Finite Group Databasebeta

← #497591

Subgroups — #497591

Order 512.

890 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#36562C211yesmaximal, characteristic
2#22513C212yesmaximal
3#13300C214yesmaximal
4#13298C212yesmaximal
5#12798C211yesmaximal, characteristic
6#12798C212yesmaximal
7C4×#354C212yesmaximal
8C4×#354C211yesmaximal, characteristic
9C2×#557C411yescharacteristic, complemented
10C2×#554C2211yescharacteristic
11#2862C411yescharacteristic, complemented
12#2859C412yescomplemented
13#2567C411yescharacteristic, complemented
14#2566C2212yes—
15#2565C412yescomplemented
16#2556—24no—
17#2542C2211yescharacteristic
18#2429—22no—
19#2360—22no—
20#2345—22no—
21C4×#124C2212yes—
22C4×#124C2211yescharacteristic
23C42×Q8C411yescharacteristic, complemented
24C2×#354C2212yes—
25C2×#354C2211yescharacteristic
26C2×#354C2211yescharacteristic
27#1996—22no—
28#1996—24no—
29#1799C2212yes—
30#1799C2214yes—
31C4×#99C2212yes—
32C4×#99C2214yes—
33#1393C2214yes—
34#1393C2212yes—
35#1393C2214yes—
36#1393C2212yes—
37#557—42no—
38#557—41no—
39#557—41no—
40#554—41no—
41#554—42no—
42#554—41no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—22no—
47C2×#125C2×C412yes—
48C2×#125C2×C412yes—
49C2×#124C2311yescharacteristic
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8D412yes—
52C2×C22⋊Q8D414yes—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×D4C2312yes—
59C2×C4×D4C2311yescharacteristic
60C22×C4⋊C4D412yes—
61C2×#99C2314yes—
62C2×#99C2312yes—
63C2×#99C2312yes—
64#399—22no—
65#398—24no—
66#396—24no—
67#395—22no—
68#391—22no—
69#391—22no—
70#391C2×C411yescharacteristic
71#391C2×C412yes—
72#390C2311yescharacteristic
73#390C2312yes—
74#389—22no—
75#389—24no—
76#389—24no—
77#389—42no—
78#388—24no—
79#388—22no—
80#387—24no—
81#387—44no—
82#385—24no—
83#384C2×C412yes—
84#384C2×C412yes—
85#382C2×C412yes—
86#382C2×C411yescharacteristic
87#382C2×C412yes—
88#380—24no—
89#380—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—42no—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4C2×C411yescharacteristic
95C4×C22⋊C4—22no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99C43C2×C412yes—
100C43C2×C411yescharacteristic
101#354—24no—
102#354—22no—
103#354Q814yescomplemented
104#342—24no—
105#342—22no—
106#342—24no—
107#342—22no—
108C22×Q8—21no—
109C22×Q8C22⋊C411yescharacteristic
110C22×Q8C22⋊C412yes—
111C22×Q8C22×C411yescharacteristic
112C22×D4C2411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D414yes—
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119C4⋊Q8—42no—
120C4⋊Q8—42no—
121#125—42no—
122#125—42no—
123#125—42no—
124#125—42no—
125#125—44no—
126#125—44no—
127#124—22no—
128#124C2×Q812yes—
129#124#4112yes—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132C22⋊Q8—42no—
133C22⋊Q8—42no—
134C22⋊Q8—48no—
135C4×Q8—22no—
136C4×Q8—21no—
137C4×Q8—21no—
138C4×Q8—42no—
139C4×Q8—42no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—44no—
143C4×Q8—42no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×D4—42no—
147C4×D4—42no—
148C4×D4—41no—
149C4×D4—41no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4C22⋊C414yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C414yes—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C412yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4C2×D414yes—
180C2×C22⋊C4C2×D412yes—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—42no—
188C2×C42—44no—
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C412yes—
194#99—22no—
195#99—24no—
196#99—22no—
197#99—24no—
198#99—22no—
199#99C2×Q814yes—
200#95—22no—
201#95—24no—
202#95—22no—
203#95—24no—
204#95—22no—
205#95—21no—
206#95—24no—
207#95—42no—
208#95—44no—
209#95—44no—
210#95—42no—
211C24C22×D412yes—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8C4×Q812yes—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4C22×Q811yescharacteristic
227C2×D4C2×#4111yescharacteristic
228C2×D42−1+412yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—84no—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C414yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C23×C411yescharacteristic
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—44no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—48no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C4⋊C4—164no—
274C4⋊C4#9912yes—
275C4⋊C4#9911yescharacteristic
276C4⋊C4#9911yescharacteristic
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—42no—
281C22⋊C4—44no—
282C22⋊C4—48no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—44no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—82no—
289C42—21no—
290C42—22no—
291C42—22no—
292C42—21no—
293C42—21no—
294C42—42no—
295C42—41no—
296C42—42no—
297C42—44no—
298C42—42no—
299C42—42no—
300C42—41no—
301C42—41no—
302C42—82no—
303C42—81no—
304C42—84no—
305C42—164no—
306C42C4×Q812yes—
307C23—22no—
308C23—22no—
309C23—42no—
310C23—42no—
311C23C22×C22⋊C411yescharacteristic
312C23#54812yes—
313C23D4×Q812yes—
314Q8—42no—
315Q8—42no—
316Q8—42no—
317Q8—42no—
318Q8—82no—
319Q8—82no—
320Q8—162no—
321D4—41no—
322D4—41no—
323D4—42no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—24no—
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—42no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—84no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4C2×#9912yes—
366C2×C4C2×#9912yes—
367C2×C4C2×#9911yescharacteristic
368C2×C4C2×#9911yescharacteristic
369C2×C4C2×C4×Q811yescharacteristic
370C2×C4#51911yescharacteristic
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—82no—
380C22Q8×C22⋊C411yescharacteristic
381C22C22×#9911yescharacteristic
382C22#288411yescharacteristic
383C4—21no—
384C4—22no—
385C4—22no—
386C4—21no—
387C4—21no—
388C4—44no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—44no—
393C4—41no—
394C4—42no—
395C4—81no—
396C4—84no—
397C4—82no—
398C4—81no—
399C4—162no—
400C4—164no—
401C4#212612yes—
402C2—21no—
403C2—21no—
404C2—42no—
405C2—42no—
406C2Q8×#9911yescharacteristic
407C2C2×#212611yescharacteristic
408C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, prime radical, cosocleC22×D4C24
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
9. C2×#557C4semidirect
11. #2862C4semidirect
12. #2859C4semidirect
13. #2567C4semidirect
15. #2565C4semidirect
23. C42×Q8C4semidirect
103. #354Q8semidirect

Maximal subgroups

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

subgroupindex
#365622
#225132
#133002
#132982
#127982
C4×#3542