Finite Group Databasebeta

← #190356

Subgroups — #190356

Order 512.

1027 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34123C211yesmaximal, characteristic
2C4×#410C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#8777C214yesmaximal, complemented
5#8776C214yesmaximal, complemented
6#8048C212yesmaximal, complemented
7#7940C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2920C2212yes—
10C2×#410—21no—
11#2356C411yescharacteristic, complemented
12#2356C411yescharacteristic, complemented
13#2353C2212yes—
14#2353C2212yes—
15#2349C411yescharacteristic, complemented
16#2349C411yescharacteristic, complemented
17#2346—21no—
18#2343—22no—
19#2339—21no—
20#2295—21no—
21#2290—22no—
22#2283C412yescomplemented
23#2283C2211yescharacteristic
24C42×D4—21no—
25C4×#117C2212yes—
26C4×#117C2211yescharacteristic
27C2×#376C412yescomplemented
28C4×#99—22no—
29C4×#99C2214yes—
30#1740—21no—
31#1465C2214yes—
32#1464—22no—
33#1461—21no—
34#1460—21no—
35#1297C2214yes—
36#1297C2214yes—
37#1284C2218yes—
38C22×#41D411yescharacteristic
39C4×#41—22no—
40C4×#41—22no—
41C4×#41—22no—
42C4×#41—22no—
43C2×C4×Q8C2×C411yescharacteristic
44C2×C4×Q8D411yescharacteristic
45C2×C4×D4—21no—
46C2×C4×D4—21no—
47C2×C4×D4C2×C411yescharacteristic
48C2×C4×D4D412yes—
49C2×C4×D4D411yescharacteristic
50C2×C4×D4C2312yes—
51C2×#117—21no—
52C2×#117—22no—
53C2×#117D411yescharacteristic
54C2×#117D412yes—
55C22×C42—22no—
56C22×C42C2×C412yes—
57C22×C42C2311yescharacteristic
58#432—22no—
59#432—22no—
60#431—22no—
61#431—22no—
62#423—22no—
63#410—28no—
64C2×#99—24no—
65#407C2314yes—
66C4×M16—22no—
67#391—21no—
68#391—21no—
69#389—21no—
70#389—21no—
71#388—22no—
72#388—22no—
73#387—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77#386—22no—
78#386—22no—
79#385—22no—
80#385—22no—
81#383—21no—
82#383C2×C412yes—
83#383C2×C412yes—
84#382—22no—
85#382C2×C411yescharacteristic
86#382C2×C411yescharacteristic
87#381—22no—
88#381C2312yes—
89#380—21no—
90#380C2×C412yes—
91#380C2×C412yes—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4C2×C411yescharacteristic
101C4×C22⋊C4—22no—
102C4×C22⋊C4—21no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4C2314yes—
108C4×C22⋊C4C2311yescharacteristic
109C4×C22⋊C4C2311yescharacteristic
110#376—22no—
111#376—22no—
112#376—22no—
113#376C2×C412yes—
114#376C2×C412yes—
115#376C2×C411yescharacteristic
116#376C2×C412yes—
117#376C2×C411yescharacteristic
118C2×#95—22no—
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121C43—21no—
122C43—22no—
123#342—24no—
124#324—24no—
125#323—24no—
126C2×#41—24no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—42no—
130C22×Q8C22⋊C411yescharacteristic
131C22×D4C22⋊C411yescharacteristic
132C22×D4C2×D412yes—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22×C412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C2×M16—24no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—44no—
155C4×D4—44no—
156#117—22no—
157#117—22no—
158#117—24no—
159#117—24no—
160#117—28no—
161#117—44no—
162#117—44no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C2×D412yes—
182C2×C22⋊C4C2×D412yes—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—24no—
197C2×C42—48no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42C22⋊C412yes—
201C2×C42C22⋊C412yes—
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C4⋊C8—42no—
212C4⋊C8—42no—
213#99—216no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216#97—42no—
217C4×C8—42no—
218#95—21no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—24no—
233#95—42no—
234#95C22×C411yescharacteristic
235#95C22×C411yescharacteristic
236#95C22×C412yes—
237#95C22×C412yes—
238#95C22×C411yescharacteristic
239#95C22×C411yescharacteristic
240C24C2×C22⋊C412yes—
241C24C22×D411yescharacteristic
242#41—48no—
243#41—48no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—42no—
252C2×D4#10414yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—81no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4#12312yes—
288C22×C4#12312yes—
289C22×C4#12312yes—
290C22×C4#12312yes—
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4112yes—
294C22×C4C2×#4112yes—
295M16—84no—
296C2×C8—44no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—48no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—28no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—82no—
318C42—24no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—44no—
326C42—44no—
327C42—44no—
328C42—42no—
329C42—48no—
330C42—42no—
331C42—42no—
332C42—84no—
333C42—81no—
334C42—81no—
335C42—84no—
336C42#11711yescharacteristic
337C42#11711yescharacteristic
338C42#11712yes—
339C42#11712yes—
340C42#11711yescharacteristic
341C42#11711yescharacteristic
342C42#12312yes—
343C42#12312yes—
344C42#12312yes—
345C42#12312yes—
346C23—22no—
347C23—22no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23C2×#10414yes—
355C23C22×C22⋊C411yescharacteristic
356C23C2×#12312yes—
357Q8—44no—
358D4—28no—
359D4—48no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—48no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—88no—
405C2×C4—88no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4#38111yescharacteristic
409C2×C4#38111yescharacteristic
410C2×C4#38111yescharacteristic
411C2×C4#38112yes—
412C2×C4#38112yes—
413C2×C4#38111yescharacteristic
414C2×C4C2×#10412yes—
415C2×C4#42112yes—
416C2×C4C2×#11711yescharacteristic
417C2×C4C2×#11711yescharacteristic
418C2×C4C2×#12312yes—
419C8—84no—
420C22—21no—
421C22—21no—
422C22—22no—
423C22—24no—
424C22—22no—
425C22—22no—
426C22—44no—
427C22—44no—
428C22—42no—
429C22—42no—
430C22—81no—
431C22—81no—
432C22C2×#38111yescharacteristic
433C22C22×#10411yescharacteristic
434C22C2×#42111yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—22no—
439C4—22no—
440C4—21no—
441C4—21no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—81no—
448C4—81no—
449C4—82no—
450C4—82no—
451C4—88no—
452C4#182614yes—
453C4#182712yes—
454C2—21no—
455C2—21no—
456C2—42no—
457C2—44no—
458C2C2×#182611yescharacteristic
459C2C2×#182711yescharacteristic
460C2#2071011yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#381
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8777C2semidirect
5. #8776C2semidirect
6. #8048C2semidirect
7. #7940C2semidirect
11. #2356C4semidirect
12. #2356C4semidirect
15. #2349C4semidirect
16. #2349C4semidirect
22. #2283C4semidirect
27. C2×#376C4semidirect

Maximal subgroups

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

subgroupindex
#341232
C4×#4102
#173662
#87772
#87762
#80482
#79402