Finite Group Databasebeta

← #6372192

Subgroups — #6372192

Order 512.

3542 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#2905C211yesmaximal, characteristic
3#33996C211yesmaximal, characteristic
4#21315C2116yesmaximal, complemented
5#21314C214yesmaximal, complemented
6#21313C214yesmaximal, complemented
7#20347C212yesmaximal
8C2×#1736C212yesmaximal
9C23×M16C2212yes—
10C2×#544—28no—
11C2×#544—24no—
12C2×#544C4116yescomplemented
13C2×#544C2218yes—
14C22×C4⋊Q8—21no—
15C22×C4⋊Q8C2212yes—
16C22×#125—21no—
17C22×#125C2212yes—
18C22×C22⋊Q8—22no—
19C22×#117C2211yescharacteristic
20C22×#117C2212yes—
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2905—28no—
24C2×#424C2218yes—
25C2×#423C2214yes—
26#2274—21no—
27#2274—21no—
28#2274C2218yes—
29#2273—22no—
30C4×#117—24no—
31C4×#117C2214yes—
32#1989C22116yescomplemented
33#1989C22116yescomplemented
34#1988C22116yescomplemented
35#1987C22116yescomplemented
36#1736C2214yes—
37#1736C2218yes—
38#1736C2214yes—
39#1736C2218yes—
40#1736C2218yes—
41#1735C2218yes—
42C2×#340C2218yes—
43C23×Q8—22no—
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C22×M16—28no—
47C22×M16—28no—
48C22×M16—28no—
49C22×M16C2314yes—
50C22×M16C2314yes—
51C23×C8C2314yes—
52#544—264no—
53#544—232no—
54#544C2×C4164yescomplemented
55C2×C4⋊Q8—216no—
56C2×C4⋊Q8—28no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—28no—
59C2×C4⋊Q8C2×C4116yescomplemented
60C2×C4⋊Q8C2318yes—
61C2×#125—216no—
62C2×#125—28no—
63C2×#125—24no—
64C2×#125—28no—
65C2×#125C2×C4116yescomplemented
66C2×#125C2318yes—
67C2×C22⋊Q8—232no—
68C2×C22⋊Q8—216no—
69C2×#117—24no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—28no—
74C2×#117C2×C418yes—
75C2×#117C2×C418yes—
76C2×#117D418yes—
77C2×#117D414yes—
78C2×#117Q8116yescomplemented
79C2×#117C2314yes—
80C2×#117C2314yes—
81C2×#117C2314yes—
82C2×#117C2314yes—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4C2×C418yes—
90C22×C4⋊C4D411yescharacteristic
91C22×C4⋊C4D411yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4D412yes—
100C22×C42—21no—
101C22×C42—22no—
102C22×C42C2312yes—
103#424—232no—
104#423—216no—
105C2×C4⋊C8C2318yes—
106C2×C4⋊C8C2318yes—
107C4×C4⋊C4—28no—
108C4×C4⋊C4—28no—
109C4×C4⋊C4C2318yes—
110C4×C4⋊C4C2318yes—
111C4×C22⋊C4—24no—
112C4×C22⋊C4—24no—
113C4×C22⋊C4—216no—
114#376—24no—
115C43—24no—
116#340C23116yes—
117#340C23132yes—
118#340C23116yes—
119C25C2×D411yescharacteristic
120C22×Q8—216no—
121C22×Q8—28no—
122C22×Q8—88no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—28no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—84no—
130C23×C4C22×C414yes—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D414yes—
136C23×C4C2×Q818yes—
137C23×C4#4118yes—
138C23×C4C2412yes—
139C23×C4C2411yescharacteristic
140C2×M16—28no—
141C2×M16—28no—
142C2×M16—28no—
143C2×M16—28no—
144C2×M16—28no—
145C2×M16—28no—
146C2×M16—432no—
147C22×C8—28no—
148C22×C8—28no—
149C22×C8—28no—
150C22×C8C2418yes—
151C4⋊Q8—264no—
152C4⋊Q8—232no—
153C4⋊Q8—216no—
154C4⋊Q8C22×C4132yes—
155#125—264no—
156#125—232no—
157#125—216no—
158#125C22×C4132yes—
159C22⋊Q8—2128no—
160#117—216no—
161#117—216no—
162#117—216no—
163#117—28no—
164#117—28no—
165#117—28no—
166#117C22⋊C4132yes—
167#117C22×C4116yes—
168#117C22×C4116yes—
169#117C2×D4116yes—
170#117C2×Q8116yes—
171#117C2×Q8116yescomplemented
172C2×C4⋊C4—28no—
173C2×C4⋊C4—216no—
174C2×C4⋊C4—216no—
175C2×C4⋊C4—216no—
176C2×C4⋊C4—216no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4C22⋊C418yes—
191C2×C4⋊C4C22⋊C418yes—
192C2×C4⋊C4C22×C418yes—
193C2×C4⋊C4C22×C4116yes—
194C2×C4⋊C4C22×C414yes—
195C2×C4⋊C4C22×C414yes—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×Q8116yescomplemented
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2412yes—
201C2×C4⋊C4C2412yes—
202C2×C4⋊C4C2418yes—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—216no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4C22⋊C4116yes—
209C2×C22⋊C4C2×D418yes—
210C2×C42—28no—
211C2×C42—216no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—28no—
216C2×C42—28no—
217C2×C42C22×C418yes—
218C2×C42C2×D414yes—
219C2×C42C2×D418yes—
220C2×C42C2414yes—
221C4⋊C8—216no—
222C4⋊C8—216no—
223#95—28no—
224#95—28no—
225C24—42no—
226C24—42no—
227C24C2×C22⋊C414yes—
228C24C22⋊Q818yes—
229C24C22×D412yes—
230C24C22×D411yescharacteristic
231C2×Q8—232no—
232C2×Q8—816no—
233C2×Q8—832no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—28no—
237C22×C4—28no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—216no—
242C22×C4—216no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—48no—
247C22×C4—44no—
248C22×C4—84no—
249C22×C4—84no—
250C22×C4—88no—
251C22×C4—816no—
252C22×C4C2×C22⋊C418yes—
253C22×C4C2×C22⋊C418yes—
254C22×C4C2×C22⋊C414yes—
255C22×C4C2×C22⋊C418yes—
256C22×C4C4×Q8132yes—
257C22×C4C22⋊Q818yes—
258C22×C4C22⋊Q818yes—
259C22×C4C22⋊Q818yes—
260C22×C4#124116yes—
261C22×C4C4⋊Q8116yes—
262C22×C4C23×C412yes—
263C22×C4C23×C414yes—
264C22×C4C22×D414yes—
265C22×C4C22×D414yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D412yes—
268C22×C4C22×D412yes—
269C22×C4C22×D414yes—
270C22×C4C22×Q814yes—
271C22×C4C22×Q814yes—
272C22×C4C22×Q814yes—
273C22×C4C2×#4114yes—
274C22×C4C2×#4114yes—
275C22×C4C2×#4114yes—
276C22×C4C2511yescharacteristic
277M16—416no—
278M16—416no—
279C2×C8—28no—
280C2×C8—28no—
281C2×C8—28no—
282C2×C8—416no—
283C4⋊C4—216no—
284C4⋊C4—28no—
285C4⋊C4—232no—
286C4⋊C4—232no—
287C4⋊C4—232no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—216no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—28no—
295C4⋊C4C2×C22⋊C4116yes—
296C4⋊C4C2×C22⋊C4116yes—
297C4⋊C4C23×C418yes—
298C4⋊C4C23×C414yes—
299C4⋊C4C23×C414yes—
300C4⋊C4C22×Q8116yes—
301C22⋊C4—216no—
302C22⋊C4—216no—
303C22⋊C4—216no—
304C22⋊C4C2×C22⋊C4132yes—
305C42—216no—
306C42—216no—
307C42—216no—
308C42—88no—
309C42—88no—
310C42C2×C22⋊C4116yes—
311C42C23×C418yes—
312C42C22×D418yes—
313C23—22no—
314C23—24no—
315C23—22no—
316C23—24no—
317C23—44no—
318C23—44no—
319C23—416no—
320C23#391116yes—
321C23C22×C22⋊C414yes—
322C23C22×C22⋊C412yes—
323C23C2×C22⋊Q814yes—
324C23C2×C22⋊Q814yes—
325C23C2×C22⋊Q814yes—
326C23C23×D411yescharacteristic
327Q8—832no—
328C2×C4—28no—
329C2×C4—28no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—216no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—416no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—44no—
344C2×C4—816no—
345C2×C4—816no—
346C2×C4#391116yes—
347C2×C4#391132yes—
348C2×C4C22×C22⋊C418yes—
349C2×C4C22×C22⋊C418yes—
350C2×C4C22×C22⋊C414yes—
351C2×C4C22×C22⋊C412yes—
352C2×C4C2×C4×Q818yes—
353C2×C4C2×C4×Q818yes—
354C2×C4C2×C4×Q818yes—
355C2×C4C2×C22⋊Q814yes—
356C2×C4C2×C22⋊Q814yes—
357C2×C4C2×C22⋊Q814yes—
358C2×C4C2×#12418yes—
359C2×C4C2×#12414yes—
360C2×C4C2×C4⋊Q814yes—
361C2×C4C2×C4⋊Q818yes—
362C2×C4C24×C411yescharacteristic
363C2×C4C23×D411yescharacteristic
364C2×C4C23×Q812yes—
365C2×C4C22×#4112yes—
366C8—48no—
367C22—28no—
368C22—28no—
369C22—28no—
370C22—22no—
371C22—24no—
372C22—22no—
373C22—24no—
374C22—416no—
375C22C2×#39118yes—
376C22C2×#39114yes—
377C22C23×C22⋊C411yescharacteristic
378C22C22×C22⋊Q812yes—
379C22#358512yes—
380C22#358612yes—
381C4—28no—
382C4—28no—
383C4—416no—
384C4—48no—
385C4—44no—
386C4—44no—
387C4C2×#39118yes—
388C4C2×#39114yes—
389C4C22×C4×Q812yes—
390C4C22×#12411yescharacteristic
391C4C22×C4⋊Q811yescharacteristic
392C2—28no—
393C2—24no—
394C2C22×#39111yescharacteristic
395C2#6008711yescharacteristic
396C2#6008811yescharacteristic
397C2#6049314yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21315C2semidirect
5. #21314C2semidirect
6. #21313C2semidirect
12. C2×#544C4semidirect
32. #1989C22semidirect
33. #1989C22semidirect
34. #1988C22semidirect
35. #1987C22semidirect
54. #544C2×C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
65. C2×#125C2×C4semidirect
78. C2×#117Q8semidirect
171. #117C2×Q8semidirect
198. C2×C4⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#29052
#339962
#213152
#213142
#213132
#203472
C2×#17362