Finite Group Databasebeta

← #190968

Subgroups — #190968

Order 512.

924 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34145C211yesmaximal, characteristic
2#21431C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#8706C212yesmaximal
5#8568C212yesmaximal, complemented
6#8564C212yesmaximal, complemented
7#8314C212yesmaximal
8#8180C211yesmaximal, characteristic, complemented
9#8178C211yesmaximal, characteristic, complemented
10#8173C211yesmaximal, characteristic, complemented
11#8171C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×#433—21no—
14#2370—22no—
15#2369C412yes—
16#2368—22no—
17#2365C412yes—
18#2360C412yes—
19#2360C412yes—
20#2296—21no—
21#2295—21no—
22#2295—21no—
23#2295C2211yescharacteristic
24#2295C2211yescharacteristic
25C42×Q8C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27C4×#117—21no—
28C4×#117C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1924—22no—
31#1917—22no—
32#1842C2212yes—
33#1832—21no—
34#1798—21no—
35#1797C2212yes—
36#1768—21no—
37#1767C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41#1744C2211yescharacteristic
42C4×#97C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#533—24no—
52#533—24no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C411yescharacteristic
58C2×C22⋊Q8C2×C411yescharacteristic
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2311yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×#117—21no—
65C2×#117D411yescharacteristic
66C2×#117D411yescharacteristic
67C22×C4⋊C4C2×C412yes—
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#433—44no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×#97—21no—
79C2×#97—22no—
80C2×C4×C8—21no—
81C2×C4×C8C2312yes—
82C2×C4×C8C2312yes—
83#391—22no—
84#391—22no—
85#391C2×C412yes—
86#391C2×C412yes—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389C2×C412yes—
92#389C2×C412yes—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#385—22no—
100#385—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4C2311yescharacteristic
117C4×C4⋊C4C2311yescharacteristic
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4C2×C412yes—
121C4×C22⋊C4D412yes—
122#376—21no—
123#376—21no—
124#376D411yescharacteristic
125#376D411yescharacteristic
126C2×#95—21no—
127C43—21no—
128C43—21no—
129C43C2311yescharacteristic
130C43C2311yescharacteristic
131#336—22no—
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137Q8⋊C8—24no—
138Q8⋊C8—24no—
139Q8⋊C8—24no—
140Q8⋊C8—24no—
141C22×Q8—21no—
142C22×Q8C22×C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C23×C4—22no—
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156#125—28no—
157C22⋊Q8—28no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C4×Q8—28no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8#4114yes—
170C4×Q8#4114yes—
171#117—28no—
172#117—22no—
173#117—22no—
174#117—24no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42C22×C412yes—
213C2×C42C22×C412yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—44no—
229C4⋊C8—44no—
230C4⋊C8—44no—
231C22⋊C8—44no—
232C22⋊C8—44no—
233#97—42no—
234#97—44no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—22no—
239C4×C8—42no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—21no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255C24C2×C22⋊C411yescharacteristic
256C2×Q8—24no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—82no—
264C2×Q8#11712yes—
265C2×Q8#11712yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—82no—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C412yes—
290C22×C4AGL(1,ℤ/8)12yes—
291C22×C4#13712yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4#11712yes—
327C4⋊C4#11712yes—
328C4⋊C4#13114yes—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C42—21no—
334C42—21no—
335C42—22no—
336C42—22no—
337C42—24no—
338C42—24no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—21no—
344C42—21no—
345C42—22no—
346C42—21no—
347C42—21no—
348C42—42no—
349C42—42no—
350C42—44no—
351C42—44no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—44no—
357C42—82no—
358C42—82no—
359C42#13714yes—
360C42C2×#4112yes—
361C42C2×#4112yes—
362C23—22no—
363C23—21no—
364C23—22no—
365C23—21no—
366C23#41712yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—24no—
371Q8—41no—
372Q8—41no—
373Q8—42no—
374Q8—84no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—82no—
413C2×C4—88no—
414C2×C4#41712yes—
415C2×C4#41914yes—
416C2×C4C2×#11711yescharacteristic
417C2×C4C2×#11711yescharacteristic
418C2×C4#52512yes—
419C2×C4#52512yes—
420C2×C4C2×#13112yes—
421C2×C4#56812yes—
422C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
423C2×C4C2×#13712yes—
424C2×C4C2×#13711yescharacteristic
425C8—44no—
426C8—82no—
427C8—82no—
428C8—82no—
429C22—22no—
430C22—24no—
431C22—41no—
432C22—42no—
433C22—41no—
434C22#228011yescharacteristic
435C22#288012yes—
436C22C2×#41911yescharacteristic
437C22C2×#41711yescharacteristic
438C22#290112yes—
439C4—22no—
440C4—22no—
441C4—24no—
442C4—41no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—44no—
448C4—42no—
449C4—41no—
450C4—42no—
451C4—42no—
452C4—82no—
453C4#292211yescharacteristic
454C4#292311yescharacteristic
455C4#292512yes—
456C4#311212yes—
457C4#311312yes—
458C2—42no—
459C2#1751212yes—
460C2#1751412yes—
461C2#2072011yescharacteristic
462C2#2074011yescharacteristic
463C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8568C2semidirect
6. #8564C2semidirect
8. #8180C2semidirect
9. #8178C2semidirect
10. #8173C2semidirect
11. #8171C2semidirect

Maximal subgroups

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

subgroupindex
#341452
#214312
#208122
#87062
#85682
#85642
#83142
#81802
#81782
#81732
#81712