Finite Group Databasebeta

← #482190

Subgroups — #482190

Order 512.

856 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35712C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#17353C211yesmaximal, characteristic
4#11853C214yesmaximal, complemented
5#11849C214yesmaximal, complemented
6#11721C212yesmaximal, complemented
7#11700C212yesmaximal, complemented
8C2×#557C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2982—21no—
11#2879—21no—
12#2633C412yes—
13#2621C2211yescharacteristic
14#2611C2212yes—
15#2599C2212yes—
16#2580C411yescharacteristic
17#2547C411yescharacteristic
18#2473—22no—
19#2460—22no—
20#2446—21no—
21#2432—21no—
22#2398C412yes—
23#2396C2211yescharacteristic
24#2367—21no—
25#2360—21no—
26#2356C411yescharacteristic
27#1822C2214yes—
28#1821—22no—
29#1742C2212yes—
30C2×#343—21no—
31#1460—21no—
32#1460—22no—
33#1458C2214yes—
34#1458C2212yes—
35#1354C2214yes—
36#1353C2214yes—
37#1346C2218yes—
38C22×#41D411yescharacteristic
39C22×M16—22no—
40#557—44no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—22no—
44C2×#125C2×C412yes—
45C2×#124—21no—
46C2×C22⋊Q8—21no—
47#520—44no—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C412yes—
55C2×C4×D4D411yescharacteristic
56C2×C4×D4D412yes—
57C2×#117C2312yes—
58C2×#117C2311yescharacteristic
59C22×C4⋊C4D412yes—
60C22×C4⋊C4D411yescharacteristic
61C8×Q8—44no—
62#431—22no—
63#431—24no—
64#408—44no—
65#407—24no—
66#400—22no—
67#400C2311yescharacteristic
68#399C2×C412yes—
69#399C2×C411yescharacteristic
70#398—22no—
71#398—21no—
72#398—21no—
73#398C2312yes—
74#397—21no—
75#397C2×C412yes—
76#397C2×C412yes—
77#397C2×C412yes—
78#396—21no—
79#395C2311yescharacteristic
80#394—22no—
81#394—22no—
82#393C2×C412yes—
83#393C2×C411yescharacteristic
84#391—21no—
85#391—21no—
86#391—21no—
87#390—21no—
88#389—21no—
89#389—22no—
90#388—22no—
91#388—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#383C2×C412yes—
102#382C2×C411yescharacteristic
103#380—22no—
104C4×C4⋊C4—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#343—22no—
112#343—24no—
113#343C2314yes—
114#324C2314yes—
115#323—24no—
116#323—22no—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—42no—
120C22×Q8—21no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22×C411yescharacteristic
123C22×D4C22⋊C411yescharacteristic
124C22×D4C22⋊C412yes—
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C2×M16—24no—
131C2×M16—42no—
132C2×M16—44no—
133C2×M16—44no—
134C22×C8—42no—
135C4⋊Q8—42no—
136C4⋊Q8—44no—
137#125—44no—
138#125—48no—
139#124—44no—
140C22⋊Q8—44no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—44no—
144C4×Q8—42no—
145C4×Q8—44no—
146C4×D4—44no—
147C4×D4—42no—
148C4×D4—42no—
149C4×D4—44no—
150#117—44no—
151#117—42no—
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—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2411yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C22×C412yes—
182C2×C22⋊C4C22×C411yescharacteristic
183C2×C22⋊C4C2×D412yes—
184C2×C22⋊C4C2×D411yescharacteristic
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D412yes—
193C4⋊C8—44no—
194C4⋊C8—42no—
195C22⋊C8—24no—
196C22⋊C8—48no—
197C4×C8—44no—
198C4×C8—42no—
199#95—22no—
200#95—21no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—21no—
207#95—22no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—22no—
212#95C22×C411yescharacteristic
213#95C22×C412yes—
214#95C22×C411yescharacteristic
215C24C2×C22⋊C412yes—
216C24C2×C22⋊C411yescharacteristic
217#41—48no—
218#41—48no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8#9914yes—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—42no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4#13514yes—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260M16—44no—
261M16—44no—
262M16—84no—
263C2×C8—44no—
264C2×C8—44no—
265C2×C8—84no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—81no—
284C4⋊C4—82no—
285C4⋊C4—81no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—82no—
296C22⋊C4—81no—
297C42—24no—
298C42—22no—
299C42—24no—
300C42—22no—
301C42—44no—
302C42—42no—
303C42—42no—
304C42—82no—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—41no—
311C23—42no—
312C23—41no—
313C23—42no—
314C23#41612yes—
315C23C22×C22⋊C411yescharacteristic
316Q8—28no—
317Q8—42no—
318Q8—42no—
319Q8—84no—
320D4—48no—
321D4—44no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—48no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—162no—
360C2×C4—162no—
361C2×C4—162no—
362C2×C4C2×#9914yes—
363C2×C4C2×#9912yes—
364C2×C4#41612yes—
365C2×C4#42112yes—
366C2×C4C2×#13512yes—
367C8—84no—
368C8—84no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—42no—
373C22—44no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—81no—
380C22—82no—
381C22C22×#9911yescharacteristic
382C22C2×#41611yescharacteristic
383C22C2×#42111yescharacteristic
384C4—22no—
385C4—24no—
386C4—41no—
387C4—42no—
388C4—41no—
389C4—42no—
390C4—41no—
391C4—42no—
392C4—41no—
393C4—81no—
394C4—81no—
395C4—82no—
396C4—81no—
397C4—82no—
398C4#151614yes—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—82no—
403C2C2×#151611yescharacteristic
404C2#3334911yescharacteristic
405C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11853C2semidirect
5. #11849C2semidirect
6. #11721C2semidirect
7. #11700C2semidirect

Maximal subgroups

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

subgroupindex
#357122
#206952
#173532
#118532
#118492
#117212
#117002