Finite Group Databasebeta

← #160646

Subgroups — #160646

Order 512.

874 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34317C211yesmaximal, characteristic
2#22734C211yesmaximal, characteristic, complemented
3#22018C211yesmaximal, characteristic, complemented
4#7602C212yesmaximal, complemented
5#7594C212yesmaximal
6#7591C211yesmaximal, characteristic, complemented
7#7585C212yesmaximal, complemented
8#7574C212yesmaximal
9#7574C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic
12C2×#554C411yescharacteristic
13C2×C8×Q8—21no—
14#2370C412yes—
15#2369C411yescharacteristic
16#2362—22no—
17#2362—21no—
18#2357C412yes—
19#2355C411yescharacteristic
20#2354C411yescharacteristic
21#2344—21no—
22#2332—22no—
23#2323—21no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×#125C2212yes—
26#2293C2211yescharacteristic
27#2293C2212yes—
28#2274—21no—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1939—22no—
33#1939—21no—
34#1916—22no—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38#1831C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2212yes—
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#554C818yescomplemented
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×C411yescharacteristic
64C2×C4×D4C2×C412yes—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—22no—
68C22×C4⋊C4—21no—
69C8×Q8—44no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4×C8—22no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80#391—21no—
81#391—21no—
82#391—22no—
83#390C2×C412yes—
84#390C2×C411yescharacteristic
85#390C2311yescharacteristic
86#389—21no—
87#389—22no—
88#389—21no—
89#389C2×C412yes—
90#389C2×C411yescharacteristic
91#389C2311yescharacteristic
92#389C2312yes—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—21no—
97#388—22no—
98#388—21no—
99#388—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#384C2×C411yescharacteristic
106#382C2×C412yes—
107#381—22no—
108#380—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—22no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—22no—
129C43C2312yes—
130C43C2311yescharacteristic
131#336—22no—
132#336—22no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—21no—
144C22×Q8C22⋊C411yescharacteristic
145C22×D4C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C4⋊Q8—22no—
155C4⋊Q8C2×C814yes—
156#125—24no—
157#125C2×C818yes—
158C22⋊Q8—28no—
159C22⋊Q8—216no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—42no—
163C4×Q8—81no—
164C4×Q8—81no—
165C4×Q8C2×C814yes—
166C4×Q8M1614yes—
167C4×D4—24no—
168C4×D4—22no—
169C4×D4C2×C818yes—
170C4×D4C2×C814yes—
171#117—44no—
172#117—42no—
173#117—42no—
174#117—82no—
175#117—82no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C412yes—
212C2×C42C22×C412yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2411yescharacteristic
223C4⋊C8—42no—
224C4⋊C8—44no—
225C4⋊C8—44no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8—82no—
229C4×C8—42no—
230C4×C8—44no—
231C4×C8—42no—
232C4×C8—81no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247C24—21no—
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—81no—
252C2×Q8—81no—
253C2×Q8C22⋊C814yes—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4C22×C812yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—41no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—28no—
301C4⋊C4—24no—
302C4⋊C4—28no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—82no—
317C4⋊C4—81no—
318C4⋊C4C22⋊C814yes—
319C4⋊C4C22⋊C818yes—
320C4⋊C4C22×C812yes—
321C4⋊C4C22×C814yes—
322C22⋊C4—24no—
323C22⋊C4—28no—
324C22⋊C4—44no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—81no—
331C22⋊C4—82no—
332C42—22no—
333C42—22no—
334C42—21no—
335C42—22no—
336C42—22no—
337C42—21no—
338C42—22no—
339C42—42no—
340C42—41no—
341C42—41no—
342C42—42no—
343C42—42no—
344C42—41no—
345C42—44no—
346C42—42no—
347C42—82no—
348C42—82no—
349C42—82no—
350C42C22×C814yes—
351C42C22×C812yes—
352C42C2×M1614yes—
353C42C2×M1612yes—
354C42AGL(1,ℤ/8)12yes—
355C42#13712yes—
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23C22×C22⋊C411yescharacteristic
364Q8—28no—
365Q8—84no—
366D4—42no—
367D4—42no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—48no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—88no—
408C2×C4C2×C22⋊C814yes—
409C2×C4C2×C22⋊C812yes—
410C2×C4C2×C22⋊C814yes—
411C2×C4C2×C22⋊C812yes—
412C2×C4#41712yes—
413C2×C4#41712yes—
414C2×C4#42112yes—
415C2×C4C23×C811yescharacteristic
416C2×C4C22×M1611yescharacteristic
417C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
418C2×C4C2×#13711yescharacteristic
419C2×C4#57611yescharacteristic
420C2×C4#57711yescharacteristic
421C8—42no—
422C8—81no—
423C8—82no—
424C8—81no—
425C8—82no—
426C8—81no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22C22×C22⋊C811yescharacteristic
435C22#288411yescharacteristic
436C22#288911yescharacteristic
437C22C2×#41711yescharacteristic
438C22#289711yescharacteristic
439C22C2×#42111yescharacteristic
440C22#290111yescharacteristic
441C4—24no—
442C4—24no—
443C4—22no—
444C4—22no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—41no—
452C4—42no—
453C4—42no—
454C4—44no—
455C4—42no—
456C4—81no—
457C4—82no—
458C4—81no—
459C4#147714yes—
460C2—82no—
461C2#1735211yescharacteristic
462C2C2×#147711yescharacteristic
463C2#1743811yescharacteristic
464C2#1760211yescharacteristic
465C2#1762811yescharacteristic
466C2#1768211yescharacteristic
467C2#1776511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C2×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. #22734C2semidirect
3. #22018C2semidirect
4. #7602C2semidirect
6. #7591C2semidirect
7. #7585C2semidirect
10. #7570C2semidirect
54. #554C8semidirect

Maximal subgroups

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

subgroupindex
#343172
#227342
#220182
#76022
#75942
#75912
#75852
#75742
#75702
#75602