Finite Group Databasebeta

← C2×#12923 #494247

Subgroups — C2×#12923 #494247

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2911C211yesmaximal, characteristic
2C2×#2159C212yesmaximal, complemented
3C22×#349C211yesmaximal, characteristic
4C2×#1809C211yesmaximal, characteristic
5#12923C218yesmaximal, direct product factor
6C2×#1386C212yesmaximal, complemented
7C23×M16C2211yescharacteristic
8C22×#117—21no—
9C22×M32C2212yes—
10#2911—24no—
11#2911—22no—
12#2911C2214yes—
13C2×#428—22no—
14C2×#428—22no—
15C2×#427—21no—
16C2×#427C2212yes—
17C22×#107C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C2×#380—21no—
20#2159—24no—
21#2159C2218yescomplemented
22C2×#367—22no—
23C2×#366—22no—
24C2×#349C418yescomplemented
25C2×#349C2214yes—
26#1809—22no—
27#1809C2214yes—
28C2×#340—21no—
29#1386C2218yescomplemented
30C24×C4D411yescharacteristic
31C22×M16—22no—
32C22×M16—22no—
33C22×M16—24no—
34C22×M16C2×C414yes—
35C22×M16D412yes—
36C22×M16Q814yes—
37C22×M16C2312yes—
38C23×C8D411yescharacteristic
39C23×C8C2311yescharacteristic
40C2×#117—22no—
41C2×#117—24no—
42C2×#117—24no—
43C2×#117—22no—
44C2×#117—42no—
45C2×#117—41no—
46C2×#117—41no—
47C2×#117—42no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C22⋊C4—41no—
53C22×C22⋊C4—41no—
54C22×C42—41no—
55C2×M32—22no—
56C2×M32—24no—
57C2×M32C2×C418yescomplemented
58C2×M32C2314yes—
59C22×C16—22no—
60#428—28no—
61#428—28no—
62#428—28no—
63#428—28no—
64#427—24no—
65#427—24no—
66#427—22no—
67#427—22no—
68#427D414yes—
69#427C2314yes—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—21no—
75C2×#107D412yes—
76C2×#107D412yes—
77C2×#107C2312yes—
78C2×#106—22no—
79C2×#106—22no—
80C2×#106—24no—
81C2×#106—21no—
82C2×#106C2312yes—
83#380—24no—
84#380—42no—
85C2×#95—41no—
86#367—28no—
87#366—28no—
88#349C2×C4116yescomplemented
89#340—24no—
90C25D811yescharacteristic
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4C4⋊C414yes—
102C23×C4D811yescharacteristic
103C23×C4D812yes—
104C23×C4Q1614yes—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C2×M16—28no—
108C2×M16—28no—
109C2×M16—24no—
110C2×M16—28no—
111C2×M16—28no—
112C2×M16—24no—
113C2×M16—24no—
114C2×M16—24no—
115C2×M16C4⋊C418yes—
116C2×M16C22×C414yes—
117C2×M16C2×Q814yes—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8C4⋊C414yes—
122C22×C8C22×C412yes—
123C22×C8C2×D412yes—
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D412yes—
126C22×C8C2×D412yes—
127C22×C8C2×Q812yes—
128C22×C8#4114yes—
129C22×C8C2411yescharacteristic
130#117—28no—
131#117—24no—
132#117—24no—
133#117—44no—
134#117—44no—
135#117—44no—
136#117—44no—
137#117—44no—
138#117—42no—
139#117—42no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—84no—
164C2×C42—42no—
165C2×C42—44no—
166C2×C42—44no—
167C2×C42—42no—
168M32—28no—
169M32—24no—
170M32C22×C418yescomplemented
171C2×C16—28no—
172C2×C16—24no—
173#107—24no—
174#107—24no—
175#107—22no—
176#107—22no—
177#107—24no—
178#107—42no—
179#107C2×D414yes—
180#107C2×D414yes—
181#106—28no—
182#106—24no—
183#106—24no—
184#106—24no—
185#106—42no—
186#95—44no—
187C24—21no—
188C24—21no—
189C24—22no—
190C24—42no—
191C24—42no—
192C24#10714yes—
193C24C2×D811yescharacteristic
194C24C2×D812yes—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—84no—
220C22×C4—84no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—82no—
228C22×C4#10714yes—
229C22×C4#10718yes—
230C22×C4C2×C4⋊C414yes—
231C22×C4C2×C4⋊C412yes—
232C22×C4#12011yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12312yes—
235C22×C4C2×D812yes—
236C22×C4C2×D811yescharacteristic
237C22×C4C2×Q1614yes—
238C22×C4C2×Q1612yes—
239C22×C4AGL(1,ℤ/8)11yescharacteristic
240C22×C4AGL(1,ℤ/8)11yescharacteristic
241C22×C4#13712yes—
242C22×C4C22×D411yescharacteristic
243M16—28no—
244M16—28no—
245M16—216no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8C2×C4⋊C414yes—
251C2×C8C2×C4⋊C412yes—
252C2×C8C4×D414yes—
253C2×C8C4×D414yes—
254C2×C8#12012yes—
255C2×C8C22⋊Q814yes—
256C2×C8C22⋊Q814yes—
257C2×C8#12312yes—
258C2×C8C23×C411yescharacteristic
259C2×C8C22×D412yes—
260C2×C8C22×Q811yescharacteristic
261C2×C8C2×#4112yes—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—82no—
275C22⋊C4—48no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—88no—
279C42—48no—
280C42—44no—
281C42—44no—
282C16—28no—
283C23—22no—
284C23—22no—
285C23—24no—
286C23—22no—
287C23—22no—
288C23—24no—
289C23—42no—
290C23—48no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—84no—
295C23—81no—
296C23—81no—
297C23C2×#10712yes—
298C23C2×#10714yes—
299C23#44711yescharacteristic
300C23#44711yescharacteristic
301C23#48012yes—
302C23C22×D811yescharacteristic
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—28no—
309C2×C4—28no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—88no—
316C2×C4—88no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—164no—
326C2×C4—162no—
327C2×C4—162no—
328C2×C4#38512yes—
329C2×C4#38512yes—
330C2×C4C2×#10714yes—
331C2×C4C2×#10712yes—
332C2×C4#42812yes—
333C2×C4#42812yes—
334C2×C4#44711yescharacteristic
335C2×C4#44711yescharacteristic
336C2×C4#45112yes—
337C2×C4#45112yes—
338C2×C4#48012yes—
339C2×C4#48412yes—
340C2×C4#48412yes—
341C2×C4C22×C4⋊C411yescharacteristic
342C2×C4C2×#12011yescharacteristic
343C2×C4C2×#12311yescharacteristic
344C2×C4C22×Q1611yescharacteristic
345C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
346C2×C4C2×#13711yescharacteristic
347C8—24no—
348C8#38514yes—
349C8C2×C4×D412yes—
350C8C2×C22⋊Q812yes—
351C22—24no—
352C22—24no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—24no—
358C22—44no—
359C22—48no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—84no—
364C22#185212yes—
365C22#185212yes—
366C22#219012yes—
367C22#219112yes—
368C22C22×#10711yescharacteristic
369C22C2×#44711yescharacteristic
370C22C2×#48011yescharacteristic
371C4—24no—
372C4—84no—
373C4—84no—
374C4—162no—
375C4—162no—
376C4#185212yes—
377C4#185212yes—
378C4C2×#38511yescharacteristic
379C4C2×#42811yescharacteristic
380C4C2×#45111yescharacteristic
381C4C2×#48411yescharacteristic
382C2—21no—
383C2—21no—
384C2—22no—
385C2—44no—
386C2#1292314yesdirect product factor
387C2C2×#185211yescharacteristic
388C2C2×#219011yescharacteristic
389C2C2×#219111yescharacteristic
subgroupG/H
center, socleC23#447
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC25D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×M16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2159C2semidirect
5. #12923C2direct (N × H)
6. C2×#1386C2semidirect
21. #2159C22semidirect
24. C2×#349C4semidirect
29. #1386C22semidirect
57. C2×M32C2×C4semidirect
88. #349C2×C4semidirect
170. M32C22×C4semidirect
386. C2#12923direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29112
C2×#21592
C22×#3492
C2×#18092
#129232
C2×#13862