Finite Group Databasebeta

← #214003

Subgroups — #214003

Order 512.

983 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35667C211yesmaximal, characteristic
2#17371C212yesmaximal, complemented
3#9452C214yesmaximal, complemented
4#9448C214yesmaximal, complemented
5#8742C212yesmaximal, complemented
6#8524C212yesmaximal
7C2×#536C411yescharacteristic
8C2×#536C2211yescharacteristic
9C22×C4⋊Q8C411yescharacteristic
10C2×#519C2211yescharacteristic
11#2983C2214yes—
12#2546C412yes—
13#2546C2212yes—
14#2545C412yes—
15#2545C2212yes—
16#2429—22no—
17#2429—22no—
18#2413—22no—
19#2413—22no—
20#2310C411yescharacteristic
21#2297C411yescharacteristic
22#2296C2211yescharacteristic
23#1558C2214yes—
24#1533C2214yes—
25#1467—22no—
26#1467C2214yes—
27#1464—22no—
28#1464C2214yes—
29#1461—22no—
30#1458—22no—
31#1285C2218yes—
32C22×#41C2×C411yescharacteristic
33C23×Q8C2×C411yescharacteristic
34C23×Q8C2×C411yescharacteristic
35#536—44no—
36#536—44no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—44no—
39C2×C4⋊Q8C2×C412yes—
40C2×C4⋊Q8C2×C412yes—
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2311yescharacteristic
44C2×#124C2×C412yes—
45C2×#124C2×C411yescharacteristic
46C2×#124C2311yescharacteristic
47C2×#123—21no—
48C2×#123—21no—
49C2×#123—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52#519—44no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8D412yes—
55C2×C4×Q8D412yes—
56C2×C4×Q8C2312yes—
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C2×#117D412yes—
60C2×#117D412yes—
61C2×#117C2311yescharacteristic
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C42C2×C411yescharacteristic
65#446—48no—
66#432—24no—
67#432C2318yes—
68#431—24no—
69#398—22no—
70#398—22no—
71#398—22no—
72#398—22no—
73#396—22no—
74#396—22no—
75#396—22no—
76#396—22no—
77#391—21no—
78#391—21no—
79#391C2×C412yes—
80#391C2×C412yes—
81#391C2×C412yes—
82#390C2×C412yes—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#386—22no—
93#385—22no—
94#381—22no—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4C2312yes—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—21no—
99#333—44no—
100#329—44no—
101#324—24no—
102#324—24no—
103#324—24no—
104#323—24no—
105C2×#41—42no—
106C2×#41—42no—
107C2×#41—42no—
108C22×Q8—24no—
109C22×Q8—21no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8C22⋊C412yes—
116C22×Q8C22×C411yescharacteristic
117C22×Q8C22×C411yescharacteristic
118C22×Q8C22×C411yescharacteristic
119C22×Q8C22×C412yes—
120C22×D4—21no—
121C22×D4C22×C411yescharacteristic
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C4⋊Q8—24no—
129C4⋊Q8—42no—
130C4⋊Q8—48no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134#124—24no—
135#124—42no—
136#124—44no—
137#123—48no—
138#123—44no—
139#123—44no—
140C22⋊Q8—48no—
141C22⋊Q8—48no—
142C4×Q8—28no—
143C4×Q8—44no—
144C4×Q8—44no—
145C4×Q8—44no—
146#117—24no—
147#117—44no—
148#117—44no—
149#117—42no—
150#117—42no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C2×D412yes—
179C2×C42—21no—
180C2×C42—42no—
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C412yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C412yes—
187C2×C42C22×C412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2411yescharacteristic
191C4⋊C8—28no—
192C4⋊C8—44no—
193C4⋊C8—44no—
194C22⋊C8—28no—
195#97—28no—
196#97—44no—
197C4×C8—44no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206C24—21no—
207C24C2×C22⋊C411yescharacteristic
208#41—88no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—416no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—42no—
228C2×Q8—88no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4#10014yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D412yes—
265C2×C8—48no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C22⋊C4—24no—
284C22⋊C4—42no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C42—22no—
293C42—24no—
294C42—24no—
295C42—42no—
296C42—41no—
297C42—41no—
298C42—42no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42H(5,𝔽2)14yes—
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23C2×#10012yes—
312C23C22×C22⋊C411yescharacteristic
313Q8—48no—
314Q8—48no—
315Q8—82no—
316Q8—82no—
317Q8—82no—
318Q8—84no—
319D4—82no—
320D4—84no—
321C2×C4—28no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—48no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4C2×#10014yes—
359C2×C4#41312yes—
360C2×C4#51812yes—
361C2×C4#53414yes—
362C8—88no—
363C22—22no—
364C22—22no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—82no—
369C22—81no—
370C22—81no—
371C22—82no—
372C22—81no—
373C22#230811yescharacteristic
374C22C22×#10011yescharacteristic
375C22C2×#41311yescharacteristic
376C4—24no—
377C4—22no—
378C4—22no—
379C4—24no—
380C4—44no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—42no—
385C4—82no—
386C4—81no—
387C4—81no—
388C4—81no—
389C2—21no—
390C2—21no—
391C2—42no—
392C2—161no—
393C2#2175211yescharacteristic
394C2#2178312yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17371C2semidirect
3. #9452C2semidirect
4. #9448C2semidirect
5. #8742C2semidirect

Maximal subgroups

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

subgroupindex
#356672
#173712
#94522
#94482
#87422
#85242