Finite Group Databasebeta

← #495294

Subgroups — #495294

Order 512.

538 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2166C211yesmaximal, characteristic, complemented
2C2×#2165C211yesmaximal, characteristic, complemented
3#20490C211yesmaximal, characteristic
4#13194C212yesmaximal
5#13163C212yesmaximal, complemented
6#13128C212yesmaximal, complemented
7#13128C212yesmaximal, complemented
8#12657C211yesmaximal, characteristic, complemented
9#12608C211yesmaximal, characteristic, complemented
10#12607C211yesmaximal, characteristic, complemented
11#12541C211yesmaximal, characteristic, complemented
12C2×#436C2211yescharacteristic
13C2×#435C411yescharacteristic
14C2×#405C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2290C411yescharacteristic
17#2166—24no—
18#2165—24no—
19C2×#364C2212yes—
20C2×#364C2212yes—
21#2150C2212yes—
22C2×#349C2212yes—
23C2×#349C2212yes—
24C2×#349C2212yes—
25C2×#349C2212yes—
26C2×#347C2211yescharacteristic
27C2×#347C2211yescharacteristic
28C2×#347C2211yescharacteristic
29C2×#347C2212yes—
30C2×#347C2211yescharacteristic
31C4×M32C2212yes—
32#1771C411yescharacteristic
33C8×C4⋊C4C2211yescharacteristic
34#1768C411yescharacteristic
35#1753C411yescharacteristic
36#1753C2211yescharacteristic
37#1753C2211yescharacteristic
38C8×C22⋊C4C411yescharacteristic
39#1750C411yescharacteristic
40C4×C22⋊C8C411yescharacteristic
41#1726C2211yescharacteristic
42#1376C2212yes—
43#1376C2212yes—
44#1375C2212yes—
45C22×M16C2×C411yescharacteristic
46C22×M16C2311yescharacteristic
47C22×M16C2311yescharacteristic
48C23×C8C2×C411yescharacteristic
49C2×C4×D4C2×C411yescharacteristic
50C2×#117C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C2×M32C2312yes—
53C2×M32C2312yes—
54C2×M32C2312yes—
55C2×M32C2312yes—
56#436—22no—
57#436—22no—
58#435—24no—
59C2×C4⋊C8C2×C411yescharacteristic
60C2×C4⋊C8C2×C411yescharacteristic
61C2×C22⋊C8C2×C411yescharacteristic
62C2×C22⋊C8C2×C411yescharacteristic
63C2×C22⋊C8C2×C411yescharacteristic
64C2×C22⋊C8C2×C411yescharacteristic
65#405—22no—
66#405—22no—
67C4×M16—22no—
68C4×M16—22no—
69C2×#97C2×C411yescharacteristic
70C2×#97C2×C411yescharacteristic
71C2×#97C2×C411yescharacteristic
72C2×#97C2311yescharacteristic
73C2×#97C2311yescharacteristic
74C2×C4×C8C2×C411yescharacteristic
75C2×C4×C8C2×C411yescharacteristic
76C2×C4×C8C2×C411yescharacteristic
77C2×C4×C8C2×C411yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#388—21no—
82#387—21no—
83#386—21no—
84#385—21no—
85C4×C4⋊C4C2×C411yescharacteristic
86C4×C22⋊C4C2×C411yescharacteristic
87C4×C22⋊C4C2×C411yescharacteristic
88#376C2×C411yescharacteristic
89#364—24no—
90#364—24no—
91C4⋊C16—22no—
92C4⋊C16—22no—
93#349—24no—
94#349—24no—
95#349—24no—
96#349—24no—
97#347—24no—
98#347—22no—
99#347—22no—
100#347—22no—
101#347D414yescomplemented
102#346—22no—
103C4×C16—22no—
104#336C2×C411yescharacteristic
105#336C2×C411yescharacteristic
106#336C2×C411yescharacteristic
107#336C2×C411yescharacteristic
108#336C2×C411yescharacteristic
109#336C2×C411yescharacteristic
110C22×D4M1611yescharacteristic
111C23×C4—21no—
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—21no—
117C2×M16—21no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—21no—
121C2×M16—21no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—41no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—41no—
128C22×C8C22×C411yescharacteristic
129C22×C8C22×C411yescharacteristic
130C22×C8C22×C411yescharacteristic
131C22×C8C22×C411yescharacteristic
132C22×C8C22×C411yescharacteristic
133C22×C8C22×C411yescharacteristic
134C22×C8C22×C411yescharacteristic
135C22×C8C2411yescharacteristic
136C4×D4—22no—
137C4×D4—22no—
138#117—22no—
139#117—41no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4M1611yescharacteristic
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4M1611yescharacteristic
147C2×C22⋊C4M1611yescharacteristic
148C2×C42—22no—
149C2×C42—21no—
150C2×C42—41no—
151C2×C42C22×C411yescharacteristic
152C2×C42C22×C411yescharacteristic
153C2×C42C22×C411yescharacteristic
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C411yescharacteristic
156M32—22no—
157M32—22no—
158M32—22no—
159M32—22no—
160M32—22no—
161M32—22no—
162M32C2×D414yes—
163M32#4114yes—
164C2×C16—22no—
165C2×C16—22no—
166C2×C16—22no—
167C2×C16—22no—
168C4⋊C8—22no—
169C4⋊C8M1614yes—
170C22⋊C8—21no—
171C22⋊C8—21no—
172C22⋊C8—21no—
173C22⋊C8—21no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176#97—22no—
177#97—22no—
178#97—22no—
179#97—22no—
180#97C2×D412yes—
181#97#4112yes—
182C4×C8—22no—
183C4×C8—21no—
184C4×C8—21no—
185C4×C8—21no—
186C4×C8—21no—
187C4×C8—22no—
188C4×C8—41no—
189C4×C8—41no—
190C4×C8#4112yes—
191C4×C8#4112yes—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197C24C2×M1611yescharacteristic
198C24C2×M1611yescharacteristic
199C2×D4—22no—
200C2×D4—81no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—81no—
220C22×C4C2×M1611yescharacteristic
221C22×C4C2×M1611yescharacteristic
222C22×C4C2×M1611yescharacteristic
223C22×C4C2×M1611yescharacteristic
224C22×C4C23×C411yescharacteristic
225M16—44no—
226M16—42no—
227M16—42no—
228M16—42no—
229M16—42no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—44no—
248C2×C8#11711yescharacteristic
249C2×C8#11711yescharacteristic
250C2×C8C4×D412yes—
251C2×C8C2×M1612yes—
252C2×C8C2×M1612yes—
253C2×C8C22×D411yescharacteristic
254C2×C8C2×#4111yescharacteristic
255C2×C8C2×#4112yes—
256C2×C8C2×#4112yes—
257C4⋊C4—22no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—81no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—42no—
273C42#11712yes—
274C42C4×D412yes—
275C42C2×M1612yes—
276C16—24no—
277C16—24no—
278C16—42no—
279C16—42no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23C22×M1611yescharacteristic
291D4—82no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—84no—
321C2×C4#43212yes—
322C2×C4#43212yes—
323C2×C4C2×#11711yescharacteristic
324C2×C4C2×C4×D411yescharacteristic
325C2×C4C4×#4111yescharacteristic
326C2×C4C22×M1611yescharacteristic
327C2×C4#56811yescharacteristic
328C2×C4#56811yescharacteristic
329C8—21no—
330C8—21no—
331C8—22no—
332C8—22no—
333C8—22no—
334C8—41no—
335C8—41no—
336C8—41no—
337C8#52912yes—
338C8#53212yes—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22C2×#43211yescharacteristic
352C22D4×M1611yescharacteristic
353C22#297011yescharacteristic
354C4—22no—
355C4—22no—
356C4—42no—
357C4—42no—
358C4—41no—
359C4—41no—
360C4—81no—
361C4—81no—
362C4#229311yescharacteristic
363C4#292311yescharacteristic
364C4D4×M1611yescharacteristic
365C4#297111yescharacteristic
366C2—21no—
367C2—21no—
368C2—81no—
369C2—81no—
370C2#2127011yescharacteristic
371C2#3342711yescharacteristic
372C2#3342811yescharacteristic
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×#432
prime radicalC22×D4M16
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2166C2semidirect
2. C2×#2165C2semidirect
5. #13163C2semidirect
6. #13128C2semidirect
7. #13128C2semidirect
8. #12657C2semidirect
9. #12608C2semidirect
10. #12607C2semidirect
11. #12541C2semidirect
101. #347D4semidirect

Maximal subgroups

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

subgroupindex
C2×#21662
C2×#21652
#204902
#131942
#131632
#131282
#126572
#126082
#126072
#125412