Finite Group Databasebeta

← #355476

Subgroups — #355476

Order 512.

1614 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#34200C211yesmaximal, characteristic
2C2×#1891C212yesmaximal
3#20397C211yesmaximal, characteristic, complemented
4#20358C211yesmaximal, characteristic, complemented
5#10937C212yesmaximal, complemented
6#10916C212yesmaximal, complemented
7#10830C212yesmaximal, complemented
8#10755C212yesmaximal, complemented
9#10727C212yesmaximal, complemented
10C2×#568C2212yes—
11C2×#525C2211yescharacteristic
12C2×#520C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C2×#413C2212yes—
15C2×#407C2214yes—
16#2873C2212yes—
17#2361—22no—
18#2360—22no—
19#2352—22no—
20D4×C4⋊C4—22no—
21#2322C414yes—
22#2317C2211yescharacteristic
23#2311C2211yescharacteristic
24C4×#120C414yes—
25#2280C2211yescharacteristic
26#2274C2211yescharacteristic
27#1891C4116yescomplemented
28C4×#101—22no—
29C4×#100—22no—
30C2×#344C2214yes—
31C2×#344C2214yes—
32#1742—22no—
33#1742C2212yes—
34C2×#343—22no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2214yes—
38C22×#41C2311yescharacteristic
39C24×C4C2×C412yes—
40C24×C4C2311yescharacteristic
41#568—24no—
42#568—44no—
43#568C2×C418yescomplemented
44C2×#131—44no—
45C22×M16—24no—
46C22×M16C2314yes—
47C22×M16C2314yes—
48#525—24no—
49#525C2×C418yes—
50C2×#123C2×C412yes—
51C2×C22⋊Q8—21no—
52C2×#121—21no—
53C2×#120C2×C412yes—
54#520—44no—
55C2×C4×Q8D412yes—
56C2×C4×D4—22no—
57C2×C4×D4—24no—
58C2×C4×D4C2×C414yes—
59C2×C4×D4D412yes—
60C2×#117D412yes—
61C2×#117C2311yescharacteristic
62C2×#117C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—44no—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71#413—24no—
72#413C2×C418yescomplemented
73C2×#101—22no—
74C2×#100—22no—
75#407—44no—
76#407C2×C4116yescomplemented
77C2×C22⋊C8—44no—
78#405—44no—
79C4×M16—24no—
80#391—21no—
81#390—21no—
82#389—24no—
83#389—22no—
84#388—24no—
85#387—24no—
86#386—24no—
87#386C2×C414yes—
88#385—24no—
89#385—24no—
90#385C2×C414yes—
91#384—21no—
92#382—21no—
93#381C2×C412yes—
94#380—22no—
95#380—21no—
96#380—21no—
97#379C2×C412yes—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C22⋊C4—24no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4C2×C414yes—
103#344—44no—
104#344C2×C4116yescomplemented
105#343—22no—
106#343—24no—
107#343—24no—
108#343—22no—
109#343—44no—
110C25C22×C411yescharacteristic
111C2×#41—24no—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41C22×C414yes—
115C2×#41#4114yes—
116C22×Q8C2×D411yescharacteristic
117C22×D4—22no—
118C22×D4C22×C412yes—
119C22×D4C2×D411yescharacteristic
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—44no—
128C23×C4—84no—
129C23×C4C22⋊C414yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C414yes—
132C23×C4C22×C412yes—
133C23×C4C22×C412yes—
134C23×C4C2×D412yes—
135C23×C4C2411yescharacteristic
136#131—416no—
137#131—416no—
138C2×M16—24no—
139C2×M16—24no—
140C2×M16—216no—
141C2×M16—24no—
142C2×M16—24no—
143C2×M16—24no—
144C2×M16—24no—
145C2×M16—24no—
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—24no—
149C2×M16—44no—
150C2×M16—48no—
151C2×M16C22×C418yes—
152C2×M16C22×C418yes—
153C22×C8—24no—
154C22×C8—24no—
155C22×C8—44no—
156#123—24no—
157#123C4218yes—
158C22⋊Q8—28no—
159#121—28no—
160#120—24no—
161#120C4218yes—
162C4×Q8—44no—
163C4×D4—216no—
164C4×D4—44no—
165C4×D4—48no—
166C4×D4—88no—
167#117—22no—
168#117—24no—
169#117—22no—
170#117—22no—
171#117—44no—
172#117C22×C414yes—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4C22⋊C414yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4C22⋊C414yes—
197C2×C22⋊C4C22×C412yes—
198C2×C22⋊C4C22×C412yes—
199C2×C22⋊C4C22×C412yes—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—24no—
203C2×C42C22⋊C414yes—
204C2×C42C2×D412yes—
205#101—28no—
206#100—28no—
207C22⋊C8—416no—
208#97—44no—
209C4×C8—44no—
210#95—22no—
211#95—22no—
212#95—24no—
213#95—24no—
214#95—24no—
215C24—21no—
216C24—22no—
217C24—21no—
218C24—22no—
219C24—42no—
220C24—42no—
221C24C2×C4212yes—
222C24C2×C22⋊C412yes—
223C24C23×C411yescharacteristic
224#41—28no—
225#41—28no—
226#41—28no—
227#41—28no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8C4×D414yes—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—28no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4C2×C4214yes—
239C2×D4#11714yes—
240C2×D4C4×D414yes—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—28no—
253C22×C4—28no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—44no—
258C22×C4—48no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—84no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—84no—
271C22×C4—88no—
272C22×C4—88no—
273C22×C4C2×C4212yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C414yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C4×D418yes—
279C22×C4C23×C412yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4H(5,𝔽2)12yes—
284C22×C4H(5,𝔽2)14yes—
285C22×C42−1+412yes—
286M16—28no—
287M16—28no—
288M16—28no—
289M16—28no—
290M16—416no—
291M16—44no—
292M16—44no—
293M16—44no—
294M16—44no—
295C2×C8—28no—
296C2×C8—28no—
297C2×C8—416no—
298C2×C8—44no—
299C2×C8—44no—
300C2×C8—44no—
301C2×C8—44no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—28no—
308C4⋊C4—44no—
309C4⋊C4—48no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—84no—
315C22⋊C4—22no—
316C22⋊C4—24no—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C22⋊C4—24no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—84no—
324C22⋊C4C2×C4214yes—
325C42—24no—
326C42—24no—
327C42—44no—
328C42—42no—
329C42—84no—
330C23—22no—
331C23—22no—
332C23—24no—
333C23—24no—
334C23—22no—
335C23—22no—
336C23—24no—
337C23—22no—
338C23—41no—
339C23—44no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—41no—
344C23—42no—
345C23—44no—
346C23—48no—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23C4×C22⋊C414yes—
353C23C22×C4211yescharacteristic
354C23C22×C22⋊C411yescharacteristic
355C23C2×#11711yescharacteristic
356C23#51812yes—
357C23#52012yes—
358Q8—28no—
359D4—28no—
360D4—28no—
361D4—28no—
362D4—416no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—28no—
366C2×C4—28no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—84no—
390C2×C4—816no—
391C2×C4—84no—
392C2×C4—164no—
393C2×C4C4×C22⋊C414yes—
394C2×C4C2×C4×D412yes—
395C2×C4C2×C4×D412yes—
396C2×C4#51814yes—
397C2×C4#52014yes—
398C2×C4#53412yes—
399C2×C4#53412yes—
400C2×C4#53511yescharacteristic
401C2×C4#53611yescharacteristic
402C2×C4#53712yes—
403C2×C4#53814yes—
404C2×C4#53914yes—
405C8—48no—
406C8—48no—
407C22—21no—
408C22—22no—
409C22—21no—
410C22—22no—
411C22—24no—
412C22—28no—
413C22—24no—
414C22—41no—
415C22—42no—
416C22—41no—
417C22—44no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—44no—
422C22—44no—
423C22—44no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—82no—
430C22C2×C4×C22⋊C411yescharacteristic
431C22#227512yes—
432C22#230811yescharacteristic
433C22#231111yescharacteristic
434C22#231812yes—
435C4—24no—
436C4—44no—
437C4—44no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—82no—
442C4—82no—
443C4#232112yes—
444C4#232212yes—
445C4#232514yes—
446C2—21no—
447C2—21no—
448C2—22no—
449C2—22no—
450C2—44no—
451C2—44no—
452C2#1378411yescharacteristic
453C2#2177811yescharacteristic
454C2#2178011yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20397C2semidirect
4. #20358C2semidirect
5. #10937C2semidirect
6. #10916C2semidirect
7. #10830C2semidirect
8. #10755C2semidirect
9. #10727C2semidirect
27. #1891C4semidirect
43. #568C2×C4semidirect
72. #413C2×C4semidirect
76. #407C2×C4semidirect
104. #344C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342002
C2×#18912
#203972
#203582
#109372
#109162
#108302
#107552
#107272