Finite Group Databasebeta

← #482193

Subgroups — #482193

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35738C211yesmaximal, characteristic
2#20691C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#11846C214yesmaximal, complemented
5#11840C214yesmaximal
6#11718C212yesmaximal, complemented
7#11691C212yesmaximal
8C2×#554C411yescharacteristic, complemented
9C2×#519C2211yescharacteristic
10#2982—21no—
11#2634C412yescomplemented
12#2628C2211yescharacteristic
13#2611C2212yes—
14#2598C2212yes—
15#2568C411yescharacteristic, complemented
16#2536C411yescharacteristic, complemented
17#2477—22no—
18#2465—22no—
19#2421—21no—
20#2409—21no—
21#2398C412yescomplemented
22#2392C2211yescharacteristic
23#2360—21no—
24#2356C411yescharacteristic, complemented
25#2352—21no—
26#2310—21no—
27#1820—22no—
28#1820C2214yes—
29#1740—21no—
30#1740C2212yes—
31#1461—21no—
32#1461—22no—
33#1458C2214yes—
34#1458C2212yes—
35#1352C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C23×Q8D411yescharacteristic
39#554—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#125C2×C412yes—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—22no—
45#519—44no—
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8D412yes—
51C2×C4×Q8D411yescharacteristic
52C2×C4×D4—21no—
53C2×C4×D4C2×C412yes—
54C2×C4×D4C2×C411yescharacteristic
55C2×#117—22no—
56C2×#117D412yes—
57C2×#117D411yescharacteristic
58C2×#117C2312yes—
59C2×#117C2311yescharacteristic
60C22×C4⋊C4—21no—
61C8×Q8—44no—
62#431—22no—
63#431—24no—
64#400—22no—
65#400C2×C412yes—
66#400C2×C412yes—
67#400C2×C412yes—
68#399C2312yes—
69#398C2×C412yes—
70#398C2×C411yescharacteristic
71#397—22no—
72#397C2311yescharacteristic
73#396—21no—
74#395—22no—
75#395—21no—
76#394—21no—
77#394C2×C412yes—
78#394C2×C411yescharacteristic
79#393—22no—
80#393—21no—
81#392C2311yescharacteristic
82#391—21no—
83#391—21no—
84#391—22no—
85#391—44no—
86#389—22no—
87#389—21no—
88#388—22no—
89#388—22no—
90#388—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#386—22no—
96#386—22no—
97#385—22no—
98#385—22no—
99#383C2×C412yes—
100#382—21no—
101#382C2×C411yescharacteristic
102#381—24no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2×C411yescharacteristic
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—21no—
110#376C2×C412yes—
111#342—22no—
112#342—24no—
113#342C2314yes—
114#324—24no—
115#324—22no—
116#323C2314yes—
117C22×Q8—24no—
118C22×Q8—21no—
119C22×Q8—24no—
120C22×Q8—42no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22⋊C412yes—
123C22×Q8C22⋊C411yescharacteristic
124C22×D4—21no—
125C22×D4C22×C411yescharacteristic
126C23×C4—21no—
127C23×C4—22no—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C4⋊Q8—42no—
131#125—44no—
132C22⋊Q8—48no—
133C22⋊Q8—44no—
134C22⋊Q8—44no—
135C4×Q8—24no—
136C4×Q8—24no—
137C4×Q8—44no—
138C4×Q8—42no—
139C4×Q8—42no—
140C4×Q8—44no—
141C4×D4—42no—
142C4×D4—44no—
143C4×D4—44no—
144#117—44no—
145#117—42no—
146#117—42no—
147#117—44no—
148#117—44no—
149#117—44no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4C2×D412yes—
181C2×C22⋊C4C2×D411yescharacteristic
182C2×C22⋊C4C2411yescharacteristic
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—42no—
186C2×C42C22⋊C412yes—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D412yes—
193C4⋊C8—44no—
194C4⋊C8—42no—
195C22⋊C8—24no—
196C4×C8—44no—
197C4×C8—42no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—21no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—21no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—48no—
212#95C22×C411yescharacteristic
213#95C22×C412yes—
214#95C22×C411yescharacteristic
215C24—21no—
216C24C22×D411yescharacteristic
217C2×Q8—24no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—42no—
225C2×Q8—48no—
226C2×Q8—48no—
227C2×Q8—42no—
228C2×Q8—82no—
229C2×Q8#9914yes—
230C2×D4—41no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—81no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4#13514yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C2×C8—44no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—84no—
281C4⋊C4—82no—
282C4⋊C4—81no—
283C22⋊C4—41no—
284C22⋊C4—42no—
285C22⋊C4—41no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—42no—
289C22⋊C4—44no—
290C22⋊C4—42no—
291C22⋊C4—44no—
292C22⋊C4—42no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—81no—
296C22⋊C4—82no—
297C42—24no—
298C42—22no—
299C42—24no—
300C42—22no—
301C42—42no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—84no—
306C42—82no—
307C23—21no—
308C23—21no—
309C23—22no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23C22×C22⋊C411yescharacteristic
319C23C2×#13512yes—
320Q8—28no—
321Q8—48no—
322Q8—44no—
323Q8—162no—
324D4—42no—
325D4—42no—
326D4—84no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—48no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—164no—
369C2×C4C2×#9914yes—
370C2×C4C2×#9912yes—
371C2×C4#41612yes—
372C2×C4#41612yes—
373C2×C4#42112yes—
374C8—84no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22C22×#9911yescharacteristic
390C22C2×#41611yescharacteristic
391C22C2×#42111yescharacteristic
392C4—22no—
393C4—24no—
394C4—42no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—42no—
399C4—41no—
400C4—42no—
401C4—81no—
402C4—82no—
403C4—81no—
404C4—82no—
405C4—81no—
406C4—164no—
407C4#151614yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—82no—
412C2C2×#151611yescharacteristic
413C2#3334911yescharacteristic
414C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#392C23
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20691C2semidirect
3. #17352C2semidirect
4. #11846C2semidirect
6. #11718C2semidirect
8. C2×#554C4semidirect
11. #2634C4semidirect
15. #2568C4semidirect
16. #2536C4semidirect
21. #2398C4semidirect
24. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#357382
#206912
#173522
#118462
#118402
#117182
#116912