Finite Group Databasebeta

← #481933

Subgroups — #481933

Order 512.

965 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35701C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic, complemented
3#17370C211yesmaximal, characteristic, complemented
4#11827C214yesmaximal, complemented
5#11821C214yesmaximal, complemented
6#11718C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×#554C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2975—21no—
11C2×#410—21no—
12#2633C2212yes—
13#2628C2211yescharacteristic
14#2611C412yescomplemented
15#2598C412yescomplemented
16#2542C411yescharacteristic, complemented
17#2535C411yescharacteristic, complemented
18#2463—22no—
19#2449—22no—
20#2423—21no—
21#2407—21no—
22#2397C2212yes—
23#2392C2211yescharacteristic
24#2361—21no—
25#2356C411yescharacteristic, complemented
26D4×C4⋊C4—21no—
27#2316—21no—
28#1799—22no—
29#1799C2214yes—
30#1740C2212yes—
31#1465C2214yes—
32#1463—22no—
33#1461—21no—
34#1458C2212yes—
35#1352C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C22×#41D411yescharacteristic
39#554—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#125C2×C412yes—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×#121—21no—
46#520—44no—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×D4—21no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4C2×C412yes—
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C2×C4×D4D411yescharacteristic
57C2×C4×D4C2312yes—
58C2×#117—22no—
59C2×#117C2×C412yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D412yes—
64#436—44no—
65#431—22no—
66#410—44no—
67C2×#99—24no—
68#407—24no—
69#400C2×C412yes—
70#399—22no—
71#399C2×C412yes—
72#399C2×C412yes—
73#399C2312yes—
74#398—22no—
75#398C2×C411yescharacteristic
76#397C2×C412yes—
77#397C2311yescharacteristic
78#396—21no—
79#395—21no—
80#394—22no—
81#394—21no—
82#394C2×C411yescharacteristic
83#393—21no—
84#392—22no—
85#392C2×C412yes—
86#392C2311yescharacteristic
87#391—21no—
88#390—21no—
89#389—22no—
90#389—21no—
91#389—44no—
92#388—22no—
93#388—22no—
94#387—22no—
95#387—22no—
96#386—22no—
97#386—22no—
98#386—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#384—21no—
104#383C2×C412yes—
105#382C2×C411yescharacteristic
106#380—21no—
107#380—21no—
108#380—24no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C411yescharacteristic
111C4×C22⋊C4—21no—
112#376C2×C412yes—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#324—24no—
117#323C2314yes—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—42no—
121C2×#41—42no—
122C22×Q8—21no—
123C22×Q8C22⋊C411yescharacteristic
124C22×D4—21no—
125C22×D4—22no—
126C22×D4C22⋊C411yescharacteristic
127C22×D4C22×C411yescharacteristic
128C22×D4C2×D412yes—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C2×M16—44no—
137C4⋊Q8—42no—
138#125—44no—
139C22⋊Q8—48no—
140C22⋊Q8—44no—
141C22⋊Q8—84no—
142#121—44no—
143C4×Q8—24no—
144C4×Q8—42no—
145C4×D4—24no—
146C4×D4—42no—
147C4×D4—44no—
148C4×D4—44no—
149C4×D4—42no—
150C4×D4—44no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—84no—
154C4×D4—84no—
155#117—42no—
156#117—44no—
157#117—44no—
158#117—44no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D412yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C22×C412yes—
196C2×C22⋊C4C2×D411yescharacteristic
197C2×C22⋊C4C2411yescharacteristic
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—42no—
201C2×C42C22⋊C412yes—
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C4⋊C8—42no—
206#99—48no—
207C22⋊C8—24no—
208C22⋊C8—44no—
209C4×C8—42no—
210#95—22no—
211#95—21no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—44no—
220#95C22×C411yescharacteristic
221#95C22×C412yes—
222#95C22×C411yescharacteristic
223C24—21no—
224C24C2×C22⋊C412yes—
225C24C22×D411yescharacteristic
226#41—48no—
227#41—48no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—42no—
232C2×Q8—82no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—82no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4AGL(1,ℤ/8)12yes—
280C22×C4#13712yes—
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D411yescharacteristic
283M16—164no—
284C2×C8—44no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—84no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—164no—
304C4⋊C4#9914yes—
305C22⋊C4—24no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C42—22no—
324C42—24no—
325C42—22no—
326C42—42no—
327C42—44no—
328C42—42no—
329C42—84no—
330C23—21no—
331C23—21no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—42no—
337C23—41no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23C22×C22⋊C411yescharacteristic
347C23C2×AGL(1,ℤ/8)11yescharacteristic
348C23C2×#13711yescharacteristic
349Q8—44no—
350Q8—162no—
351D4—44no—
352D4—44no—
353D4—42no—
354D4—44no—
355D4—44no—
356D4—42no—
357D4—44no—
358D4—88no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—42no—
373C2×C4—48no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4—164no—
400C2×C4—162no—
401C2×C4C2×#9914yes—
402C2×C4C2×#9912yes—
403C2×C4#41712yes—
404C2×C4#41712yes—
405C2×C4#42112yes—
406C8—84no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—41no—
412C22—41no—
413C22—42no—
414C22—42no—
415C22—44no—
416C22—41no—
417C22—41no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—81no—
423C22—82no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22C22×#9911yescharacteristic
429C22C2×#41711yescharacteristic
430C22C2×#42111yescharacteristic
431C4—24no—
432C4—22no—
433C4—42no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—42no—
438C4—41no—
439C4—42no—
440C4—81no—
441C4—82no—
442C4—82no—
443C4—81no—
444C4—164no—
445C4#151314yes—
446C2—21no—
447C2—21no—
448C2—41no—
449C2—41no—
450C2—82no—
451C2—82no—
452C2C2×#151311yescharacteristic
453C2#3334911yescharacteristic
454C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35701C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20634C2semidirect
3. #17370C2semidirect
4. #11827C2semidirect
5. #11821C2semidirect
6. #11718C2semidirect
7. #11691C2semidirect
8. C2×#554C4semidirect
14. #2611C4semidirect
15. #2598C4semidirect
16. #2542C4semidirect
17. #2535C4semidirect
25. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#357012
#206342
#173702
#118272
#118212
#117182
#116912