Finite Group Databasebeta

← C2×#8856 #192062

Subgroups — C2×#8856 #192062

Order 512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2C2×#1852C212yesmaximal
3C22×#324C211yesmaximal, characteristic
4C2×#1441C211yesmaximal, characteristic
5#8856C218yesmaximal, direct product factor
6C2×#1294C212yesmaximal
7C22×#117—21no—
8C23×C4⋊C4C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C22×#107—22no—
11C22×C22⋊C8C2212yes—
12#2316—24no—
13#2316—22no—
14#2316C2214yes—
15C2×#389—22no—
16C2×#389—22no—
17C2×#380—21no—
18C2×#380C2212yes—
19C22×#95—21no—
20C22×#95—21no—
21C22×#95C2211yescharacteristic
22#1852—24no—
23#1852C2218yes—
24C2×#340—22no—
25C2×#324C418yescomplemented
26C2×#324C2214yes—
27#1441—22no—
28#1441C2214yes—
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—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—44no—
51C22×C4⋊C4C2×C414yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4Q814yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C42—22no—
60C2×#107—28no—
61C2×#107—48no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8C2×C418yes—
65C2×C22⋊C8C2314yes—
66#389—28no—
67#389—28no—
68#389—28no—
69#389—28no—
70#380—24no—
71#380—22no—
72#380—22no—
73#380—24no—
74#380D414yes—
75#380C2314yes—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—24no—
79C2×#95—21no—
80C2×#95—24no—
81C2×#95—24no—
82C2×#95—44no—
83C2×#95—44no—
84C2×#95C2312yes—
85#340—28no—
86#324C2×C4116yescomplemented
87C25C2×D411yescharacteristic
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—41no—
113C23×C4—41no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4C4⋊C414yes—
120C23×C4C4⋊C414yes—
121C23×C4D814yes—
122C23×C4Q1614yes—
123C23×C4C22×C412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×Q812yes—
130C23×C4#4114yes—
131C23×C4C2411yescharacteristic
132C22×C8—28no—
133C22×C8—24no—
134C22×C8—44no—
135C22×C8—44no—
136#117—48no—
137#117—416no—
138#117—48no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4C4⋊C418yes—
167C2×C4⋊C4C22×C414yes—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4C2×Q814yes—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C42—28no—
176C2×C42—48no—
177#107—416no—
178C22⋊C8—28no—
179C22⋊C8—24no—
180C22⋊C8C22×C418yes—
181#95—28no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—48no—
186#95—42no—
187#95—48no—
188C24—22no—
189C24—22no—
190C24—22no—
191C24—22no—
192C24C2×C4⋊C412yes—
193C24#12011yescharacteristic
194C24#12011yescharacteristic
195C24#12312yes—
196C24C2×D812yes—
197C24C2×Q1612yes—
198C24AGL(1,ℤ/8)12yes—
199C24#13712yes—
200C24C22×D411yescharacteristic
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—28no—
215C22×C4—28no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—44no—
231C22×C4—48no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—48no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—48no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—84no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4—84no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—84no—
276C22×C4#107116yescomplemented
277C22×C4C2×C4⋊C414yes—
278C22×C4C2×C4⋊C414yes—
279C22×C4C2×C4⋊C412yes—
280C22×C4C4×D414yes—
281C22×C4C4×D414yes—
282C22×C4#12012yes—
283C22×C4C22⋊Q814yes—
284C22×C4C22⋊Q814yes—
285C22×C4#12312yes—
286C22×C4C2×D814yes—
287C22×C4C2×Q1614yes—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D412yes—
290C22×C4C22×Q811yescharacteristic
291C22×C4C2×#4112yes—
292C2×C8—28no—
293C2×C8—44no—
294C2×C8—416no—
295C2×C8—44no—
296C4⋊C4—28no—
297C4⋊C4—28no—
298C4⋊C4—216no—
299C4⋊C4—28no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—48no—
313C42—416no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—24no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—24no—
323C23—24no—
324C23—24no—
325C23—24no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23#38512yes—
335C23#38512yes—
336C23C2×#10714yes—
337C23#42814yes—
338C23#44712yes—
339C23#44712yes—
340C23#45112yes—
341C23#45112yes—
342C23#45412yes—
343C23#45612yes—
344C23B(4,𝔽2)12yes—
345C23#45812yes—
346C23#48014yes—
347C23#48414yes—
348C23C22×C4⋊C411yescharacteristic
349C23C2×#12011yescharacteristic
350C23C2×#12311yescharacteristic
351C23C22×D811yescharacteristic
352C23C22×Q1611yescharacteristic
353C23C2×AGL(1,ℤ/8)11yescharacteristic
354C23C2×#13711yescharacteristic
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—216no—
358C2×C4—216no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—82no—
394C2×C4—84no—
395C2×C4—82no—
396C2×C4—816no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—88no—
402C2×C4—82no—
403C2×C4—88no—
404C2×C4—88no—
405C2×C4—88no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4#38514yes—
409C2×C4C2×#10718yescomplemented
410C2×C4C2×C4×D412yes—
411C2×C4C2×C22⋊Q812yes—
412C8—48no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—28no—
420C22—28no—
421C22—28no—
422C22—28no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—21no—
427C22—21no—
428C22—22no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22#159814yes—
436C22#160514yes—
437C22#161014yes—
438C22#161714yes—
439C22#185214yes—
440C22#185214yes—
441C22#185714yes—
442C22#186714yes—
443C22C2×#38511yescharacteristic
444C22C22×#10711yescharacteristic
445C22C2×#42811yescharacteristic
446C22C2×#44711yescharacteristic
447C22C2×#45111yescharacteristic
448C22C2×#45411yescharacteristic
449C22C2×#45611yescharacteristic
450C22C2×B(4,𝔽2)11yescharacteristic
451C22C2×#45811yescharacteristic
452C22C2×#48011yescharacteristic
453C22C2×#48411yescharacteristic
454C4—28no—
455C4—22no—
456C4—22no—
457C4—24no—
458C4—44no—
459C4—84no—
460C4—84no—
461C4—84no—
462C4—84no—
463C2—21no—
464C2—21no—
465C2—24no—
466C2—21no—
467C2—21no—
468C2#885618yesdirect product factor
469C2C2×#159811yescharacteristic
470C2C2×#160511yescharacteristic
471C2C2×#161011yescharacteristic
472C2C2×#161711yescharacteristic
473C2C2×#185211yescharacteristic
474C2C2×#185711yescharacteristic
475C2C2×#186711yescharacteristic
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. #8856C2direct (N × H)
25. C2×#324C4semidirect
86. #324C2×C4semidirect
276. C22×C4#107semidirect
409. C2×C4C2×#107semidirect
468. C2#8856direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23162
C2×#18522
C22×#3242
C2×#14412
#88562
C2×#12942