Finite Group Databasebeta

← #162232

Subgroups — #162232

Order 512.

1076 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2861C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic, complemented
3C4×#432C211yesmaximal, characteristic, complemented
4#7696C214yesmaximal, complemented
5#7676C214yesmaximal, complemented
6#7575C212yesmaximal, complemented
7#7572C212yesmaximal
8C22×#125C411yescharacteristic
9C22×#124C411yescharacteristic
10C2×#432—21no—
11C2×#431—21no—
12#2861—21no—
13#2861—21no—
14#2861—21no—
15#2861—21no—
16#2861C412yes—
17#2861C412yes—
18#2861C2212yes—
19#2861C2212yes—
20C2×#391C411yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#390C411yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#388—21no—
25C2×#388—21no—
26C2×#388—21no—
27C2×#388—21no—
28C2×C43C2211yescharacteristic
29#1842—21no—
30#1831C2212yescomplemented
31#1797—21no—
32#1797C2212yes—
33#1767C2212yescomplemented
34C4×C4⋊C8—21no—
35C4×C4⋊C8—21no—
36C4×C4⋊C8C2212yescomplemented
37#1749—21no—
38C4×C22⋊C8C2212yes—
39#1745C2212yescomplemented
40C4×#97—21no—
41#1272C2218yescomplemented
42#1270C2218yes—
43C23×Q8C2×C411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C2×#125—21no—
46C2×#125—21no—
47C2×#125—44no—
48C2×#125C2×C412yes—
49C2×#125C2×C412yes—
50C2×#124—21no—
51C2×#124—21no—
52C2×#124—44no—
53C2×#124C2×C412yes—
54C2×#124C2×C412yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72#432—28no—
73#431—44no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×#97—22no—
81C2×#97—22no—
82C2×C4×C8—22no—
83C2×C4×C8—22no—
84#391—21no—
85#391—21no—
86#391—21no—
87#391—21no—
88#391C2×C412yes—
89#391C2×C412yes—
90#391C2×C412yes—
91#391C2312yes—
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390C2×C412yes—
97#390C2×C412yes—
98#390C2×C412yes—
99#390C2312yes—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C43C2×C411yescharacteristic
121C43C2×C411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43C2311yescharacteristic
127C43C2311yescharacteristic
128#336—22no—
129#336—22no—
130#336C2314yes—
131#336C2314yes—
132C25—21no—
133C22×Q8—21no—
134C22×Q8—24no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8C22×C412yes—
138C22×D4—21no—
139C22×D4—24no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4C22×C412yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C22×C8—24no—
154C22×C8—24no—
155#125—44no—
156#125—44no—
157#125—44no—
158#125—44no—
159#124—44no—
160#124—44no—
161#124—44no—
162#124—44no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2411yescharacteristic
230C4⋊C8—28no—
231C4⋊C8—44no—
232C4⋊C8—44no—
233C4⋊C8—44no—
234C22⋊C8—28no—
235C22⋊C8—44no—
236#97—28no—
237#97—44no—
238C4×C8—44no—
239C4×C8—44no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248C24—22no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24C2×C22⋊C411yescharacteristic
257C2×Q8—24no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—88no—
264C2×D4—24no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—88no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4#10414yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4#13114yes—
314C22×C4AGL(1,ℤ/8)12yes—
315C22×C4#13712yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C2×C8—28no—
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—44no—
323C2×C8—44no—
324C2×C8—44no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C42—22no—
347C42—22no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42#10014yes—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—84no—
388C23—84no—
389C23—84no—
390C23—84no—
391C23—84no—
392C23—84no—
393C23—84no—
394C23#40812yes—
395C23#41712yes—
396C23C2×#10412yes—
397C23C22×C22⋊C411yescharacteristic
398Q8—84no—
399Q8—84no—
400D4—84no—
401D4—84no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4#40812yes—
453C2×C4#41012yes—
454C2×C4C2×#10012yes—
455C2×C4C2×#10012yes—
456C2×C4C2×#10012yes—
457C2×C4#41712yes—
458C2×C4C2×#10412yes—
459C2×C4C2×#10412yes—
460C2×C4#42112yes—
461C2×C4C2×#13112yes—
462C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
463C2×C4C2×#13711yescharacteristic
464C2×C4#57512yes—
465C8—48no—
466C8—84no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—168no—
481C22#148314yes—
482C22C2×#40811yescharacteristic
483C22C2×#41011yescharacteristic
484C22C22×#10011yescharacteristic
485C22C2×#41711yescharacteristic
486C22#289611yescharacteristic
487C22C22×#10411yescharacteristic
488C22C2×#42111yescharacteristic
489C4—22no—
490C4—22no—
491C4—22no—
492C4—22no—
493C4—22no—
494C4—21no—
495C4—21no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4#146914yes—
507C2—24no—
508C2—162no—
509C2C2×#146911yescharacteristic
510C2C2×#148311yescharacteristic
511C2#1748011yescharacteristic
512C2#1753711yescharacteristic
513C2#1758111yescharacteristic
514C2#1764711yescharacteristic
515C2#1775011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20808C2semidirect
3. C4×#432C2semidirect
4. #7696C2semidirect
5. #7676C2semidirect
6. #7575C2semidirect
30. #1831C22semidirect
33. #1767C22semidirect
36. C4×C4⋊C8C22semidirect
39. #1745C22semidirect
41. #1272C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
#208082
C4×#4322
#76962
#76762
#75752
#75722