Finite Group Databasebeta

← #162485

Subgroups — #162485

Order 512.

823 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35209C211yesmaximal, characteristic
2#22828C211yesmaximal, characteristic, complemented
3#20570C211yesmaximal, characteristic, complemented
4#7712C212yesmaximal
5#7703C212yesmaximal, complemented
6#7692C212yesmaximal
7#7683C212yesmaximal, complemented
8#7588C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic
12C2×#541—21no—
13C2×#446—21no—
14#2860C2212yes—
15#2859C2212yes—
16#2664—22no—
17#2552C411yescharacteristic
18#2546C411yescharacteristic
19#2542C411yescharacteristic
20#2538C411yescharacteristic
21#2519—22no—
22#2512C412yes—
23#2506C412yes—
24#2489—21no—
25#2429—21no—
26#2323—21no—
27C4×C4⋊Q8C2211yescharacteristic
28#2293C2211yescharacteristic
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1916—22no—
32#1839C2212yes—
33#1831C2211yescharacteristic
34#1771—22no—
35#1770—21no—
36C8×C4⋊C4—21no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1745C2212yes—
41#1744C2212yes—
42#1744C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53#541—44no—
54#541—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#126—22no—
58C2×#126—22no—
59C2×#125—21no—
60C2×#125—22no—
61C2×#125—22no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×#117—21no—
71#446—28no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×#97—22no—
77C2×#97—22no—
78C2×C4×C8—21no—
79C2×C4×C8—22no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#400—22no—
83#399—22no—
84#398—22no—
85#398—22no—
86#398—22no—
87#397—22no—
88#396—22no—
89#396—22no—
90#391—21no—
91#391—21no—
92#390C2×C412yes—
93#390C2×C412yes—
94#390C2×C411yescharacteristic
95#390C2311yescharacteristic
96#389—21no—
97#389—21no—
98#389—22no—
99#389—22no—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2×C412yes—
104#389C2×C411yescharacteristic
105#389C2312yes—
106#389C2311yescharacteristic
107#388—22no—
108#388—22no—
109#388—22no—
110#388—21no—
111#388—21no—
112#388—22no—
113#387—22no—
114#384C2×C411yescharacteristic
115#384C2×C412yes—
116#384C2×C412yes—
117#382—22no—
118#380—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D412yes—
128C4×C22⋊C4—21no—
129C43C2311yescharacteristic
130C43C2312yes—
131#336—21no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—21no—
136#336—22no—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143C22×Q8—21no—
144C22×Q8C22⋊C411yescharacteristic
145C22×D4C22×C411yescharacteristic
146C23×C4—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C4⋊Q8—24no—
154C4⋊Q8—44no—
155#126—44no—
156#126—44no—
157#126—48no—
158#125—44no—
159#125—44no—
160#125—44no—
161#125—48no—
162C22⋊Q8—44no—
163C22⋊Q8—44no—
164C22⋊Q8—44no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×D4—24no—
172#117—42no—
173#117—42no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42C22×C412yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2411yescharacteristic
223C4⋊C8—24no—
224C4⋊C8—28no—
225C4⋊C8—42no—
226C4⋊C8—42no—
227#97—28no—
228#97—82no—
229C4×C8—24no—
230C4×C8—44no—
231C4×C8—42no—
232C4×C8—81no—
233#95—21no—
234#95—21no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—22no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245C24—21no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—24no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—42no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—81no—
317C4⋊C4—82no—
318C4⋊C4#10414yes—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—44no—
328C22⋊C4—81no—
329C22⋊C4—82no—
330C42—24no—
331C42—22no—
332C42—22no—
333C42—21no—
334C42—22no—
335C42—21no—
336C42—22no—
337C42—22no—
338C42—24no—
339C42—42no—
340C42—41no—
341C42—41no—
342C42—42no—
343C42—41no—
344C42—42no—
345C42—44no—
346C42#13114yes—
347C42AGL(1,ℤ/8)12yes—
348C42AGL(1,ℤ/8)12yes—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23C22×C22⋊C411yescharacteristic
357Q8—44no—
358Q8—84no—
359D4—82no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—84no—
400C2×C4#40812yes—
401C2×C4#40812yes—
402C2×C4#41812yes—
403C2×C4#41812yes—
404C2×C4C2×#10414yes—
405C2×C4C2×#10412yes—
406C2×C4C2×#13112yes—
407C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
408C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
409C2×C4#57711yescharacteristic
410C2×C4#57811yescharacteristic
411C8—42no—
412C8—44no—
413C8—42no—
414C8—42no—
415C8—81no—
416C8—82no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22C2×#40811yescharacteristic
425C22#288411yescharacteristic
426C22#288911yescharacteristic
427C22C2×#41811yescharacteristic
428C22#289811yescharacteristic
429C22C22×#10411yescharacteristic
430C22#290111yescharacteristic
431C4—22no—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—22no—
438C4—24no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4—41no—
443C4—81no—
444C4—82no—
445C4—81no—
446C4—81no—
447C4—82no—
448C4#148414yes—
449C2—161no—
450C2#1739111yescharacteristic
451C2C2×#148411yescharacteristic
452C2#1748411yescharacteristic
453C2#1754911yescharacteristic
454C2#1763111yescharacteristic
455C2#1765311yescharacteristic
456C2#1775911yescharacteristic
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. #22828C2semidirect
3. #20570C2semidirect
5. #7703C2semidirect
7. #7683C2semidirect
8. #7588C2semidirect
10. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#352092
#228282
#205702
#77122
#77032
#76922
#76832
#75882
#75742
#75702
#75602