Finite Group Databasebeta

← C2×#11773 #481594

Subgroups — C2×#11773 #481594

Order 512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×#1466C211yesmaximal, characteristic
4#11773C218yesmaximal, direct product factor
5C2×#1344C214yesmaximal, complemented
6C22×#123—21no—
7C22×C4×D4C2211yescharacteristic
8C23×C4⋊C4C411yescharacteristic, complemented
9C2×#432—21no—
10C22×#99—21no—
11#2486—22no—
12#2486—22no—
13#2486C414yescomplemented
14#2486C2214yes—
15C2×#398C412yescomplemented
16C2×#397C2212yes—
17C2×#389—21no—
18C2×#381—21no—
19C2×#380—21no—
20C2×#380C411yescharacteristic, complemented
21C2×C4×C22⋊C4—21no—
22C4×#99—22no—
23C4×#99C2214yes—
24C2×#342C2212yes—
25#1466—22no—
26#1466C2214yes—
27C2×#323—21no—
28C2×#323C2212yes—
29#1344C22116yescomplemented
30C23×D4D411yescharacteristic
31C24×C4—21no—
32C24×C4C2×C411yescharacteristic
33C24×C4D411yescharacteristic
34C2×#123—24no—
35C2×#123—24no—
36C2×#123—48no—
37C2×C4×D4—41no—
38C2×C4×D4—41no—
39C2×C4×D4—42no—
40C2×C4×D4C2×C412yes—
41C2×C4×D4D412yes—
42C2×C4×D4D412yes—
43C2×C4×D4C2312yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—42no—
49C22×C4⋊C4—41no—
50C22×C4⋊C4—41no—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C414yescomplemented
54C22×C22⋊C4—22no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4D411yescharacteristic
58C22×C22⋊C4C2311yescharacteristic
59C22×C42—41no—
60C22×C42D411yescharacteristic
61#432—24no—
62#432—42no—
63C2×C4⋊C8—41no—
64C2×#99—28no—
65C2×#99—24no—
66C2×C22⋊C8—22no—
67C2×#97—41no—
68#398—24no—
69#398C2×C418yescomplemented
70#397—22no—
71#397—22no—
72#397C2×C414yes—
73#397C2314yes—
74#389—24no—
75#389—24no—
76#381—24no—
77#381—24no—
78#380—22no—
79#380—24no—
80#380—24no—
81#380C2×C414yescomplemented
82C4×C22⋊C4—24no—
83C4×C22⋊C4—42no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—41no—
87C2×#95C2×C412yes—
88#342—22no—
89#342C2314yes—
90#323—24no—
91#323—22no—
92#323C2314yes—
93C25C22⋊C411yescharacteristic
94C25C2×D411yescharacteristic
95C22×D4—24no—
96C22×D4—24no—
97C22×D4—41no—
98C22×D4—41no—
99C22×D4C22⋊C412yes—
100C22×D4C2×D412yes—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—44no—
111C23×C4—44no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—44no—
116C23×C4—44no—
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C412yes—
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C22×C8—42no—
127#123—416no—
128#123—416no—
129C4×D4—42no—
130C4×D4—42no—
131C4×D4—42no—
132C4×D4—42no—
133C4×D4—42no—
134C4×D4—42no—
135C4×D4—42no—
136C4×D4—42no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—48no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22×C414yes—
156C2×C4⋊C4C22×C412yes—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—48no—
170C2×C22⋊C4C22⋊C412yes—
171C2×C22⋊C4C22×C411yescharacteristic
172C2×C22⋊C4C2×D411yescharacteristic
173C2×C22⋊C4C2×D412yes—
174C2×C22⋊C4C2×D411yescharacteristic
175C2×C22⋊C4C2411yescharacteristic
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—44no—
180C2×C42—84no—
181C2×C42C22⋊C412yes—
182C2×C42C2×D412yes—
183C4⋊C8—44no—
184#99—216no—
185C22⋊C8—24no—
186C22⋊C8—42no—
187#97—44no—
188#95—28no—
189#95—24no—
190#95—22no—
191#95—24no—
192#95—44no—
193#95C22×C414yes—
194C24—21no—
195C24—21no—
196C24—21no—
197C24—22no—
198C24—21no—
199C24—22no—
200C24—42no—
201C24—42no—
202C24—42no—
203C24—42no—
204C24#10414yes—
205C24C2×C22⋊C411yescharacteristic
206C24C2×C22⋊C411yescharacteristic
207C24C2×C22⋊C412yes—
208C24AGL(1,ℤ/8)14yes—
209C24C22×D411yescharacteristic
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—48no—
217C2×D4—48no—
218C22×C4—24no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—48no—
261C22×C4—44no—
262C22×C4—84no—
263C22×C4—82no—
264C22×C4—88no—
265C22×C4—88no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4#9912yes—
269C22×C4#9911yescharacteristic
270C22×C4#9911yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C2×C8—44no—
279C2×C8—84no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—48no—
284C4⋊C4—44no—
285C4⋊C4—416no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C22⋊C4—28no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—48no—
296C22⋊C4—48no—
297C22⋊C4—44no—
298C22⋊C4—48no—
299C22⋊C4—48no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—88no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—21no—
309C23—21no—
310C23—22no—
311C23—24no—
312C23—24no—
313C23—24no—
314C23—21no—
315C23—21no—
316C23—24no—
317C23—42no—
318C23—48no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—48no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—48no—
327C23—41no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—81no—
334C23—84no—
335C23C2×#9911yescharacteristic
336C23C2×#9911yescharacteristic
337C23#41812yes—
338C23#41812yes—
339C23C2×#10412yes—
340C23C2×#10414yes—
341C23C22×C22⋊C411yescharacteristic
342C23C2×AGL(1,ℤ/8)12yes—
343D4—48no—
344D4—48no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—88no—
385C2×C4—816no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—816no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4#35214yes—
392C2×C4C2×#9912yes—
393C2×C4C2×#9912yes—
394C8—84no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—24no—
400C22—24no—
401C22—22no—
402C22—24no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—44no—
407C22—44no—
408C22—48no—
409C22—42no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—42no—
414C22—41no—
415C22—48no—
416C22—48no—
417C22—81no—
418C22—81no—
419C22—84no—
420C22#150712yes—
421C22#150712yes—
422C22C2×#35212yes—
423C22#212912yes—
424C22C22×#9911yescharacteristic
425C22C2×#41811yescharacteristic
426C22C22×#10411yescharacteristic
427C4—22no—
428C4—41no—
429C4—41no—
430C4—42no—
431C4—41no—
432C4—41no—
433C4—42no—
434C4—84no—
435C4—88no—
436C4—84no—
437C4C2×#35214yes—
438C2—21no—
439C2—21no—
440C2—22no—
441C2—42no—
442C2—44no—
443C2—42no—
444C2#1177314yesdirect product factor
445C2C2×#150711yescharacteristic
446C2C22×#35211yescharacteristic
447C2C2×#212911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11773C2direct (N × H)
5. C2×#1344C2semidirect
8. C23×C4⋊C4C4semidirect
13. #2486C4semidirect
15. C2×#398C4semidirect
20. C2×#380C4semidirect
29. #1344C22semidirect
53. C22×C4⋊C4C2×C4semidirect
69. #398C2×C4semidirect
81. #380C2×C4semidirect
444. C2#11773direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24862
C2×C4×#992
C2×#14662
#117732
C2×#13442