Finite Group Databasebeta

← #493036

Subgroups — #493036

Order 512.

1362 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34211C211yesmaximal, characteristic
2C22×#354C212yesmaximal
3#20392C212yesmaximal
4#12627C214yesmaximal, complemented
5C4×#354C214yesmaximal, complemented
6#12582C212yesmaximal
7C22×#124C2211yescharacteristic
8C2×#519C411yescharacteristic, complemented
9C2×#518C2211yescharacteristic
10C22×#99C2214yes—
11#2367—24no—
12#2342—22no—
13#2342—22no—
14#2326C412yescomplemented
15#2326C2212yes—
16#2310C411yescharacteristic, complemented
17#2309C2211yescharacteristic
18C4×#124C412yescomplemented
19C4×#124C2212yes—
20#2285C411yescharacteristic, complemented
21#2273—22no—
22C4×#117C411yescharacteristic, complemented
23C2×#354C4116yescomplemented
24C2×#354C2214yes—
25C2×#354C2214yes—
26C4×#99—24no—
27C4×#99C2218yes—
28C2×#342—22no—
29#1393C2218yes—
30C23×Q8C2×C411yescharacteristic
31C23×D4C2311yescharacteristic
32C2×#124—22no—
33C2×#124—24no—
34C2×#124C2×C418yes—
35C2×#124C2×C411yescharacteristic
36C2×#124C2×C411yescharacteristic
37C2×#124C2311yescharacteristic
38C2×#124C2311yescharacteristic
39#519—44no—
40#518—44no—
41C2×C4×Q8—22no—
42C2×C4×Q8C2×C412yescomplemented
43C2×C4×Q8C2×C412yescomplemented
44C2×C4×D4C2×C412yes—
45C2×C4×D4D414yes—
46C2×C4×D4C2312yes—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117C2×C411yescharacteristic
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C22×C22⋊C4—22no—
54C22×C22⋊C4—24no—
55C22×C22⋊C4D412yes—
56C22×C22⋊C4D412yes—
57C22×C42C2×C411yescharacteristic
58C2×#99—24no—
59C2×#99—28no—
60C2×#99C2×C4116yes—
61C2×#99C2318yes—
62#391—22no—
63#391C2×C412yescomplemented
64#391C2×C412yescomplemented
65#390—22no—
66#390C2×C412yes—
67#390C2312yes—
68#388—24no—
69#388—24no—
70#388—24no—
71#388—24no—
72#385—24no—
73#385—24no—
74#384C2×C412yescomplemented
75#382C2×C412yescomplemented
76#381—22no—
77#380—22no—
78#379—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C22⋊C4—24no—
82C4×C22⋊C4—24no—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—28no—
86C4×C22⋊C4—48no—
87#376C2×C412yescomplemented
88C43C2×C412yescomplemented
89#354C2×C4116yescomplemented
90#354C2×C4116yescomplemented
91#342—28no—
92C25C2×D412yes—
93C22×Q8—21no—
94C22×Q8—22no—
95C22×Q8—42no—
96C22×Q8—42no—
97C22×Q8C4214yescomplemented
98C22×Q8C22×C411yescharacteristic
99C22×Q8C22×C411yescharacteristic
100C22×D4—42no—
101C22×D4—42no—
102C22×D4C22×C414yes—
103C22×D4C22×C411yescharacteristic
104C22×D4C2×D412yes—
105C22×D4#4114yes—
106C22×D4C2411yescharacteristic
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—44no—
110C23×C4C22⋊C414yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D414yes—
113#124—22no—
114#124—22no—
115#124—216no—
116#124—22no—
117#124—22no—
118#124C4214yescomplemented
119#124C4214yescomplemented
120#124C22×C414yes—
121#124C22×C414yes—
122C4×Q8—44no—
123C4×Q8—44no—
124C4×Q8—48no—
125C4×D4—44no—
126C4×D4—44no—
127C4×D4—48no—
128#117—24no—
129#117—42no—
130#117—44no—
131#117—44no—
132#117—42no—
133#117—42no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4C22⋊C412yes—
139C2×C4⋊C4C22×C412yes—
140C2×C4⋊C4C22×C412yes—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—216no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—216no—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—48no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4C22⋊C411yescharacteristic
157C2×C22⋊C4C22⋊C412yes—
158C2×C22⋊C4C22⋊C414yes—
159C2×C22⋊C4C22⋊C411yescharacteristic
160C2×C22⋊C4C2×D414yes—
161C2×C42—21no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—48no—
169C2×C42C4214yescomplemented
170C2×C42C22⋊C412yes—
171C2×C42C22×C412yes—
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C412yes—
175#99—216no—
176#99—28no—
177#99C22×C4116yes—
178#95—22no—
179#95—22no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—48no—
184C24—24no—
185C24—24no—
186C24—84no—
187C24C2×C22⋊C412yes—
188C24C2×C22⋊C411yescharacteristic
189C24C4×D418yes—
190C24C22×D412yes—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—24no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—48no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—48no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8C2×C4212yes—
203C2×Q8C2×C4212yes—
204C2×D4—41no—
205C2×D4—42no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—44no—
209C2×D4—44no—
210C2×D4—42no—
211C2×D4—48no—
212C2×D4—41no—
213C2×D4C2×C4212yes—
214C2×D4#11712yes—
215C2×D4C4×D418yes—
216C2×D4C23×C412yes—
217C2×D4C2×#4112yes—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—28no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—48no—
233C22×C4—416no—
234C22×C4—82no—
235C22×C4—84no—
236C22×C4—84no—
237C22×C4—168no—
238C22×C4C2×C4212yes—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C414yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4#11712yes—
244C22×C4C23×C411yescharacteristic
245C4⋊C4—24no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—84no—
252C22⋊C4—28no—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—28no—
256C22⋊C4—24no—
257C22⋊C4—22no—
258C22⋊C4—28no—
259C22⋊C4—216no—
260C22⋊C4—44no—
261C22⋊C4—48no—
262C22⋊C4—416no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—84no—
266C22⋊C4—84no—
267C22⋊C4#9914yes—
268C42—21no—
269C42—21no—
270C42—24no—
271C42—24no—
272C42—42no—
273C42—42no—
274C42—42no—
275C42—44no—
276C42—44no—
277C42—82no—
278C42—82no—
279C42—168no—
280C42C2×C4212yes—
281C42C2×C4212yes—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—28no—
287C23—28no—
288C23—42no—
289C23—44no—
290C23—44no—
291C23—88no—
292C23—88no—
293C23—164no—
294C23C4×C22⋊C414yes—
295C23C4×C22⋊C412yes—
296C23C2×#9912yes—
297C23#41012yes—
298C23C22×C22⋊C411yescharacteristic
299C23C2×C4×D414yes—
300Q8—42no—
301Q8—42no—
302Q8—42no—
303Q8—42no—
304Q8—48no—
305D4—42no—
306D4—42no—
307D4—48no—
308D4—42no—
309D4—42no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—48no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—416no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—88no—
339C2×C4—84no—
340C2×C4—816no—
341C2×C4—164no—
342C2×C4—1616no—
343C2×C4C4×C22⋊C412yes—
344C2×C4C2×#9914yes—
345C2×C4C22×C4211yescharacteristic
346C2×C4C2×#11711yescharacteristic
347C22—22no—
348C22—24no—
349C22—24no—
350C22—24no—
351C22—42no—
352C22—42no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—44no—
357C22—88no—
358C22—82no—
359C22—82no—
360C22—88no—
361C22C4×#9914yes—
362C22C2×C4×C22⋊C411yescharacteristic
363C22C22×#9911yescharacteristic
364C22C2×#41011yescharacteristic
365C4—24no—
366C4—21no—
367C4—22no—
368C4—21no—
369C4—42no—
370C4—42no—
371C4—44no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4—82no—
376C4—816no—
377C4—82no—
378C2—21no—
379C2—21no—
380C2—24no—
381C2—44no—
382C2—44no—
383C2C2×C4×#9911yescharacteristic
384C2#3336112yes—
subgroupG/H
center, socleC22C22×#99
commutatorC2×C4C22×C42
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #12627C2semidirect
5. C4×#354C2semidirect
8. C2×#519C4semidirect
14. #2326C4semidirect
16. #2310C4semidirect
18. C4×#124C4semidirect
20. #2285C4semidirect
22. C4×#117C4semidirect
23. C2×#354C4semidirect
42. C2×C4×Q8C2×C4semidirect
43. C2×C4×Q8C2×C4semidirect
63. #391C2×C4semidirect
64. #391C2×C4semidirect
74. #384C2×C4semidirect
75. #382C2×C4semidirect
87. #376C2×C4semidirect
88. C43C2×C4semidirect
89. #354C2×C4semidirect
90. #354C2×C4semidirect
97. C22×Q8C42semidirect
118. #124C42semidirect
119. #124C42semidirect
169. C2×C42C42semidirect

Maximal subgroups

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

subgroupindex
#342112
C22×#3542
#203922
#126272
C4×#3542
#125822