Finite Group Databasebeta

← #481930

Subgroups — #481930

Order 512.

839 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35749C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic
3#17371C211yesmaximal, characteristic
4#11835C214yesmaximal, complemented
5#11831C214yesmaximal, complemented
6#11721C212yesmaximal
7#11700C212yesmaximal
8C2×#557C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2983—21no—
11#2905—21no—
12C22×#101—21no—
13#2634C2212yes—
14#2621C2211yescharacteristic
15#2611C412yes—
16#2599C412yes—
17#2581C411yescharacteristic
18#2546C411yescharacteristic
19#2475—22no—
20#2462—22no—
21#2446—21no—
22#2434—21no—
23#2399C2212yes—
24#2396C2211yescharacteristic
25#2365—21no—
26#2362—21no—
27#2356C411yescharacteristic
28#1801—22no—
29#1801C2214yes—
30#1742C2212yes—
31#1467C2214yes—
32#1464—22no—
33#1461—21no—
34#1458C2212yes—
35#1354C2214yes—
36#1353C2214yes—
37#1346C2218yes—
38C23×Q8D411yescharacteristic
39C22×M16—22no—
40#557—44no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—22no—
45C2×#125C2×C412yes—
46C2×C22⋊Q8—21no—
47#519—44no—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8C2312yes—
57C2×#117C2×C412yes—
58C2×#117D412yes—
59C2×#117D411yescharacteristic
60C2×#117C2311yescharacteristic
61#446—44no—
62#432—24no—
63#431—22no—
64#424—44no—
65#423—24no—
66C2×#101—24no—
67C2×#101—44no—
68#400—22no—
69#400C2×C412yes—
70#400C2311yescharacteristic
71#399C2×C411yescharacteristic
72#398—21no—
73#398—21no—
74#398—22no—
75#398C2×C412yes—
76#398C2×C412yes—
77#398C2312yes—
78#397—21no—
79#397C2×C412yes—
80#396—21no—
81#395—22no—
82#395—22no—
83#395C2×C412yes—
84#395C2311yescharacteristic
85#393C2×C411yescharacteristic
86#391—21no—
87#391—21no—
88#391—21no—
89#391—21no—
90#389—21no—
91#389—22no—
92#389—22no—
93#388—22no—
94#388—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#383C2×C412yes—
105#382C2×C411yescharacteristic
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C411yescharacteristic
111C4×C22⋊C4—21no—
112#376C2×C412yes—
113#343—22no—
114#343C2314yes—
115#324—24no—
116#324C2314yes—
117#323—22no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C411yescharacteristic
126C22×Q8C2×D412yes—
127C23×C4C22⋊C412yes—
128C23×C4C2×D411yescharacteristic
129C2×M16—24no—
130C2×M16—42no—
131C2×M16—44no—
132C2×M16—44no—
133C22×C8—42no—
134C4⋊Q8—42no—
135C4⋊Q8—44no—
136C4⋊Q8—44no—
137#125—44no—
138#125—48no—
139C22⋊Q8—42no—
140C22⋊Q8—42no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—44no—
144C4×Q8—44no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×Q8—44no—
148C4×Q8—42no—
149C4×Q8—44no—
150#117—24no—
151#117—42no—
152#117—44no—
153#117—44no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2411yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C22×C411yescharacteristic
182C2×C22⋊C4C2×D411yescharacteristic
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C4⋊C8—48no—
195C4⋊C8—42no—
196C4⋊C8—44no—
197#101—48no—
198C22⋊C8—24no—
199#97—44no—
200C4×C8—42no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—21no—
205#95—22no—
206#95—21no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—22no—
216#95C22×C411yescharacteristic
217#95C22×C412yes—
218#95C22×C411yescharacteristic
219C24C2×C22⋊C411yescharacteristic
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—48no—
232C2×Q8—48no—
233C2×Q8—41no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4AGL(1,ℤ/8)12yes—
256C22×C4#13712yes—
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—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4#9914yes—
290C22⋊C4—24no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—44no—
296C42—22no—
297C42—24no—
298C42—24no—
299C42—24no—
300C42—22no—
301C42—42no—
302C42—42no—
303C42—44no—
304C42—44no—
305C42—42no—
306C42—42no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23#41712yes—
312C23C22×C22⋊C411yescharacteristic
313Q8—44no—
314Q8—44no—
315Q8—42no—
316Q8—42no—
317Q8—44no—
318Q8—44no—
319Q8—84no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—48no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4C2×#9914yes—
361C2×C4C2×#9912yes—
362C2×C4#41712yes—
363C2×C4#42112yes—
364C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
365C2×C4C2×#13711yescharacteristic
366C8—84no—
367C8—84no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—42no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22C22×#9911yescharacteristic
376C22C2×#41711yescharacteristic
377C22C2×#42111yescharacteristic
378C4—24no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—41no—
383C4—42no—
384C4—42no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—42no—
389C4—42no—
390C4—81no—
391C4—82no—
392C4—81no—
393C4—82no—
394C4#151314yes—
395C2—21no—
396C2—21no—
397C2—42no—
398C2C2×#151311yescharacteristic
399C2#3334911yescharacteristic
400C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11835C2semidirect
5. #11831C2semidirect

Maximal subgroups

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

subgroupindex
#357492
#206392
#173712
#118352
#118312
#117212
#117002