Finite Group Databasebeta

← #380050

Subgroups — #380050

Order 512.

1408 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34634C211yesmaximal, characteristic
2C2×#2298C211yesmaximal, characteristic
3C2×#1995C211yesmaximal, characteristic, complemented
4C2×#1994C211yesmaximal, characteristic, complemented
5#20627C211yesmaximal, characteristic
6#11226C214yesmaximal, complemented
7#11201C214yesmaximal
8#10840C211yesmaximal, characteristic, complemented
9#10839C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#407C2212yes—
15#2629C414yescomplemented
16#2628C414yescomplemented
17#2627—22no—
18#2626—22no—
19#2396—24no—
20#2298—24no—
21#2298—24no—
22#2296C2212yes—
23C2×#388—22no—
24C2×#387—22no—
25#2285C2211yescharacteristic
26#2280C2212yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2212yes—
29C2×#376C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1995—24no—
32#1994—24no—
33#1798—24no—
34C2×#339C2214yes—
35C2×#339C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38#1728C2212yes—
39C24×C4C2311yescharacteristic
40C22×M16C2312yes—
41C22×M16C2312yes—
42#533—22no—
43#533—22no—
44#533—22no—
45#533—22no—
46C2×C4⋊Q8C2×C412yes—
47C2×#125C2×C412yes—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—22no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117C2×C412yes—
55C2×#117C2×C412yes—
56C2×#117D414yes—
57C2×#117C2312yes—
58C2×#117C2312yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D412yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D412yes—
63C22×C42—22no—
64C22×C42C2312yes—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#423—22no—
69#423—22no—
70#423—22no—
71#423—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74#407—44no—
75C2×C22⋊C8—24no—
76#400—24no—
77#399—24no—
78#398—28no—
79#396—28no—
80#395—24no—
81#393—24no—
82#389C2×C418yes—
83#388—216no—
84#387—216no—
85#385—24no—
86#384C2×C412yes—
87#382C2×C412yes—
88#381—22no—
89#380—22no—
90#380C2×C418yes—
91#379—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—24no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—24no—
99#376—22no—
100#376D414yes—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C43—22no—
105C43D414yes—
106#339—28no—
107#339—28no—
108#339—44no—
109#339—44no—
110#336—24no—
111C25C2×D411yescharacteristic
112C22×Q8—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4C22×C412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2411yescharacteristic
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—24no—
137C22×C8—22no—
138C22×C8—22no—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141#125—24no—
142#125—24no—
143C22⋊Q8—44no—
144C22⋊Q8—44no—
145C4×Q8—44no—
146C4×Q8—44no—
147#117—24no—
148#117—24no—
149#117—24no—
150#117—24no—
151#117—28no—
152#117—28no—
153#117—22no—
154#117—22no—
155#117—22no—
156#117—22no—
157#117—24no—
158#117—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4C22⋊C414yes—
168C2×C4⋊C4C22⋊C414yes—
169C2×C4⋊C4C22×C414yes—
170C2×C4⋊C4C22×C414yes—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4C22⋊C414yes—
179C2×C22⋊C4C22⋊C414yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—28no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—48no—
198C2×C42C22×C412yes—
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D414yes—
202C2×C42C2×D414yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209C4⋊C8—44no—
210C4⋊C8—44no—
211C22⋊C8—48no—
212#95—28no—
213#95—28no—
214#95—28no—
215#95—24no—
216#95—28no—
217#95—28no—
218C24—21no—
219C24—21no—
220C24—22no—
221C24—22no—
222C24—44no—
223C24C2×C22⋊C412yes—
224C24C22×D411yescharacteristic
225C2×Q8—44no—
226C2×Q8—84no—
227C22×C4—28no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C414yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4#12014yes—
270C22×C4#12314yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4H(5,𝔽2)12yes—
278C22×C42−1+412yes—
279C22×C42−1+412yes—
280C22×C42−1+412yes—
281M16—44no—
282M16—44no—
283M16—44no—
284M16—44no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—416no—
297C4⋊C4—48no—
298C4⋊C4—44no—
299C4⋊C4—84no—
300C4⋊C4—88no—
301C22⋊C4—28no—
302C22⋊C4—28no—
303C22⋊C4—416no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—88no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—24no—
313C42—24no—
314C42—24no—
315C42—24no—
316C42—416no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42C4×D412yes—
326C42C4×D412yes—
327C42C4×D412yes—
328C42C4×D412yes—
329C42#12114yes—
330C42#12114yes—
331C23—24no—
332C23—22no—
333C23—22no—
334C23—28no—
335C23—22no—
336C23—44no—
337C23—44no—
338C23—42no—
339C23—42no—
340C23—84no—
341C23—84no—
342C23C22×C22⋊C411yescharacteristic
343C23C2×#12011yescharacteristic
344C23C2×#12311yescharacteristic
345C23#53612yes—
346C23#53712yes—
347Q8—88no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—28no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—28no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—48no—
377C2×C4—48no—
378C2×C4—44no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4#38614yes—
382C2×C4#38614yes—
383C2×C4C2×C4×D411yescharacteristic
384C2×C4C2×C4×D411yescharacteristic
385C2×C4#51912yes—
386C2×C4#52012yes—
387C2×C4C2×#12111yescharacteristic
388C2×C4C2×#12111yescharacteristic
389C2×C4#53511yescharacteristic
390C2×C4#53612yes—
391C2×C4#53612yes—
392C2×C4#53611yescharacteristic
393C2×C4#53611yescharacteristic
394C2×C4#53611yescharacteristic
395C2×C4#53712yes—
396C2×C4#53712yes—
397C2×C4#54112yes—
398C2×C4#54112yes—
399C2×C4#54212yes—
400C2×C4#54212yes—
401C8—82no—
402C8—82no—
403C22—24no—
404C22—24no—
405C22—22no—
406C22—22no—
407C22—44no—
408C22—44no—
409C22—42no—
410C22—44no—
411C22—84no—
412C22—84no—
413C22C2×#38611yescharacteristic
414C22#231011yescharacteristic
415C22#231111yescharacteristic
416C22#261612yes—
417C22#262512yes—
418C22#290114yes—
419C4—24no—
420C4—24no—
421C4—48no—
422C4—42no—
423C4—44no—
424C4—84no—
425C4—84no—
426C4—84no—
427C4#232312yes—
428C4#232312yes—
429C4#262011yescharacteristic
430C4#262011yescharacteristic
431C4#262111yescharacteristic
432C4#262111yescharacteristic
433C2—22no—
434C2—22no—
435C2—82no—
436C2#1425811yescharacteristic
437C2#2098012yes—
438C2#2184612yes—
439C2#2185812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
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
3. C2×#1995C2semidirect
4. C2×#1994C2semidirect
6. #11226C2semidirect
8. #10840C2semidirect
9. #10839C2semidirect
15. #2629C4semidirect
16. #2628C4semidirect

Maximal subgroups

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

subgroupindex
#346342
C2×#22982
C2×#19952
C2×#19942
#206272
#112262
#112012
#108402
#108392