Finite Group Databasebeta

← #108185

Subgroups — #108185

Order 512.

1226 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2313C211yesmaximal, characteristic
2C2×#1463C211yesmaximal, characteristic
3#13870C211yesmaximal, characteristic
4#10733C212yesmaximal
5C2×#1284C212yesmaximal
6C22×#117C2211yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407C2212yes—
9#2313—24no—
10#2313—22no—
11#2313C414yescomplemented
12C2×#387—24no—
13C2×#381—21no—
14C2×#381—21no—
15C2×#381C2211yescharacteristic
16C2×#380—21no—
17C2×C4×C22⋊C4C2212yes—
18C2×#339—24no—
19#1463—22no—
20#1463C414yescomplemented
21C2×#324—22no—
22#1441C414yescomplemented
23#1436—22no—
24C22⋊#95—21no—
25#1284—24no—
26C24×C4D411yescharacteristic
27C24×C4D411yescharacteristic
28C24×C4C2311yescharacteristic
29C22×M16—22no—
30C2×#117—22no—
31C2×#117—22no—
32C2×#117—44no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4D412yes—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4Q812yes—
45C22×C22⋊C4—21no—
46C22×C22⋊C4—22no—
47C22×C22⋊C4—21no—
48C22×C42—21no—
49C22×C42D414yes—
50#407—22no—
51#407—22no—
52#407—22no—
53#407C2×C414yes—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56#387—216no—
57#387—216no—
58#381—24no—
59#381—24no—
60#381—22no—
61#381—22no—
62#381—44no—
63#380—24no—
64#380—24no—
65C4×C22⋊C4—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4C2×C414yes—
68C4×C22⋊C4D414yes—
69C2×#95—22no—
70C2×#95—24no—
71C2×#95—22no—
72C2×#95—24no—
73C2×#95—24no—
74C2×#95—21no—
75C2×#95C2×C412yes—
76#339—48no—
77#324—28no—
78C25C2×D411yescharacteristic
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—24no—
89C23×C4—22no—
90C23×C4—42no—
91C23×C4—44no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C412yes—
99C23×C4C4⋊C412yes—
100C23×C4C4⋊C412yes—
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2×Q811yescharacteristic
109C23×C4#4112yes—
110C23×C4#4112yes—
111C2×M16—24no—
112C2×M16—42no—
113C2×M16—44no—
114C2×M16—44no—
115C22×C8—42no—
116#117—28no—
117#117—48no—
118#117—44no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4—82no—
136C2×C4⋊C4C4214yes—
137C2×C4⋊C4C22⋊C414yes—
138C2×C4⋊C4C4⋊C414yes—
139C2×C4⋊C4C4⋊C414yes—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—48no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C42—24no—
152C2×C42—44no—
153C2×C42—42no—
154C2×C42—44no—
155C2×C42—44no—
156C2×C42C22⋊C418yes—
157C22⋊C8—24no—
158C22⋊C8—24no—
159C22⋊C8—42no—
160C22⋊C8—42no—
161#95—28no—
162#95—216no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—48no—
167#95—48no—
168#95—42no—
169C24—24no—
170C24—22no—
171C24—22no—
172C24—42no—
173C24C2×C22⋊C411yescharacteristic
174C24C2×C22⋊C411yescharacteristic
175C24C2×C4⋊C411yescharacteristic
176C24#12011yescharacteristic
177C24#12011yescharacteristic
178C24#12311yescharacteristic
179C24#12311yescharacteristic
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—28no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—48no—
224C22×C4—48no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—82no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—82no—
239C22×C4#9514yes—
240C22×C4#9514yes—
241C22×C4C2×C4211yescharacteristic
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C4⋊C411yescharacteristic
245C22×C4C2×C4⋊C411yescharacteristic
246C22×C4#11712yes—
247C22×C4C4×D412yes—
248C22×C4C4×D412yes—
249C22×C4C4×D412yes—
250C22×C4C4×D412yes—
251C22×C4#12012yes—
252C22×C4#12112yes—
253C22×C4#12112yes—
254C22×C4C22⋊Q812yes—
255C22×C4C22⋊Q812yes—
256C22×C4#12312yes—
257M16—48no—
258M16—84no—
259C2×C8—44no—
260C2×C8—82no—
261C2×C8—82no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—28no—
265C4⋊C4—28no—
266C4⋊C4—28no—
267C4⋊C4—44no—
268C4⋊C4—88no—
269C22⋊C4—48no—
270C22⋊C4—48no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C42—48no—
276C42—48no—
277C42—84no—
278C23—21no—
279C23—21no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—48no—
300C23—48no—
301C23C2×#9511yescharacteristic
302C23#37911yescharacteristic
303C23#38111yescharacteristic
304C23#38511yescharacteristic
305C23#38511yescharacteristic
306C23#38611yescharacteristic
307C23#38611yescharacteristic
308C23#42114yes—
309C23B(4,𝔽2)12yes—
310C23#45812yes—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
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—44no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—44no—
348C2×C4—48no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—88no—
353C2×C4—88no—
354C2×C4—84no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4C4×C22⋊C412yes—
363C2×C4#38012yes—
364C2×C4#38512yes—
365C2×C4#38612yes—
366C8—84no—
367C22—24no—
368C22—24no—
369C22—24no—
370C22—24no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—44no—
389C22—42no—
390C22—42no—
391C22C22⋊#9511yescharacteristic
392C22#173312yes—
393C22#179012yes—
394C22#186712yes—
395C22#188911yescharacteristic
396C22#189011yescharacteristic
397C22#189712yes—
398C22#189712yes—
399C22#191212yes—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—22no—
404C4—44no—
405C4—44no—
406C4—88no—
407C4—84no—
408C4—162no—
409C2—22no—
410C2—22no—
411C2—22no—
412C2—22no—
413C2—41no—
414C2—41no—
415C2#948012yes—
416C2#948212yes—
417C2#1072011yescharacteristic
418C2#1074111yescharacteristic
419C2#1074511yescharacteristic
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
11. #2313C4semidirect
20. #1463C4semidirect
22. #1441C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23132
C2×#14632
#138702
#107332
C2×#12842