Finite Group Databasebeta

← #493729

Subgroups — #493729

Order 512.

1263 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34275C211yesmaximal, characteristic
2C2×#2134C212yesmaximal
3#20640C212yesmaximal, complemented
4#12803C214yesmaximal, complemented
5#12801C214yesmaximal, complemented
6#12585C212yesmaximal, complemented
7C2×#535C2211yescharacteristic
8C2×#520C2211yescharacteristic
9C2×#518C411yescharacteristic
10#2905—22no—
11C2×#413C2214yes—
12#2363—22no—
13#2361—22no—
14#2358—24no—
15#2328C412yes—
16#2326C2212yes—
17#2317C411yescharacteristic
18C4×#127C412yes—
19C4×#124C2212yes—
20#2297C2211yescharacteristic
21C2×#390C411yescharacteristic
22C4×#117C411yescharacteristic
23#2134—28no—
24C2×#355C2218yes—
25#1801C2218yescomplemented
26#1800—24no—
27C2×#343—22no—
28#1394C2214yescomplemented
29#1394C2214yescomplemented
30C22×#41C2311yescharacteristic
31C23×D4C2×C411yescharacteristic
32C22×M16—24no—
33#535—24no—
34#535D414yes—
35#535Q814yes—
36C2×#127C2×C412yes—
37C2×#124C2312yes—
38C2×#121—22no—
39C2×#120—22no—
40#520—44no—
41#518—44no—
42C2×C4×Q8C2312yes—
43C2×C4×D4—24no—
44C2×C4×D4—21no—
45C2×C4×D4—21no—
46C2×C4×D4C2×C412yes—
47C2×C4×D4C2×C412yes—
48C2×C4×D4C2×C412yes—
49C2×C4×D4D414yes—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117D412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4D412yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C42C2×C411yescharacteristic
59#424—48no—
60#423—28no—
61#413—28no—
62#413—28no—
63C2×#101C2318yes—
64C2×#100—24no—
65#391C2311yescharacteristic
66#391C2311yescharacteristic
67#390—22no—
68#390—24no—
69#390C2×C411yescharacteristic
70#390C2×C412yes—
71#390C2×C412yes—
72#390C2×C411yescharacteristic
73#389—22no—
74#388—24no—
75#388—24no—
76#386—22no—
77#386—24no—
78#386—24no—
79#385—22no—
80#385—24no—
81#384C2×C412yes—
82#382C2×C412yes—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—24no—
88#376C2×C412yes—
89C43C2×C412yes—
90#355—216no—
91#343—28no—
92C25—21no—
93C2×#41—42no—
94C2×#41—42no—
95C2×#41C2×D412yes—
96C2×#41C2×Q812yes—
97C2×#41#4114yes—
98C22×Q8C2411yescharacteristic
99C22×D4—24no—
100C22×D4—21no—
101C22×D4—22no—
102C22×D4—42no—
103C22×D4—42no—
104C22×D4C4⋊C414yes—
105C22×D4C22×C412yes—
106C22×D4C22×C411yescharacteristic
107C22×D4C2×D412yes—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C414yes—
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D412yes—
115C2×M16—28no—
116C2×M16—44no—
117C2×M16—48no—
118C2×M16—48no—
119C22×C8—44no—
120#127—24no—
121#127C4⋊C418yes—
122#124—24no—
123#124C2×D414yes—
124#124C2×Q814yes—
125#121—28no—
126#121—28no—
127#120—28no—
128#120—28no—
129C4×Q8—44no—
130C4×D4—44no—
131C4×D4—48no—
132C4×D4—44no—
133C4×D4—44no—
134C4×D4—44no—
135C4×D4—44no—
136C4×D4—88no—
137#117—22no—
138#117—24no—
139#117—28no—
140#117—42no—
141#117C4⋊C414yes—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22⋊C411yescharacteristic
149C2×C4⋊C4C22×C411yescharacteristic
150C2×C4⋊C4C22×C411yescharacteristic
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C2×D414yes—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—48no—
163C2×C22⋊C4—84no—
164C2×C22⋊C4C22⋊C412yes—
165C2×C22⋊C4C22⋊C414yes—
166C2×C42—21no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—42no—
171C2×C42C22×C412yes—
172C2×C42C22×C412yes—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D414yes—
176C4⋊C8—416no—
177#101—28no—
178#101—28no—
179#100—48no—
180#95—22no—
181#95—22no—
182#95—24no—
183#95—22no—
184#95—22no—
185C24—22no—
186C24—24no—
187C24—42no—
188C24—42no—
189C24—82no—
190C24—82no—
191C24C2×C22⋊C411yescharacteristic
192C24C2×C22⋊C412yes—
193#41—44no—
194#41—44no—
195#41—48no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8C22×D411yescharacteristic
199C2×Q8C22×Q811yescharacteristic
200C2×Q8C2×#4112yes—
201C2×D4—216no—
202C2×D4—22no—
203C2×D4—24no—
204C2×D4—41no—
205C2×D4—42no—
206C2×D4—41no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—48no—
211C2×D4—48no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—84no—
215C2×D4—84no—
216C2×D4C2×C4⋊C414yes—
217C2×D4C2×C4⋊C412yes—
218C2×D4#11712yes—
219C2×D4#12114yes—
220C2×D4C22⋊Q814yes—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—81no—
237C22×C4—84no—
238C22×C4—84no—
239C22×C4—82no—
240C22×C4—84no—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C414yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C4⋊C412yes—
246C22×C4#11712yes—
247C22×C4#12114yes—
248C22×C4C22⋊Q814yes—
249C22×C4C23×C411yescharacteristic
250C22×C4C22×D412yes—
251M16—48no—
252M16—48no—
253M16—88no—
254C2×C8—48no—
255C2×C8—88no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—42no—
260C4⋊C4—44no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—84no—
264C4⋊C4#9914yes—
265C22⋊C4—24no—
266C22⋊C4—28no—
267C22⋊C4—28no—
268C22⋊C4—24no—
269C22⋊C4—28no—
270C22⋊C4—44no—
271C22⋊C4—48no—
272C22⋊C4—44no—
273C22⋊C4—42no—
274C22⋊C4—44no—
275C22⋊C4—82no—
276C22⋊C4—84no—
277C42—22no—
278C42—28no—
279C42—24no—
280C42—41no—
281C42—41no—
282C42—44no—
283C42—42no—
284C42—42no—
285C42—82no—
286C42C2×C4⋊C414yes—
287C23—21no—
288C23—22no—
289C23—21no—
290C23—22no—
291C23—24no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—44no—
297C23—44no—
298C23—48no—
299C23—42no—
300C23—42no—
301C23—88no—
302C23—164no—
303C23—162no—
304C23—162no—
305C23#38012yes—
306C23#38014yes—
307C23C22×C22⋊C411yescharacteristic
308Q8—41no—
309Q8—41no—
310Q8—42no—
311D4—42no—
312D4—42no—
313D4—44no—
314D4—44no—
315D4—44no—
316D4—44no—
317D4—48no—
318D4—816no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—28no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—28no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—84no—
346C2×C4—164no—
347C2×C4—162no—
348C2×C4—164no—
349C2×C4—164no—
350C2×C4#38012yes—
351C2×C4C2×#9914yes—
352C2×C4C2×#9912yes—
353C2×C4#41012yes—
354C2×C4C22×C4⋊C411yescharacteristic
355C2×C4C2×#11711yescharacteristic
356C2×C4C2×#12112yes—
357C2×C4C2×C22⋊Q812yes—
358C8—88no—
359C22—21no—
360C22—21no—
361C22—22no—
362C22—24no—
363C22—44no—
364C22—42no—
365C22—42no—
366C22—48no—
367C22—42no—
368C22—41no—
369C22—42no—
370C22—41no—
371C22—44no—
372C22—82no—
373C22—84no—
374C22—82no—
375C22—82no—
376C22—168no—
377C22C2×#38011yescharacteristic
378C22C22×#9911yescharacteristic
379C22C2×#41011yescharacteristic
380C4—24no—
381C4—22no—
382C4—21no—
383C4—21no—
384C4—22no—
385C4—42no—
386C4—44no—
387C4—44no—
388C4—42no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4#179914yes—
393C2—21no—
394C2—21no—
395C2—42no—
396C2—44no—
397C2—84no—
398C2C2×#179911yescharacteristic
399C2#3337011yescharacteristic
400C2#3337311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20640C2semidirect
4. #12803C2semidirect
5. #12801C2semidirect
6. #12585C2semidirect
25. #1801C22semidirect
28. #1394C22semidirect
29. #1394C22semidirect

Maximal subgroups

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

subgroupindex
#342752
C2×#21342
#206402
#128032
#128012
#125852