Finite Group Databasebeta

← #98799

Subgroups — #98799

Order 512.

830 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#391C211yesmaximal, characteristic
2#10832C212yesmaximal
3C2×#1272C212yesmaximal
4#7025C212yesmaximal
5C22×C4×Q8C411yescharacteristic
6C22×C4⋊C8—22no—
7C2×#391C411yescharacteristic
8C2×#391C2211yescharacteristic
9C2×C4×C4⋊C4C411yescharacteristic
10C2×C43C2211yescharacteristic
11C2×#336—22no—
12C2×#336C2214yes—
13#1436—22no—
14C4⋊#95C411yescharacteristic
15C4⋊#95C2211yescharacteristic
16C4×#95—22no—
17#1272—24no—
18C23×Q8C2×C411yescharacteristic
19C23×C8—24no—
20C2×C4×Q8—22no—
21C2×C4×Q8—22no—
22C2×C4×Q8—42no—
23C2×C4×Q8—42no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C42—21no—
29C22×C42—21no—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42D412yes—
33C22×C42D412yes—
34C22×C42C2311yescharacteristic
35C2×C4⋊C8—28no—
36C2×C4⋊C8—44no—
37C2×C4⋊C8—44no—
38#391—22no—
39#391—22no—
40#391C814yescomplemented
41#391D412yes—
42#391Q812yes—
43C4×C4⋊C4—42no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—21no—
48C2×#95—22no—
49C43—42no—
50#336—24no—
51#336—24no—
52#336—44no—
53C22×Q8—21no—
54C22×Q8—21no—
55C22×Q8—21no—
56C22×Q8—41no—
57C22×Q8—41no—
58C22×Q8—41no—
59C22×Q8—41no—
60C22×Q8C4⋊C412yes—
61C22×Q8C2×C812yes—
62C22×Q8M1612yes—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C412yes—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D412yes—
72C22×C8—28no—
73C22×C8—44no—
74C22×C8—44no—
75C22×C8—44no—
76C22×C8—44no—
77C22×C8—44no—
78C22×C8—44no—
79C4×Q8—48no—
80C4×Q8—48no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4C4⋊C412yes—
94C2×C4⋊C4C2×C812yes—
95C2×C4⋊C4M1612yes—
96C2×C42—22no—
97C2×C42—24no—
98C2×C42—22no—
99C2×C42—24no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—24no—
105C2×C42—41no—
106C2×C42—42no—
107C2×C42—41no—
108C2×C42—41no—
109C2×C42—42no—
110C2×C42—41no—
111C2×C42—42no—
112C2×C42—42no—
113C2×C42—44no—
114C2×C42—42no—
115C2×C42C2×C812yes—
116C2×C42M1612yes—
117C2×C42SD1614yes—
118C2×C42Q1614yes—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×Q811yescharacteristic
121C2×C42#4112yes—
122C4⋊C8—416no—
123#95—28no—
124#95—24no—
125#95—24no—
126#95—42no—
127#95—44no—
128#95—42no—
129#95—42no—
130C24C2×C22⋊C411yescharacteristic
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8C4⋊C814yescomplemented
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4C22⋊C814yes—
189C22×C4#9912yes—
190C22×C4#10012yes—
191C22×C4#10314yes—
192C22×C4#10314yes—
193C22×C4#10414yes—
194C22×C4C2×C4⋊C411yescharacteristic
195C22×C4#11711yescharacteristic
196C22×C4#12112yes—
197C22×C4C22⋊Q812yes—
198C22×C4C22×C811yescharacteristic
199C22×C4C2×M1611yescharacteristic
200C22×C4C2×M1611yescharacteristic
201C22×C4C2×M1611yescharacteristic
202C22×C4C2×SD1612yes—
203C22×C4C2×Q1612yes—
204C22×C4AGL(1,ℤ/8)12yes—
205C22×C4#13712yes—
206C2×C8—48no—
207C2×C8—48no—
208C2×C8—48no—
209C2×C8—84no—
210C2×C8—84no—
211C2×C8—84no—
212C2×C8—84no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—48no—
220C42—28no—
221C42—28no—
222C42—24no—
223C42—24no—
224C42—42no—
225C42—42no—
226C42—44no—
227C42—44no—
228C42—41no—
229C42—41no—
230C42—42no—
231C42—48no—
232C42—84no—
233C23#38011yescharacteristic
234C23C2×C22⋊C811yescharacteristic
235C23#40711yescharacteristic
236C23C2×#9911yescharacteristic
237C23#41011yescharacteristic
238C23C2×#10011yescharacteristic
239C23#41311yescharacteristic
240C23C2×#10312yes—
241C23#41712yes—
242C23C2×#10412yes—
243C23#42112yes—
244Q8—44no—
245Q8—44no—
246Q8—44no—
247Q8—44no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4Q8⋊C818yes—
310C2×C4#33314yes—
311C2×C4C2×C4⋊C811yescharacteristic
312C2×C4#42311yescharacteristic
313C2×C4#43111yescharacteristic
314C2×C4#43211yescharacteristic
315C2×C4#46012yes—
316C2×C4#46212yes—
317C2×C4#47512yes—
318C2×C4#47712yes—
319C8—88no—
320C22#145911yescharacteristic
321C22#146011yescharacteristic
322C22#146211yescharacteristic
323C22#146411yescharacteristic
324C22C2×Q8⋊C812yes—
325C22#147712yes—
326C22#148712yes—
327C22#148912yes—
328C22#150312yes—
329C22#150412yes—
330C22#151212yes—
331C22#152012yes—
332C22#152511yescharacteristic
333C22#153212yes—
334C22#153311yescharacteristic
335C22C2×#33311yescharacteristic
336C22#155012yes—
337C22#156011yescharacteristic
338C22#179711yescharacteristic
339C22#179911yescharacteristic
340C22#180011yescharacteristic
341C22#180312yes—
342C22#180612yes—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—28no—
352C4—24no—
353C4—44no—
354C4—42no—
355C4—82no—
356C4—82no—
357C4—82no—
358C2#757111yescharacteristic
359C2#758912yes—
360C2#760111yescharacteristic
361C2#761512yes—
362C2#763111yescharacteristic
363C2#764311yescharacteristic
364C2#855911yescharacteristic
365C2#856412yes—
366C2#857812yes—
367C2#858311yescharacteristic
368C2#858711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
40. #391C8semidirect
146. C2×Q8C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3912
#108322
C2×#12722
#70252