Finite Group Databasebeta

← #237815

Subgroups — #237815

Order 512.

1082 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#35167C211yesmaximal, characteristic
2#20734C212yesmaximal
3#13925C211yesmaximal, characteristic
4#10343C214yesmaximal
5#10341C214yesmaximal
6#8026C212yesmaximal
7#7908C211yesmaximal, characteristic
8C2×#544—21no—
9C22×C4×Q8C2211yescharacteristic
10C2×#431C2212yes—
11#2857—22no—
12#2659C2214yes—
13#2552—22no—
14#2551—22no—
15#2550—24no—
16#2316—21no—
17#2285C2212yes—
18C2×#382C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1824C2214yes—
21#1803—22no—
22#1803—22no—
23#1787C2214yes—
24C4×#103—24no—
25#1446C2212yes—
26#1445C2212yes—
27#1436—22no—
28C4⋊#95—21no—
29C4⋊#95—21no—
30#1294C2218yes—
31#1280C2214yes—
32C23×Q8C2311yescharacteristic
33#544—44no—
34#544—44no—
35C2×C4⋊Q8—24no—
36C2×C4⋊Q8—22no—
37C2×#125—24no—
38C2×C22⋊Q8—24no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8—24no—
43C2×C4×Q8Q814yescomplemented
44C2×#117—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C42—21no—
51C22×C42D412yes—
52C22×C42C2311yescharacteristic
53#431—24no—
54#431—24no—
55C2×C4⋊C8—22no—
56C2×#103—24no—
57C2×#103—24no—
58C2×C22⋊C8C2314yes—
59C2×C4×C8—22no—
60#398—28no—
61#395—28no—
62#391—24no—
63#389—24no—
64#389—24no—
65#389—24no—
66#389—24no—
67#384—22no—
68#384D412yes—
69#384D412yes—
70#382—22no—
71#382—24no—
72#380—24no—
73#380—22no—
74#380C2314yes—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4D414yes—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95C2312yes—
85C2×#95C2312yes—
86#340—24no—
87#340—24no—
88C22×Q8—24no—
89C22×Q8—21no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—42no—
93C22×Q8—42no—
94C22×Q8C2×Q812yes—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2411yescharacteristic
103C22×C8—24no—
104C4⋊Q8—416no—
105C4⋊Q8—84no—
106#125—416no—
107C22⋊Q8—48no—
108C22⋊Q8—48no—
109C4×Q8—28no—
110C4×Q8—28no—
111C4×Q8—44no—
112C4×Q8—44no—
113#117—44no—
114#117—44no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4SD1614yes—
138C2×C4⋊C4Q1614yes—
139C2×C4⋊C4C2×D414yes—
140C2×C4⋊C4C2×D414yes—
141C2×C4⋊C4C2×Q812yes—
142C2×C22⋊C4—24no—
143C2×C42—24no—
144C2×C42—22no—
145C2×C42—24no—
146C2×C42—22no—
147C2×C42—42no—
148C2×C42—42no—
149C2×C42—82no—
150C2×C42C2×D412yes—
151C2×C42C2×D412yes—
152C2×C42C2×Q812yes—
153C2×C42#4114yes—
154C2×C42#4112yes—
155C4⋊C8—44no—
156#103—48no—
157#103—48no—
158C22⋊C8—24no—
159C22⋊C8—24no—
160C4×C8—44no—
161#95—28no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—44no—
167#95—44no—
168#95—42no—
169#95—82no—
170#95—82no—
171C24C22×D411yescharacteristic
172C2×Q8—24no—
173C2×Q8—24no—
174C2×Q8—24no—
175C2×Q8—44no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—44no—
179C2×Q8—48no—
180C2×Q8—48no—
181C2×Q8—42no—
182C2×Q8—84no—
183C2×Q8—84no—
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—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4#12011yescharacteristic
220C22×C4#12011yescharacteristic
221C22×C4#12012yes—
222C22×C4C22⋊Q812yes—
223C22×C4C22⋊Q812yes—
224C22×C4#12412yes—
225C22×C4#12412yes—
226C22×C4C2×SD1614yes—
227C22×C4C2×SD1612yes—
228C22×C4C2×Q1614yes—
229C22×C4C2×Q1612yes—
230C22×C4AGL(1,ℤ/8)12yes—
231C22×C4#13712yes—
232C22×C4C22×D412yes—
233C22×C4C22×Q811yescharacteristic
234C22×C4C2×#4112yes—
235C22×C4C2×#4111yescharacteristic
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C4⋊C4—28no—
240C4⋊C4—28no—
241C4⋊C4—24no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—48no—
246C4⋊C4—416no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C42—44no—
259C42—48no—
260C42—48no—
261C42—82no—
262C42C22⋊Q814yes—
263C42C22⋊Q814yes—
264C42#12414yes—
265C42#12414yes—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23C2×#12011yescharacteristic
271C23C2×C22⋊Q811yescharacteristic
272C23C2×#12411yescharacteristic
273C23C22×SD1611yescharacteristic
274C23C22×Q1611yescharacteristic
275C23C2×AGL(1,ℤ/8)11yescharacteristic
276C23C2×#13711yescharacteristic
277Q8—28no—
278Q8—44no—
279Q8—44no—
280Q8—44no—
281Q8—168no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
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—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4#39312yes—
331C2×C4#39312yes—
332C2×C4#39314yes—
333C2×C4#45014yes—
334C2×C4#45114yes—
335C2×C4#47514yes—
336C2×C4#47714yes—
337C2×C4#48714yes—
338C2×C4#48914yes—
339C2×C4C2×C22⋊Q812yes—
340C2×C4C2×#12412yes—
341C8—84no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22C2×#39311yescharacteristic
355C22C2×#45011yescharacteristic
356C22C2×#45111yescharacteristic
357C22#301911yescharacteristic
358C22#302011yescharacteristic
359C22#302511yescharacteristic
360C22#302511yescharacteristic
361C22C2×#47511yescharacteristic
362C22C2×#47711yescharacteristic
363C22C2×#48711yescharacteristic
364C22C2×#48911yescharacteristic
365C4—22no—
366C4—22no—
367C4—22no—
368C4—22no—
369C4—44no—
370C4—44no—
371C4—44no—
372C4—84no—
373C4—84no—
374C4—82no—
375C4—162no—
376C4#202418yes—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2#1821111yescharacteristic
382C2#1821311yescharacteristic
383C2#1821911yescharacteristic
384C2#1822111yescharacteristic
385C2C2×#202411yescharacteristic
386C2#2346211yescharacteristic
387C2#2346711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
43. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#207342
#139252
#103432
#103412
#80262
#79082