Finite Group Databasebeta

← #497592

Subgroups — #497592

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39618C211yesmaximal, characteristic
2Q8×#99C212yesmaximal, complemented
3#13300C214yesmaximal, complemented
4#13299C212yesmaximal, complemented
5#12799C212yesmaximal
6#12797C211yesmaximal, characteristic
7#12628C212yesmaximal
8C4×#353C211yesmaximal, characteristic
9C2×#554C411yescharacteristic, complemented
10C2×D4×Q8C2211yescharacteristic
11#2862C411yescharacteristic, complemented
12#2692—24no—
13#2660C412yescomplemented
14#2566C2212yes—
15#2563C412yescomplemented
16#2563C2211yescharacteristic
17#2542C411yescharacteristic, complemented
18#2417—22no—
19#2409—22no—
20Q8×C22⋊C4—22no—
21#2328C2212yes—
22C4×#127C2211yescharacteristic
23#2295C411yescharacteristic, complemented
24C2×#353C2211yescharacteristic
25C2×#353C2212yes—
26C2×#353C2211yescharacteristic
27#1996—22no—
28#1996—24no—
29#1799C2212yes—
30#1799C2214yes—
31C4×#99C2212yes—
32C4×#99C2214yes—
33#1393C2212yes—
34#1393C2214yes—
35#1393C2214yes—
36#1393C2212yes—
37C23×Q8D412yes—
38#554—42no—
39#554—42no—
40D4×Q8—42no—
41D4×Q8—42no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#127C2311yescharacteristic
46C2×#125C2×C412yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8D412yes—
51C2×C22⋊Q8D414yes—
52C2×C4×Q8—42no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×D4—22no—
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4C2×C412yes—
57C2×C4×D4C2311yescharacteristic
58C2×C4×D4C2312yes—
59C22×C4⋊C4—21no—
60C2×#99C2314yes—
61C2×#99C2312yes—
62C2×#99C2312yes—
63#399—22no—
64#398—24no—
65#396—24no—
66#395—24no—
67#393—24no—
68#393—22no—
69#391—22no—
70#391—21no—
71#391—22no—
72#391—42no—
73#391—44no—
74#390—22no—
75#390C2×C411yescharacteristic
76#390C2×C412yes—
77#390C2311yescharacteristic
78#390C2312yes—
79#389—21no—
80#387—22no—
81#386—24no—
82#386—24no—
83#385—24no—
84#384C2×C411yescharacteristic
85#384C2×C412yes—
86#382C2×C412yes—
87#382C2×C412yes—
88#382C2×C412yes—
89#380—22no—
90#380—24no—
91#380—22no—
92#380—24no—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4C2×C411yescharacteristic
95C4×C22⋊C4—22no—
96C4×C22⋊C4—24no—
97#376C2×C412yes—
98C43C2×C411yescharacteristic
99#353—22no—
100#353—24no—
101#353Q814yescomplemented
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×D4—22no—
115C22×D4C22×C411yescharacteristic
116C22×D4C2411yescharacteristic
117C23×C4—21no—
118C23×C4—22no—
119C23×C4C2×D412yes—
120C23×C4C2×D414yes—
121C4⋊Q8—44no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8—41no—
126#127—22no—
127#127C2×Q812yes—
128#127#4112yes—
129#125—44no—
130C22⋊Q8—42no—
131C22⋊Q8—42no—
132C22⋊Q8—44no—
133C22⋊Q8—44no—
134C22⋊Q8—48no—
135C22⋊Q8—42no—
136C22⋊Q8—42no—
137C22⋊Q8—84no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—88no—
141C4×D4—44no—
142C4×D4—41no—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—44no—
147C4×D4—84no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C414yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4C2×D414yes—
177C2×C22⋊C4C2×D412yes—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—44no—
181C2×C42—42no—
182C2×C42C22×C412yes—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C412yes—
187#99—22no—
188#99—22no—
189#99—24no—
190#99—24no—
191#99—22no—
192#99C2×Q814yes—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—21no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—42no—
201#95—44no—
202#95—42no—
203#95—44no—
204C24—21no—
205C24C22×D412yes—
206C2×Q8—24no—
207C2×Q8—22no—
208C2×Q8—24no—
209C2×Q8—22no—
210C2×Q8—42no—
211C2×Q8—44no—
212C2×Q8—42no—
213C2×Q8—48no—
214C2×Q8—84no—
215C2×Q8—82no—
216C2×Q8—84no—
217C2×Q8—88no—
218C2×Q8#9912yes—
219C2×Q8#9912yes—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—21no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—82no—
228C2×D4—82no—
229C2×D4C4×Q812yescomplemented
230C2×D4C22×Q811yescharacteristic
231C2×D4C2×#4111yescharacteristic
232C2×D42−1+412yes—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—84no—
251C22×C4—84no—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C414yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C23×C411yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—24no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—44no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—84no—
276C4⋊C4—88no—
277C4⋊C4—84no—
278C22⋊C4—24no—
279C22⋊C4—42no—
280C22⋊C4—44no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—44no—
284C22⋊C4—42no—
285C22⋊C4—44no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—84no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C22⋊C4—84no—
292C42—21no—
293C42—24no—
294C42—21no—
295C42—21no—
296C42—41no—
297C42—42no—
298C42—42no—
299C42—81no—
300C42—88no—
301C42—84no—
302C42C4×Q812yescomplemented
303C23—22no—
304C23—22no—
305C23—22no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—41no—
311C23—41no—
312C23C22×C22⋊C411yescharacteristic
313C23#54812yes—
314C23D4×Q812yes—
315Q8—24no—
316Q8—42no—
317Q8—48no—
318Q8—44no—
319Q8—82no—
320Q8—84no—
321Q8—162no—
322D4—41no—
323D4—41no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—84no—
328D4—84no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—48no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4—84no—
366C2×C4—82no—
367C2×C4—84no—
368C2×C4—82no—
369C2×C4—164no—
370C2×C4C2×#9914yes—
371C2×C4C2×#9911yescharacteristic
372C2×C4C2×#9911yescharacteristic
373C2×C4C2×C4×Q811yescharacteristic
374C2×C4#51911yescharacteristic
375C22—21no—
376C22—21no—
377C22—22no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—82no—
386C22—82no—
387C22—81no—
388C22—81no—
389C22Q8×C22⋊C411yescharacteristic
390C22C22×#9911yescharacteristic
391C22#288411yescharacteristic
392C4—21no—
393C4—24no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—44no—
398C4—42no—
399C4—42no—
400C4—41no—
401C4—81no—
402C4—82no—
403C4—84no—
404C4—81no—
405C4—82no—
406C4—164no—
407C4—162no—
408C4#212712yes—
409C2—21no—
410C2—21no—
411C2—42no—
412C2—42no—
413C2—82no—
414C2Q8×#9911yescharacteristic
415C2C2×#212711yescharacteristic
416C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#127C23
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
2. Q8×#99C2semidirect
3. #13300C2semidirect
4. #13299C2semidirect
9. C2×#554C4semidirect
11. #2862C4semidirect
13. #2660C4semidirect
15. #2563C4semidirect
17. #2542C4semidirect
23. #2295C4semidirect
101. #353Q8semidirect
229. C2×D4C4×Q8semidirect
302. C42C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#396182
Q8×#992
#133002
#132992
#127992
#127972
#126282
C4×#3532