Finite Group Databasebeta

← #204709

Subgroups — #204709

Order 512.

1098 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic, complemented
3C4×#432C211yesmaximal, characteristic, complemented
4C2×#1535C214yesmaximal, complemented
5#9688C214yesmaximal, complemented
6#9686C212yesmaximal, complemented
7#9685C212yesmaximal, complemented
8C2×#529C2212yes—
9C2×#432C2212yes—
10C2×#432C2212yes—
11#2861C2212yes—
12#2860—21no—
13#2860—21no—
14#2859C2211yescharacteristic
15#2858C2211yescharacteristic
16#2857C411yescharacteristic
17#2856C411yescharacteristic
18#2854—21no—
19#2854—21no—
20#2850C412yes—
21C4×C4⋊Q8C411yescharacteristic
22C4×#127C411yescharacteristic
23C4×#124C412yes—
24C4×#121—21no—
25C4×#121—21no—
26#2293—21no—
27#2293—21no—
28C2×C43C2211yescharacteristic
29#1842—21no—
30#1831—21no—
31#1797—21no—
32#1797—21no—
33#1767—21no—
34#1767C2212yes—
35#1767C2212yes—
36C4×C4⋊C8—21no—
37#1749—21no—
38C4×C22⋊C8—21no—
39#1745C2212yes—
40C4×#97C2212yes—
41#1535—216no—
42C2×#330C2218yes—
43C2×#329C2218yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#127C2×C411yescharacteristic
46#529—28no—
47#529—28no—
48C2×#126—22no—
49C2×#125C2312yes—
50C2×#124C2×C412yes—
51C2×#124C2312yes—
52C2×#123—22no—
53C2×C22⋊Q8C2×C412yes—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121C2×C412yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4C2×C412yes—
62C2×#117—22no—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42C2311yescharacteristic
70#432—24no—
71#432—28no—
72#432D418yescomplemented
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×#97—22no—
80C2×#97C2314yes—
81C2×#97C2314yes—
82C2×C4×C8—22no—
83#391C2×C412yes—
84#391C2×C412yes—
85#390—21no—
86#390—21no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390—21no—
91#390—21no—
92#390—21no—
93#390C2×C412yes—
94#390C2×C412yes—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
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#386—22no—
112#386—22no—
113#384C2×C411yescharacteristic
114#384C2×C411yescharacteristic
115#382C2×C412yes—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C43C2×C411yescharacteristic
123C43C2×C411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128C43C2311yescharacteristic
129C43C2311yescharacteristic
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#330—216no—
135#329—216no—
136C22×Q8C22×C412yes—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C22×C412yes—
141C23×C4—22no—
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C22×C8—24no—
150C22×C8—24no—
151C4⋊Q8—42no—
152#127—42no—
153#126—44no—
154#126—44no—
155#125—24no—
156#125—24no—
157#124—24no—
158#124—24no—
159#124—44no—
160#123—48no—
161C22⋊Q8—28no—
162#121—28no—
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—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
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×C42—22no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
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×D414yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4114yes—
231C2×C42C2411yescharacteristic
232C4⋊C8—28no—
233C4⋊C8—44no—
234C4⋊C8—44no—
235C4⋊C8—44no—
236C22⋊C8—28no—
237C22⋊C8—44no—
238#97—28no—
239#97—24no—
240#97—44no—
241#97C2×D418yes—
242C4×C8—44no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249C24—22no—
250C24C2×C22⋊C411yescharacteristic
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×D4—24no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
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×C4C4×D414yes—
302C22×C4#12114yes—
303C22×C4#12412yes—
304C22×C4#12712yes—
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C2×C8—28no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—44no—
313C2×C8—44no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—44no—
364C42C4×D414yes—
365C42#12114yes—
366C42#12412yes—
367C42#12712yes—
368C42C22×D412yes—
369C42C2×#4112yes—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23#39012yes—
384C23C22×C22⋊C411yescharacteristic
385Q8—84no—
386D4—84no—
387D4—84no—
388D4—88no—
389D4—84no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
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—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—88no—
438C2×C4#39012yes—
439C2×C4#39012yes—
440C2×C4#39012yes—
441C2×C4#41312yes—
442C2×C4#41312yes—
443C2×C4#42112yes—
444C2×C4#42112yes—
445C2×C4#42112yes—
446C2×C4#42112yes—
447C2×C4C2×C4×D412yes—
448C2×C4C2×#12112yes—
449C2×C4C2×#12411yescharacteristic
450C2×C4C2×#12711yescharacteristic
451C8—48no—
452C8—84no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22C2×#39011yescharacteristic
468C22C2×#41311yescharacteristic
469C22C2×#41311yescharacteristic
470C22C2×#42111yescharacteristic
471C22C2×#42111yescharacteristic
472C22C2×#42111yescharacteristic
473C22C2×#42111yescharacteristic
474C4—21no—
475C4—22no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—22no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—82no—
489C4—82no—
490C4—82no—
491C4#153514yescomplemented
492C2—42no—
493C2—162no—
494C2C2×#153511yescharacteristic
495C2#1766511yescharacteristic
496C2#1767011yescharacteristic
497C2#1767711yescharacteristic
498C2#2120911yescharacteristic
499C2#2121811yescharacteristic
500C2#2125211yescharacteristic
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. #20810C2semidirect
3. C4×#432C2semidirect
4. C2×#1535C2semidirect
5. #9688C2semidirect
6. #9686C2semidirect
7. #9685C2semidirect
72. #432D4semidirect
491. C4#1535semidirect

Maximal subgroups

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

subgroupindex
#366312
#208102
C4×#4322
C2×#15352
#96882
#96862
#96852