Finite Group Databasebeta

← #380289

Subgroups — #380289

Order 512.

1530 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2C2×#1820C211yesmaximal, characteristic
3#14128C212yesmaximal
4#13958C211yesmaximal, characteristic
5#11344C214yesmaximal
6#11287C214yesmaximal
7#10726C212yesmaximal
8C22×#117C2211yescharacteristic
9C22×#117C2212yes—
10#2317—22no—
11#2317—22no—
12#2316—21no—
13#2316—21no—
14#2315C2214yes—
15#2296—24no—
16C2×#385C2214yes—
17#2285—22no—
18C2×#381C2211yescharacteristic
19C2×#381C2212yes—
20C22×#95C2211yescharacteristic
21#1820C418yescomplemented
22C2×#342C2218yes—
23C2×#342C2214yes—
24#1445—21no—
25#1445—21no—
26#1444—22no—
27#1442—22no—
28#1436—24no—
29#1434—22no—
30C22⋊#95C2214yes—
31C22⋊#95C2212yes—
32C22⋊#95C2212yes—
33C24×C4C2311yescharacteristic
34C24×C4C2311yescharacteristic
35C24×C4C2311yescharacteristic
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—24no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—48no—
44C2×#117C2×C414yes—
45C2×#117Q814yes—
46C22×C4⋊C4—24no—
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D412yes—
50C22×C4⋊C4D412yes—
51C22×C22⋊C4C2312yes—
52C22×C22⋊C4C2314yes—
53C22×C22⋊C4C2314yes—
54C22×C22⋊C4C2312yes—
55C22×C42D412yes—
56C22×C42D414yes—
57#389—24no—
58#389—24no—
59#389—24no—
60#389—24no—
61#385—28no—
62#385—28no—
63#385—48no—
64#384—22no—
65#384—22no—
66#382—24no—
67#381—24no—
68#381—24no—
69#381C2×C418yes—
70#380—24no—
71#380—24no—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—24no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—24no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—24no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—24no—
89#342—48no—
90#342C2×C4116yes—
91C25C2411yescharacteristic
92C23×C4—24no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4—81no—
105C23×C4—81no—
106C23×C4C22×C412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D414yes—
112C23×C4C2×D414yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×Q812yes—
116C23×C4#4114yes—
117#117—28no—
118#117—28no—
119#117—28no—
120#117—28no—
121#117—48no—
122#117—48no—
123#117—48no—
124#117—48no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—48no—
140C2×C4⋊C4—48no—
141C2×C4⋊C4—84no—
142C2×C4⋊C4C4⋊C414yescomplemented
143C2×C4⋊C4C4⋊C414yescomplemented
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4C22×C418yes—
157C2×C22⋊C4C22×C414yes—
158C2×C22⋊C4C2×Q814yes—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—24no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—44no—
165C2×C42—44no—
166C2×C42C4⋊C418yescomplemented
167#95—28no—
168#95—28no—
169#95—28no—
170#95—28no—
171#95—48no—
172#95—44no—
173#95—44no—
174#95—48no—
175#95—41no—
176#95—41no—
177#95—42no—
178#95—42no—
179#95—42no—
180#95—82no—
181#95—82no—
182#95—84no—
183C24—41no—
184C24—41no—
185C24—41no—
186C24—41no—
187C24C23×C411yescharacteristic
188C24C22×D411yescharacteristic
189C24C22×D412yes—
190C24C22×Q811yescharacteristic
191C24C2×#4112yes—
192C24H(5,𝔽2)14yes—
193C242−1+414yes—
194C22×C4—28no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—216no—
204C22×C4—28no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—28no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—88no—
256C22×C4—88no—
257C22×C4C2×C4⋊C412yes—
258C22×C4C2×C4⋊C412yes—
259C22×C4C2×C4⋊C414yes—
260C22×C4C2×C4⋊C414yes—
261C22×C4C4×D414yes—
262C22×C4C4×D414yes—
263C22×C4#12011yescharacteristic
264C22×C4#12012yes—
265C22×C4#12011yescharacteristic
266C22×C4C22⋊Q814yes—
267C22×C4C22⋊Q814yes—
268C22×C4#12312yes—
269C22×C4#12312yes—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—84no—
277C4⋊C4—84no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C4⋊C4—816no—
281C22⋊C4—28no—
282C22⋊C4—28no—
283C22⋊C4—216no—
284C22⋊C4—48no—
285C22⋊C4—48no—
286C22⋊C4—48no—
287C22⋊C4—48no—
288C42—28no—
289C42—28no—
290C42—48no—
291C42—48no—
292C23—21no—
293C23—21no—
294C23—22no—
295C23—22no—
296C23—21no—
297C23—22no—
298C23—21no—
299C23—22no—
300C23—44no—
301C23—44no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—42no—
307C23—42no—
308C23—44no—
309C23—44no—
310C23C22×C4⋊C411yescharacteristic
311C23C2×C4×D412yes—
312C23#52014yes—
313C23C2×#12011yescharacteristic
314C23C2×C22⋊Q812yes—
315C23C2×#12311yescharacteristic
316C23#53414yes—
317C23#53512yes—
318C23#53612yes—
319C23#53614yes—
320C23#53918yes—
321C23#54414yes—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—44no—
350C2×C4—48no—
351C2×C4—44no—
352C2×C4—48no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—82no—
369C2×C4—84no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4#38512yes—
385C2×C4#38512yes—
386C2×C4#38514yes—
387C22—21no—
388C22—22no—
389C22—21no—
390C22—22no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—22no—
395C22—24no—
396C22—22no—
397C22—24no—
398C22—22no—
399C22—24no—
400C22—22no—
401C22—24no—
402C22—22no—
403C22—44no—
404C22—48no—
405C22—44no—
406C22C2×#38511yescharacteristic
407C22#231712yes—
408C22#232214yes—
409C22#261311yescharacteristic
410C22#261411yescharacteristic
411C22#262312yes—
412C22#262714yes—
413C22#302511yescharacteristic
414C22#302511yescharacteristic
415C22#302711yescharacteristic
416C22#302811yescharacteristic
417C4—22no—
418C4—24no—
419C4—22no—
420C4—44no—
421C4—44no—
422C4—88no—
423C4—82no—
424C4—84no—
425C4—84no—
426C4—82no—
427C4—162no—
428C2—24no—
429C2—22no—
430C2—22no—
431C2—24no—
432C2#1426311yescharacteristic
433C2#2088811yescharacteristic
434C2#2089011yescharacteristic
435C2#2963611yescharacteristic
436C2#2963711yescharacteristic
437C2#2965811yescharacteristic
438C2#2965911yescharacteristic
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
21. #1820C4semidirect
142. C2×C4⋊C4C4⋊C4semidirect
143. C2×C4⋊C4C4⋊C4semidirect
166. C2×C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#340522
C2×#18202
#141282
#139582
#113442
#112872
#107262