Finite Group Databasebeta

← #204605

Subgroups — #204605

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2#20463C211yesmaximal, characteristic, complemented
3C2×#1521C214yesmaximal, complemented
4#14688C211yesmaximal, characteristic, complemented
5#9684C212yesmaximal
6#9683C212yesmaximal, complemented
7#9682C214yesmaximal, complemented
8C2×#529C2212yes—
9C2×#431C2212yes—
10#2861C412yescomplemented
11#2860—21no—
12#2860—21no—
13#2859C411yescharacteristic, complemented
14#2858C411yescharacteristic, complemented
15#2857C2211yescharacteristic
16#2856C2211yescharacteristic
17#2854—21no—
18#2854—21no—
19#2850C2212yes—
20C4×C4⋊Q8C411yescharacteristic, complemented
21C4×#127C411yescharacteristic, complemented
22C4×#124C412yescomplemented
23C4×#121—21no—
24C4×#121—21no—
25#2293—21no—
26#2293—21no—
27C2×#383C2212yes—
28C2×C43C2211yescharacteristic
29#1767—21no—
30#1767—21no—
31#1749C2212yes—
32#1749C2212yes—
33#1744—21no—
34C4×#97—21no—
35#1521—216no—
36C2×#328C2218yes—
37C2×#327C2218yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41#1434—21no—
42#1434—21no—
43#1434C2212yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#127C2×C411yescharacteristic
46#529—28no—
47#529—28no—
48C2×#126—22no—
49C2×#125C2×C412yes—
50C2×#124C2×C412yes—
51C2×#124C2×C412yes—
52C2×#123—22no—
53C2×C22⋊Q8C2312yes—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121C2312yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4C2×C412yes—
62C2×#117—22no—
63C22×C4⋊C4—22no—
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#431—24no—
72#431D418yescomplemented
73C2×C4⋊C8—22no—
74C2×C22⋊C8C2314yes—
75C2×#97—22no—
76C2×#97—22no—
77C2×C4×C8—22no—
78#391C2×C412yes—
79#391C2×C412yes—
80#390—21no—
81#390—21no—
82#390—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390—21no—
88#390C2×C412yes—
89#390C2×C412yes—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#386—22no—
107#386—22no—
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#383—28no—
111#382C2×C412yes—
112#380—21no—
113#380—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95C2314yes—
124C43C2×C411yescharacteristic
125C43C2×C411yescharacteristic
126C43C2×C411yescharacteristic
127C43C2×C411yescharacteristic
128C43D411yescharacteristic
129C43D411yescharacteristic
130C43D411yescharacteristic
131C43D411yescharacteristic
132#336—22no—
133#336—22no—
134#328—216no—
135#327—216no—
136C22×Q8C22×C412yes—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C22×C412yes—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—24no—
151C4⋊Q8—42no—
152#127—42no—
153#126—48no—
154#125—28no—
155#124—28no—
156#124—44no—
157#123—44no—
158#123—44no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161#121—24no—
162#121—24no—
163#121—44no—
164#121—44no—
165C4×Q8—28no—
166C4×D4—28no—
167C4×D4—44no—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4—44no—
171#117—48no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C22⋊C4—22no—
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×C42—22no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
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×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D414yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4114yes—
235C4⋊C8—28no—
236C22⋊C8—24no—
237C22⋊C8C2×D418yes—
238#97—44no—
239#97—44no—
240C4×C8—28no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—28no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253C24—22no—
254C24C22×D411yescharacteristic
255C2×Q8—24no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×D4—24no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—42no—
270C2×D4—42no—
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—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
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×C4C2×C22⋊C411yescharacteristic
314C22×C4#12412yes—
315C22×C4#12412yes—
316C22×C4#12712yes—
317C22×C4#12712yes—
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D412yes—
321C22×C4C2×#4112yes—
322C2×C8—28no—
323C2×C8—44no—
324C2×C8—44no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—88no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—44no—
375C42C4×D414yes—
376C42C4×D414yes—
377C42#12114yes—
378C42#12114yes—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23C22×C22⋊C411yescharacteristic
391C23C2×#12411yescharacteristic
392C23C2×#12711yescharacteristic
393C23#57512yes—
394C23#57512yes—
395Q8—84no—
396D4—84no—
397D4—84no—
398D4—84no—
399D4—84no—
400D4—84no—
401C2×C4—28no—
402C2×C4—24no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—24no—
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—21no—
421C2×C4—21no—
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—21no—
435C2×C4—21no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—88no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4#39012yes—
461C2×C4#39012yes—
462C2×C4#39012yes—
463C2×C4#39012yes—
464C2×C4#41012yes—
465C2×C4#41012yes—
466C2×C4#42112yes—
467C2×C4#42112yes—
468C2×C4C2×C4×D412yes—
469C2×C4C2×#12112yes—
470C8—48no—
471C22—21no—
472C22—21no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—21no—
478C22—21no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22C2×#39011yescharacteristic
487C22C2×#41011yescharacteristic
488C22C2×#41011yescharacteristic
489C22#289611yescharacteristic
490C22#289611yescharacteristic
491C22C2×#42111yescharacteristic
492C22C2×#42111yescharacteristic
493C22#313512yes—
494C22#315012yes—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—22no—
504C4—22no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—82no—
510C4—82no—
511C4—82no—
512C4—164no—
513C2—22no—
514C2—22no—
515C2—162no—
516C2#1757911yescharacteristic
517C2#1758011yescharacteristic
518C2#1759311yescharacteristic
519C2#1759411yescharacteristic
520C2#2119911yescharacteristic
521C2#2121211yescharacteristic
522C2#2125211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20463C2semidirect
3. C2×#1521C2semidirect
4. #14688C2semidirect
6. #9683C2semidirect
7. #9682C2semidirect
10. #2861C4semidirect
13. #2859C4semidirect
14. #2858C4semidirect
20. C4×C4⋊Q8C4semidirect
21. C4×#127C4semidirect
22. C4×#124C4semidirect
72. #431D4semidirect

Maximal subgroups

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

subgroupindex
#366312
#204632
C2×#15212
#146882
#96842
#96832
#96822