Finite Group Databasebeta

← C2×#11817 #481866

Subgroups — C2×#11817 #481866

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C2×#1821C211yesmaximal, characteristic
3C2×#1463C211yesmaximal, characteristic
4#11817C218yesmaximal, direct product factor
5C2×#1354C212yesmaximal, complemented
6C2×#1353C212yesmaximal, complemented
7C22×C22⋊Q8C411yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407—21no—
10C2×#407—21no—
11#2627—22no—
12#2627—22no—
13#2627C414yes—
14#2627C2214yes—
15C2×#400C411yescharacteristic
16C2×#400C2211yescharacteristic
17C2×#398—21no—
18C2×#398—21no—
19C2×#395C411yescharacteristic
20C2×#395C2211yescharacteristic
21C2×#385—21no—
22C2×#381—21no—
23C2×#380C411yescharacteristic
24#1821—22no—
25#1821C2214yes—
26C2×#343—21no—
27C2×#343C2212yes—
28#1463—22no—
29#1463C2214yes—
30C2×#324—21no—
31C2×#324C2212yes—
32#1354C2218yescomplemented
33#1353C2218yescomplemented
34C23×Q8C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4D411yescharacteristic
38C22×M16—22no—
39C22×M16—41no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—42no—
42C2×C22⋊Q8—41no—
43C2×C22⋊Q8—41no—
44C2×C22⋊Q8C2×C414yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—41no—
50C22×C4⋊C4—41no—
51C22×C4⋊C4—42no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4D412yes—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63#407—24no—
64#407—24no—
65#407—42no—
66#407—42no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8—41no—
70C2×C22⋊C8—41no—
71#400—22no—
72#400—21no—
73#400—21no—
74#400C2×C412yes—
75#400C2×C414yes—
76#400C2312yes—
77#398—24no—
78#398—24no—
79#398—24no—
80#398—24no—
81#395—22no—
82#395—21no—
83#395—21no—
84#395C2×C412yes—
85#395C2×C414yes—
86#395C2312yes—
87#385—24no—
88#385—24no—
89#381—24no—
90#381—24no—
91#380—22no—
92#380C2×C414yes—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101#343—22no—
102#343—24no—
103#343C2314yes—
104#324—24no—
105#324—22no—
106#324C2314yes—
107C25C22⋊C411yescharacteristic
108C22×Q8—21no—
109C22×Q8—41no—
110C22×Q8—41no—
111C22×Q8—82no—
112C22×Q8C22×C412yes—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—42no—
128C23×C4—82no—
129C23×C4—82no—
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C2×M16—24no—
143C2×M16—42no—
144C2×M16—44no—
145C2×M16—44no—
146C2×M16—44no—
147C2×M16—84no—
148C22×C8—42no—
149C22×C8—42no—
150C22⋊Q8—44no—
151C22⋊Q8—44no—
152C22⋊Q8—44no—
153C22⋊Q8—44no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2411yescharacteristic
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C22⋊C8—24no—
201C22⋊C8—42no—
202C22⋊C8—44no—
203C22⋊C8—44no—
204C22⋊C8—44no—
205#95—24no—
206#95—22no—
207#95—24no—
208#95—22no—
209#95—24no—
210#95—22no—
211#95—21no—
212#95—21no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95C22×C412yes—
217#95C22×C412yes—
218C24—22no—
219C24—22no—
220C24—42no—
221C24—42no—
222C24#9911yescharacteristic
223C24#9911yescharacteristic
224C24#9912yes—
225C24C2×C22⋊C411yescharacteristic
226C24C2×C22⋊C412yes—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—84no—
232C2×Q8—84no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—48no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C412yes—
304C22×C4#13714yes—
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308M16—44no—
309M16—44no—
310M16—84no—
311M16—88no—
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—84no—
315C2×C8—84no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—48no—
328C4⋊C4—42no—
329C4⋊C4—84no—
330C4⋊C4—84no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C22⋊C4—48no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—24no—
345C23—22no—
346C23—22no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—48no—
352C23—48no—
353C23C2×#9912yes—
354C23C2×#9912yes—
355C23C2×#9911yescharacteristic
356C23C2×#9911yescharacteristic
357C23#41912yes—
358C23#42112yes—
359C23C22×C22⋊C411yescharacteristic
360Q8—84no—
361Q8—84no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4#41912yes—
423C2×C4C2×#13712yes—
424C8—84no—
425C8—84no—
426C22—24no—
427C22—24no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—44no—
439C22—44no—
440C22—48no—
441C22—44no—
442C22—44no—
443C22#151112yes—
444C22#151112yes—
445C22#212112yes—
446C22#213312yes—
447C22C22×#9911yescharacteristic
448C22C2×#41911yescharacteristic
449C22C2×#42111yescharacteristic
450C4—22no—
451C4—22no—
452C4—41no—
453C4—42no—
454C4—41no—
455C4—42no—
456C4—41no—
457C4—41no—
458C4—42no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—82no—
463C2—21no—
464C2—21no—
465C2—22no—
466C2—42no—
467C2—42no—
468C2#1181714yesdirect product factor
469C2C2×#151111yescharacteristic
470C2C2×#212111yescharacteristic
471C2C2×#213311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11817C2direct (N × H)
5. C2×#1354C2semidirect
6. C2×#1353C2semidirect
32. #1354C22semidirect
33. #1353C22semidirect
468. C2#11817direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26272
C2×#18212
C2×#14632
#118172
C2×#13542
C2×#13532