Finite Group Databasebeta

← #493727

Subgroups — #493727

Order 512.

1224 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2C2×#2132C212yesmaximal
3#20640C212yesmaximal, complemented
4#12803C214yesmaximal, complemented
5#12802C214yesmaximal, complemented
6#12587C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#520C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2905—22no—
11C2×#413C2214yes—
12#2361—24no—
13#2354—22no—
14D4×C4⋊C4—22no—
15#2326C412yes—
16#2326C2212yes—
17#2311C411yescharacteristic
18#2311C2211yescharacteristic
19C4×C4⋊Q8C2212yes—
20C4×#127C412yes—
21C2×#384C411yescharacteristic
22C4×#117C411yescharacteristic
23#2132—28no—
24C2×#356C2214yes—
25C2×#355C2214yes—
26#1801C2218yes—
27#1800—24no—
28C2×#343—22no—
29#1394C2218yes—
30C22×#41C2×C411yescharacteristic
31C22×#41C2311yescharacteristic
32C22×M16—24no—
33#532—24no—
34#532D414yes—
35#532Q814yes—
36C2×C4⋊Q8C2311yescharacteristic
37C2×#127C2×C411yescharacteristic
38C2×#124C2×C411yescharacteristic
39C2×#124C2311yescharacteristic
40C2×#121—22no—
41#520—44no—
42#520—44no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2312yes—
45C2×C4×D4—22no—
46C2×C4×D4—24no—
47C2×C4×D4—22no—
48C2×C4×D4C2×C412yes—
49C2×C4×D4C2×C412yes—
50C2×C4×D4D414yes—
51C2×#117—21no—
52C2×#117C2×C411yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117D412yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4D412yes—
60C22×C42C2×C411yescharacteristic
61#424—48no—
62#423—28no—
63#413—28no—
64#413—28no—
65C2×#101C2318yes—
66C2×#100—24no—
67#391C2×C412yes—
68#391C2312yes—
69#390—24no—
70#390C2×C412yes—
71#390C2×C412yes—
72#389—24no—
73#388—24no—
74#388—24no—
75#386—22no—
76#386—22no—
77#385—24no—
78#385—24no—
79#384—22no—
80#384C2×C412yes—
81#384C2×C412yes—
82#380—22no—
83#380—22no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—24no—
87C4×C22⋊C4—22no—
88#376C2×C412yes—
89C43C2×C412yes—
90#356—28no—
91#355—28no—
92#343—28no—
93C2×#41—22no—
94C2×#41—42no—
95C2×#41—42no—
96C2×#41—42no—
97C2×#41—42no—
98C2×#41C4⋊C414yes—
99C2×#41C2×D412yes—
100C2×#41C2×Q812yes—
101C2×#41#4114yes—
102C22×Q8C22×C411yescharacteristic
103C22×Q8C2411yescharacteristic
104C22×D4—22no—
105C22×D4—21no—
106C22×D4C22×C411yescharacteristic
107C22×D4C22×C411yescharacteristic
108C22×D4C2×D412yes—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4C22⋊C412yes—
114C23×C4C22⋊C414yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D412yes—
117C2×M16—28no—
118C2×M16—44no—
119C2×M16—48no—
120C2×M16—48no—
121C22×C8—44no—
122C4⋊Q8—22no—
123C4⋊Q8C2×D412yes—
124C4⋊Q8C2×Q812yes—
125#127—22no—
126#127C4⋊C414yes—
127#124—22no—
128#124—22no—
129#124C4⋊C414yes—
130#124C2×D412yes—
131#124C2×Q812yes—
132#121—28no—
133#121—28no—
134C4×Q8—44no—
135C4×Q8—44no—
136C4×D4—28no—
137C4×D4—28no—
138C4×D4—44no—
139C4×D4—48no—
140C4×D4—44no—
141C4×D4—48no—
142C4×D4—88no—
143#117—24no—
144#117—28no—
145#117—44no—
146#117—42no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—84no—
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C2×D414yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4C22⋊C412yes—
171C2×C22⋊C4C22⋊C414yes—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42C4⋊C414yes—
178C2×C42C22×C412yes—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C412yes—
182C2×C42C2×D414yes—
183C4⋊C8—416no—
184#101—28no—
185#101—28no—
186#100—48no—
187#95—22no—
188#95—22no—
189#95—24no—
190#95—24no—
191C24—21no—
192C24C2×C22⋊C412yes—
193C24C2×C22⋊C411yescharacteristic
194#41—44no—
195#41—44no—
196#41—48no—
197#41—48no—
198#41—48no—
199C2×Q8—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8C2×C4⋊C412yes—
205C2×Q8C22×D411yescharacteristic
206C2×Q8C22×Q811yescharacteristic
207C2×Q8C2×#4112yes—
208C2×D4—28no—
209C2×D4—24no—
210C2×D4—21no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—41no—
214C2×D4—42no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4C2×C4⋊C412yes—
222C2×D4C2×C4⋊C412yes—
223C2×D4#11712yes—
224C2×D4#12114yes—
225C2×D4C22⋊Q814yes—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—82no—
246C22×C4—84no—
247C22×C4—82no—
248C22×C4—84no—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C414yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C4⋊C412yes—
254C22×C4#11712yes—
255C22×C4#12114yes—
256C22×C4C22⋊Q814yes—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D412yes—
259M16—48no—
260M16—48no—
261M16—88no—
262C2×C8—48no—
263C2×C8—88no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—48no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—84no—
276C4⋊C4#9914yes—
277C22⋊C4—28no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—28no—
281C22⋊C4—24no—
282C22⋊C4—42no—
283C22⋊C4—44no—
284C22⋊C4—84no—
285C22⋊C4—84no—
286C42—21no—
287C42—21no—
288C42—24no—
289C42—28no—
290C42—24no—
291C42—42no—
292C42—44no—
293C42—44no—
294C42—44no—
295C42C2×C4⋊C412yes—
296C42C2×C4⋊C412yes—
297C23—21no—
298C23—22no—
299C23—21no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—41no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—41no—
309C23—41no—
310C23#38014yes—
311C23#38012yes—
312C23C22×C22⋊C411yescharacteristic
313Q8—41no—
314Q8—41no—
315Q8—42no—
316Q8—42no—
317Q8—42no—
318D4—42no—
319D4—42no—
320D4—44no—
321D4—44no—
322D4—42no—
323D4—416no—
324D4—42no—
325D4—48no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—28no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—48no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—162no—
356C2×C4—164no—
357C2×C4—164no—
358C2×C4—162no—
359C2×C4#38012yes—
360C2×C4C2×#9914yes—
361C2×C4C2×#9912yes—
362C2×C4#41012yes—
363C2×C4C22×C4⋊C411yescharacteristic
364C2×C4C2×#11711yescharacteristic
365C2×C4C2×#12112yes—
366C2×C4C2×C22⋊Q812yes—
367C8—88no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—24no—
372C22—42no—
373C22—44no—
374C22—42no—
375C22—44no—
376C22—41no—
377C22—42no—
378C22—41no—
379C22—42no—
380C22—42no—
381C22—82no—
382C22—82no—
383C22—81no—
384C22—81no—
385C22C2×#38011yescharacteristic
386C22C22×#9911yescharacteristic
387C22C2×#41011yescharacteristic
388C4—24no—
389C4—24no—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—42no—
395C4—44no—
396C4—42no—
397C4—42no—
398C4—82no—
399C4—82no—
400C4#179914yes—
401C2—21no—
402C2—21no—
403C2—42no—
404C2—44no—
405C2—82no—
406C2C2×#179911yescharacteristic
407C2#3337211yescharacteristic
408C2#3337311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#532C22
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. #12802C2semidirect
6. #12587C2semidirect

Maximal subgroups

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

subgroupindex
#342072
C2×#21322
#206402
#128032
#128022
#125872