Finite Group Databasebeta

← #109696

Subgroups — #109696

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20534C211yesmaximal, characteristic
2#14298C211yesmaximal, characteristic
3C4×#343C211yesmaximal, characteristic
4#8033C212yesmaximal
5C2×#1288C212yesmaximal
6C2×#432C2212yes—
7C2×C4×M16C2212yes—
8C2×#383C2211yescharacteristic
9C2×#382C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1771—24no—
12#1749—21no—
13C4×C22⋊C8—21no—
14C2×#343—21no—
15C2×#343—21no—
16#1444—21no—
17#1444C411yescharacteristic
18#1444C411yescharacteristic
19#1442—21no—
20#1442C411yescharacteristic
21#1442C411yescharacteristic
22#1436—21no—
23#1436—21no—
24#1295—24no—
25#1288C418yescomplemented
26C22×M16—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4D411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42C2311yescharacteristic
36#432—22no—
37#432—22no—
38#432C2×C414yes—
39#432Q814yes—
40C2×C4⋊C8—22no—
41C2×C22⋊C8—22no—
42C4×M16—22no—
43C4×M16—22no—
44C4×M16—22no—
45C4×M16C2×C414yes—
46C2×#97—22no—
47C2×#97—22no—
48C2×#97D414yes—
49C2×C4×C8—22no—
50C2×C4×C8—22no—
51#383—22no—
52#383C2×C414yes—
53#382—22no—
54#382—22no—
55C4×C4⋊C4—22no—
56C4×C4⋊C4—22no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95C2×C411yescharacteristic
68C2×#95C2×C411yescharacteristic
69C2×#95C2×C411yescharacteristic
70C2×#95C2×C411yescharacteristic
71#343—24no—
72#343—24no—
73#336—22no—
74#336—22no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C2×M16—24no—
87C2×M16—24no—
88C2×M16—24no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—42no—
92C22×C8—24no—
93C22×C8—42no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—81no—
105C2×C4⋊C4—81no—
106C2×C4⋊C4—81no—
107C2×C4⋊C4—81no—
108C2×C4⋊C4—81no—
109C2×C4⋊C4—81no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—81no—
115C2×C42—81no—
116C2×C42C22×C412yes—
117C2×C42C2×D412yes—
118C2×C42C2×Q812yes—
119C2×C42#4112yes—
120C2×C42#4112yes—
121C2×C42#4112yes—
122C4⋊C8—24no—
123C4⋊C8—24no—
124C22⋊C8—24no—
125C22⋊C8—24no—
126#97—24no—
127#97—24no—
128#97—42no—
129#97—82no—
130#97C4⋊C418yescomplemented
131C4×C8—24no—
132C4×C8—42no—
133C4×C8—82no—
134#95—21no—
135#95—21no—
136#95—24no—
137#95—24no—
138#95—22no—
139#95—22no—
140#95—42no—
141#95—42no—
142#95—41no—
143#95—42no—
144#95—41no—
145#95—81no—
146#95—81no—
147#95—81no—
148#95—81no—
149#95C4212yes—
150#95C4212yes—
151C24C2×C22⋊C411yescharacteristic
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
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—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4#10412yes—
207C22×C4#10412yes—
208C22×C4C2×C4211yescharacteristic
209C22×C4#11711yescharacteristic
210C22×C4C4×D412yes—
211C22×C4C4×D412yes—
212C22×C4C4×D412yes—
213C22×C4C4×Q812yes—
214C22×C4#12011yescharacteristic
215C22×C4#12112yes—
216C22×C4C22⋊Q812yes—
217C22×C4#12311yescharacteristic
218C22×C4#12411yescharacteristic
219C22×C4C4⋊Q811yescharacteristic
220M16—28no—
221M16—44no—
222M16—44no—
223M16—44no—
224C2×C8—28no—
225C2×C8—44no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C42—42no—
237C42—42no—
238C42—81no—
239C42—81no—
240C42C2×C4⋊C412yes—
241C42#11712yes—
242C42C4×D412yes—
243C42C4×Q812yes—
244C42C22⋊Q812yes—
245C42#12312yes—
246C42#12512yes—
247C42#12612yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23C4×C22⋊C411yescharacteristic
253C23#38611yescharacteristic
254C23#39111yescharacteristic
255C23C2×#10411yescharacteristic
256C23C2×#10411yescharacteristic
257C23#42111yescharacteristic
258C23#42111yescharacteristic
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4C4×C4⋊C412yes—
327C2×C4#38512yes—
328C2×C4#38712yes—
329C2×C4#38712yes—
330C2×C4#38812yes—
331C2×C4#38912yes—
332C8—48no—
333C8—48no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22#143611yescharacteristic
347C22C4×#10411yescharacteristic
348C22#175911yescharacteristic
349C22#187111yescharacteristic
350C22#187111yescharacteristic
351C22#191111yescharacteristic
352C22#191211yescharacteristic
353C22#191411yescharacteristic
354C22#191511yescharacteristic
355C22#198611yescharacteristic
356C22#198611yescharacteristic
357C4—22no—
358C4—22no—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—42no—
364C4—42no—
365C4—82no—
366C4—82no—
367C4—81no—
368C4—81no—
369C4—161no—
370C4—161no—
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2#918711yescharacteristic
376C2#918811yescharacteristic
377C2#918911yescharacteristic
378C2#919011yescharacteristic
379C2#1088611yescharacteristic
380C2#1088811yescharacteristic
381C2#1091611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1288C4semidirect
130. #97C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#205342
#142982
C4×#3432
#80332
C2×#12882