Finite Group Databasebeta

← #392432

Subgroups — #392432

Order 512.

1637 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#518C211yesmaximal, characteristic
2C2×#1828C212yesmaximal
3C2×#1799C212yesmaximal
4#10854C214yesmaximal, complemented
5#10829C214yesmaximal, complemented
6#10768C212yesmaximal
7C2×#518C2211yescharacteristic
8C2×#518C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C22×#99C2214yes—
11C2×#380C2214yes—
12#2275—24no—
13#2274C412yescomplemented
14#2273C412yescomplemented
15#2273C2212yes—
16C42×D4—24no—
17C4×#117C412yescomplemented
18C2×C4×C22⋊C4C412yescomplemented
19C2×C4×C22⋊C4C2212yes—
20#1828—28no—
21#1799—24no—
22#1799—24no—
23C2×#342C2218yes—
24C2×#342C2214yes—
25C2×#342C2218yes—
26C23×D4C2311yescharacteristic
27C24×C4C2×C411yescharacteristic
28C24×C4C2311yescharacteristic
29#518—24no—
30#518—28no—
31#518D414yes—
32#518Q814yes—
33C2×C4×D4—24no—
34C2×C4×D4—24no—
35C2×C4×D4—24no—
36C2×C4×D4—24no—
37C2×C4×D4D414yes—
38C2×C4×D4D414yes—
39C2×#117C2×C412yes—
40C2×#117C2×C412yes—
41C22×C4⋊C4—24no—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2318yes—
47C22×C22⋊C4C2311yescharacteristic
48C22×C22⋊C4C2312yes—
49C22×C22⋊C4C2312yes—
50C22×C42—22no—
51C22×C42—22no—
52C22×C42C2×C411yescharacteristic
53C2×#99—28no—
54C2×#99—28no—
55C2×#99—28no—
56#380—24no—
57#380—24no—
58#380—24no—
59#380—24no—
60C4×C4⋊C4—24no—
61C4×C4⋊C4—24no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C22⋊C4—22no—
65C4×C22⋊C4—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—24no—
69C4×C22⋊C4—24no—
70C4×C22⋊C4—24no—
71C4×C22⋊C4—24no—
72C4×C22⋊C4C2×C414yes—
73C4×C22⋊C4C2×C414yes—
74C4×C22⋊C4C2×C414yes—
75C4×C22⋊C4C2×C414yes—
76#376—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—24no—
80C43—22no—
81#342—28no—
82#342—216no—
83#342—216no—
84C25C22×C411yescharacteristic
85C25C2411yescharacteristic
86C22×D4—24no—
87C22×D4—24no—
88C22×D4C2×D414yes—
89C22×D4C2×D412yes—
90C22×D4C2×Q812yes—
91C22×D4#4114yes—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—28no—
99C23×C4—82no—
100C23×C4—82no—
101C23×C4C22⋊C412yes—
102C23×C4C22⋊C412yes—
103C23×C4C22×C411yescharacteristic
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C411yescharacteristic
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4#4114yes—
111C23×C4#4114yes—
112C4×D4—232no—
113C4×D4—216no—
114C4×D4—216no—
115C4×D4—28no—
116C4×D4—28no—
117C4×D4—28no—
118C4×D4—28no—
119#117—24no—
120#117—28no—
121#117C4⋊C418yescomplemented
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—88no—
128C2×C4⋊C4C22⋊C414yes—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—28no—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—28no—
145C2×C22⋊C4—28no—
146C2×C22⋊C4C22⋊C414yes—
147C2×C22⋊C4C4⋊C418yescomplemented
148C2×C22⋊C4C22×C414yes—
149C2×C22⋊C4C22×C414yes—
150C2×C22⋊C4C2×D414yes—
151C2×C22⋊C4C2×D414yes—
152C2×C22⋊C4C2×Q814yes—
153C2×C22⋊C4#4118yes—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—84no—
163C2×C42—84no—
164C2×C42C22⋊C412yes—
165C2×C42C22⋊C412yes—
166#99—416no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—44no—
172#95—44no—
173C24—22no—
174C24—22no—
175C24—81no—
176C24—81no—
177C24—81no—
178C24—81no—
179C24C2×C22⋊C412yes—
180C24C2×C4⋊C412yes—
181C24#11712yes—
182C24C23×C411yescharacteristic
183C24C22×D412yes—
184C24C22×D411yescharacteristic
185C24C22×Q811yescharacteristic
186C24C2×#4114yes—
187C24C2×#4112yes—
188C24C2×#4114yes—
189C2×D4—28no—
190C2×D4—28no—
191C2×D4—28no—
192C2×D4—28no—
193C2×D4—28no—
194C2×D4—28no—
195C2×D4#12118yes—
196C2×D4C22⋊Q818yes—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—28no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—28no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—44no—
212C22×C4—48no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—48no—
216C22×C4—48no—
217C22×C4—48no—
218C22×C4—84no—
219C22×C4—81no—
220C22×C4—82no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—88no—
235C22×C4—88no—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C4⋊C412yes—
243C22×C4#11712yes—
244C22×C4#11712yes—
245C22×C4#11712yes—
246C22×C4#11714yes—
247C22×C4#12314yes—
248C22×C4#12314yes—
249C22×C4#12314yes—
250C22×C4#12314yes—
251C4⋊C4—28no—
252C4⋊C4—28no—
253C4⋊C4—28no—
254C4⋊C4—88no—
255C4⋊C4—88no—
256C22⋊C4—28no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—28no—
262C22⋊C4—28no—
263C22⋊C4—28no—
264C22⋊C4—28no—
265C22⋊C4—48no—
266C22⋊C4—48no—
267C22⋊C4—48no—
268C22⋊C4—48no—
269C22⋊C4C2×C4⋊C418yes—
270C22⋊C4#11718yes—
271C42—28no—
272C42—28no—
273C42—22no—
274C42—24no—
275C42—22no—
276C42—48no—
277C42—84no—
278C42—84no—
279C42—84no—
280C23—22no—
281C23—22no—
282C23—24no—
283C23—24no—
284C23—42no—
285C23—44no—
286C23—41no—
287C23—42no—
288C23—44no—
289C23—41no—
290C23—42no—
291C23—42no—
292C23—82no—
293C23—82no—
294C23—81no—
295C23—81no—
296C23—82no—
297C23—84no—
298C23—82no—
299C23—82no—
300C23—84no—
301C23#38014yes—
302C23C22×C22⋊C411yescharacteristic
303C23C22×C4⋊C411yescharacteristic
304C23C2×#11712yes—
305C23C2×#11711yescharacteristic
306C23C2×#11712yes—
307C23C2×#12112yes—
308C23C2×C22⋊Q812yes—
309C23C2×#12314yes—
310C23#52514yes—
311C23#52914yes—
312C23#52914yes—
313C23#53214yes—
314C23#53314yes—
315D4—216no—
316D4—216no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—28no—
323C2×C4—24no—
324C2×C4—28no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—28no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—84no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—88no—
349C2×C4—88no—
350C2×C4—88no—
351C2×C4—88no—
352C2×C4—88no—
353C2×C4—88no—
354C2×C4—168no—
355C2×C4—168no—
356C2×C4#38014yes—
357C2×C4#38111yescharacteristic
358C2×C4#38112yes—
359C2×C4#38114yes—
360C2×C4#38111yescharacteristic
361C22—21no—
362C22—24no—
363C22—22no—
364C22—21no—
365C22—24no—
366C22—22no—
367C22—24no—
368C22—42no—
369C22—42no—
370C22—44no—
371C22—44no—
372C22—44no—
373C22—41no—
374C22—41no—
375C22—42no—
376C22—44no—
377C22—44no—
378C22—44no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—84no—
383C22—84no—
384C22C2×#38011yescharacteristic
385C22C2×#38111yescharacteristic
386C22#228011yescharacteristic
387C22#228312yes—
388C22#228512yes—
389C22#285414yes—
390C22#285514yes—
391C4—24no—
392C4—21no—
393C4—22no—
394C4—24no—
395C4—48no—
396C4—48no—
397C4—42no—
398C4—42no—
399C4—168no—
400C4#182812yescomplemented
401C2—21no—
402C2—21no—
403C2—24no—
404C2—22no—
405C2—22no—
406C2—24no—
407C2—44no—
408C2#1468511yescharacteristic
409C2C2×#182811yescharacteristic
410C2#2071011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10854C2semidirect
5. #10829C2semidirect
13. #2274C4semidirect
14. #2273C4semidirect
17. C4×#117C4semidirect
18. C2×C4×C22⋊C4C4semidirect
121. #117C4⋊C4semidirect
147. C2×C22⋊C4C4⋊C4semidirect
400. C4#1828semidirect

Maximal subgroups

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

subgroupindex
C4×#5182
C2×#18282
C2×#17992
#108542
#108292
#107682