Finite Group Databasebeta

← #109672

Subgroups — #109672

Order 512.

888 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20526C211yesmaximal, characteristic
2#13908C211yesmaximal, characteristic
3#10846C211yesmaximal, characteristic
4#8023C212yesmaximal
5C2×#1294C212yesmaximal
6C22×C22⋊C8C2212yes—
7C2×#380—21no—
8C2×#380C2212yes—
9C2×#380C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11C22×#95C2211yescharacteristic
12#1752—24no—
13#1749—22no—
14C2×#342—21no—
15C2×#342—21no—
16#1444C412yescomplemented
17#1442C412yescomplemented
18#1436—22no—
19C22⋊#95—21no—
20#1294—24no—
21#1294C418yescomplemented
22C24×C4—21no—
23C24×C4C2311yescharacteristic
24C23×C8—22no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—22no—
27C22×C4⋊C4C2×C412yes—
28C22×C22⋊C4—22no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42D412yes—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8C2×C414yes—
37C2×C22⋊C8D414yes—
38C2×C22⋊C8Q814yes—
39C2×#97—24no—
40#380—22no—
41#380—22no—
42#380—22no—
43#380—22no—
44#380—22no—
45#380—44no—
46#380C2×C414yes—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C2×#95—21no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—21no—
55C2×#95—22no—
56C2×#95C2×C412yes—
57C2×#95C2×C414yes—
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C411yescharacteristic
60#342—42no—
61#342—42no—
62#336—24no—
63C25C2×D411yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—24no—
67C23×C4—22no—
68C23×C4—44no—
69C23×C4—41no—
70C23×C4—42no—
71C23×C4—41no—
72C23×C4—42no—
73C23×C4—82no—
74C23×C4—82no—
75C23×C4C22⋊C412yes—
76C23×C4C22⋊C412yes—
77C23×C4C22×C411yescharacteristic
78C23×C4C22×C412yes—
79C23×C4C2×D412yes—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×Q812yes—
82C23×C4#4112yes—
83C23×C4#4112yes—
84C23×C4#4112yes—
85C22×C8—24no—
86C22×C8—24no—
87C22×C8—24no—
88C22×C8—84no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—44no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—82no—
95C2×C4⋊C4—84no—
96C2×C4⋊C4—82no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—41no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—82no—
118C22⋊C8—24no—
119C22⋊C8—24no—
120C22⋊C8—24no—
121C22⋊C8C4⋊C418yescomplemented
122#97—84no—
123#95—21no—
124#95—21no—
125#95—24no—
126#95—24no—
127#95—24no—
128#95—44no—
129#95—44no—
130#95—41no—
131#95—41no—
132#95—42no—
133#95—82no—
134#95C4212yes—
135#95C4212yes—
136C24—42no—
137C24—42no—
138C24C2×C22⋊C411yescharacteristic
139C24C4×D412yes—
140C24#12112yes—
141C24#12411yescharacteristic
142C24C4⋊Q811yescharacteristic
143C24AGL(1,ℤ/8)11yescharacteristic
144C24AGL(1,ℤ/8)11yescharacteristic
145C24AGL(1,ℤ/8)12yes—
146C24#13711yescharacteristic
147C24#13711yescharacteristic
148C24#13712yes—
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—48no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—44no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4—84no—
189C22×C4—88no—
190C22×C4—84no—
191C22×C4—84no—
192C22×C4—84no—
193C22×C4—84no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4C2×C4211yescharacteristic
197C22×C4C2×C4⋊C412yes—
198C22×C4#11712yes—
199C22×C4#11711yescharacteristic
200C22×C4C4×D412yes—
201C22×C4C4×D412yes—
202C22×C4C4×D412yes—
203C22×C4C4×Q812yes—
204C22×C4C4×Q812yes—
205C22×C4#12011yescharacteristic
206C22×C4C22⋊Q812yes—
207C22×C4C22⋊Q812yes—
208C22×C4#12311yescharacteristic
209C22×C4#12312yes—
210C22×C4#12512yes—
211C22×C4#12612yes—
212C2×C8—28no—
213C2×C8—48no—
214C2×C8—84no—
215C2×C8—84no—
216C4⋊C4—84no—
217C4⋊C4—88no—
218C4⋊C4—84no—
219C4⋊C4—84no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C42—44no—
231C42—82no—
232C23—22no—
233C23—24no—
234C23—22no—
235C23—24no—
236C23—42no—
237C23—42no—
238C23—41no—
239C23—41no—
240C23—42no—
241C23—44no—
242C23—42no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—44no—
247C23—42no—
248C23C4×C22⋊C411yescharacteristic
249C23#38611yescharacteristic
250C23#38812yes—
251C23#38912yes—
252C23#39111yescharacteristic
253C23#41711yescharacteristic
254C23#41711yescharacteristic
255C23#41811yescharacteristic
256C23#41911yescharacteristic
257C23#44012yes—
258C23#44012yes—
259C23#44112yes—
260C23#44212yes—
261C23#46812yes—
262C23#46812yes—
263C23#46912yes—
264C23#47012yes—
265C23#48912yes—
266C23#49011yescharacteristic
267C23#49111yescharacteristic
268C23#50111yescharacteristic
269C23#50112yes—
270C23#50111yescharacteristic
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—88no—
304C2×C4—88no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—84no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—84no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—84no—
319C2×C4—164no—
320C2×C4—164no—
321C2×C4C4×C4⋊C412yes—
322C2×C4#38512yes—
323C2×C4#38712yes—
324C2×C4#38712yes—
325C8—48no—
326C22—21no—
327C22—22no—
328C22—22no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—22no—
335C22—22no—
336C22—24no—
337C22—24no—
338C22—22no—
339C22—22no—
340C22—24no—
341C22—24no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—42no—
351C22—41no—
352C22#143611yescharacteristic
353C22#176211yescharacteristic
354C22#176311yescharacteristic
355C22#188412yes—
356C22#189411yescharacteristic
357C22#189511yescharacteristic
358C22#191312yes—
359C22#193512yes—
360C22#193612yes—
361C22#194312yes—
362C22#194412yes—
363C22#199211yescharacteristic
364C22#199311yescharacteristic
365C4—22no—
366C4—22no—
367C4—22no—
368C4—22no—
369C4—44no—
370C4—82no—
371C4—82no—
372C4—82no—
373C4—164no—
374C4—162no—
375C2—21no—
376C2—22no—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2#938512yes—
384C2#938612yes—
385C2#1088111yescharacteristic
386C2#1089311yescharacteristic
387C2#1091511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1444C4semidirect
17. #1442C4semidirect
21. #1294C4semidirect
121. C22⋊C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#205262
#139082
#108462
#80232
C2×#12942