Finite Group Databasebeta

← #190972

Subgroups — #190972

Order 512.

919 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34151C211yesmaximal, characteristic
2#21431C211yesmaximal, characteristic, complemented
3#20786C211yesmaximal, characteristic, complemented
4#8708C212yesmaximal, complemented
5#8570C212yesmaximal, complemented
6#8565C212yesmaximal, complemented
7#8315C212yesmaximal
8#8181C211yesmaximal, characteristic, complemented
9#8179C211yesmaximal, characteristic, complemented
10#8173C211yesmaximal, characteristic, complemented
11#8171C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13#2374—22no—
14#2373—22no—
15#2370C412yes—
16#2367C412yes—
17#2363C412yes—
18#2362C412yes—
19#2298—21no—
20#2295C2211yescharacteristic
21#2293—21no—
22#2293C2211yescharacteristic
23#2290—21no—
24C42×Q8C2211yescharacteristic
25C42×D4C2211yescharacteristic
26C4×#117—21no—
27C4×#117C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1924—22no—
30#1917—22no—
31#1840—21no—
32#1831C2212yes—
33#1818—21no—
34#1797C2212yes—
35#1767C2211yescharacteristic
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2212yes—
38#1750—21no—
39C4×C22⋊C8C2212yes—
40#1746—21no—
41#1744C2211yescharacteristic
42C4×#97C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#529—24no—
52#529—24no—
53C2×#126—21no—
54C2×#125C2×C411yescharacteristic
55C2×#124C2×C411yescharacteristic
56C2×#123—21no—
57C2×C22⋊Q8C2×C411yescharacteristic
58C2×#121C2×C411yescharacteristic
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2311yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2311yescharacteristic
64C2×#117—21no—
65C2×#117—21no—
66C2×#117C2×C412yes—
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C2×C4⋊C8—22no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×#97—22no—
77C2×#97—21no—
78C2×C4×C8—21no—
79C2×C4×C8C2312yes—
80C2×C4×C8C2312yes—
81#391C2×C412yes—
82#390—22no—
83#390C2×C412yes—
84#389—22no—
85#389C2×C412yes—
86#389C2×C412yes—
87#388—22no—
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#386—22no—
98#385—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4C2311yescharacteristic
110C4×C4⋊C4C2311yescharacteristic
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4D412yes—
120#376—21no—
121#376—21no—
122#376D411yescharacteristic
123#376D411yescharacteristic
124C43—21no—
125C43—21no—
126C43C2311yescharacteristic
127C43C2311yescharacteristic
128#336—22no—
129#336—22no—
130#336—21no—
131#336—21no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136Q8⋊C8—24no—
137Q8⋊C8—24no—
138D4⋊C8—24no—
139D4⋊C8—24no—
140C22×Q8C22×C411yescharacteristic
141C22×D4—21no—
142C22×D4C22×C411yescharacteristic
143C23×C4—21no—
144C23×C4—22no—
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152#126—28no—
153#125—22no—
154#125—22no—
155#124—22no—
156#124—22no—
157#123—28no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160#121—22no—
161#121—22no—
162C4×Q8—22no—
163C4×Q8—24no—
164C4×Q8—42no—
165C4×Q8#4114yes—
166C4×D4—28no—
167C4×D4—24no—
168C4×D4—22no—
169C4×D4—42no—
170C4×D4#4114yes—
171#117—28no—
172#117—24no—
173#117—41no—
174#117—41no—
175#117—42no—
176#117—44no—
177#117—42no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C412yes—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42C22×C412yes—
215C2×C42C22×C412yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2411yescharacteristic
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
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—21no—
248#95—21no—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253C24—21no—
254C24C2×C22⋊C411yescharacteristic
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8#11712yes—
259C2×D4—24no—
260C2×D4—21no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—82no—
264C2×D4#11712yes—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—82no—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C2×C8—24no—
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—
307C2×C8—42no—
308C2×C8—42no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—82no—
323C4⋊C4#11712yes—
324C4⋊C4#11712yes—
325C4⋊C4#13114yes—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—82no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—22no—
341C42—24no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—22no—
351C42—22no—
352C42—42no—
353C42—42no—
354C42—44no—
355C42—44no—
356C42—44no—
357C42—41no—
358C42—41no—
359C42—42no—
360C42—82no—
361C42—82no—
362C42C2×#4112yes—
363C42C2×#4112yes—
364C23—22no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—22no—
369C23—21no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—42no—
374C23C22×C22⋊C411yescharacteristic
375Q8—22no—
376Q8—22no—
377Q8—42no—
378D4—24no—
379D4—41no—
380D4—41no—
381D4—84no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—82no—
423C2×C4—88no—
424C2×C4C2×#11711yescharacteristic
425C2×C4C2×#11711yescharacteristic
426C2×C4#52512yes—
427C2×C4#52512yes—
428C2×C4C2×#13112yes—
429C2×C4#56812yes—
430C2×C4#57511yescharacteristic
431C2×C4#57511yescharacteristic
432C2×C4#57512yes—
433C8—44no—
434C8—82no—
435C8—82no—
436C8—82no—
437C22—41no—
438C22—42no—
439C22—41no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—41no—
444C22—42no—
445C22—82no—
446C22—82no—
447C22#228011yescharacteristic
448C22#288012yes—
449C22#289611yescharacteristic
450C22#289611yescharacteristic
451C22#290112yes—
452C4—22no—
453C4—41no—
454C4—42no—
455C4—41no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—44no—
460C4—42no—
461C4—42no—
462C4—41no—
463C4—42no—
464C4—82no—
465C4#292211yescharacteristic
466C4#292311yescharacteristic
467C4#292512yes—
468C4#311112yes—
469C4#311112yes—
470C2—81no—
471C2—82no—
472C2#1751312yes—
473C2#1751612yes—
474C2#2072011yescharacteristic
475C2#2074211yescharacteristic
476C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×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
2. #21431C2semidirect
3. #20786C2semidirect
4. #8708C2semidirect
5. #8570C2semidirect
6. #8565C2semidirect
8. #8181C2semidirect
9. #8179C2semidirect
10. #8173C2semidirect
11. #8171C2semidirect

Maximal subgroups

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

subgroupindex
#341512
#214312
#207862
#87082
#85702
#85652
#83152
#81812
#81792
#81732
#81712