Finite Group Databasebeta

← C2×#11777 #481599

Subgroups — C2×#11777 #481599

Order 512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4#11777C218yesmaximal, direct product factor
5C2×#1344C214yesmaximal, complemented
6C22×C4⋊Q8C411yescharacteristic, complemented
7C22×#124—21no—
8C22×C4×D4C2211yescharacteristic
9C22×#99—21no—
10C2×#407—21no—
11#2547—22no—
12#2547—22no—
13#2547C414yescomplemented
14#2547C2214yes—
15C2×#397C412yescomplemented
16C2×#397C2212yes—
17C2×#391—21no—
18C2×#380—21no—
19C2×#380—21no—
20C2×C4×C4⋊C4C411yescharacteristic, complemented
21C2×C4×C22⋊C4—21no—
22C4×#99—22no—
23C4×#99C2214yes—
24C2×#342C2212yes—
25#1465—22no—
26#1465C2214yes—
27C2×#324—21no—
28C2×#323C2212yes—
29#1344C22116yescomplemented
30C23×Q8—21no—
31C23×D4D411yescharacteristic
32C24×C4D411yescharacteristic
33C24×C4D411yescharacteristic
34C22×M16—41no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—42no—
37C2×C4⋊Q8—41no—
38C2×C4⋊Q8—41no—
39C2×C4⋊Q8C2×C414yescomplemented
40C2×#124—24no—
41C2×#124—24no—
42C2×#124—48no—
43C2×C4×D4—41no—
44C2×C4×D4—41no—
45C2×C4×D4—42no—
46C2×C4×D4C2×C412yes—
47C2×C4×D4D412yes—
48C2×C4×D4D412yes—
49C2×C4×D4C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4D411yescharacteristic
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2311yescharacteristic
59C22×C42—21no—
60C22×C42—41no—
61C22×C42C2×C411yescharacteristic
62C2×#99—28no—
63C2×#99—24no—
64#407—24no—
65#407—42no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—41no—
68#397—24no—
69#397—22no—
70#397—22no—
71#397C2×C414yes—
72#397C2×C418yescomplemented
73#397C2314yes—
74#391—24no—
75#391—24no—
76#380—24no—
77#380—24no—
78#380—24no—
79#380—24no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4C2×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#324—24no—
91#323—22no—
92#323C2314yes—
93C25C22⋊C411yescharacteristic
94C25C2×D411yescharacteristic
95C22×Q8—24no—
96C22×Q8—44no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8—44no—
100C22×D4—24no—
101C22×D4—24no—
102C22×D4—41no—
103C22×D4—41no—
104C22×D4C22⋊C412yes—
105C22×D4C2×D412yes—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—44no—
115C23×C4—44no—
116C23×C4—42no—
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C412yes—
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C2×M16—44no—
127C2×M16—84no—
128C22×C8—42no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133#124—416no—
134#124—416no—
135C4×D4—42no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141C4×D4—42no—
142C4×D4—42no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22×C414yes—
156C2×C4⋊C4C2×D412yes—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—48no—
170C2×C22⋊C4C22⋊C412yes—
171C2×C22⋊C4C22×C412yes—
172C2×C22⋊C4C22×C411yescharacteristic
173C2×C22⋊C4C2×D411yescharacteristic
174C2×C22⋊C4C2×D411yescharacteristic
175C2×C22⋊C4C2411yescharacteristic
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—24no—
180C2×C42—44no—
181C2×C42—44no—
182C2×C42—84no—
183C2×C42C22⋊C412yes—
184C2×C42C22×C412yes—
185#99—216no—
186C22⋊C8—24no—
187C22⋊C8—42no—
188C22⋊C8—44no—
189#95—24no—
190#95—28no—
191#95—22no—
192#95—24no—
193#95—44no—
194#95C22×C414yes—
195C24—21no—
196C24—21no—
197C24—22no—
198C24—21no—
199C24—22no—
200C24—21no—
201C24—22no—
202C24—42no—
203C24—42no—
204C24—42no—
205C24—42no—
206C24C2×C22⋊C411yescharacteristic
207C24C2×C22⋊C412yes—
208C24C2×C22⋊C411yescharacteristic
209C24AGL(1,ℤ/8)14yes—
210C24C22×D411yescharacteristic
211C2×Q8—48no—
212C2×Q8—48no—
213C2×Q8—48no—
214C2×Q8—88no—
215C2×Q8—88no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—48no—
223C2×D4—48no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—84no—
263C22×C4—84no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4#9912yes—
269C22×C4#9911yescharacteristic
270C22×C4#9911yescharacteristic
271C22×C4#10414yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279M16—88no—
280C2×C8—44no—
281C2×C8—84no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C22⋊C4—28no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—48no—
297C22⋊C4—416no—
298C22⋊C4—44no—
299C22⋊C4—48no—
300C22⋊C4—48no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—48no—
306C42—88no—
307C23—22no—
308C23—22no—
309C23—21no—
310C23—21no—
311C23—24no—
312C23—21no—
313C23—21no—
314C23—24no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—41no—
319C23—42no—
320C23—41no—
321C23—41no—
322C23—42no—
323C23—48no—
324C23—41no—
325C23—48no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—81no—
334C23—84no—
335C23—81no—
336C23—81no—
337C23—84no—
338C23C2×#9911yescharacteristic
339C23C2×#9911yescharacteristic
340C23#41812yes—
341C23#41812yes—
342C23C2×#10412yes—
343C23C22×C22⋊C411yescharacteristic
344C23C2×AGL(1,ℤ/8)12yes—
345Q8—816no—
346D4—48no—
347D4—48no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
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—82no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—88no—
387C2×C4—84no—
388C2×C4—816no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4#35414yes—
393C2×C4C2×#9912yes—
394C2×C4C2×#9912yes—
395C2×C4C2×#10414yes—
396C8—84no—
397C22—24no—
398C22—24no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—22no—
405C22—42no—
406C22—42no—
407C22—44no—
408C22—48no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—81no—
417C22—81no—
418C22—84no—
419C22—82no—
420C22—84no—
421C22—84no—
422C22#150712yes—
423C22#150712yes—
424C22#212312yes—
425C22C2×#35412yes—
426C22C22×#9911yescharacteristic
427C22C2×#41811yescharacteristic
428C22C22×#10411yescharacteristic
429C4—22no—
430C4—24no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—42no—
436C4—84no—
437C4—88no—
438C4—84no—
439C4C2×#35414yes—
440C2—21no—
441C2—21no—
442C2—22no—
443C2—42no—
444C2—42no—
445C2—82no—
446C2#1177714yesdirect product factor
447C2C2×#150711yescharacteristic
448C2C2×#212311yescharacteristic
449C2C22×#35411yescharacteristic
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. #11777C2direct (N × H)
5. C2×#1344C2semidirect
6. C22×C4⋊Q8C4semidirect
13. #2547C4semidirect
15. C2×#397C4semidirect
20. C2×C4×C4⋊C4C4semidirect
29. #1344C22semidirect
39. C2×C4⋊Q8C2×C4semidirect
72. #397C2×C4semidirect
81. C4×C4⋊C4C2×C4semidirect
446. C2#11777direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25472
C2×C4×#992
C2×#14652
#117772
C2×#13442