Finite Group Databasebeta

← C2×#9027 #194716

Subgroups — C2×#9027 #194716

Order 512.

1886 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3C2×#1823C212yesmaximal
4C22×#324C211yesmaximal, characteristic
5#9027C218yesmaximal, direct product factor
6C2×#1294C212yesmaximal
7C22×#117—21no—
8C23×C4⋊C4—21no—
9C23×C4⋊C4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12C2×#398—24no—
13C2×#398—22no—
14C2×#398C418yescomplemented
15C2×#398C2214yes—
16#2316—24no—
17#2316—22no—
18#2316C2214yes—
19C2×#389—24no—
20C2×#380—21no—
21C2×#380C2212yes—
22C22×#95—21no—
23C22×#95C2211yescharacteristic
24#1823—24no—
25#1823C2218yes—
26C2×#340—24no—
27C2×#324—24no—
28C2×#324C2214yes—
29#1294C2218yes—
30C24×C4—21no—
31C24×C4—21no—
32C24×C4D411yescharacteristic
33C24×C4D411yescharacteristic
34C24×C4C2311yescharacteristic
35C23×C8—22no—
36C2×#117—24no—
37C2×#117—24no—
38C2×#117—44no—
39C2×#117—44no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—28no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—44no—
53C22×C4⋊C4—44no—
54C22×C4⋊C4C2×C414yes—
55C22×C4⋊C4C2×C414yes—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4D414yes—
58C22×C4⋊C4D414yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C42—22no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8C2314yes—
67#398—216no—
68#398—28no—
69#398C2×C4116yescomplemented
70#389—216no—
71#389—216no—
72#380—24no—
73#380—22no—
74#380—22no—
75#380—24no—
76#380D414yes—
77#380C2314yes—
78C2×#95—22no—
79C2×#95—28no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—24no—
83C2×#95—22no—
84C2×#95C2×C414yes—
85C2×#95C2312yes—
86#340—216no—
87#324—28no—
88C25C2×D411yescharacteristic
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—28no—
95C23×C4—24no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—44no—
104C23×C4—44no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—44no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4C22⋊C414yes—
116C23×C4C22⋊C414yes—
117C23×C4D814yes—
118C23×C4SD1614yes—
119C23×C4C22×C412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4#4114yes—
127C23×C4C2411yescharacteristic
128C22×C8—24no—
129C22×C8—48no—
130C22×C8—44no—
131#117—48no—
132#117—416no—
133#117—48no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—216no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—48no—
153C2×C4⋊C4—432no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—84no—
162C2×C4⋊C4C22⋊C418yes—
163C2×C4⋊C4C22×C414yes—
164C2×C4⋊C4C22×C414yes—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×D418yes—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C42—28no—
172C2×C42—48no—
173C22⋊C8—24no—
174C22⋊C8—24no—
175C22⋊C8—44no—
176#95—28no—
177#95—216no—
178#95—24no—
179#95—24no—
180#95—22no—
181#95—24no—
182#95C22×C414yes—
183C24—24no—
184C24—24no—
185C24C2×C22⋊C412yes—
186C24#12011yescharacteristic
187C24#12011yescharacteristic
188C24#12312yes—
189C24C2×D812yes—
190C24C2×SD1612yes—
191C24AGL(1,ℤ/8)12yes—
192C24#13712yes—
193C24C22×D411yescharacteristic
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—28no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—28no—
206C22×C4—28no—
207C22×C4—28no—
208C22×C4—216no—
209C22×C4—28no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—28no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—48no—
222C22×C4—42no—
223C22×C4—416no—
224C22×C4—416no—
225C22×C4—48no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—48no—
230C22×C4—44no—
231C22×C4—48no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—48no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—48no—
246C22×C4—84no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—84no—
253C22×C4—84no—
254C22×C4#102116yes—
255C22×C4C2×C22⋊C414yes—
256C22×C4C2×C22⋊C414yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C4×D414yes—
259C22×C4C4×D414yes—
260C22×C4#12012yes—
261C22×C4#12114yes—
262C22×C4#12114yes—
263C22×C4#12312yes—
264C22×C4C2×D814yes—
265C22×C4C2×SD1614yes—
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D412yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4112yes—
270C2×C8—48no—
271C2×C8—44no—
272C2×C8—88no—
273C4⋊C4—28no—
274C4⋊C4—28no—
275C4⋊C4—216no—
276C4⋊C4—28no—
277C4⋊C4—48no—
278C4⋊C4—416no—
279C4⋊C4—44no—
280C4⋊C4—48no—
281C4⋊C4—432no—
282C4⋊C4—84no—
283C4⋊C4—88no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—48no—
287C42—416no—
288C23—28no—
289C23—28no—
290C23—28no—
291C23—28no—
292C23—24no—
293C23—24no—
294C23—24no—
295C23—24no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23#38612yes—
301C23#38612yes—
302C23C2×#10214yes—
303C23#41714yes—
304C23#44712yes—
305C23#44712yes—
306C23#44812yes—
307C23#44812yes—
308C23#45614yes—
309C23#45814yes—
310C23#48014yes—
311C23#48114yes—
312C23C22×C22⋊C411yescharacteristic
313C23C2×#12011yescharacteristic
314C23C2×#12311yescharacteristic
315C23C22×D811yescharacteristic
316C23C22×SD1611yescharacteristic
317C23C2×AGL(1,ℤ/8)11yescharacteristic
318C23C2×#13711yescharacteristic
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—216no—
323C2×C4—216no—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—28no—
332C2×C4—24no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—432no—
341C2×C4—48no—
342C2×C4—48no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—48no—
353C2×C4—44no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—816no—
359C2×C4—88no—
360C2×C4—88no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4#38614yes—
364C2×C4C2×#10218yes—
365C2×C4C2×C4×D412yes—
366C2×C4C2×#12112yes—
367C8—84no—
368C22—28no—
369C22—28no—
370C22—28no—
371C22—28no—
372C22—22no—
373C22—22no—
374C22—24no—
375C22—22no—
376C22—22no—
377C22—24no—
378C22—21no—
379C22—21no—
380C22—22no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22#161018yes—
385C22#161118yes—
386C22#187414yes—
387C22#187414yes—
388C22#188614yes—
389C22#189014yes—
390C22C2×#38611yescharacteristic
391C22C22×#10211yescharacteristic
392C22C2×#41711yescharacteristic
393C22C2×#44711yescharacteristic
394C22C2×#44811yescharacteristic
395C22C2×#45612yes—
396C22C2×#45812yes—
397C22C2×#48011yescharacteristic
398C22C2×#48111yescharacteristic
399C4—28no—
400C4—22no—
401C4—22no—
402C4—24no—
403C4—48no—
404C4—44no—
405C4—84no—
406C4—88no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—24no—
412C2#902718yesdirect product factor
413C2C2×#161012yes—
414C2C2×#161112yes—
415C2C2×#187411yescharacteristic
416C2C2×#188611yescharacteristic
417C2C2×#189011yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9027C2direct (N × H)
14. C2×#398C4semidirect
69. #398C2×C4semidirect
412. C2#9027direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3982
C2×#23162
C2×#18232
C22×#3242
#90272
C2×#12942