Finite Group Databasebeta

← #481883

Subgroups — #481883

Order 512.

841 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36040C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#17372C211yesmaximal, characteristic
4#11819C214yesmaximal, complemented
5#11816C214yesmaximal, complemented
6#11724C212yesmaximal, complemented
7#11721C212yesmaximal, complemented
8C2×#557C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2983—21no—
11#2879—21no—
12#2628C2212yes—
13#2628C2211yescharacteristic
14#2580C411yescharacteristic
15#2580C412yes—
16#2563C411yescharacteristic
17#2563C412yes—
18#2454—21no—
19#2454—22no—
20#2442—22no—
21#2442—21no—
22#2396C2212yes—
23#2396C2211yescharacteristic
24#2351—21no—
25#2351C411yescharacteristic
26#2341—21no—
27#1822C2214yes—
28#1821—22no—
29#1742C2212yes—
30C2×#343—21no—
31#1466—22no—
32#1464C2214yes—
33#1460—21no—
34#1458C2212yes—
35#1354C2218yes—
36#1354C2214yes—
37#1353C2214yes—
38C22×#41D411yescharacteristic
39C22×M16—22no—
40#557—44no—
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—22no—
44C2×#125—21no—
45#520—44no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C412yes—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8D411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C412yes—
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D412yes—
56C2×#117C2312yes—
57C2×#117C2311yescharacteristic
58C22×C4⋊C4D412yes—
59C22×C4⋊C4D411yescharacteristic
60#446—44no—
61#432—24no—
62#431—22no—
63#408—44no—
64#407—24no—
65#400—22no—
66#400—21no—
67#400C2311yescharacteristic
68#399—22no—
69#399—21no—
70#399C2×C412yes—
71#399C2×C411yescharacteristic
72#398—22no—
73#398—21no—
74#396—22no—
75#396—21no—
76#395C2311yescharacteristic
77#395C2312yes—
78#393C2×C412yes—
79#393C2×C411yescharacteristic
80#393C2×C412yes—
81#393C2×C412yes—
82#393C2×C412yes—
83#390—21no—
84#390—22no—
85#390—21no—
86#390—22no—
87#389—22no—
88#389—21no—
89#389—22no—
90#389—21no—
91#389—21no—
92#389—22no—
93#385—22no—
94#385—22no—
95#385—22no—
96#382—22no—
97#382—21no—
98#382C2×C412yes—
99#382C2×C411yescharacteristic
100#380—22no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111#343—22no—
112#343—24no—
113#343C2314yes—
114#324C2314yes—
115#323—22no—
116#323—24no—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—42no—
120C22×Q8C22⋊C411yescharacteristic
121C22×Q8C22×C412yes—
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—44no—
136C4⋊Q8—42no—
137#125—48no—
138#125—44no—
139C4×Q8—24no—
140C4×Q8—24no—
141C4×Q8—44no—
142C4×Q8—42no—
143C4×Q8—44no—
144C4×Q8—44no—
145C4×D4—44no—
146C4×D4—42no—
147C4×D4—42no—
148C4×D4—44no—
149#117—42no—
150#117—44no—
151C2×C4⋊C4—22no—
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—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2411yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4C22×C412yes—
183C2×C22⋊C4C22×C411yescharacteristic
184C2×C22⋊C4C2×D412yes—
185C2×C22⋊C4C2×D411yescharacteristic
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42C22×C412yes—
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C4⋊C8—42no—
195C4⋊C8—44no—
196C22⋊C8—24no—
197C22⋊C8—48no—
198#97—44no—
199C4×C8—42no—
200#95—21no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—21no—
207#95—21no—
208#95—22no—
209#95—21no—
210#95—21no—
211#95C22×C411yescharacteristic
212#95C22×C412yes—
213#95C22×C411yescharacteristic
214C24C2×C22⋊C412yes—
215C24C2×C22⋊C411yescharacteristic
216#41—48no—
217#41—48no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—42no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—81no—
224C2×Q8#9914yes—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—42no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4#13714yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258M16—44no—
259M16—44no—
260M16—84no—
261C2×C8—44no—
262C2×C8—44no—
263C2×C8—84no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—81no—
281C4⋊C4—82no—
282C4⋊C4—81no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—81no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—81no—
291C22⋊C4—82no—
292C22⋊C4—81no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4—82no—
297C42—22no—
298C42—24no—
299C42—24no—
300C42—22no—
301C42—44no—
302C42—44no—
303C42—42no—
304C42—42no—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—41no—
311C23—42no—
312C23—41no—
313C23—42no—
314C23#41912yes—
315C23C22×C22⋊C411yescharacteristic
316Q8—28no—
317Q8—84no—
318Q8—82no—
319D4—48no—
320D4—44no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—48no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—162no—
359C2×C4—162no—
360C2×C4—162no—
361C2×C4C2×#9914yes—
362C2×C4C2×#9912yes—
363C2×C4#41912yes—
364C2×C4#42112yes—
365C2×C4C2×#13712yes—
366C8—84no—
367C8—84no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—42no—
372C22—44no—
373C22—41no—
374C22—41no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—81no—
379C22—82no—
380C22C22×#9911yescharacteristic
381C22C2×#41911yescharacteristic
382C22C2×#42111yescharacteristic
383C4—22no—
384C4—24no—
385C4—41no—
386C4—41no—
387C4—42no—
388C4—41no—
389C4—41no—
390C4—42no—
391C4—81no—
392C4—82no—
393C4—81no—
394C4—82no—
395C4—81no—
396C4—81no—
397C4#151114yes—
398C2—21no—
399C2—21no—
400C2—42no—
401C2—82no—
402C2C2×#151111yescharacteristic
403C2#3335111yescharacteristic
404C2#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. #11819C2semidirect
5. #11816C2semidirect
6. #11724C2semidirect
7. #11721C2semidirect

Maximal subgroups

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

subgroupindex
#360402
#206952
#173722
#118192
#118162
#117242
#117212