Finite Group Databasebeta

← #353620

Subgroups — #353620

Order 512.

2190 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2905C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3#33996C211yesmaximal, characteristic
4C22×H(3,ℤ/4)C214yesmaximal
5#10836C218yesmaximal, complemented
6C23×M16C2212yes—
7C22×#117C2212yes—
8C22×#117C2212yes—
9C22×#117C2211yescharacteristic
10#2905—24no—
11#2905—24no—
12C2×#424C418yescomplemented
13C2×#423C2214yes—
14#2317—24no—
15#2316—22no—
16#2315—22no—
17#2296C418yescomplemented
18#2285C2214yes—
19#2274—24no—
20#2274—22no—
21#2273—22no—
22C4×#117C418yescomplemented
23C4×#117C2214yes—
24C2×H(3,ℤ/4)—216no—
25C2×H(3,ℤ/4)C22116yes—
26C24×C4C2311yescharacteristic
27C22×M16—24no—
28C22×M16—24no—
29C22×M16—28no—
30C22×M16—28no—
31C22×M16C2×C414yes—
32C22×M16C2314yes—
33C23×C8C2×C414yes—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—24no—
37C2×#117—24no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—28no—
41C2×#117—28no—
42C2×#117C2×C414yes—
43C2×#117C2×C414yes—
44C2×#117C2×C414yes—
45C2×#117C2×C414yes—
46C2×#117C2×C414yes—
47C2×#117D418yes—
48C2×#117D414yes—
49C2×#117D414yes—
50C2×#117Q818yes—
51C2×#117Q814yes—
52C2×#117C2314yes—
53C2×#117C2314yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4D412yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D412yes—
60C22×C42C2×C414yes—
61C22×C42C2×C414yes—
62C22×C42C2312yes—
63#424—232no—
64#423—28no—
65#423—28no—
66C2×C4⋊C8C2×C4116yescomplemented
67#389—216no—
68#385—216no—
69#384—24no—
70#382—24no—
71#380—28no—
72C4×C4⋊C4—216no—
73C4×C4⋊C4—28no—
74C4×C4⋊C4C2×C4116yescomplemented
75C4×C22⋊C4—28no—
76C4×C22⋊C4—216no—
77#376C2×C418yescomplemented
78C43C2×C418yescomplemented
79H(3,ℤ/4)—232no—
80C25C2×D411yescharacteristic
81C23×C4—24no—
82C23×C4—24no—
83C23×C4C22×C412yes—
84C23×C4C22×C412yes—
85C23×C4C22×C412yes—
86C23×C4C2×D414yes—
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D412yes—
89C23×C4C2×D412yes—
90C23×C4C2×D414yes—
91C23×C4C2×Q812yes—
92C23×C4C2×Q814yes—
93C23×C4#4114yes—
94C23×C4#4114yes—
95C23×C4C2411yescharacteristic
96C2×M16—28no—
97C2×M16—216no—
98C2×M16—28no—
99C2×M16—24no—
100C2×M16—24no—
101C2×M16—28no—
102C2×M16—48no—
103C2×M16—48no—
104C2×M16—48no—
105C2×M16—48no—
106C22×C8—28no—
107C22×C8—216no—
108C22×C8C22×C418yes—
109#117—28no—
110#117—28no—
111#117—28no—
112#117—28no—
113#117—28no—
114#117—24no—
115#117—24no—
116#117—216no—
117#117—28no—
118#117—28no—
119#117—48no—
120#117—44no—
121#117—44no—
122#117C42116yes—
123#117C22⋊C4116yes—
124#117C4⋊C4116yes—
125#117C4⋊C4116yes—
126#117C4⋊C4116yescomplemented
127#117C2×D418yes—
128#117C2×Q818yes—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—216no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4C22⋊C418yes—
139C2×C4⋊C4C22⋊C418yes—
140C2×C4⋊C4C4⋊C418yes—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—48no—
148C2×C22⋊C4C22⋊C418yes—
149C2×C22⋊C4C22⋊C418yes—
150C2×C22⋊C4C4⋊C418yes—
151C2×C42—24no—
152C2×C42—28no—
153C2×C42—216no—
154C2×C42—24no—
155C2×C42—216no—
156C2×C42C4⋊C4116yescomplemented
157C2×C42C22×C418yes—
158C2×C42C22×C418yes—
159C2×C42C22×C414yes—
160C2×C42C22×C414yes—
161C2×C42C22×C414yes—
162C2×C42C2×D414yes—
163C2×C42C2×D414yes—
164C2×C42C2×Q814yes—
165C4⋊C8—232no—
166#95—216no—
167#95—216no—
168C24—82no—
169C24C2×C22⋊C412yes—
170C24C2×C22⋊C412yes—
171C24C2×C4⋊C412yes—
172C24#12114yes—
173C24C22⋊Q814yes—
174C24C22×D411yescharacteristic
175C22×C4—28no—
176C22×C4—216no—
177C22×C4—28no—
178C22×C4—48no—
179C22×C4—84no—
180C22×C4—88no—
181C22×C4—88no—
182C22×C4C2×C4214yes—
183C22×C4C2×C22⋊C418yes—
184C22×C4C2×C22⋊C418yes—
185C22×C4C2×C22⋊C414yes—
186C22×C4C2×C22⋊C412yes—
187C22×C4C2×C22⋊C412yes—
188C22×C4C2×C4⋊C418yes—
189C22×C4C2×C4⋊C414yes—
190C22×C4C2×C4⋊C414yes—
191C22×C4C2×C4⋊C414yes—
192C22×C4C2×C4⋊C414yes—
193C22×C4C2×C4⋊C412yes—
194C22×C4#11714yes—
195C22×C4#11714yes—
196C22×C4C4×D418yes—
197C22×C4C4×D418yes—
198C22×C4C4×Q818yes—
199C22×C4C4×Q818yes—
200C22×C4#12114yes—
201C22×C4#12114yes—
202C22×C4#12114yes—
203C22×C4C22⋊Q814yes—
204C22×C4C22⋊Q814yes—
205C22×C4C22⋊Q814yes—
206C22×C4#12414yes—
207C22×C4#12414yes—
208C22×C4#12518yes—
209C22×C4#12714yes—
210C22×C4C4⋊Q818yes—
211C22×C4C4⋊Q814yes—
212C22×C4C23×C411yescharacteristic
213C22×C4C23×C411yescharacteristic
214C22×C4C23×C411yescharacteristic
215C22×C4C22×D411yescharacteristic
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D412yes—
218C22×C4C22×Q811yescharacteristic
219C22×C4C22×Q812yes—
220C22×C4C2×#4112yes—
221C22×C4C2×#4112yes—
222M16—48no—
223M16—44no—
224M16—44no—
225M16—416no—
226C2×C8—28no—
227C2×C8—216no—
228C2×C8—416no—
229C4⋊C4—28no—
230C4⋊C4—28no—
231C4⋊C4—216no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—48no—
235C4⋊C4—816no—
236C4⋊C4#95132yes—
237C22⋊C4—28no—
238C22⋊C4—28no—
239C22⋊C4—216no—
240C22⋊C4—48no—
241C22⋊C4—48no—
242C22⋊C4#95132yes—
243C42—28no—
244C42—216no—
245C42—216no—
246C42—48no—
247C42—816no—
248C42C2×C4218yes—
249C42C2×C22⋊C418yes—
250C42C2×C4⋊C4116yes—
251C42C2×C4⋊C418yes—
252C42C2×C4⋊C418yes—
253C23—44no—
254C23—44no—
255C23—42no—
256C23—44no—
257C23—84no—
258C23—84no—
259C23C2×#9514yes—
260C23#38014yes—
261C23#38014yes—
262C23#38918yes—
263C23#39014yes—
264C23#39114yes—
265C23C22×C22⋊C411yescharacteristic
266C23C22×C22⋊C411yescharacteristic
267C23C22×C4⋊C411yescharacteristic
268C23C2×#12112yes—
269C23C2×C22⋊Q812yes—
270C2×C4—216no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—48no—
274C2×C4—416no—
275C2×C4—416no—
276C2×C4—48no—
277C2×C4—816no—
278C2×C4—816no—
279C2×C4C2×#95116yes—
280C2×C4C2×#9514yes—
281C2×C4C4×C4⋊C418yes—
282C2×C4C4×C4⋊C418yes—
283C2×C4#38014yes—
284C2×C4#38014yes—
285C2×C4#38218yes—
286C2×C4#38418yes—
287C2×C4#38918yes—
288C2×C4#38918yes—
289C2×C4#38918yes—
290C2×C4#39014yes—
291C2×C4#39018yes—
292C2×C4#39114yes—
293C2×C4#39118yes—
294C2×C4C22×C4211yescharacteristic
295C2×C4C22×C22⋊C411yescharacteristic
296C2×C4C22×C4⋊C412yes—
297C2×C4C22×C4⋊C411yescharacteristic
298C2×C4C22×C4⋊C411yescharacteristic
299C2×C4C2×#11712yes—
300C2×C4C2×C4×D412yes—
301C2×C4C2×C4×D412yes—
302C2×C4C2×C4×Q812yes—
303C2×C4C2×C4×Q812yes—
304C2×C4C2×#12112yes—
305C2×C4C2×C22⋊Q812yes—
306C2×C4C2×#12411yescharacteristic
307C2×C4C2×#12411yescharacteristic
308C2×C4C2×#12512yes—
309C2×C4C2×#12711yescharacteristic
310C2×C4C2×C4⋊Q812yes—
311C2×C4C2×C4⋊Q811yescharacteristic
312C8—48no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—28no—
317C22—44no—
318C22—44no—
319C22—42no—
320C22—42no—
321C22—44no—
322C22—48no—
323C22—44no—
324C22C4⋊#9518yes—
325C22C22×#9511yescharacteristic
326C22C2×#38012yes—
327C22C2×#38912yes—
328C22C2×#39011yescharacteristic
329C22C2×#39111yescharacteristic
330C4—216no—
331C4—416no—
332C4—88no—
333C4C4⋊#9518yes—
334C4C2×C4×C4⋊C412yes—
335C4C2×#38211yescharacteristic
336C4C2×#38411yescharacteristic
337C4C2×#38912yes—
338C4C2×#39011yescharacteristic
339C4C2×#39111yescharacteristic
340C2—22no—
341C2—21no—
342C2—28no—
343C2—22no—
344C2—21no—
345C2C2×C4⋊#9511yescharacteristic
346C2#2033812yes—
subgroupG/H
center, commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC22C22×#95
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10836C2semidirect
12. C2×#424C4semidirect
17. #2296C4semidirect
22. C4×#117C4semidirect
66. C2×C4⋊C8C2×C4semidirect
74. C4×C4⋊C4C2×C4semidirect
77. #376C2×C4semidirect
78. C43C2×C4semidirect
126. #117C4⋊C4semidirect
156. C2×C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29052
#340522
#339962
C22×H(3,ℤ/4)2
#108362