Finite Group Databasebeta

← #136603

Subgroups — #136603

Order 512.

538 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#466C211yesmaximal, characteristic
2C22×#335C211yesmaximal, characteristic, complemented
3C4×C22⋊C16C211yesmaximal, characteristic, complemented
4C2×#1331C212yesmaximal, complemented
5#7171C212yesmaximal, complemented
6C2×#466C411yescharacteristic
7C2×C4×D8C2211yescharacteristic
8C22×C4⋊C8—21no—
9C22×C4×C8C2211yescharacteristic
10C4×#121—21no—
11C2×C22⋊C16—21no—
12C2×C4×C16C2212yes—
13#1970C411yescharacteristic
14#1942C411yescharacteristic
15#1934—21no—
16#1893—21no—
17#1808C411yescharacteristic
18C4×#107C2211yescharacteristic
19C4×#102—21no—
20C2×#335—21no—
21C2×#335—24no—
22C2×#335C2212yes—
23#1374—21no—
24#1331—24no—
25C22×D8C2×C411yescharacteristic
26C23×C8—41no—
27C23×C8C2×C411yescharacteristic
28C23×C8D411yescharacteristic
29C2×#121—21no—
30C2×C4×D4—21no—
31C2×C4×D4—41no—
32C22×C42D411yescharacteristic
33C22×C16—22no—
34#466—22no—
35#466C814yescomplemented
36C4×D8—22no—
37C4×D8—22no—
38C2×#107C2×C411yescharacteristic
39C2×#107C2×C411yescharacteristic
40C2×#107C2×C411yescharacteristic
41C2×C4⋊C8—24no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—42no—
44C2×C4⋊C8—41no—
45C2×#102—21no—
46C2×#102—21no—
47C2×#102—21no—
48C2×C4×C8—22no—
49C2×C4×C8—22no—
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8C2311yescharacteristic
54#390—21no—
55#389—21no—
56#388—41no—
57#386—41no—
58#380—21no—
59C4×C4⋊C4—21no—
60C4×C22⋊C4—41no—
61C22⋊C16—24no—
62C4×C16—22no—
63C4×C16—22no—
64#340—21no—
65#340—21no—
66#340—21no—
67#335—24no—
68#335—22no—
69#335D414yescomplemented
70C22×D4—21no—
71C22×D4—41no—
72C23×C4—41no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4D811yescharacteristic
75C23×C4SD1611yescharacteristic
76C2×D8—22no—
77C2×D8—21no—
78C2×D8—22no—
79C2×D8C2×C812yescomplemented
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—41no—
85C22×C8—44no—
86C22×C8—42no—
87C22×C8—82no—
88C22×C8—82no—
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8C2×C812yes—
92C22×C8M1612yes—
93C22×C8C22×C411yescharacteristic
94C22×C8C2×D411yescharacteristic
95#121—24no—
96#121—42no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—82no—
102C4×D4—82no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—41no—
117C2×C42C22⋊C411yescharacteristic
118C2×C42D811yescharacteristic
119C2×C42SD1611yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C16—24no—
122C2×C16—42no—
123C2×C16—42no—
124#107—21no—
125#107—22no—
126#107—22no—
127#107C2×C812yescomplemented
128C4⋊C8—24no—
129C4⋊C8—44no—
130C4⋊C8—42no—
131#102—24no—
132#102—42no—
133#102—42no—
134C4×C8—22no—
135C4×C8—22no—
136C4×C8—21no—
137C4×C8—21no—
138C4×C8C2×D412yes—
139C4×C8#4112yes—
140#95—41no—
141#95—41no—
142#95—41no—
143C24—41no—
144C24#10211yescharacteristic
145C2×D4—22no—
146C2×D4—44no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—82no—
153C2×D4—82no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4C22⋊C812yes—
176C22×C4#10211yescharacteristic
177C22×C4#10211yescharacteristic
178C22×C4#10211yescharacteristic
179C22×C4#10412yes—
180C22×C4D1612yes—
181C22×C4SD3212yes—
182C22×C4C2×C22⋊C411yescharacteristic
183C22×C4C2×D811yescharacteristic
184C22×C4C2×SD1611yescharacteristic
185D8—24no—
186D8—42no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—24no—
192C2×C8—24no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—44no—
197C2×C8—44no—
198C2×C8—88no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8C22⋊C812yes—
202C2×C8C4×D412yes—
203C2×C8#12112yes—
204C2×C8C22×C811yescharacteristic
205C2×C8C2×M1611yescharacteristic
206C2×C8#13112yes—
207C4⋊C4—22no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—82no—
214C22⋊C4—44no—
215C22⋊C4—82no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C22⋊C4—81no—
219C22⋊C4—81no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—81no—
227C42—81no—
228C42#12011yescharacteristic
229C42#12311yescharacteristic
230C42AGL(1,ℤ/8)11yescharacteristic
231C42#13711yescharacteristic
232C16—44no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—42no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23—81no—
243C23—81no—
244C23D4⋊C812yes—
245C23#35712yes—
246C23#35912yes—
247C23C2×#10211yescharacteristic
248D4—42no—
249D4—42no—
250D4—88no—
251D4—81no—
252D4—81no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—161no—
282C2×C4—161no—
283C2×C4—161no—
284C2×C4—161no—
285C2×C4D4⋊C812yes—
286C2×C4#35712yes—
287C2×C4#38611yescharacteristic
288C2×C4C2×C22⋊C811yescharacteristic
289C2×C4#40811yescharacteristic
290C2×C4#41711yescharacteristic
291C2×C4C2×#10411yescharacteristic
292C2×C4#42111yescharacteristic
293C2×C4#44711yescharacteristic
294C2×C4#44811yescharacteristic
295C2×C4#48011yescharacteristic
296C2×C4#48111yescharacteristic
297C2×C4C2×D1611yescharacteristic
298C2×C4C2×SD3211yescharacteristic
299C2×C4#50911yescharacteristic
300C2×C4#51011yescharacteristic
301C8—22no—
302C8—22no—
303C8—41no—
304C8—84no—
305C8C8×D412yes—
306C8#43512yes—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—82no—
314C22—82no—
315C22—82no—
316C22—161no—
317C22—161no—
318C22—161no—
319C22—161no—
320C22#133114yescomplemented
321C22C2×D4⋊C811yescharacteristic
322C22#148011yescharacteristic
323C22#187411yescharacteristic
324C22C2×#35711yescharacteristic
325C22#213911yescharacteristic
326C22C2×#35911yescharacteristic
327C22#214311yescharacteristic
328C4—21no—
329C4—21no—
330C4—22no—
331C4—22no—
332C4—41no—
333C4—82no—
334C4—161no—
335C4—161no—
336C4#147711yescharacteristic
337C4#148311yescharacteristic
338C4#186911yescharacteristic
339C4#187111yescharacteristic
340C4#218411yescharacteristic
341C4#218511yescharacteristic
342C4#223211yescharacteristic
343C4#223311yescharacteristic
344C2—24no—
345C2—162no—
346C2#893911yescharacteristic
347C2C2×#133111yescharacteristic
348C2#1159211yescharacteristic
349C2#1164511yescharacteristic
350C2#1164811yescharacteristic
351C2#1299011yescharacteristic
352C2#1299111yescharacteristic
subgroupG/H
centerC22×C4C2×D8
commutatorC2×C8C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×#102
prime radicalC2×#466C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#335C2semidirect
3. C4×C22⋊C16C2semidirect
4. C2×#1331C2semidirect
5. #7171C2semidirect
35. #466C8semidirect
69. #335D4semidirect
79. C2×D8C2×C8semidirect
127. #107C2×C8semidirect
320. C22#1331semidirect

Maximal subgroups

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

subgroupindex
C4×#4662
C22×#3352
C4×C22⋊C162
C2×#13312
#71712