Finite Group Databasebeta

← #480707

Subgroups — #480707

Order 512.

855 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35618C211yesmaximal, characteristic
2#17372C212yesmaximal, complemented
3#11554C214yesmaximal, complemented
4#11552C214yesmaximal, complemented
5#11516C212yesmaximal, complemented
6#11507C212yesmaximal, complemented
7C2×#537C411yescharacteristic
8C2×#520C2211yescharacteristic
9#2983—22no—
10#2504C412yes—
11#2503C412yes—
12#2500C2212yes—
13#2499C2212yes—
14#2444—22no—
15#2444—22no—
16#2421—22no—
17#2420—22no—
18#2317C2211yescharacteristic
19#2311C411yescharacteristic
20#2297C411yescharacteristic
21#2297C2211yescharacteristic
22C2×#382C411yescharacteristic
23#1466—22no—
24#1466—22no—
25#1464C2214yes—
26#1464C2214yes—
27#1460—22no—
28#1458C2214yes—
29#1325C2218yes—
30#1319C2214yes—
31#1318C2214yes—
32C22×#41C2×C411yescharacteristic
33C22×#41C2×C411yescharacteristic
34#537—44no—
35C2×C22⋊Q8—21no—
36C2×C22⋊Q8C2×C412yes—
37C2×#121—21no—
38C2×#121—22no—
39C2×#121C2×C412yes—
40#520—44no—
41C2×C4×Q8D412yes—
42C2×C4×D4D412yes—
43C2×C4×D4D412yes—
44C2×C4×D4D412yes—
45C2×#117C2×C411yescharacteristic
46C2×#117C2311yescharacteristic
47C2×#117C2312yes—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C412yes—
51C22×C42C2×C411yescharacteristic
52#446—48no—
53#432—24no—
54#432—24no—
55#431—24no—
56#400—22no—
57#400—22no—
58#397—22no—
59#397—22no—
60#397—22no—
61#397—22no—
62#394—22no—
63#394—22no—
64#391—21no—
65#391C2×C412yes—
66#391C2×C411yescharacteristic
67#391C2311yescharacteristic
68#390—21no—
69#390C2×C412yes—
70#390C2×C411yescharacteristic
71#390C2311yescharacteristic
72#389—22no—
73#388—22no—
74#388—22no—
75#387—22no—
76#387—22no—
77#386—22no—
78#386—21no—
79#386—21no—
80#386—22no—
81#386—22no—
82#386—22no—
83#385—22no—
84#384C2312yes—
85#382—22no—
86#382C2×C412yes—
87#382C2×C412yes—
88#382C2×C412yes—
89#381—22no—
90#381—22no—
91#380—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4C2×C412yes—
94C4×C22⋊C4C2×C412yes—
95C2×#95—21no—
96C2×#95—21no—
97#324C2318yes—
98#323—24no—
99#323—24no—
100#323—24no—
101C2×#41—24no—
102C2×#41—42no—
103C2×#41—42no—
104C2×#41—42no—
105C2×#41—42no—
106C2×#41—42no—
107C22×Q8C22⋊C411yescharacteristic
108C22×Q8C22×C411yescharacteristic
109C22×D4—21no—
110C22×D4—22no—
111C22×D4C22⋊C412yes—
112C22×D4C22⋊C411yescharacteristic
113C22×D4C22×C411yescharacteristic
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C412yes—
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C412yes—
123C22⋊Q8—44no—
124C22⋊Q8—44no—
125#121—44no—
126#121—48no—
127#121—44no—
128C4×Q8—28no—
129C4×D4—44no—
130C4×D4—44no—
131C4×D4—44no—
132#117—42no—
133#117—44no—
134#117—42no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C22×C411yescharacteristic
149C2×C4⋊C4C2411yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4C2×D412yes—
160C2×C22⋊C4C2×D412yes—
161C2×C22⋊C4C2×D412yes—
162C2×C42—21no—
163C2×C42—42no—
164C2×C42C22×C412yes—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C4⋊C8—44no—
171C4⋊C8—44no—
172C4⋊C8—44no—
173C22⋊C8—28no—
174#97—44no—
175#97—44no—
176C4×C8—44no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189C24—21no—
190C24C2×C22⋊C412yes—
191C24C2×C22⋊C411yescharacteristic
192#41—416no—
193#41—88no—
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8#9914yes—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—41no—
205C2×D4—42no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—44no—
213C2×D4—82no—
214C2×D4—82no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D412yes—
250C2×C8—48no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—44no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—82no—
270C22⋊C4—82no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C42—24no—
276C42—24no—
277C42—24no—
278C42—24no—
279C42—41no—
280C42—41no—
281C42—42no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—41no—
288C23—41no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23#41312yes—
297C23C22×C22⋊C411yescharacteristic
298Q8—44no—
299Q8—82no—
300D4—48no—
301D4—44no—
302D4—82no—
303D4—84no—
304D4—164no—
305C2×C4—28no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—48no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—161no—
339C2×C4—161no—
340C2×C4—162no—
341C2×C4—162no—
342C2×C4—162no—
343C2×C4C2×#9914yes—
344C2×C4C2×#9912yes—
345C2×C4#56812yes—
346C8—88no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—82no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22#287911yescharacteristic
365C22C22×#9911yescharacteristic
366C22C2×#41311yescharacteristic
367C4—22no—
368C4—22no—
369C4—24no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4—42no—
374C4—82no—
375C4—81no—
376C4—81no—
377C4—82no—
378C4—81no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—82no—
383C2—161no—
384C2#1739211yescharacteristic
385C2#3336211yescharacteristic
386C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#537C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
2. #17372C2semidirect
3. #11554C2semidirect
4. #11552C2semidirect
5. #11516C2semidirect
6. #11507C2semidirect

Maximal subgroups

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

subgroupindex
#356182
#173722
#115542
#115522
#115162
#115072