Finite Group Databasebeta

← #480889

Subgroups — #480889

Order 512.

838 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38958C211yesmaximal, characteristic
2#17418C212yesmaximal
3#11564C212yesmaximal, complemented
4#11556C214yesmaximal, complemented
5#11531C211yesmaximal, characteristic, complemented
6#11510C212yesmaximal, complemented
7#11477C211yesmaximal, characteristic, complemented
8#11434C212yesmaximal, complemented
9C2×#555C2211yescharacteristic
10C2×D4×Q8C411yescharacteristic
11#2924—22no—
12#2862C411yescharacteristic
13#2700—24no—
14#2620C412yes—
15#2567C411yescharacteristic
16#2547C2211yescharacteristic
17#2499C2212yes—
18#2439—22no—
19#2436C412yes—
20#2420—22no—
21#2310C2212yes—
22C4×#124C2211yescharacteristic
23#2293C411yescharacteristic
24#1471—22no—
25#1469—24no—
26#1465C2212yes—
27#1464C2214yes—
28#1459C2212yes—
29#1458C2214yes—
30#1324C2212yes—
31#1324C2212yes—
32#1319C2212yes—
33#1319C2214yes—
34#1319C2214yes—
35#1319C2211yescharacteristic
36#1319C2211yescharacteristic
37C23×Q8C2×C412yes—
38#555—42no—
39#555—42no—
40D4×Q8—42no—
41D4×Q8—42no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2×C412yes—
47C2×#124D414yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8C2×C412yes—
50C2×#121C2×C412yes—
51C2×#121D412yes—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×D4—21no—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×#117C2×C412yes—
57C2×#117D412yes—
58C2×#117C2312yes—
59#433—44no—
60#432—24no—
61#431—24no—
62#424—42no—
63#407—22no—
64C2×C22⋊C8—22no—
65#405—42no—
66#397—24no—
67#397—22no—
68#396—24no—
69#395—22no—
70#394—24no—
71#393—24no—
72#391—22no—
73#391—24no—
74#391C2×C412yes—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2311yescharacteristic
78#391C2312yes—
79#390—21no—
80#390C2×C411yescharacteristic
81#389—22no—
82#389—21no—
83#388—22no—
84#388—22no—
85#387—24no—
86#386—24no—
87#384C2×C412yes—
88#382—22no—
89#382C2311yescharacteristic
90#381—24no—
91#380—22no—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C22⋊C4—22no—
94C4×C22⋊C4C2×C412yes—
95C43C2×C411yescharacteristic
96#324C2312yes—
97#324C2314yes—
98#324C2312yes—
99#323—24no—
100#323—22no—
101#323—24no—
102#323—22no—
103C22×Q8—22no—
104C22×Q8—24no—
105C22×Q8—42no—
106C22×Q8—44no—
107C22×Q8—82no—
108C22×Q8C22⋊C412yes—
109C22×Q8C22×C412yes—
110C22×Q8C22×C411yescharacteristic
111C22×D4C22⋊C411yescharacteristic
112C22×D4C22⋊C414yes—
113C22×D4C22×C411yescharacteristic
114C23×C4—22no—
115C23×C4C22×C412yes—
116C23×C4C2×D412yes—
117C2×M16—42no—
118C22×C8—42no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—42no—
124C4⋊Q8—42no—
125#124—22no—
126#124—44no—
127#124—44no—
128#124—41no—
129#124—82no—
130C22⋊Q8—48no—
131C22⋊Q8—44no—
132#121—44no—
133#121—42no—
134#121—42no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×D4—22no—
138C4×D4—22no—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—42no—
142C4×D4—42no—
143#117—22no—
144#117—22no—
145#117—24no—
146#117—44no—
147#117—82no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22×C411yescharacteristic
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C2411yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊C414yes—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C2×D414yes—
169C2×C22⋊C4C2×D412yes—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C412yes—
177C2×C42C2×D414yes—
178C4⋊C8—44no—
179C4⋊C8—44no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—24no—
183C22⋊C8—42no—
184C22⋊C8—42no—
185#97—44no—
186C4×C8—44no—
187#95—22no—
188#95—21no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—22no—
193#95—22no—
194#95—24no—
195C24C2×C22⋊C412yes—
196C24C2×C22⋊C412yes—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×Q8—44no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—44no—
207C2×Q8—82no—
208C2×Q8—82no—
209C2×Q8—88no—
210C2×Q8—88no—
211C2×Q8#13112yes—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—21no—
215C2×D4—21no—
216C2×D4—42no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—44no—
220C2×D4—84no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—82no—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C414yes—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C23×C411yescharacteristic
242C22×C4C22×D412yes—
243M16—84no—
244C2×C8—42no—
245C2×C8—44no—
246C2×C8—42no—
247C2×C8—84no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—44no—
259C4⋊C4—82no—
260C4⋊C4—84no—
261C4⋊C4#9912yes—
262C4⋊C4#9912yes—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—44no—
267C22⋊C4—42no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—82no—
271C22⋊C4—84no—
272C22⋊C4—82no—
273C22⋊C4—82no—
274C42—24no—
275C42—24no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—41no—
281C42—42no—
282C42—42no—
283C42—41no—
284C42—41no—
285C42—81no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23C22×C22⋊C411yescharacteristic
297Q8—42no—
298Q8—42no—
299Q8—88no—
300Q8—82no—
301Q8—82no—
302Q8—162no—
303D4—42no—
304D4—42no—
305D4—81no—
306D4—81no—
307D4—164no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—24no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4—88no—
341C2×C4C2×#9914yes—
342C2×C4C2×#9911yescharacteristic
343C2×C4C2×#9911yescharacteristic
344C2×C4C2×#13111yescharacteristic
345C2×C4#56811yescharacteristic
346C8—82no—
347C8—82no—
348C8—84no—
349C22—21no—
350C22—21no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—82no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22#288011yescharacteristic
362C22C22×#9911yescharacteristic
363C22#288811yescharacteristic
364C4—21no—
365C4—21no—
366C4—24no—
367C4—21no—
368C4—42no—
369C4—42no—
370C4—44no—
371C4—41no—
372C4—81no—
373C4—82no—
374C4—81no—
375C4—82no—
376C4—81no—
377C4#212912yes—
378C2—21no—
379C2—21no—
380C2—82no—
381C2—82no—
382C2#1741111yescharacteristic
383C2C2×#212911yescharacteristic
384C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#555 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11564C2semidirect
4. #11556C2semidirect
5. #11531C2semidirect
6. #11510C2semidirect
7. #11477C2semidirect
8. #11434C2semidirect

Maximal subgroups

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

subgroupindex
#389582
#174182
#115642
#115562
#115312
#115102
#114772
#114342