Finite Group Databasebeta

← #373729

Subgroups — #373729

Order 512.

1546 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34177C211yesmaximal, characteristic
2C2×#1820C211yesmaximal, characteristic
3#14263C212yesmaximal
4#13795C211yesmaximal, characteristic
5#11344C212yesmaximal
6#11287C212yesmaximal
7#10918C214yesmaximal
8#10736C212yesmaximal
9C22×#117C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12#2343—24no—
13#2343—24no—
14#2315C2211yescharacteristic
15#2315C2211yescharacteristic
16#2313—22no—
17C2×#385C2214yes—
18#2283—22no—
19C2×#381C2212yes—
20C2×#381C2212yes—
21#2273C2212yes—
22#1820C418yescomplemented
23C2×#342C2214yes—
24C2×#342C2214yes—
25C2×#342C2212yes—
26C2×#342C2212yes—
27#1444—24no—
28#1442—24no—
29C4⋊#95—22no—
30C4⋊#95—22no—
31C22⋊#95C2212yes—
32C22⋊#95C2214yes—
33C22⋊#95C2212yes—
34C24×C4C2312yes—
35C24×C4C2311yescharacteristic
36C2×#117—24no—
37C2×#117—24no—
38C2×#117—24no—
39C2×#117—24no—
40C2×#117—42no—
41C2×#117C2×C414yes—
42C2×#117Q814yes—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—42no—
50C22×C4⋊C4D412yes—
51C22×C22⋊C4C2312yes—
52C22×C22⋊C4C2312yes—
53C22×C22⋊C4C2314yes—
54C22×C22⋊C4C2312yes—
55C22×C22⋊C4C2311yescharacteristic
56C22×C22⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42D412yes—
59#387—24no—
60#387—24no—
61#387—24no—
62#387—24no—
63#385—22no—
64#385—24no—
65#385—22no—
66#385—28no—
67#385—44no—
68#383—24no—
69#382—22no—
70#382—22no—
71#381—22no—
72#381—24no—
73#381—22no—
74#381—24no—
75#381C2×C418yes—
76#380—22no—
77#380—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C22⋊C4—24no—
81C4×C22⋊C4—22no—
82C4×C22⋊C4—24no—
83C4×C22⋊C4—22no—
84#376—24no—
85C2×#95—24no—
86C2×#95—24no—
87C2×#95—24no—
88C2×#95—24no—
89C2×#95—28no—
90C2×#95D414yes—
91C2×#95D414yes—
92#342—28no—
93#342—44no—
94#342C2×C418yes—
95#342C2×C418yes—
96C25C2411yescharacteristic
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—24no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—24no—
107C23×C4—84no—
108C23×C4C22×C412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D414yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D414yes—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×Q812yes—
118C23×C4#4114yes—
119C23×C4#4114yes—
120#117—216no—
121#117—216no—
122#117—416no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—416no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—48no—
140C2×C4⋊C4—88no—
141C2×C4⋊C4C4⋊C418yescomplemented
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4C22×C418yes—
163C2×C22⋊C4C22×C412yes—
164C2×C22⋊C4C22×C412yes—
165C2×C22⋊C4C2×Q812yes—
166C2×C22⋊C4C2×Q812yes—
167C2×C22⋊C4#4114yes—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42C4⋊C418yescomplemented
177#95—24no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—216no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—48no—
187#95—84no—
188#95—84no—
189#95—88no—
190C24—42no—
191C24—42no—
192C24C23×C411yescharacteristic
193C24C22×D411yescharacteristic
194C24C22×D412yes—
195C24C22×Q811yescharacteristic
196C24C2×#4112yes—
197C24C2×#4112yes—
198C24C2×#4112yes—
199C24H(5,𝔽2)12yes—
200C242−1+412yes—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—24no—
213C22×C4—28no—
214C22×C4—28no—
215C22×C4—28no—
216C22×C4—28no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—28no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—28no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—28no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—48no—
237C22×C4—48no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4—48no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—84no—
255C22×C4—88no—
256C22×C4—84no—
257C22×C4C2×C4⋊C412yes—
258C22×C4C2×C4⋊C412yes—
259C22×C4C2×C4⋊C414yes—
260C22×C4C2×C4⋊C414yes—
261C22×C4C4×D418yes—
262C22×C4#12012yes—
263C22×C4#12012yes—
264C22×C4C22⋊Q818yes—
265C22×C4#12312yes—
266C22×C4#12312yes—
267C4⋊C4—216no—
268C4⋊C4—48no—
269C4⋊C4—48no—
270C4⋊C4—48no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—816no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—28no—
281C22⋊C4—28no—
282C22⋊C4—28no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C42—28no—
293C42—28no—
294C42—48no—
295C42—84no—
296C42—84no—
297C23—22no—
298C23—24no—
299C23—22no—
300C23—24no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—44no—
306C23—44no—
307C23—42no—
308C23—42no—
309C23—44no—
310C23C22×C4⋊C411yescharacteristic
311C23C2×C4×D412yes—
312C23C4×#4112yes—
313C23#52012yes—
314C23C2×#12011yescharacteristic
315C23C2×C22⋊Q812yes—
316C23C2×#12311yescharacteristic
317C23#52511yescharacteristic
318C23#52511yescharacteristic
319C23#53312yes—
320C23#53511yescharacteristic
321C23#53611yescharacteristic
322C23#54412yes—
323C23#54614yes—
324C23#54714yes—
325C23#55114yes—
326C23#55214yes—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—216no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—216no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—48no—
350C2×C4—48no—
351C2×C4—416no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—84no—
368C2×C4—88no—
369C2×C4—84no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—84no—
373C2×C4—88no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—88no—
378C2×C4—84no—
379C2×C4#38514yes—
380C2×C4#38514yes—
381C22—22no—
382C22—28no—
383C22—22no—
384C22—22no—
385C22—24no—
386C22—22no—
387C22—24no—
388C22—21no—
389C22—24no—
390C22—21no—
391C22—21no—
392C22—24no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—44no—
399C22—44no—
400C22C2×#38511yescharacteristic
401C22#229611yescharacteristic
402C22#231711yescharacteristic
403C22#235914yes—
404C22#240411yescharacteristic
405C22#240611yescharacteristic
406C22#243814yes—
407C22#248211yescharacteristic
408C22#248411yescharacteristic
409C22#302712yes—
410C22#302812yes—
411C4—24no—
412C4—24no—
413C4—44no—
414C4—44no—
415C4—84no—
416C4—88no—
417C4—84no—
418C4—84no—
419C4—82no—
420C4—82no—
421C2—21no—
422C2—24no—
423C2—21no—
424C2—24no—
425C2—21no—
426C2—21no—
427C2#1412811yescharacteristic
428C2#2089012yes—
429C2#2398811yescharacteristic
430C2#2398911yescharacteristic
431C2#2411512yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1820C4semidirect
141. C2×C4⋊C4C4⋊C4semidirect
176. C2×C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341772
C2×#18202
#142632
#137952
#113442
#112872
#109182
#107362