Finite Group Databasebeta

← #497593

Subgroups — #497593

Order 512.

946 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39618C211yesmaximal, characteristic
2#22513C212yesmaximal, complemented
3#13299C214yesmaximal, complemented
4#13298C212yesmaximal, complemented
5#12799C212yesmaximal
6#12797C211yesmaximal, characteristic
7#12628C212yesmaximal
8C4×#353C211yesmaximal, characteristic
9C2×#554C2211yescharacteristic
10C2×D4×Q8C411yescharacteristic, complemented
11#2862C411yescharacteristic, complemented
12#2692—24no—
13#2660C412yescomplemented
14#2566C412yescomplemented
15#2563C411yescharacteristic, complemented
16#2563C2212yes—
17#2542C2211yescharacteristic
18#2417—22no—
19#2409—22no—
20#2345—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×Q8—21no—
38#554—42no—
39#554—42no—
40D4×Q8—42no—
41D4×Q8—42no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#127C2311yescharacteristic
46C2×#125C2×C412yes—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8D412yes—
51C2×C22⋊Q8D414yes—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×D4—22no—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2×C412yes—
56C2×C4×D4C2311yescharacteristic
57C2×C4×D4C2312yes—
58C22×C4⋊C4D412yes—
59C2×#99C2314yes—
60C2×#99C2312yes—
61C2×#99C2312yes—
62#399—22no—
63#398—24no—
64#396—24no—
65#395—24no—
66#393—22no—
67#393—24no—
68#391—22no—
69#391—21no—
70#391—22no—
71#390—22no—
72#390C2×C411yescharacteristic
73#390C2×C412yes—
74#390C2311yescharacteristic
75#390C2312yes—
76#389—21no—
77#389—42no—
78#387—22no—
79#387—44no—
80#386—24no—
81#386—24no—
82#385—24no—
83#384C2×C411yescharacteristic
84#384C2×C412yes—
85#382C2×C412yes—
86#382C2×C412yes—
87#382C2×C412yes—
88#380—24no—
89#380—22no—
90#380—24no—
91#380—22no—
92C4×C4⋊C4—42no—
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C4⋊C4C2×C412yes—
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—21no—
107C22×Q8—22no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—84no—
111C22×Q8C22⋊C411yescharacteristic
112C22×Q8C22⋊C412yes—
113C22×D4—22no—
114C22×D4C22×C411yescharacteristic
115C22×D4C2411yescharacteristic
116C23×C4—21no—
117C23×C4—22no—
118C23×C4C2×D412yes—
119C23×C4C2×D414yes—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—44no—
125#127—22no—
126#127C2×Q812yes—
127#127#4112yes—
128#125—44no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C22⋊Q8—42no—
132C22⋊Q8—42no—
133C22⋊Q8—48no—
134C22⋊Q8—42no—
135C22⋊Q8—42no—
136C22⋊Q8—84no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×D4—44no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—41no—
144C4×D4—44no—
145C4×D4—84no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C414yes—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C412yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4C2×D414yes—
176C2×C22⋊C4C2×D412yes—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—42no—
180C2×C42—44no—
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—24no—
190#99—22no—
191#99C2×Q814yescomplemented
192#95—22no—
193#95—22no—
194#95—22no—
195#95—21no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—42no—
200#95—44no—
201#95—44no—
202#95—42no—
203C24—21no—
204C24C22×D412yes—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—84no—
214C2×Q8—82no—
215C2×Q8—88no—
216C2×Q8—88no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—82no—
225C2×D4—82no—
226C2×D4C4×Q812yes—
227C2×D4C22×Q811yescharacteristic
228C2×D4C2×#4111yescharacteristic
229C2×D42−1+412yes—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C414yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C23×C411yescharacteristic
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—44no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C4⋊C4—164no—
274C4⋊C4#9912yes—
275C4⋊C4#9912yes—
276C22⋊C4—24no—
277C22⋊C4—42no—
278C22⋊C4—44no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—44no—
282C22⋊C4—42no—
283C22⋊C4—44no—
284C22⋊C4—42no—
285C22⋊C4—44no—
286C22⋊C4—84no—
287C22⋊C4—82no—
288C22⋊C4—82no—
289C22⋊C4—84no—
290C42—21no—
291C42—24no—
292C42—21no—
293C42—21no—
294C42—41no—
295C42—42no—
296C42—42no—
297C42—81no—
298C42—84no—
299C42—164no—
300C42C4×Q812yes—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—41no—
309C23—41no—
310C23C22×C22⋊C411yescharacteristic
311C23#54812yes—
312C23D4×Q812yes—
313Q8—42no—
314Q8—42no—
315Q8—82no—
316Q8—88no—
317Q8—82no—
318Q8—162no—
319D4—41no—
320D4—41no—
321D4—42no—
322D4—42no—
323D4—42no—
324D4—84no—
325D4—84no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—44no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—88no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—82no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—82no—
367C2×C4C2×#9914yes—
368C2×C4C2×#9911yescharacteristic
369C2×C4C2×#9911yescharacteristic
370C2×C4C2×C4×Q811yescharacteristic
371C2×C4#51911yescharacteristic
372C22—21no—
373C22—21no—
374C22—22no—
375C22—22no—
376C22—42no—
377C22—42no—
378C22—44no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—82no—
383C22—82no—
384C22—81no—
385C22—81no—
386C22Q8×C22⋊C411yescharacteristic
387C22C22×#9911yescharacteristic
388C22#288411yescharacteristic
389C4—21no—
390C4—24no—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—44no—
397C4—41no—
398C4—81no—
399C4—82no—
400C4—84no—
401C4—81no—
402C4—82no—
403C4—162no—
404C4—164no—
405C4#212812yes—
406C2—21no—
407C2—21no—
408C2—42no—
409C2—42no—
410C2—82no—
411C2Q8×#9911yescharacteristic
412C2C2×#212811yescharacteristic
413C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#554 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22513C2semidirect
3. #13299C2semidirect
4. #13298C2semidirect
10. C2×D4×Q8C4semidirect
11. #2862C4semidirect
13. #2660C4semidirect
14. #2566C4semidirect
15. #2563C4semidirect
23. #2295C4semidirect
101. #353Q8semidirect
191. #99C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#396182
#225132
#132992
#132982
#127992
#127972
#126282
C4×#3532