Finite Group Databasebeta

← #380039

Subgroups — #380039

Order 512.

1283 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60123C211yesmaximal, characteristic
2#34947C211yesmaximal, characteristic
3#21306C212yesmaximal, complemented
4#20632C211yesmaximal, characteristic
5#11212C214yesmaximal, complemented
6#11160C214yesmaximal
7#10851C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2923—24no—
11#2922—24no—
12#2921—22no—
13#2920—22no—
14#2905C2212yes—
15C2×#410C2212yes—
16#2879C2212yes—
17C2×C4×M16C2212yes—
18#2568C411yescharacteristic, complemented
19#2567C411yescharacteristic, complemented
20#2566C412yescomplemented
21#2565C412yescomplemented
22#2564—22no—
23#2563C411yescharacteristic, complemented
24#2542C411yescharacteristic, complemented
25#2417—22no—
26#2396—22no—
27C4×C22⋊Q8—22no—
28#2297C2212yes—
29#2285C2212yes—
30#2285C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1986—24no—
33#1986—24no—
34#1799—22no—
35#1799—22no—
36#1740C2212yes—
37#1733C2218yes—
38C2×#339C2218yes—
39C22×#41C2311yescharacteristic
40C22×M16C2314yes—
41#533—22no—
42#533—22no—
43#533—22no—
44#533—22no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125C2×C412yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C4×#41—22no—
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C2×C4×D4D412yes—
63C2×#117D412yes—
64C2×#117C2314yes—
65C2×#117C2312yes—
66C2×#117C2311yescharacteristic
67C22×C4⋊C4—21no—
68C22×C42D412yes—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71#433—216no—
72#432—28no—
73#431—28no—
74#424—28no—
75#423—24no—
76#410—44no—
77C2×#99—24no—
78#408—28no—
79#407—24no—
80C4×M16—22no—
81C4×M16—22no—
82C4×M16D414yes—
83C4×M16D414yes—
84C2×#97—22no—
85C2×C4×C8—22no—
86#400—22no—
87#399—22no—
88#398—24no—
89#396—24no—
90#395—22no—
91#393—22no—
92#391—22no—
93#391C2×C412yes—
94#391C2×C412yes—
95#390C2×C412yes—
96#390C2×C412yes—
97#389—22no—
98#387—24no—
99#386—24no—
100#385—24no—
101#384—22no—
102#384C2×C412yes—
103#384C2×C412yes—
104#382—22no—
105#382C2×C412yes—
106#382C2×C412yes—
107#380—22no—
108#380—24no—
109#380—22no—
110#380—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C22⋊C4—24no—
120C2×#95—21no—
121#342—24no—
122#339—216no—
123#339—48no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41—24no—
128C2×#41—22no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8C22×C411yescharacteristic
132C22×D4C22×C411yescharacteristic
133C22×D4C2×D412yes—
134C23×C4—22no—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C2×M16—24no—
142C2×M16—24no—
143C2×M16—24no—
144C2×M16—24no—
145C2×M16—24no—
146C2×M16C2×D418yes—
147C22×C8—24no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152#125—24no—
153#125—24no—
154C22⋊Q8—48no—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158C22⋊Q8—42no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—41no—
164C4×Q8—42no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×Q8—41no—
168C4×Q8—42no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177#117—22no—
178#117—22no—
179#117—24no—
180#117—24no—
181#117—24no—
182#117—24no—
183#117—28no—
184#117—28no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C2×D412yes—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42C22⋊C412yes—
221C2×C42C22⋊C412yes—
222C2×C42C22×C412yes—
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4112yes—
232C2×C42#4112yes—
233C4⋊C8—216no—
234#99—48no—
235C22⋊C8—216no—
236#97—24no—
237#97—24no—
238C4×C8—24no—
239C4×C8—24no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—42no—
248#95—42no—
249C24C2×C22⋊C412yes—
250C24C22×D411yescharacteristic
251#41—44no—
252#41—44no—
253#41—44no—
254#41—44no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—44no—
259C2×Q8—41no—
260C2×Q8—42no—
261C2×Q8—41no—
262C2×Q8—82no—
263C2×D4—24no—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—81no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D412yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4112yes—
306C22×C4H(5,𝔽2)12yes—
307C22×C42−1+414yes—
308C22×C42−1+412yes—
309M16—28no—
310M16—28no—
311M16—44no—
312M16—44no—
313C2×C8—28no—
314C2×C8—44no—
315C2×C8—44no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—48no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—48no—
333C4⋊C4—84no—
334C4⋊C4—84no—
335C4⋊C4—84no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—48no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—84no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—24no—
351C42—24no—
352C42—42no—
353C42—42no—
354C42—44no—
355C42—48no—
356C42C4×D412yes—
357C42C4×D412yes—
358C42C4×D412yes—
359C42C4×D412yes—
360C42#12011yescharacteristic
361C42#12012yes—
362C42#12011yescharacteristic
363C42#12311yescharacteristic
364C42#12311yescharacteristic
365C42#12312yes—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23C22×C22⋊C411yescharacteristic
374C23C2×#12112yes—
375C23#53511yescharacteristic
376C23#53612yes—
377C23#53611yescharacteristic
378Q8—44no—
379Q8—82no—
380Q8—84no—
381Q8—82no—
382D4—44no—
383D4—44no—
384D4—44no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4#38611yescharacteristic
422C2×C4#38611yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4#38612yes—
425C2×C4#38612yes—
426C2×C4#38611yescharacteristic
427C2×C4C2×C4×D412yes—
428C2×C4#51912yes—
429C2×C4#52012yes—
430C2×C4C2×#12011yescharacteristic
431C2×C4C2×#12311yescharacteristic
432C2×C4#53614yes—
433C2×C4#53612yes—
434C2×C4#53714yes—
435C2×C4#53712yes—
436C2×C4#54114yes—
437C2×C4#54214yes—
438C8—48no—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—22no—
443C22—41no—
444C22—41no—
445C22—42no—
446C22—44no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—82no—
451C22C2×#38611yescharacteristic
452C22#231011yescharacteristic
453C22#231111yescharacteristic
454C22#262012yes—
455C22#262112yes—
456C4—21no—
457C4—22no—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—44no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4—164no—
469C4#188412yes—
470C4#232314yes—
471C4#261612yes—
472C4#262512yes—
473C2—21no—
474C2—22no—
475C2—21no—
476C2—22no—
477C2—82no—
478C2#1425811yescharacteristic
479C2C2×#188411yescharacteristic
480C2#2094711yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21306C2semidirect
5. #11212C2semidirect
7. #10851C2semidirect
18. #2568C4semidirect
19. #2567C4semidirect
20. #2566C4semidirect
21. #2565C4semidirect
23. #2563C4semidirect
24. #2542C4semidirect

Maximal subgroups

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

subgroupindex
#601232
#349472
#213062
#206322
#112122
#111602
#108512