Finite Group Databasebeta

← #482137

Subgroups — #482137

Order 512.

907 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36040C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic
3#17351C211yesmaximal, characteristic
4#11900C214yesmaximal, complemented
5#11897C214yesmaximal, complemented
6#11717C212yesmaximal, complemented
7#11715C212yesmaximal, complemented
8C2×#557C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2972—21no—
11C2×#410—21no—
12#2628C412yescomplemented
13#2628C2211yescharacteristic
14#2580C411yescharacteristic, complemented
15#2580C2212yes—
16#2563C411yescharacteristic, complemented
17#2563C2212yes—
18#2454—22no—
19#2454—21no—
20#2442—21no—
21#2442—22no—
22#2396C412yescomplemented
23#2396C2211yescharacteristic
24#2351—21no—
25#2351C411yescharacteristic, complemented
26#2341—21no—
27#2316—21no—
28#1799—22no—
29#1799C2214yes—
30#1740C2212yes—
31#1461—21no—
32#1459C2214yes—
33#1458C2212yes—
34C2×#324—22no—
35#1352C2214yes—
36#1351C2218yes—
37#1351C2214yes—
38C22×#41D411yescharacteristic
39#557—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C412yes—
42C2×#125—22no—
43C2×#125—21no—
44#520—44no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8C2×C412yes—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4D412yes—
53C2×C4×D4D411yescharacteristic
54C2×C4×D4C2312yes—
55C2×#117—22no—
56C2×#117C2×C412yes—
57C2×#117C2311yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D412yes—
60C8×D4—44no—
61#431—22no—
62#410—44no—
63C2×#99—24no—
64C2×C22⋊C8—24no—
65#400—22no—
66#400—21no—
67#400C2×C411yescharacteristic
68#399—22no—
69#399—21no—
70#399C2×C412yes—
71#399C2311yescharacteristic
72#398—22no—
73#398—21no—
74#396—22no—
75#396—21no—
76#395C2×C411yescharacteristic
77#395C2×C412yes—
78#393C2×C412yes—
79#393C2×C412yes—
80#393C2×C412yes—
81#393C2311yescharacteristic
82#393C2312yes—
83#390—21no—
84#390—21no—
85#390—22no—
86#390—22no—
87#389—21no—
88#389—22no—
89#389—21no—
90#389—22no—
91#389—22no—
92#389—21no—
93#389—44no—
94#385—22no—
95#385—22no—
96#385—22no—
97#382—21no—
98#382—22no—
99#382C2×C411yescharacteristic
100#382C2×C412yes—
101#380—22no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—21no—
106#380—24no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C4⋊C4C2×C412yes—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#324—24no—
117#323C2314yes—
118C2×#41—42no—
119C2×#41—42no—
120C2×#41SD1614yes—
121C2×#41Q1614yes—
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22×C411yescharacteristic
124C22×Q8C22×C412yes—
125C22×D4C22⋊C411yescharacteristic
126C22×D4C2×D412yes—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C22×C8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—42no—
135#125—48no—
136#125—44no—
137C4×Q8—24no—
138C4×Q8—42no—
139C4×Q8—44no—
140C4×Q8—42no—
141C4×Q8—44no—
142C4×Q8—44no—
143C4×D4—24no—
144C4×D4—42no—
145C4×D4—44no—
146C4×D4—44no—
147#117—42no—
148#117—44no—
149#117—44no—
150#117—44no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2×D412yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22×C412yes—
187C2×C22⋊C4C2×D411yescharacteristic
188C2×C22⋊C4C2411yescharacteristic
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—42no—
194C2×C42C22⋊C412yes—
195C2×C42C22×C412yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C4⋊C8—42no—
199#99—48no—
200C22⋊C8—24no—
201C22⋊C8—44no—
202C4×C8—42no—
203#95—21no—
204#95—22no—
205#95—21no—
206#95—21no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—22no—
212#95—21no—
213#95—22no—
214#95—44no—
215#95C22×C411yescharacteristic
216#95C22×C411yescharacteristic
217#95C22×C412yes—
218C24C2×C22⋊C412yes—
219C24C22×D411yescharacteristic
220#41—48no—
221#41—48no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—81no—
227C2×Q8—82no—
228C2×Q8#10314yes—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4#10314yes—
234C2×D4C2×SD1614yes—
235C2×D4C2×Q1614yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4#10318yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×SD1612yes—
262C22×C4C2×Q1612yes—
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C2×C8—44no—
266C2×C8—164no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—84no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—81no—
286C4⋊C4—82no—
287C4⋊C4—81no—
288C4⋊C4—81no—
289C4⋊C4—164no—
290C4⋊C4#9914yes—
291C22⋊C4—24no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—82no—
298C22⋊C4—81no—
299C22⋊C4—82no—
300C22⋊C4—81no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C42—22no—
305C42—24no—
306C42—22no—
307C42—42no—
308C42—44no—
309C42—44no—
310C42—42no—
311C42—42no—
312C42—84no—
313C23—22no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—42no—
318C23—41no—
319C23—42no—
320C23—41no—
321C23C2×#10314yes—
322C23C22×C22⋊C411yescharacteristic
323C23C22×SD1611yescharacteristic
324C23C22×Q1611yescharacteristic
325Q8—28no—
326Q8—84no—
327Q8—82no—
328D4—28no—
329D4—44no—
330D4—44no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—162no—
373C2×C4C2×#9914yes—
374C2×C4C2×#9912yes—
375C2×C4C2×#10312yes—
376C2×C4C2×#10314yes—
377C2×C4C2×#10312yes—
378C2×C4#42112yes—
379C8—84no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—41no—
386C22—41no—
387C22—44no—
388C22—41no—
389C22—42no—
390C22—41no—
391C22—42no—
392C22—81no—
393C22—82no—
394C22C22×#9911yescharacteristic
395C22C22×#10311yescharacteristic
396C22C2×#42111yescharacteristic
397C4—24no—
398C4—22no—
399C4—24no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—42no—
404C4—41no—
405C4—81no—
406C4—82no—
407C4—81no—
408C4—82no—
409C4—81no—
410C4—81no—
411C4—164no—
412C4#152014yes—
413C2—21no—
414C2—21no—
415C2—41no—
416C2—41no—
417C2—44no—
418C2C2×#152011yescharacteristic
419C2#3335111yescharacteristic
420C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11900C2semidirect
5. #11897C2semidirect
6. #11717C2semidirect
7. #11715C2semidirect
8. C2×#557C4semidirect
12. #2628C4semidirect
14. #2580C4semidirect
16. #2563C4semidirect
22. #2396C4semidirect
25. #2351C4semidirect

Maximal subgroups

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

subgroupindex
#360402
#206342
#173512
#119002
#118972
#117172
#117152