Finite Group Databasebeta

← #373722

Subgroups — #373722

Order 512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34178C211yesmaximal, characteristic
2#20845C212yesmaximal
3C2×#1821C211yesmaximal, characteristic
4#13795C211yesmaximal, characteristic
5#11342C212yesmaximal, complemented
6#11296C212yesmaximal, complemented
7#10899C214yesmaximal, complemented
8#10720C212yesmaximal
9C22×#117C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C2×#428C2214yes—
13C2×#407C2212yes—
14#2345—24no—
15#2344—24no—
16#2316C2211yescharacteristic
17#2316C2211yescharacteristic
18#2313—22no—
19#2280—22no—
20#2280C2212yes—
21#2274C2212yes—
22#1854—24no—
23#1854—24no—
24#1821C418yescomplemented
25C2×#343C2212yes—
26C2×#343C2212yes—
27#1736C2214yes—
28C2×#339C2214yes—
29C2×#339C2214yes—
30C4⋊#95—22no—
31C4⋊#95C2212yes—
32C4⋊#95C2212yes—
33C22⋊#95—22no—
34C24×C4D412yes—
35C24×C4C2311yescharacteristic
36C22×M16C2314yes—
37C2×#117—24no—
38C2×#117—24no—
39C2×#117—24no—
40C2×#117—24no—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—42no—
44C2×#117—42no—
45C2×#117C2312yes—
46C2×#117C2312yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×C414yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4Q814yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C42—21no—
60C22×C42C2312yes—
61#428—28no—
62#428—44no—
63C2×#107—24no—
64C2×#106—24no—
65#407—24no—
66#407C2×C418yes—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69#389—24no—
70#389—24no—
71#387—28no—
72#387—28no—
73#381—24no—
74#381—22no—
75#380—24no—
76#380—22no—
77#380—22no—
78#380D414yes—
79#380D414yes—
80#379—22no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—24no—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—24no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87C2×#95—24no—
88C2×#95—24no—
89#343C2×C418yes—
90#343C2×C418yes—
91#340—28no—
92#339—28no—
93#339—44no—
94C25C2×D411yescharacteristic
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—84no—
105C23×C4C4⋊C418yes—
106C23×C4C22×C412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×Q812yes—
113C23×C4#4114yes—
114C23×C4C2411yescharacteristic
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16C22×C418yes—
121C22×C8—24no—
122#117—24no—
123#117—24no—
124#117—24no—
125#117—24no—
126#117—416no—
127#117—48no—
128#117—48no—
129#117—88no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4C4⋊C418yes—
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C22×C412yes—
151C2×C4⋊C4C2×D414yes—
152C2×C4⋊C4C2×D414yes—
153C2×C4⋊C4C2×Q812yes—
154C2×C4⋊C4C2×Q812yes—
155C2×C4⋊C4#4114yes—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—42no—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—28no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42#4114yes—
177#107—48no—
178#106—48no—
179C22⋊C8—28no—
180C22⋊C8—28no—
181#95—28no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—84no—
187#95—84no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—42no—
193C24—42no—
194C24C2×C4⋊C412yes—
195C24#12011yescharacteristic
196C24#12011yescharacteristic
197C24#12312yes—
198C24C22×D411yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—28no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—28no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—28no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—48no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—88no—
237C22×C4—84no—
238C22×C4—88no—
239C22×C4C2×C4⋊C412yes—
240C22×C4C2×C4⋊C414yes—
241C22×C4C2×C4⋊C414yes—
242C22×C4C4×D414yes—
243C22×C4C4×D414yes—
244C22×C4#12012yes—
245C22×C4C22⋊Q814yes—
246C22×C4C22⋊Q814yes—
247C22×C4#12312yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D412yes—
250C22×C4C22×Q811yescharacteristic
251C22×C4C2×#4112yes—
252C22×C4C2×#4112yes—
253C22×C4C2×#4112yes—
254C22×C4H(5,𝔽2)12yes—
255C22×C42−1+412yes—
256M16—28no—
257M16—28no—
258M16—44no—
259M16—44no—
260C2×C8—28no—
261C2×C8—44no—
262C2×C8—44no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—216no—
268C4⋊C4—24no—
269C4⋊C4—28no—
270C4⋊C4—24no—
271C4⋊C4—28no—
272C4⋊C4—48no—
273C4⋊C4—44no—
274C4⋊C4—84no—
275C4⋊C4—84no—
276C22⋊C4—48no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—48no—
280C22⋊C4—48no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C42—28no—
286C42—48no—
287C42—44no—
288C42—44no—
289C42—44no—
290C42—88no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—44no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23#38512yes—
314C23#38512yes—
315C23C22×C4⋊C411yescharacteristic
316C23C2×#12011yescharacteristic
317C23C2×#12311yescharacteristic
318C23#52511yescharacteristic
319C23#52511yescharacteristic
320C23#53511yescharacteristic
321C23#53611yescharacteristic
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—28no—
329C2×C4—28no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—416no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—48no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—88no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—84no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4#38514yes—
361C2×C4C2×C4×D412yes—
362C2×C4C4×#4112yes—
363C2×C4#52012yes—
364C2×C4C2×C22⋊Q812yes—
365C2×C4#53312yes—
366C2×C4#54412yes—
367C2×C4#54614yes—
368C2×C4#54714yes—
369C2×C4#55114yes—
370C2×C4#55214yes—
371C8—48no—
372C22—22no—
373C22—22no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—44no—
387C22—44no—
388C22—44no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22C2×#38511yescharacteristic
395C22#229611yescharacteristic
396C22#231711yescharacteristic
397C22#240411yescharacteristic
398C22#240611yescharacteristic
399C22#248211yescharacteristic
400C22#248411yescharacteristic
401C22#301912yes—
402C22#302012yes—
403C4—24no—
404C4—44no—
405C4—44no—
406C4—84no—
407C4—84no—
408C4—84no—
409C4—84no—
410C4#235914yes—
411C4#243814yes—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—42no—
417C2—42no—
418C2#1412811yescharacteristic
419C2#2085611yescharacteristic
420C2#2085711yescharacteristic
421C2#2395112yes—
422C2#2407912yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11342C2semidirect
6. #11296C2semidirect
7. #10899C2semidirect
24. #1821C4semidirect

Maximal subgroups

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

subgroupindex
#341782
#208452
C2×#18212
#137952
#113422
#112962
#108992
#107202