Finite Group Databasebeta

← #379879

Subgroups — #379879

Order 512.

1701 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34635C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic, complemented
3C2×#1897C211yesmaximal, characteristic
4C2×#1896C211yesmaximal, characteristic
5#20683C211yesmaximal, characteristic, complemented
6#11226C214yesmaximal, complemented
7#11075C214yesmaximal, complemented
8#10720C211yesmaximal, characteristic, complemented
9#10719C211yesmaximal, characteristic, complemented
10C2×#525C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#104C2212yes—
15C22×#104C2212yes—
16C2×#407C2211yescharacteristic
17C2×#407C2211yescharacteristic
18#2625C414yescomplemented
19#2624C414yescomplemented
20#2623—22no—
21#2622—22no—
22#2395—24no—
23#2297C2212yes—
24#2296—24no—
25#2296—24no—
26#2296C2212yes—
27C2×#389—22no—
28C2×#385—22no—
29#2280C2211yescharacteristic
30C2×C4×C4⋊C4C2214yes—
31#1897—24no—
32#1896—24no—
33#1818—24no—
34C2×#339C2212yes—
35C2×#339C2212yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38#1728C2214yes—
39C22×#41C2311yescharacteristic
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C22×M16C2312yes—
43C22×M16C2312yes—
44#525—22no—
45#525—22no—
46#525—22no—
47#525—22no—
48C2×#123—21no—
49C2×C22⋊Q8C2×C412yes—
50C2×#121C2×C412yes—
51C2×#120—21no—
52C2×C4×D4—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—44no—
59C2×#117C2×C414yes—
60C2×#117D414yes—
61C2×#117D414yes—
62C2×#117C2311yescharacteristic
63C2×#117C2312yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4D412yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4D412yes—
69C22×C42—21no—
70C22×C42—22no—
71C22×C42—44no—
72C22×C42D414yes—
73C22×C42C2312yes—
74C22×C42C2312yes—
75C22×C42C2312yes—
76C2×#104—24no—
77C2×#104—24no—
78C2×#104—44no—
79C2×#104—44no—
80#407—22no—
81#407—22no—
82#407—22no—
83#407—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86#400—24no—
87#399—24no—
88#398—24no—
89#397—24no—
90#397—28no—
91#394—28no—
92#391C2×C414yescomplemented
93#390C2×C414yescomplemented
94#389—216no—
95#389C2×C418yescomplemented
96#386—24no—
97#385—216no—
98#385—24no—
99#381—21no—
100#380C2×C414yes—
101#379—21no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—28no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—24no—
107C2×#95—24no—
108#339—24no—
109#339—24no—
110#339—44no—
111#339—44no—
112#336—28no—
113C25C2×D411yescharacteristic
114C2×#41—21no—
115C2×#41—21no—
116C2×#41—21no—
117C2×#41—21no—
118C2×#41—24no—
119C2×#41—24no—
120C22×Q8C22×C411yescharacteristic
121C22×D4—21no—
122C22×D4—22no—
123C22×D4C22×C411yescharacteristic
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—48no—
131C23×C4—44no—
132C23×C4C22×C412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D414yes—
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4C2411yescharacteristic
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—24no—
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—24no—
149C22×C8—22no—
150C22×C8—22no—
151#123—42no—
152#123—42no—
153C22⋊Q8—24no—
154C22⋊Q8—24no—
155#121—24no—
156#121—24no—
157#120—42no—
158#120—42no—
159C4×D4—44no—
160C4×D4—44no—
161#117—28no—
162#117—28no—
163#117—28no—
164#117—28no—
165#117—21no—
166#117—21no—
167#117—21no—
168#117—21no—
169#117—24no—
170#117—24no—
171#117—48no—
172#117—48no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—216no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—416no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4C22⋊C414yes—
182C2×C4⋊C4C22⋊C414yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C414yes—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4C22⋊C418yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—48no—
210C2×C42—44no—
211C2×C42—48no—
212C2×C42C22×C414yes—
213C2×C42C2×D414yes—
214C2×C42C2×D414yes—
215C2×C42C2×D414yes—
216#104—48no—
217#104—48no—
218C22⋊C8—44no—
219C22⋊C8—44no—
220#95—216no—
221#95—24no—
222#95—28no—
223#95—28no—
224C24—22no—
225C24—22no—
226C24—21no—
227C24—44no—
228C24C2×C22⋊C412yes—
229C24#12012yes—
230C24#12312yes—
231C24C22×D411yescharacteristic
232#41—42no—
233#41—42no—
234#41—42no—
235#41—42no—
236#41—44no—
237#41—44no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—24no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—24no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—84no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—48no—
269C22×C4—416no—
270C22×C4—48no—
271C22×C4—44no—
272C22×C4—432no—
273C22×C4—44no—
274C22×C4—48no—
275C22×C4—44no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—44no—
279C22×C4C2×C22⋊C418yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C4×D414yes—
282C22×C4C4×D414yes—
283C22×C4#12012yes—
284C22×C4#12114yes—
285C22×C4#12114yes—
286C22×C4#12312yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4H(5,𝔽2)14yes—
294C22×C42−1+414yes—
295M16—42no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—44no—
300M16—44no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—44no—
304C2×C8—44no—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—28no—
310C4⋊C4—432no—
311C4⋊C4—48no—
312C4⋊C4—44no—
313C4⋊C4—88no—
314C22⋊C4—216no—
315C22⋊C4—48no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—88no—
323C22⋊C4—84no—
324C42—24no—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—24no—
332C42—48no—
333C42—41no—
334C42—41no—
335C42—44no—
336C42—44no—
337C42—44no—
338C23—24no—
339C23—24no—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—24no—
344C23—24no—
345C23—44no—
346C23—44no—
347C23—48no—
348C23—48no—
349C23—42no—
350C23—44no—
351C23—41no—
352C23#38614yes—
353C23C22×C22⋊C411yescharacteristic
354C23C2×#12011yescharacteristic
355C23C2×#12311yescharacteristic
356C23#53512yes—
357C23#53612yes—
358Q8—42no—
359Q8—42no—
360D4—42no—
361D4—42no—
362D4—88no—
363D4—81no—
364D4—81no—
365D4—82no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—28no—
371C2×C4—28no—
372C2×C4—28no—
373C2×C4—28no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—28no—
377C2×C4—28no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—416no—
381C2×C4—44no—
382C2×C4—432no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—88no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4#38614yes—
396C2×C4C2×C4×D411yescharacteristic
397C2×C4C2×C4×D411yescharacteristic
398C2×C4#52014yes—
399C2×C4C2×#12111yescharacteristic
400C2×C4C2×#12111yescharacteristic
401C2×C4#53412yes—
402C2×C4#53412yes—
403C2×C4#53612yes—
404C2×C4#53612yes—
405C2×C4#53714yes—
406C2×C4#53914yes—
407C2×C4#53914yes—
408C8—82no—
409C8—82no—
410C22—24no—
411C22—24no—
412C22—22no—
413C22—22no—
414C22—24no—
415C22—24no—
416C22—24no—
417C22—44no—
418C22—44no—
419C22—48no—
420C22—48no—
421C22—84no—
422C22—81no—
423C22—82no—
424C22C2×#38611yescharacteristic
425C22#231112yes—
426C22#261311yescharacteristic
427C22#261411yescharacteristic
428C22#262312yes—
429C22#290114yes—
430C4—24no—
431C4—48no—
432C4—48no—
433C4—84no—
434C4—84no—
435C4—81no—
436C4—82no—
437C4#232212yes—
438C4#232212yes—
439C4#261812yes—
440C4#261812yes—
441C2—21no—
442C2—21no—
443C2—22no—
444C2—44no—
445C2—161no—
446C2#1425611yescharacteristic
447C2#2098012yes—
448C2#2190114yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2296C2semidirect
5. #20683C2semidirect
6. #11226C2semidirect
7. #11075C2semidirect
8. #10720C2semidirect
9. #10719C2semidirect
18. #2625C4semidirect
19. #2624C4semidirect
92. #391C2×C4semidirect
93. #390C2×C4semidirect
95. #389C2×C4semidirect

Maximal subgroups

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

subgroupindex
#346352
C2×#22962
C2×#18972
C2×#18962
#206832
#112262
#110752
#107202
#107192