Finite Group Databasebeta

← #224645

Subgroups — #224645

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36601C211yesmaximal, characteristic
2#21345C212yesmaximal, complemented
3#9690C212yesmaximal, complemented
4#9688C212yesmaximal
5#9687C212yesmaximal
6#8584C212yesmaximal, complemented
7#8579C211yesmaximal, characteristic
8#8227C212yesmaximal, complemented
9#8224C211yesmaximal, characteristic
10#2861C2212yes—
11#2859C2211yescharacteristic
12#2596—22no—
13#2581C411yescharacteristic
14#2580C411yescharacteristic
15#2573—22no—
16#2512C412yes—
17#2511C412yes—
18#2469—22no—
19#2373—22no—
20#2357C411yescharacteristic
21#2356C411yescharacteristic
22C4×#125C2212yes—
23C4×#124C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25#1984—22no—
26#1939—22no—
27#1917—22no—
28#1917—22no—
29#1767C2212yes—
30#1767C2212yes—
31#1767C2212yes—
32#1745C2212yes—
33#1745C2212yes—
34C4×#97C2212yes—
35C2×#330C2212yes—
36C2×#330C2212yes—
37C2×#330C2212yes—
38C2×#330C2212yes—
39C2×#329C2212yes—
40C2×#329C2211yescharacteristic
41C2×#329C2211yescharacteristic
42#1295C2211yescharacteristic
43#1295C2211yescharacteristic
44#1295C2212yes—
45C2×#126—22no—
46C2×#125C2312yes—
47C2×#124C2311yescharacteristic
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8D412yes—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×#97—22no—
56C2×#97C2312yes—
57C2×#97C2312yes—
58C2×#97C2312yes—
59C2×#97C2312yes—
60C2×C4×C8—22no—
61#400—22no—
62#400—21no—
63#400—22no—
64#399—21no—
65#397—22no—
66#396—22no—
67#395—21no—
68#393—21no—
69#391C2×C412yes—
70#391C2×C412yes—
71#391C2×C411yescharacteristic
72#390C2×C412yes—
73#390C2×C412yes—
74#390C2×C411yescharacteristic
75#389—22no—
76#389—22no—
77#389C2×C412yes—
78#389C2×C412yes—
79#388—22no—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#385—22no—
93#384C2×C412yes—
94#383—21no—
95#382C2×C411yescharacteristic
96#382C2×C411yescharacteristic
97#382C2311yescharacteristic
98#381—21no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4C2×C411yescharacteristic
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—21no—
110#376—21no—
111C43C2312yes—
112C43C2311yescharacteristic
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#330—24no—
121#330—24no—
122#330—42no—
123#330—42no—
124#329—24no—
125#329—41no—
126#329—41no—
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C22×C411yescharacteristic
129C22×D4C22×C411yescharacteristic
130C23×C4—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135#126—48no—
136#125—22no—
137#125—22no—
138#125—22no—
139#125—22no—
140#124—22no—
141#124—21no—
142#124—21no—
143C4×Q8—42no—
144C4×Q8—44no—
145C4×Q8—42no—
146C4×D4—42no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C22⋊C4—21no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C412yes—
175C2×C42C22×C412yes—
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2411yescharacteristic
184C4⋊C8—44no—
185C4⋊C8—44no—
186C4⋊C8—44no—
187#97—22no—
188#97—22no—
189#97—22no—
190#97—22no—
191#97—22no—
192#97—22no—
193#97—22no—
194#97—44no—
195#97#4114yes—
196C4×C8—44no—
197#95—21no—
198#95—22no—
199#95—21no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—21no—
206#95—21no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95C22⋊C411yescharacteristic
212#95C22⋊C411yescharacteristic
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215C24—21no—
216C2×Q8—24no—
217C2×Q8—21no—
218C2×Q8—21no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—41no—
225C2×D4—41no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—82no—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C412yes—
240C22×C4C23×C411yescharacteristic
241C22×C4C22×D412yes—
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C22⋊C4—42no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—44no—
280C22⋊C4—82no—
281C22⋊C4—82no—
282C42—22no—
283C42—24no—
284C42—42no—
285C42—42no—
286C42—42no—
287C42—42no—
288C42—41no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42—44no—
293C42—42no—
294C42—82no—
295C42C2×#4112yes—
296C42H(5,𝔽2)11yescharacteristic
297C422−1+411yescharacteristic
298C23—41no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23C22×C22⋊C411yescharacteristic
304Q8—44no—
305Q8—81no—
306Q8—81no—
307D4—81no—
308D4—81no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—162no—
344C2×C4#41312yes—
345C2×C4#42114yes—
346C2×C4C4×#4111yescharacteristic
347C2×C4#52011yescharacteristic
348C2×C4#54812yes—
349C2×C4#55012yes—
350C8—44no—
351C8—44no—
352C8—82no—
353C8—82no—
354C22—81no—
355C22—81no—
356C22—82no—
357C22—81no—
358C22—82no—
359C22#234111yescharacteristic
360C22C2×#41311yescharacteristic
361C22#288911yescharacteristic
362C22C2×#42112yes—
363C22#290112yes—
364C4—21no—
365C4—21no—
366C4—24no—
367C4—22no—
368C4—22no—
369C4—22no—
370C4—41no—
371C4—42no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—82no—
379C4—82no—
380C2—161no—
381C2#1767811yescharacteristic
382C2#1768212yes—
383C2#1768611yescharacteristic
384C2#2253711yescharacteristic
385C2#2255112yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21345C2semidirect
3. #9690C2semidirect
6. #8584C2semidirect
8. #8227C2semidirect

Maximal subgroups

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

subgroupindex
#366012
#213452
#96902
#96882
#96872
#85842
#85792
#82272
#82242