Finite Group Databasebeta

← #355475

Subgroups — #355475

Order 512.

1636 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34199C211yesmaximal, characteristic
2C2×#1890C212yesmaximal
3#20392C211yesmaximal, characteristic
4#20354C211yesmaximal, characteristic
5#10936C212yesmaximal, complemented
6#10915C212yesmaximal, complemented
7#10829C212yesmaximal, complemented
8#10754C212yesmaximal
9#10726C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C2×#518C2212yes—
12C2×#518C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#99C2212yes—
15#2359—24no—
16#2344—22no—
17#2342—22no—
18#2322C414yescomplemented
19#2315C2211yescharacteristic
20#2308C2211yescharacteristic
21C4×#123C414yescomplemented
22C2×#381C2211yescharacteristic
23C2×#381C2214yes—
24#2273C2211yescharacteristic
25#2273C2212yes—
26#1890C4116yescomplemented
27C4×#99—24no—
28#1740—22no—
29C2×#342—22no—
30C2×#342C2212yes—
31C2×#342C2212yes—
32C2×#342C2212yes—
33C2×#342C2214yes—
34C2×#342C2214yes—
35C2×#342C2214yes—
36C23×D4C2311yescharacteristic
37C24×C4C2311yescharacteristic
38C2×#123—22no—
39C2×#123—24no—
40C2×#123C2×C412yes—
41C2×#123C2×C418yes—
42C2×#123C2×C412yes—
43#518—24no—
44#518—44no—
45#518—44no—
46#518C2×C418yes—
47C2×C4×D4—44no—
48C2×C4×D4C2×C414yes—
49C2×C4×D4D414yes—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—44no—
54C2×#117D412yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2×C412yescomplemented
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D412yes—
60C22×C22⋊C4C2311yescharacteristic
61C22×C22⋊C4C2311yescharacteristic
62C22×C22⋊C4C2314yes—
63C22×C22⋊C4C2314yes—
64C22×C22⋊C4C2311yescharacteristic
65C22×C22⋊C4C2311yescharacteristic
66C22×C22⋊C4C2311yescharacteristic
67C22×C42C2×C412yescomplemented
68C2×#99—24no—
69C2×#99—24no—
70C2×#99C2×C418yes—
71#389—24no—
72#388—24no—
73#388—24no—
74#387—24no—
75#387—24no—
76#386—22no—
77#386C2×C414yes—
78#385—24no—
79#385—22no—
80#385C2×C414yescomplemented
81#381—21no—
82#381—22no—
83#381—44no—
84#381C2×C412yescomplemented
85#381C2×C412yescomplemented
86#381C2×C4116yescomplemented
87#380—22no—
88#379—21no—
89C4×C4⋊C4—24no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—44no—
95C4×C22⋊C4C2×C414yescomplemented
96C2×#95—22no—
97C2×#95—44no—
98#342—24no—
99#342—24no—
100#342—24no—
101#342—44no—
102#342—44no—
103#342C2×C4116yescomplemented
104C25C2×D411yescharacteristic
105C25C2411yescharacteristic
106C22×D4—24no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4C22×C412yes—
110C22×D4C22×C414yes—
111C22×D4C2×D412yes—
112C22×D4#4114yes—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—42no—
124C23×C4C22⋊C414yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C414yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C412yes—
129C23×C4C22×C412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132#123—24no—
133#123—24no—
134#123—216no—
135#123C4218yescomplemented
136#123C4218yescomplemented
137C4×D4—416no—
138C4×D4—48no—
139C4×D4—48no—
140C4×D4—416no—
141#117—28no—
142#117—44no—
143#117—48no—
144#117—48no—
145#117—44no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4C22⋊C414yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4C22⋊C414yes—
183C2×C22⋊C4C22×C418yes—
184C2×C22⋊C4C22×C418yes—
185C2×C22⋊C4C22×C412yes—
186C2×C22⋊C4C22×C412yes—
187C2×C22⋊C4C22×C414yes—
188C2×C22⋊C4C22×C412yes—
189C2×C22⋊C4C2×D412yes—
190C2×C22⋊C4C2×D412yes—
191C2×C22⋊C4C2×D412yes—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—44no—
195C2×C42—42no—
196C2×C42—44no—
197C2×C42C22⋊C414yes—
198#99—216no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—416no—
205#95—44no—
206C24—22no—
207C24—21no—
208C24—21no—
209C24—82no—
210C24—82no—
211C24C2×C22⋊C412yes—
212C24C4×D414yes—
213C24C23×C411yescharacteristic
214C24C23×C412yes—
215C24C22×D411yescharacteristic
216C24C22×D411yescharacteristic
217C24C2×#4112yes—
218C24H(5,𝔽2)14yes—
219C24H(5,𝔽2)12yes—
220C242−1+412yes—
221C2×D4—24no—
222C2×D4—28no—
223C2×D4—24no—
224C2×D4—28no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—28no—
228C2×D4—28no—
229C2×D4C2×C4214yes—
230C2×D4#11714yes—
231C2×D4C4×D418yes—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—28no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—28no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—84no—
276C22×C4—88no—
277C22×C4—88no—
278C22×C4—164no—
279C22×C4C2×C4212yes—
280C22×C4C2×C4212yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C414yes—
283C22×C4C2×C22⋊C414yes—
284C22×C4C4×D414yes—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—84no—
291C22⋊C4—22no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—28no—
298C22⋊C4—28no—
299C22⋊C4—28no—
300C22⋊C4—28no—
301C22⋊C4—28no—
302C22⋊C4—24no—
303C22⋊C4—28no—
304C22⋊C4—44no—
305C22⋊C4—48no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—44no—
310C22⋊C4—41no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—84no—
315C22⋊C4C2×C4214yes—
316C42—44no—
317C42—48no—
318C42—48no—
319C42—84no—
320C42—164no—
321C23—22no—
322C23—22no—
323C23—24no—
324C23—24no—
325C23—42no—
326C23—42no—
327C23—48no—
328C23—42no—
329C23—44no—
330C23—84no—
331C23—82no—
332C23—82no—
333C23—82no—
334C23—82no—
335C23—84no—
336C23—161no—
337C23—161no—
338C23C4×C22⋊C414yes—
339C23C22×C4211yescharacteristic
340C23C22×C22⋊C411yescharacteristic
341C23C2×#11711yescharacteristic
342C23C2×C4×D412yes—
343C23C2×C4×D412yes—
344C23#51812yes—
345C23#51814yes—
346C23#52012yes—
347C23#52014yes—
348C23#53412yes—
349C23#53412yes—
350C23#53511yescharacteristic
351C23#53611yescharacteristic
352C23#53712yes—
353C23#53814yes—
354C23#53914yes—
355D4—28no—
356D4—216no—
357D4—28no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—44no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—48no—
386C2×C4—48no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—88no—
391C2×C4—88no—
392C2×C4—84no—
393C2×C4—88no—
394C2×C4—88no—
395C2×C4—164no—
396C2×C4—162no—
397C2×C4—164no—
398C2×C4—164no—
399C2×C4—168no—
400C2×C4C4×C22⋊C414yes—
401C22—22no—
402C22—24no—
403C22—24no—
404C22—24no—
405C22—24no—
406C22—44no—
407C22—42no—
408C22—42no—
409C22—44no—
410C22—44no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—41no—
418C22—44no—
419C22—44no—
420C22—84no—
421C22—82no—
422C22—84no—
423C22C2×C4×C22⋊C411yescharacteristic
424C22#227512yes—
425C22#230811yescharacteristic
426C22#231111yescharacteristic
427C22#231812yes—
428C22#232112yes—
429C22#232212yes—
430C22#232514yes—
431C4—24no—
432C4—44no—
433C4—44no—
434C4—44no—
435C4—42no—
436C4—42no—
437C4—82no—
438C4—88no—
439C4—88no—
440C4—82no—
441C2—21no—
442C2—22no—
443C2—21no—
444C2—24no—
445C2—22no—
446C2—22no—
447C2—22no—
448C2—44no—
449C2#1378411yescharacteristic
450C2#2177712yes—
subgroupG/H
center, socleC22#2308
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10936C2semidirect
6. #10915C2semidirect
7. #10829C2semidirect
9. #10726C2semidirect
18. #2322C4semidirect
21. C4×#123C4semidirect
26. #1890C4semidirect
56. C22×C4⋊C4C2×C4semidirect
67. C22×C42C2×C4semidirect
80. #385C2×C4semidirect
84. #381C2×C4semidirect
85. #381C2×C4semidirect
86. #381C2×C4semidirect
95. C4×C22⋊C4C2×C4semidirect
103. #342C2×C4semidirect
135. #123C42semidirect
136. #123C42semidirect

Maximal subgroups

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

subgroupindex
#341992
C2×#18902
#203922
#203542
#109362
#109152
#108292
#107542
#107262