Finite Group Databasebeta

← #95654

Subgroups — #95654

Order 512.

886 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20714C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic
3#14683C211yesmaximal, characteristic
4#7024C214yesmaximal, complemented
5C23×C42C2211yescharacteristic
6C2×#431C412yescomplemented
7C2×#423—21no—
8C2×#407—21no—
9C2×#383—21no—
10#1817—21no—
11#1797—21no—
12#1797—21no—
13#1797—21no—
14#1797—21no—
15#1797C412yescomplemented
16#1797C2212yes—
17#1779—21no—
18C4×C22⋊C8C412yescomplemented
19C4×C22⋊C8C2212yes—
20#1728C412yescomplemented
21#1434—21no—
22#1434C412yes—
23#1434C2212yes—
24C22⋊#95—21no—
25C22⋊#95—21no—
26C22⋊#95C412yes—
27C24×C4C2×C411yescharacteristic
28C24×C4D411yescharacteristic
29C24×C4D411yescharacteristic
30C22×M16—22no—
31C22×C22⋊C4—22no—
32C22×C42—44no—
33C22×C42C2×C411yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42C2×C411yescharacteristic
36C22×C42C2×C411yescharacteristic
37C22×C42C2×C411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42C22×C42Q811yescharacteristic
43C22×C42Q811yescharacteristic
44C22×C42C2311yescharacteristic
45#431—44no—
46#423—44no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50#407—44no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C4×C8C2×C414yes—
57#383—44no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95C2×C414yes—
64#336—22no—
65#336—22no—
66#336—22no—
67#336C2×C414yes—
68C25C22⋊C411yescharacteristic
69C23×C4—24no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—24no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4C4211yescharacteristic
81C23×C4C4211yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C4⋊C411yescharacteristic
90C23×C4C4⋊C411yescharacteristic
91C23×C4C4⋊C411yescharacteristic
92C23×C4C4⋊C411yescharacteristic
93C23×C4C4⋊C411yescharacteristic
94C23×C4C4⋊C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C2×M16—44no—
99C2×M16—44no—
100C2×M16—44no—
101C22×C8—24no—
102C22×C8—24no—
103C2×C22⋊C4—44no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—44no—
106C2×C42—24no—
107C2×C42—24no—
108C2×C42—24no—
109C2×C42—42no—
110C2×C42—42no—
111C2×C42—42no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42C4211yescharacteristic
119C2×C42C4211yescharacteristic
120C2×C42C22⋊C411yescharacteristic
121C2×C42C22⋊C411yescharacteristic
122C2×C42C22⋊C411yescharacteristic
123C2×C42C22⋊C411yescharacteristic
124C2×C42C4⋊C411yescharacteristic
125C2×C42C4⋊C411yescharacteristic
126C2×C42C4⋊C411yescharacteristic
127C2×C42C4⋊C411yescharacteristic
128C2×C42C4⋊C411yescharacteristic
129C2×C42C4⋊C411yescharacteristic
130C2×C42C22×C411yescharacteristic
131C2×C42C22×C411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×Q811yescharacteristic
134C4⋊C8—44no—
135C4⋊C8—44no—
136C4⋊C8—44no—
137C22⋊C8—44no—
138C22⋊C8—44no—
139C22⋊C8—44no—
140C22⋊C8—44no—
141C22⋊C8—44no—
142C4×C8—44no—
143#95—28no—
144#95—44no—
145#95—44no—
146#95—44no—
147#95—44no—
148#95—42no—
149#95—42no—
150C24—22no—
151C24—22no—
152C24—22no—
153C24—22no—
154C24—22no—
155C24—41no—
156C24—41no—
157C24#9511yescharacteristic
158C24#9511yescharacteristic
159C24C2×C22⋊C411yescharacteristic
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
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—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—48no—
215C22×C4—416no—
216C22×C4—48no—
217C22×C4—416no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4#9511yescharacteristic
226C22×C4#9511yescharacteristic
227C22×C4#9511yescharacteristic
228C22×C4#9511yescharacteristic
229C22×C4#9511yescharacteristic
230C22×C4#9511yescharacteristic
231C22×C4C2×C4211yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C4⋊C411yescharacteristic
235C22×C4C2×C4⋊C411yescharacteristic
236C22×C4C2×C4⋊C411yescharacteristic
237C22×C4#13114yes—
238C22×C4AGL(1,ℤ/8)12yes—
239C22×C4#13712yes—
240M16—88no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—44no—
246C2×C8—44no—
247C22⋊C4—88no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—41no—
258C42—41no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42—41no—
263C42#9914yes—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—28no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—44no—
285C23—44no—
286C23—44no—
287C23H(3,ℤ/4)12yes—
288C23#34412yes—
289C23C2×#9511yescharacteristic
290C23#40812yes—
291C23#41712yes—
292C23#42112yes—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—28no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4H(3,ℤ/4)12yes—
350C2×C4#34212yes—
351C2×C4#34212yes—
352C2×C4#40512yes—
353C2×C4C2×#9912yes—
354C2×C4#41811yescharacteristic
355C2×C4#41911yescharacteristic
356C2×C4#42412yes—
357C2×C4#42812yes—
358C8—84no—
359C8—84no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—416no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22#149514yes—
382C22#172911yescharacteristic
383C22C2×H(3,ℤ/4)11yescharacteristic
384C22C2×H(3,ℤ/4)11yescharacteristic
385C22#173311yescharacteristic
386C22#173611yescharacteristic
387C22C2×#34211yescharacteristic
388C22C2×#34411yescharacteristic
389C4—22no—
390C4—22no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—164no—
403C2—24no—
404C2—42no—
405C2—42no—
406C2#785811yescharacteristic
407C2#789911yescharacteristic
408C2#792711yescharacteristic
409C2#794611yescharacteristic
410C2#796011yescharacteristic
411C2#797511yescharacteristic
412C2#798311yescharacteristic
subgroupG/H
center, socleC23C2×#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7024C2semidirect
6. C2×#431C4semidirect
15. #1797C4semidirect
18. C4×C22⋊C8C4semidirect
20. #1728C4semidirect

Maximal subgroups

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

subgroupindex
#207142
#206262
#146832
#70242