋊C414yes—
216C4×D4C2×D414yes—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42C2411yescharacteristic
251#106—28no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—41no—
261C4⋊C8C2×D414yes—
262#103—28no—
263#102—28no—
264C22⋊C8—28no—
265C22⋊C8—28no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—41no—
273C22⋊C8—41no—
274C22⋊C8—41no—
275C22⋊C8C22×C414yes—
276#97—22no—
277#97—22no—
278#97—22no—
279#97—41no—
280C4×C8—22no—
281C4×C8—22no—
282C4×C8—22no—
283C4×C8—22no—
284C4×C8—41no—
285C4×C8—41no—
286C24—22no—
287C24C4×D411yescharacteristic
288#41—48no—
289#41—84no—
290#41—84no—
291#41—84no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—44no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—82no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—44no—
316C2×D4—82no—
317C2×D4—82no—
318C2×D4—84no—
319C2×D4C2×C22⋊C414yes—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—44no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—84no—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12311yescharacteristic
349C22×C4#12311yescharacteristic
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4111yescharacteristic
353SD16—416no—
354M16—44no—
355M16—88no—
356M16—82no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—28no—
361C2×C8—44no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—44no—
365C2×C8—88no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4C2×C22⋊C414yes—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42C2×C22⋊C411yescharacteristic
387C42C2×C22⋊C411yescharacteristic
388C42C4×D411yescharacteristic
389C42C4×D411yescharacteristic
390C42#12011yescharacteristic
391C42#12011yescharacteristic
392C42#12111yescharacteristic
393C42#12111yescharacteristic
394C42#12111yescharacteristic
395C42#12111yescharacteristic
396C42#12311yescharacteristic
397C42#12311yescharacteristic
398C42C22×D411yescharacteristic
399C42C22×D411yescharacteristic
400C42C22×D411yescharacteristic
401C42C2×#4111yescharacteristic
402C23—22no—
403C23—22no—
404C23—24no—
405C23—41no—
406C23—44no—
407C23—82no—
408C23—82no—
409C23#38611yescharacteristic
410C23#38611yescharacteristic
411C23C2×C4×D411yescharacteristic
412Q8—24no—
413Q8—24no—
414Q8—42no—
415Q8—42no—
416Q8—44no—
417Q8—44no—
418Q8—44no—
419Q8—82no—
420D4—44no—
421D4—48no—
422D4—44no—
423D4—44no—
424D4—82no—
425D4—82no—
426D4—82no—
427D4—84no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—48no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—84no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#38611yescharacteristic
461C2×C4#38611yescharacteristic
462C2×C4#38611yescharacteristic
463C2×C4#38611yescharacteristic
464C2×C4C22×C22⋊C411yescharacteristic
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C2×#12011yescharacteristic
467C2×C4C2×#12111yescharacteristic
468C2×C4C2×#12111yescharacteristic
469C2×C4C2×#12311yescharacteristic
470C2×C4#57714yes—
471C8—44no—
472C8—44no—
473C8—44no—
474C8—82no—
475C8—82no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—44no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—164no—
488C22#188112yes—
489C22C2×#38611yescharacteristic
490C22#294912yes—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—44no—
501C4—44no—
502C4—82no—
503C4#189112yes—
504C4#300912yes—
505C4#304212yes—
506C2—22no—
507C2—22no—
508C2—41no—
509C2—84no—
510C2#2093311yescharacteristic
511C2C2×#188111yescharacteristic
512C2C2×#189111yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ 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
3. #21820C2semidirect
5. #17449C2semidirect
6. #9047C2semidirect
7. #8076C2semidirect
8. #7922C2semidirect
32. #1985C22semidirect
40. #1487C22semidirect
42. #1301C22semidirect
43. #1281C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31032
#602372
#218202
#177172
#174492
#90472
#80762
#79222