Finite Group Databasebeta

← #497594

Subgroups — #497594

Order 512.

943 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#39616C211yesmaximal, characteristic
2Q8×#99C212yesmaximal
3#13300C214yesmaximal
4#13298C212yesmaximal
5#12800C212yesmaximal
6#12798C211yesmaximal, characteristic
7#12627C212yesmaximal
8C4×#354C211yesmaximal, characteristic
9C2×#557C411yescharacteristic, complemented
10C2×D4×Q8C2211yescharacteristic
11#2868C411yescharacteristic, complemented
12#2720—24no—
13#2660C412yescomplemented
14#2568C411yescharacteristic, complemented
15#2567C412yescomplemented
16#2566C2212yes—
17#2542C2211yescharacteristic
18#2434—22no—
19#2429—22no—
20Q8×C22⋊C4—22no—
21#2326C2212yes—
22C4×#124C2211yescharacteristic
23#2295C411yescharacteristic, complemented
24C2×#354C2211yescharacteristic
25C2×#354C2212yes—
26C2×#354C2211yescharacteristic
27#1996—22no—
28#1996—24no—
29#1799C2214yes—
30#1799C2212yes—
31C4×#99C2212yes—
32C4×#99C2214yes—
33#1393C2214yes—
34#1393C2212yes—
35#1393C2214yes—
36#1393C2212yes—
37C23×Q8D412yes—
38#557—42no—
39#557—42no—
40D4×Q8—42no—
41D4×Q8—42no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—22no—
48C2×#125C2×C412yes—
49C2×#124C2311yescharacteristic
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8D412yes—
52C2×C22⋊Q8D414yes—
53C2×C4×Q8—21no—
54C2×C4×Q8—42no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×D4C2312yes—
59C2×C4×D4C2311yescharacteristic
60C2×#99C2312yes—
61C2×#99C2314yes—
62C2×#99C2312yes—
63#400—22no—
64#399—22no—
65#398—24no—
66#396—24no—
67#395—24no—
68#395—24no—
69#391—22no—
70#391—21no—
71#391—42no—
72#391—44no—
73#391C2×C411yescharacteristic
74#391C2×C412yes—
75#390C2311yescharacteristic
76#390C2312yes—
77#389—24no—
78#389—21no—
79#388—24no—
80#388—22no—
81#387—22no—
82#385—24no—
83#385—24no—
84#384—22no—
85#384C2×C412yes—
86#384C2×C412yes—
87#384C2×C412yes—
88#382C2×C412yes—
89#382C2×C411yescharacteristic
90#380—22no—
91#380—24no—
92C4×C4⋊C4C2×C412yes—
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—24no—
97#376C2×C412yes—
98C43C2×C411yescharacteristic
99#354—22no—
100#354—24no—
101#354Q814yescomplemented
102#342—24no—
103#342—22no—
104#342—24no—
105#342—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—24no—
109C22×Q8—21no—
110C22×Q8—84no—
111C22×Q8C22⋊C412yes—
112C22×Q8C22⋊C414yes—
113C22×Q8C22⋊C411yescharacteristic
114C22×Q8C22×C411yescharacteristic
115C22×D4C2411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D414yes—
118C4⋊Q8—44no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—42no—
124C4⋊Q8—42no—
125C4⋊Q8—84no—
126#125—44no—
127#125—48no—
128#124—22no—
129#124C2×Q812yes—
130#124#4112yes—
131C22⋊Q8—42no—
132C22⋊Q8—42no—
133C22⋊Q8—44no—
134C22⋊Q8—44no—
135C22⋊Q8—84no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—44no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—42no—
142C4×Q8—42no—
143C4×Q8—88no—
144C4×D4—44no—
145C4×D4—41no—
146C4×D4—41no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C414yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4C2×D414yes—
174C2×C22⋊C4C2×D412yes—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—44no—
180C2×C42—42no—
181C2×C42C22×C412yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C412yes—
186#99—22no—
187#99—22no—
188#99—24no—
189#99—22no—
190#99—24no—
191#99C2×Q814yes—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—21no—
198#95—24no—
199#95—24no—
200#95—42no—
201#95—44no—
202#95—42no—
203#95—44no—
204C24C22×D412yes—
205C2×Q8—21no—
206C2×Q8—24no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—24no—
210C2×Q8—42no—
211C2×Q8—44no—
212C2×Q8—44no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—48no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—84no—
219C2×Q8—84no—
220C2×Q8—88no—
221C2×Q8#9912yes—
222C2×Q8#9912yes—
223C2×Q8C4×Q812yescomplemented
224C2×D4—41no—
225C2×D4—41no—
226C2×D4C22×Q811yescharacteristic
227C2×D4C2×#4111yescharacteristic
228C2×D42−1+412yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—84no—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C414yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C23×C411yescharacteristic
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—24no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—84no—
268C4⋊C4—84no—
269C4⋊C4—84no—
270C4⋊C4—88no—
271C4⋊C4—84no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—44no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—44no—
281C22⋊C4—42no—
282C22⋊C4—44no—
283C22⋊C4—84no—
284C42—21no—
285C42—24no—
286C42—21no—
287C42—21no—
288C42—41no—
289C42—42no—
290C42—42no—
291C42—44no—
292C42—42no—
293C42—41no—
294C42—41no—
295C42—88no—
296C42—84no—
297C42C4×Q812yescomplemented
298C23—22no—
299C23—22no—
300C23—42no—
301C23—42no—
302C23C22×C22⋊C411yescharacteristic
303C23#54812yes—
304C23D4×Q812yes—
305Q8—24no—
306Q8—42no—
307Q8—42no—
308Q8—42no—
309Q8—48no—
310Q8—44no—
311Q8—82no—
312Q8—84no—
313Q8—82no—
314Q8—82no—
315D4—41no—
316D4—41no—
317D4—42no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—42no—
329C2×C4—48no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—84no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—84no—
358C2×C4—82no—
359C2×C4C2×#9911yescharacteristic
360C2×C4C2×#9914yes—
361C2×C4C2×#9911yescharacteristic
362C2×C4C2×C4×Q811yescharacteristic
363C2×C4#51911yescharacteristic
364C22—21no—
365C22—21no—
366C22—22no—
367C22—22no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—82no—
373C22Q8×C22⋊C411yescharacteristic
374C22C22×#9911yescharacteristic
375C22#288411yescharacteristic
376C4—21no—
377C4—24no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—42no—
382C4—44no—
383C4—42no—
384C4—42no—
385C4—41no—
386C4—42no—
387C4—82no—
388C4—84no—
389C4—82no—
390C4—81no—
391C4—164no—
392C4—162no—
393C4#212712yes—
394C2—21no—
395C2—21no—
396C2—42no—
397C2—42no—
398C2Q8×#9911yescharacteristic
399C2C2×#212711yescharacteristic
400C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, prime radical, cosocleC22×D4C24
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×D4×Q8 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C2×#557C4semidirect
11. #2868C4semidirect
13. #2660C4semidirect
14. #2568C4semidirect
15. #2567C4semidirect
23. #2295C4semidirect
101. #354Q8semidirect
223. C2×Q8C4×Q8semidirect
297. C42C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#396162
Q8×#992
#133002
#132982
#128002
#127982
#126272
C4×#3542