Finite Group Databasebeta

← #165668

Subgroups — #165668

Order 512.

748 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36539C211yesmaximal, characteristic
2#22837C212yesmaximal, complemented
3#7718C212yesmaximal, complemented
4#7697C212yesmaximal, complemented
5#7674C212yesmaximal, complemented
6#7635C211yesmaximal, characteristic, complemented
7#7623C212yesmaximal, complemented
8#7601C211yesmaximal, characteristic, complemented
9#7583C212yesmaximal, complemented
10#2864C2211yescharacteristic
11#2854C2212yes—
12#2602—22no—
13#2594—22no—
14#2576—22no—
15#2536C411yescharacteristic
16#2535C411yescharacteristic
17#2494C412yes—
18#2426C412yes—
19#2374—22no—
20#2357C411yescharacteristic
21#2351C411yescharacteristic
22C4×#124C2211yescharacteristic
23C4×C22⋊Q8C2212yes—
24C42×D4C2211yescharacteristic
25#1955—22no—
26#1924—22no—
27#1923—22no—
28#1831C2212yes—
29#1817C2212yes—
30#1817C2212yes—
31#1771—22no—
32#1767C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1271C2212yes—
39#1271C2212yes—
40#1271C2212yes—
41#1271C2212yes—
42#1270C2211yescharacteristic
43#1270C2212yes—
44#1270C2211yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#123—22no—
47C2×C22⋊Q8C2×C412yes—
48C2×#121C2×C412yes—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C412yes—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C22×C42C2312yes—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×#97—22no—
62C2×#97—22no—
63C2×C4×C8—22no—
64#401—22no—
65#398—21no—
66#397—21no—
67#397—22no—
68#396—22no—
69#394—21no—
70#394—22no—
71#392—21no—
72#391C2311yescharacteristic
73#390—22no—
74#390C2×C412yes—
75#390C2311yescharacteristic
76#389—22no—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2312yes—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#386—22no—
92#386—22no—
93#385—22no—
94#382—21no—
95#382C2×C411yescharacteristic
96#382C2×C411yescharacteristic
97#382C2×C411yescharacteristic
98#381—21no—
99#380C2×C412yes—
100#380C2×C412yes—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4D412yes—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4C2×C412yes—
110C4×C22⋊C4D412yes—
111C4×C22⋊C4D412yes—
112C43C2311yescharacteristic
113#336—22no—
114#336—22no—
115#336—22no—
116#336C2312yes—
117#336C2312yes—
118#336C2312yes—
119#336C2312yes—
120C22×Q8C22×C411yescharacteristic
121C22×D4—22no—
122C22×D4C22⋊C411yescharacteristic
123C22×D4C22×C411yescharacteristic
124C23×C4—21no—
125C23×C4C22×C412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132#124—22no—
133#124—41no—
134#123—44no—
135#123—44no—
136C22⋊Q8—24no—
137C22⋊Q8—42no—
138#121—82no—
139C4×Q8—42no—
140C4×Q8—82no—
141C4×D4—42no—
142C4×D4—42no—
143C4×D4—42no—
144C4×D4—42no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C411yescharacteristic
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C22⋊C412yes—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—82no—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2411yescharacteristic
181C4⋊C8—44no—
182C4⋊C8—44no—
183C22⋊C8—44no—
184C22⋊C8—44no—
185C22⋊C8—82no—
186C22⋊C8—82no—
187#97—44no—
188#97—82no—
189C4×C8—82no—
190#95—21no—
191#95—22no—
192#95—22no—
193#95—21no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—21no—
198#95—21no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95C22⋊C411yescharacteristic
203#95C22⋊C412yes—
204#95C22⋊C412yes—
205#95C22⋊C411yescharacteristic
206C24—21no—
207C24C2×C22⋊C412yes—
208C2×Q8—21no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8#13112yes—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—41no—
215C2×D4—42no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—44no—
219C2×D4—82no—
220C2×D4—82no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—82no—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D412yes—
248C2×C8—24no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C22⋊C4—24no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—42no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—82no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C42—21no—
288C42—21no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—41no—
296C42—42no—
297C42—42no—
298C42—82no—
299C42—82no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23—42no—
312C23C22×C22⋊C411yescharacteristic
313Q8—42no—
314Q8—42no—
315D4—44no—
316D4—81no—
317D4—81no—
318D4—164no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
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—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—162no—
358C2×C4#41312yes—
359C2×C4C2×#13111yescharacteristic
360C2×C4#56811yescharacteristic
361C2×C4#57512yes—
362C2×C4#57512yes—
363C8—44no—
364C8—82no—
365C8—82no—
366C8—82no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—82no—
378C22—82no—
379C22#288011yescharacteristic
380C22#288311yescharacteristic
381C22C2×#41311yescharacteristic
382C22#289612yes—
383C22#290112yes—
384C4—21no—
385C4—21no—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4—82no—
398C4—82no—
399C2—82no—
400C2—161no—
401C2#1742411yescharacteristic
402C2#1751612yes—
403C2#1762512yes—
404C2#1768611yescharacteristic
405C2#1775711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2535C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22837C2semidirect
3. #7718C2semidirect
4. #7697C2semidirect
5. #7674C2semidirect
6. #7635C2semidirect
7. #7623C2semidirect
8. #7601C2semidirect
9. #7583C2semidirect

Maximal subgroups

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

subgroupindex
#365392
#228372
#77182
#76972
#76742
#76352
#76232
#76012
#75832