Finite Group Databasebeta

← #204674

Subgroups — #204674

Order 512.

1176 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C4×#432C211yesmaximal, characteristic, complemented
3C2×#1767C211yesmaximal, characteristic, complemented
4C2×#1532C214yesmaximal, complemented
5#9688C214yesmaximal, complemented
6#9686C212yesmaximal, complemented
7#9685C212yesmaximal, complemented
8C2×#529C2212yes—
9C2×#432C2212yes—
10C22×C4⋊C8—21no—
11C22×#97C2212yes—
12#2861C2212yes—
13#2860—21no—
14#2860—21no—
15#2859C2211yescharacteristic
16#2858C2211yescharacteristic
17#2857C411yescharacteristic
18#2856C411yescharacteristic
19#2854—21no—
20#2854—21no—
21#2850C412yes—
22C4×C4⋊Q8C411yescharacteristic
23C4×#127C411yescharacteristic
24C4×#124C412yes—
25C4×#121—21no—
26C4×#121—21no—
27#2293—21no—
28#2293—21no—
29C2×C43C2211yescharacteristic
30#1767—21no—
31#1767—21no—
32#1767—21no—
33#1767C2212yes—
34#1767C2212yes—
35C4×C4⋊C8—21no—
36#1749—21no—
37C4×C22⋊C8—21no—
38#1745C2212yes—
39C2×#336—21no—
40C4×#97C2212yes—
41#1532—216no—
42C2×#330C2218yes—
43C2×#329C2218yes—
44C23×C8—22no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#127C2×C411yescharacteristic
47#529—28no—
48#529—28no—
49C2×#126—22no—
50C2×#125C2312yes—
51C2×#124C2×C412yes—
52C2×#124C2312yes—
53C2×#123—22no—
54C2×C22⋊Q8C2×C412yes—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121C2×C412yes—
58C2×C4×Q8C2×C412yes—
59C2×C4×D4—22no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C2×#117—22no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42C2311yescharacteristic
71#432—24no—
72#432D418yescomplemented
73C2×C4⋊C8—28no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×#97—28no—
79C2×#97—22no—
80C2×#97—22no—
81C2×#97C2314yes—
82C2×#97C2314yes—
83C2×C4×C8—22no—
84#391C2×C412yes—
85#391C2×C412yes—
86#390—21no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390—21no—
91#390—21no—
92#390—21no—
93#390—21no—
94#390C2×C412yes—
95#390C2×C412yes—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
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#386—22no—
113#386—22no—
114#384C2×C411yescharacteristic
115#384C2×C411yescharacteristic
116#382C2×C412yes—
117#380—21no—
118#380—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C43C2×C411yescharacteristic
124C43C2×C411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128C43D411yescharacteristic
129C43C2311yescharacteristic
130C43C2311yescharacteristic
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#330—216no—
136#329—216no—
137C22×Q8C22×C412yes—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4C22×C412yes—
142C23×C4—22no—
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C22×C8—28no—
151C22×C8—24no—
152C22×C8—24no—
153C22×C8—44no—
154C22×C8—44no—
155C4⋊Q8—42no—
156#127—42no—
157#126—48no—
158#125—24no—
159#125—24no—
160#124—24no—
161#124—24no—
162#124—44no—
163#123—44no—
164#123—44no—
165C22⋊Q8—28no—
166#121—28no—
167#121—44no—
168#121—44no—
169C4×Q8—28no—
170C4×D4—28no—
171C4×D4—44no—
172C4×D4—44no—
173C4×D4—48no—
174#117—44no—
175#117—44no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C42—24no—
200C2×C42—22no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
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×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D414yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4114yes—
235C2×C42C2411yescharacteristic
236C4⋊C8—28no—
237C4⋊C8—28no—
238C4⋊C8—28no—
239C22⋊C8—28no—
240#97—28no—
241#97—24no—
242#97—44no—
243#97—44no—
244#97C2×D418yes—
245C4×C8—44no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252C24—22no—
253C24C2×C22⋊C411yescharacteristic
254C2×Q8—24no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×D4—24no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C22×C4—22no—
270C22×C4—22no—
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—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4#10414yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C4×D414yes—
305C22×C4#12114yes—
306C22×C4#12412yes—
307C22×C4#12712yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C2×C8—28no—
312C2×C8—28no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—416no—
317C2×C8—44no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C22⋊C4—44no—
332C22⋊C4—44no—
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—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—44no—
368C42C4×D414yes—
369C42#12114yes—
370C42#12412yes—
371C42#12712yes—
372C42C22×D412yes—
373C42C2×#4112yes—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23#39012yes—
386C23C2×#10412yes—
387C23#42112yes—
388C23C22×C22⋊C411yescharacteristic
389Q8—84no—
390D4—84no—
391D4—84no—
392D4—88no—
393D4—84no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
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—24no—
413C2×C4—24no—
414C2×C4—28no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—88no—
442C2×C4#39012yes—
443C2×C4#39012yes—
444C2×C4#39012yes—
445C2×C4#41312yes—
446C2×C4#41312yes—
447C2×C4C2×#10412yes—
448C2×C4C2×#10412yes—
449C2×C4#42112yes—
450C2×C4#42112yes—
451C2×C4C2×C4×D412yes—
452C2×C4C2×#12112yes—
453C2×C4C2×#12411yescharacteristic
454C2×C4C2×#12711yescharacteristic
455C8—48no—
456C8—48no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22#198014yes—
471C22C2×#39011yescharacteristic
472C22C2×#41311yescharacteristic
473C22C2×#41311yescharacteristic
474C22C22×#10411yescharacteristic
475C22C2×#42111yescharacteristic
476C22C2×#42111yescharacteristic
477C22C2×#42111yescharacteristic
478C4—21no—
479C4—21no—
480C4—22no—
481C4—22no—
482C4—22no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—82no—
493C4—82no—
494C4—82no—
495C4#153214yescomplemented
496C2—24no—
497C2—162no—
498C2C2×#153211yescharacteristic
499C2#1764911yescharacteristic
500C2#1766811yescharacteristic
501C2#1767711yescharacteristic
502C2#2120911yescharacteristic
503C2#2121811yescharacteristic
504C2C2×#198011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#432C2semidirect
3. C2×#1767C2semidirect
4. C2×#1532C2semidirect
5. #9688C2semidirect
6. #9686C2semidirect
7. #9685C2semidirect
72. #432D4semidirect
495. C4#1532semidirect

Maximal subgroups

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

subgroupindex
#366312
C4×#4322
C2×#17672
C2×#15322
#96882
#96862
#96852