Finite Group Databasebeta

← #160908

Subgroups — #160908

Order 512.

882 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34317C211yesmaximal, characteristic
2#22736C211yesmaximal, characteristic, complemented
3#20570C211yesmaximal, characteristic, complemented
4#7636C212yesmaximal, complemented
5#7628C212yesmaximal
6#7619C212yesmaximal, complemented
7#7614C212yesmaximal
8#7591C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic
12C2×#554C411yescharacteristic
13C2×#446—21no—
14C2×#446—21no—
15#2370C412yes—
16#2369C411yescharacteristic
17#2362—22no—
18#2362—21no—
19#2357C412yes—
20#2355C411yescharacteristic
21#2354C411yescharacteristic
22#2344—21no—
23#2332—22no—
24#2323—21no—
25C4×C4⋊Q8C2211yescharacteristic
26C4×#125C2212yes—
27#2293C2211yescharacteristic
28#2293C2212yes—
29#2274—21no—
30C42×Q8C2211yescharacteristic
31#1940—22no—
32#1939—21no—
33#1831C2211yescharacteristic
34#1831C2212yes—
35#1771—22no—
36#1770—21no—
37#1767C2212yes—
38#1767C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1745C2212yes—
42#1744C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2211yescharacteristic
45#1271C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53#554—24no—
54#554—24no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—22no—
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C411yescharacteristic
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—22no—
68C22×C4⋊C4—21no—
69#446—28no—
70#446—44no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—21no—
77C2×#97—22no—
78C2×#97—22no—
79C2×#97—22no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83#391—21no—
84#391—21no—
85#391—22no—
86#390C2×C412yes—
87#390C2×C411yescharacteristic
88#390C2311yescharacteristic
89#389—21no—
90#389—22no—
91#389—21no—
92#389C2×C412yes—
93#389C2×C411yescharacteristic
94#389C2311yescharacteristic
95#389C2312yes—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—21no—
100#388—22no—
101#388—21no—
102#388—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#384C2×C411yescharacteristic
109#382C2×C412yes—
110#381—22no—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C4×C22⋊C4—22no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—22no—
132C43C2312yes—
133C43C2311yescharacteristic
134#336—22no—
135#336—21no—
136#336—21no—
137#336—22no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2311yescharacteristic
144C22×Q8—21no—
145C22×Q8C22⋊C411yescharacteristic
146C22×D4C22×C411yescharacteristic
147C23×C4—21no—
148C23×C4—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C4⋊Q8—22no—
156C4⋊Q8M1614yes—
157#125—24no—
158#125—24no—
159C22⋊Q8—28no—
160C22⋊Q8—216no—
161C4×Q8—24no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—81no—
167C4×Q8—81no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—22no—
171C4×D4M1614yes—
172#117—44no—
173#117—42no—
174#117—42no—
175#117—82no—
176#117—82no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42C22×C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C412yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224C4⋊C8—24no—
225C4⋊C8—28no—
226C4⋊C8—44no—
227C4⋊C8—44no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230#97—28no—
231#97—44no—
232#97—82no—
233C4×C8—24no—
234C4×C8—42no—
235C4×C8—81no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250C24—21no—
251C2×Q8—22no—
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—81no—
256C2×Q8—81no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4C2×M1612yes—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—24no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—42no—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—82no—
321C4⋊C4—81no—
322C4⋊C4#10414yes—
323C4⋊C4C2×M1614yes—
324C4⋊C4C2×M1612yes—
325C22⋊C4—28no—
326C22⋊C4—24no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—81no—
334C22⋊C4—82no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—21no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—22no—
344C42—24no—
345C42—42no—
346C42—41no—
347C42—41no—
348C42—42no—
349C42—41no—
350C42—42no—
351C42—44no—
352C42—82no—
353C42—82no—
354C42—82no—
355C42C2×M1612yes—
356C42C2×M1612yes—
357C42AGL(1,ℤ/8)12yes—
358C42#13712yes—
359C23—22no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23C22×C22⋊C411yescharacteristic
367Q8—44no—
368Q8—84no—
369D4—42no—
370D4—42no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—48no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—88no—
412C2×C4#40712yes—
413C2×C4#40712yes—
414C2×C4#41712yes—
415C2×C4#41712yes—
416C2×C4C2×#10414yes—
417C2×C4C2×#10412yes—
418C2×C4C22×M1611yescharacteristic
419C2×C4C22×M1611yescharacteristic
420C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
421C2×C4C2×#13711yescharacteristic
422C2×C4#57611yescharacteristic
423C2×C4#57711yescharacteristic
424C8—42no—
425C8—44no—
426C8—42no—
427C8—42no—
428C8—81no—
429C8—82no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22C2×#40711yescharacteristic
438C22#288411yescharacteristic
439C22#288911yescharacteristic
440C22C2×#41711yescharacteristic
441C22#289711yescharacteristic
442C22C22×#10411yescharacteristic
443C22#290111yescharacteristic
444C4—22no—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—24no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—44no—
457C4—42no—
458C4—41no—
459C4—81no—
460C4—82no—
461C4—81no—
462C4#148914yes—
463C2—82no—
464C2#1737111yescharacteristic
465C2C2×#148911yescharacteristic
466C2#1745311yescharacteristic
467C2#1755011yescharacteristic
468C2#1762811yescharacteristic
469C2#1765311yescharacteristic
470C2#1776511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22736C2semidirect
3. #20570C2semidirect
4. #7636C2semidirect
6. #7619C2semidirect
8. #7591C2semidirect
10. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#343172
#227362
#205702
#76362
#76282
#76192
#76142
#75912
#75742
#75702
#75602