Finite Group Databasebeta

← #162604

Subgroups — #162604

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic, complemented
3C4×#423C211yesmaximal, characteristic, complemented
4#7685C212yesmaximal, complemented
5#7684C212yesmaximal
6#7672C214yesmaximal, complemented
7#7614C214yesmaximal, complemented
8C2×#529—21no—
9C2×#423—21no—
10C2×C4×M16—21no—
11#2861—21no—
12#2860—21no—
13#2860C2212yes—
14#2859C411yescharacteristic
15#2858C411yescharacteristic
16#2857C2211yescharacteristic
17#2856C2211yescharacteristic
18#2854—21no—
19#2854C412yes—
20#2850—21no—
21C4×C4⋊Q8C411yescharacteristic
22C4×#127C411yescharacteristic
23C4×#124—21no—
24C4×#121—21no—
25C4×#121C412yes—
26#2293—21no—
27#2293C2212yes—
28C2×C43C2211yescharacteristic
29#1839C2212yes—
30#1831C2212yes—
31#1817—21no—
32#1797—21no—
33#1767—21no—
34#1767C2212yes—
35C4×C4⋊C8—21no—
36C4×C4⋊C8—21no—
37C4×C4⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39#1744—21no—
40#1728C2212yes—
41#1271C2218yes—
42#1270C2218yes—
43C22×M16—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#127C2×C411yescharacteristic
46#529—44no—
47#529—44no—
48C2×#126—22no—
49C2×#125—21no—
50C2×#124—21no—
51C2×#124—21no—
52C2×#123—22no—
53C2×C22⋊Q8—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121C2×C412yes—
57C2×C4×Q8—21no—
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#423—28no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8—22no—
75C4×M16—44no—
76C2×#97—22no—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79#391—21no—
80#391—21no—
81#390—21no—
82#390—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390C2×C412yes—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2312yes—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2312yes—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#386—22no—
108#386—22no—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#382—21no—
112#380—21no—
113#380C2×C412yes—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43C2311yescharacteristic
125C43C2311yescharacteristic
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336C2314yes—
131#336C2314yes—
132C22×Q8—21no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C22×C412yes—
137C23×C4—22no—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C2×M16—28no—
146C2×M16—44no—
147C2×M16—44no—
148C22×C8—24no—
149C22×C8—24no—
150C4⋊Q8—42no—
151#127—42no—
152#126—44no—
153#126—44no—
154#125—44no—
155#124—44no—
156#124—44no—
157#123—48no—
158C22⋊Q8—42no—
159C22⋊Q8—42no—
160#121—44no—
161#121—42no—
162#121—42no—
163#121—44no—
164C4×Q8—44no—
165C4×D4—44no—
166C4×D4—44no—
167C4×D4—48no—
168C4×D4—44no—
169#117—44no—
170#117—44no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
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—22no—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C22⋊C4—22no—
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×C42—24no—
195C2×C42—22no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42C22⋊C411yescharacteristic
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×C42D814yes—
214C2×C42SD1614yes—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
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×C42C2×D411yescharacteristic
230C2×C42C2411yescharacteristic
231C4⋊C8—28no—
232C4⋊C8—28no—
233C4⋊C8—44no—
234C22⋊C8—44no—
235#97—44no—
236C4×C8—44no—
237C4×C8—44no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244C24—22no—
245C24C2×C22⋊C411yescharacteristic
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×D4—24no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4#10214yes—
286C22×C4#10214yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
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×D812yes—
299C22×C4C2×SD1612yes—
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303M16—416no—
304C2×C8—28no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C4⋊C4—24no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—44no—
359C42#10214yes—
360C42#10214yes—
361C42#10414yes—
362C42C2×D812yes—
363C42C2×D812yes—
364C42C2×SD1612yes—
365C42C2×SD1612yes—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23C2×#10212yes—
380C23C22×C22⋊C411yescharacteristic
381Q8—84no—
382D4—84no—
383D4—84no—
384D4—88no—
385D4—84no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
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—24no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—88no—
432C2×C4#41012yes—
433C2×C4#41312yes—
434C2×C4C2×#10212yes—
435C2×C4C2×#10212yes—
436C2×C4C2×#10212yes—
437C2×C4C2×#10212yes—
438C2×C4C2×#10212yes—
439C2×C4C2×#10412yes—
440C2×C4C2×#10412yes—
441C2×C4C2×#10412yes—
442C2×C4#42112yes—
443C2×C4#56812yes—
444C2×C4C22×D811yescharacteristic
445C2×C4C22×SD1611yescharacteristic
446C2×C4#57512yes—
447C8—48no—
448C8—84no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22#287911yescharacteristic
464C22C2×#41011yescharacteristic
465C22C2×#41311yescharacteristic
466C22C22×#10211yescharacteristic
467C22#289611yescharacteristic
468C22C22×#10411yescharacteristic
469C22C2×#42111yescharacteristic
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—21no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—82no—
485C4—82no—
486C4—82no—
487C4#150214yes—
488C2—42no—
489C2—162no—
490C2#1739911yescharacteristic
491C2#1748511yescharacteristic
492C2#1750311yescharacteristic
493C2C2×#150211yescharacteristic
494C2#1757911yescharacteristic
495C2#1764911yescharacteristic
496C2#1770411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C22
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. #20777C2semidirect
3. C4×#423C2semidirect
4. #7685C2semidirect
6. #7672C2semidirect
7. #7614C2semidirect

Maximal subgroups

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

subgroupindex
#366312
#207772
C4×#4232
#76852
#76842
#76722
#76142