Finite Group Databasebeta

← #497648

Subgroups — #497648

Order 512.

984 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39167C211yesmaximal, characteristic
2Q8×#99C212yesmaximal, complemented
3#13225C214yesmaximal
4#13224C212yesmaximal, complemented
5#12800C212yesmaximal, complemented
6#12797C211yesmaximal, characteristic
7#12627C212yesmaximal, complemented
8C4×#353C211yesmaximal, characteristic
9C2×#556C411yescharacteristic, complemented
10C2×D4×Q8C2211yescharacteristic
11#2862C411yescharacteristic, complemented
12#2723—24no—
13#2640C412yescomplemented
14#2567C412yescomplemented
15#2547C2211yescharacteristic
16#2544C2212yes—
17#2542C411yescharacteristic, complemented
18#2415—22no—
19#2409—22no—
20Q8×C22⋊C4—22no—
21#2326C2212yes—
22C4×#127C2211yescharacteristic
23#2295C411yescharacteristic, complemented
24C2×#354C2214yes—
25C2×#354C2214yes—
26C2×#353C2212yes—
27C2×#353C2212yes—
28#1996—22no—
29#1996—24no—
30#1799C2214yes—
31#1799C2212yes—
32C4×#99C2212yes—
33C4×#99C2214yes—
34#1393C2211yescharacteristic
35#1393C2212yes—
36#1393C2211yescharacteristic
37C23×Q8D412yes—
38#556—42no—
39#556—42no—
40D4×Q8—42no—
41D4×Q8—42no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#127C2311yescharacteristic
46C2×#126—22no—
47C2×#124C2×C412yes—
48C2×#124C2312yes—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8D412yes—
52C2×C22⋊Q8D414yes—
53C2×C4×Q8—42no—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×D4—22no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2311yescharacteristic
59C2×C4×D4C2312yes—
60C2×#117—21no—
61C2×#99C2312yes—
62C2×#99C2314yes—
63C2×#99C2312yes—
64#399—22no—
65#397—24no—
66#397—22no—
67#395—24no—
68#394—24no—
69#393—24no—
70#391—22no—
71#391—21no—
72#391—42no—
73#391—44no—
74#391C2×C412yes—
75#391C2×C412yes—
76#391C2×C412yes—
77#390—22no—
78#390C2×C411yescharacteristic
79#390C2×C412yes—
80#390C2311yescharacteristic
81#389—24no—
82#389—21no—
83#388—24no—
84#387—22no—
85#386—24no—
86#385—24no—
87#384C2×C411yescharacteristic
88#384C2×C412yes—
89#382—22no—
90#380—22no—
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#354—28no—
100#354—44no—
101#353—24no—
102#353—42no—
103#342—24no—
104#342—22no—
105#342—24no—
106#342—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—24no—
110C22×Q8—84no—
111C22×Q8C22⋊C412yes—
112C22×Q8C22⋊C414yes—
113C22×Q8C22⋊C411yescharacteristic
114C22×Q8C22×C412yes—
115C22×D4—22no—
116C22×D4C22×C411yescharacteristic
117C22×D4C2411yescharacteristic
118C23×C4—21no—
119C23×C4C2×D412yes—
120C23×C4C2×D414yes—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—44no—
124C4⋊Q8—41no—
125C4⋊Q8—41no—
126#127—22no—
127#127—21no—
128#127—21no—
129#126—48no—
130#124—24no—
131#124—22no—
132#124—22no—
133#124—44no—
134C22⋊Q8—42no—
135C22⋊Q8—42no—
136C22⋊Q8—44no—
137C22⋊Q8—44no—
138C22⋊Q8—42no—
139C22⋊Q8—42no—
140C22⋊Q8—84no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—44no—
144C4×Q8—88no—
145C4×D4—41no—
146C4×D4—41no—
147C4×D4—44no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—84no—
151#117—42no—
152#117—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C414yes—
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4C2×D414yes—
180C2×C22⋊C4C2×D412yes—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—44no—
185C2×C42—42no—
186C2×C42C22×C412yes—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191#99—22no—
192#99—22no—
193#99—24no—
194#99—24no—
195#99—22no—
196#99#4114yes—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—21no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—42no—
205#95—44no—
206#95—42no—
207#95—44no—
208C24—21no—
209C24C22×D412yes—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8—24no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—48no—
221C2×Q8—84no—
222C2×Q8—84no—
223C2×Q8—88no—
224C2×Q8#9912yes—
225C2×Q8#9912yes—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—82no—
236C2×D4C2×#4112yes—
237C2×D4H(5,𝔽2)11yescharacteristic
238C2×D42−1+411yescharacteristic
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—84no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C414yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C23×C411yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—44no—
277C4⋊C4—82no—
278C4⋊C4—84no—
279C4⋊C4—88no—
280C4⋊C4—84no—
281C22⋊C4—24no—
282C22⋊C4—44no—
283C22⋊C4—42no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—42no—
291C22⋊C4—44no—
292C22⋊C4—42no—
293C22⋊C4—44no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4—84no—
297C42—21no—
298C42—21no—
299C42—24no—
300C42—22no—
301C42—21no—
302C42—22no—
303C42—21no—
304C42—41no—
305C42—42no—
306C42—44no—
307C42—42no—
308C42—88no—
309C42—84no—
310C23—22no—
311C23—22no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23C22×C22⋊C411yescharacteristic
320C23#54812yes—
321C23#55012yes—
322Q8—24no—
323Q8—42no—
324Q8—48no—
325Q8—44no—
326Q8—82no—
327Q8—84no—
328Q8—82no—
329Q8—82no—
330D4—42no—
331D4—41no—
332D4—41no—
333D4—81no—
334D4—81no—
335D4—84no—
336D4—84no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—48no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—84no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4C2×#9914yes—
379C2×C4C2×#9911yescharacteristic
380C2×C4C2×#9911yescharacteristic
381C2×C4C4×#4111yescharacteristic
382C2×C4#52011yescharacteristic
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22#234111yescharacteristic
398C22C22×#9911yescharacteristic
399C22#288411yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—24no—
404C4—42no—
405C4—44no—
406C4—42no—
407C4—42no—
408C4—41no—
409C4—82no—
410C4—81no—
411C4—84no—
412C4—82no—
413C4—81no—
414C4—162no—
415C4—164no—
416C4#212712yes—
417C2—21no—
418C2—21no—
419C2—42no—
420C2—42no—
421C2—161no—
422C2#2253411yescharacteristic
423C2C2×#212711yescharacteristic
424C2#3336011yescharacteristic
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
4. #13224C2semidirect
5. #12800C2semidirect
7. #12627C2semidirect
9. C2×#556C4semidirect
11. #2862C4semidirect
13. #2640C4semidirect
14. #2567C4semidirect
17. #2542C4semidirect
23. #2295C4semidirect

Maximal subgroups

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

subgroupindex
#391672
Q8×#992
#132252
#132242
#128002
#127972
#126272
C4×#3532