Finite Group Databasebeta

← #98923

Subgroups — #98923

Order 512.

490 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×C22⋊Q8C211yesmaximal, characteristic
2#10862C211yesmaximal, characteristic
3#8596C211yesmaximal, characteristic
4#8572C212yesmaximal
5#7039C212yesmaximal
6C2×C8×Q8C411yescharacteristic, complemented
7C22×#107—21no—
8C22×C4×C8C2211yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10#1984C411yescharacteristic, complemented
11#1939C411yescharacteristic, complemented
12#1916—21no—
13#1850—21no—
14#1797C2212yes—
15#1797C2211yescharacteristic
16C8×C4⋊C4—21no—
17C8×C4⋊C4C411yescharacteristic, complemented
18C8×C22⋊C4—21no—
19C2×#340—21no—
20C2×#335—21no—
21C2×#328—22no—
22C4⋊#95C2212yes—
23#1294—22no—
24#1277—21no—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C22⋊Q8D411yescharacteristic
28C2×C4×Q8C2×C411yescharacteristic
29C22×C4⋊C4—22no—
30C22×C42C2311yescharacteristic
31C8×Q8—42no—
32C2×#107—24no—
33C2×#107—42no—
34C2×#107—42no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C22⋊C8—21no—
41C2×C22⋊C8—22no—
42C2×C4×C8—21no—
43C2×C4×C8—21no—
44C2×C4×C8—42no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49#391Q811yescharacteristic
50#389Q811yescharacteristic
51#387—21no—
52#385—21no—
53#380D411yescharacteristic
54C4×C4⋊C4—21no—
55C4×C4⋊C4C2×C411yescharacteristic
56C4×C22⋊C4—21no—
57C2×#95—22no—
58C2×#95—22no—
59#340—42no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—22no—
66#335—42no—
67#328—44no—
68C22×Q8C4⋊C411yescharacteristic
69C23×C4—42no—
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C23×C4#4111yescharacteristic
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—42no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8SD1612yes—
87C22×C8Q1612yes—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C22⋊Q8D812yes—
91C22⋊Q8Q1612yes—
92C4×Q8—41no—
93C4×Q8—41no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4C4⋊C411yescharacteristic
104C2×C4⋊C4C4⋊C411yescharacteristic
105C2×C4⋊C4C4⋊C411yescharacteristic
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42C22×C411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×Q811yescharacteristic
115C2×C42#4111yescharacteristic
116#107—48no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—41no—
130C4×C8—41no—
131C4×C8—42no—
132C4×C8—42no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—42no—
139#95—42no—
140#95—82no—
141C24#12111yescharacteristic
142C2×Q8—21no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8#10712yescomplemented
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—44no—
161C22×C4—82no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C2×C4⋊C411yescharacteristic
169C22×C4#11711yescharacteristic
170C22×C4#12111yescharacteristic
171C22×C4C22⋊Q811yescharacteristic
172C22×C4C22⋊Q811yescharacteristic
173C22×C4C2×D811yescharacteristic
174C22×C4C2×SD1611yescharacteristic
175C22×C4C2×Q1611yescharacteristic
176C22×C4C2×Q1611yescharacteristic
177C22×C4#13511yescharacteristic
178C22×C4#13511yescharacteristic
179C22×C4AGL(1,ℤ/8)11yescharacteristic
180C22×C4#13711yescharacteristic
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—24no—
184C2×C8—24no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8—82no—
202C2×C8#9912yes—
203C2×C8#10312yes—
204C2×C8#10312yes—
205C2×C8#10412yes—
206C4⋊C4—21no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4#10712yescomplemented
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—41no—
227C42—41no—
228C42—41no—
229C42—41no—
230C42—41no—
231C42—41no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23#38011yescharacteristic
237C23#46011yescharacteristic
238C23#46211yescharacteristic
239C23#46611yescharacteristic
240C23#46711yescharacteristic
241C23#47211yescharacteristic
242C23#47311yescharacteristic
243Q8—42no—
244Q8—42no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—84no—
284C2×C4—84no—
285C2×C4—84no—
286C2×C4—162no—
287C2×C4—162no—
288C2×C4—162no—
289C2×C4—162no—
290C2×C4C2×#9911yescharacteristic
291C2×C4#41011yescharacteristic
292C2×C4C2×#10311yescharacteristic
293C2×C4#41711yescharacteristic
294C2×C4C2×#10411yescharacteristic
295C2×C4#42111yescharacteristic
296C2×C4C2×#10711yescharacteristic
297C2×C4#42711yescharacteristic
298C2×C4#46611yescharacteristic
299C2×C4#46711yescharacteristic
300C2×C4#47511yescharacteristic
301C2×C4#47711yescharacteristic
302C8—22no—
303C8—41no—
304C8—41no—
305C8—81no—
306C8—81no—
307C8—162no—
308C8#32814yescomplemented
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22#166811yescharacteristic
322C22#167011yescharacteristic
323C22#167411yescharacteristic
324C22#167611yescharacteristic
325C22#179911yescharacteristic
326C22#180311yescharacteristic
327C22#180611yescharacteristic
328C22#180811yescharacteristic
329C22#181211yescharacteristic
330C22#181411yescharacteristic
331C22#181611yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—42no—
339C4—41no—
340C4—41no—
341C4—42no—
342C4—81no—
343C4—81no—
344C4—164no—
345C4C2×#32811yescharacteristic
346C4#150911yescharacteristic
347C4#151311yescharacteristic
348C4#151811yescharacteristic
349C2—21no—
350C2—21no—
351C2—21no—
352C2—21no—
353C2#857211yescharacteristic
354C2#860911yescharacteristic
355C2#862711yescharacteristic
356C2#864511yescharacteristic
357C2#866711yescharacteristic
358C2#867711yescharacteristic
359C2#869511yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×C8×Q8C4semidirect
10. #1984C4semidirect
11. #1939C4semidirect
17. C8×C4⋊C4C4semidirect
145. C2×Q8#107semidirect
217. C4⋊C4#107semidirect
308. C8#328semidirect

Maximal subgroups

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

subgroupindex
C8×C22⋊Q82
#108622
#85962
#85722
#70392