Finite Group Databasebeta

← C2×#11432 #480000

Subgroups — C2×#11432 #480000

Order 512.

1822 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2311C211yesmaximal, characteristic
2C22×#324C212yesmaximal, complemented
3#11432C218yesmaximal, direct product factor
4C2×#1319C212yesmaximal, complemented
5C2×#1318C212yesmaximal, complemented
6C23×#41C411yescharacteristic
7C22×#117C411yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C22×C22⋊C8—22no—
10#2311—24no—
11#2311C414yes—
12#2311C2214yes—
13C2×#391C411yescharacteristic
14C2×#391C2211yescharacteristic
15C2×#390C411yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#386—22no—
18C2×#380—22no—
19C2×#324—24no—
20C2×#324—22no—
21C2×#324C2214yes—
22C2×#324C2218yescomplemented
23#1319C2218yescomplemented
24#1318C2218yescomplemented
25C22×#41—22no—
26C22×#41—24no—
27C22×#41—42no—
28C22×#41—42no—
29C22×#41—44no—
30C22×#41C818yescomplemented
31C22×#41C2×C414yes—
32C23×Q8C2×C411yescharacteristic
33C23×D4—21no—
34C23×D4C2×C411yescharacteristic
35C24×C4—21no—
36C24×C4C2×C411yescharacteristic
37C24×C4D412yes—
38C23×C8—42no—
39C2×#117—22no—
40C2×#117—44no—
41C2×#117C2×C414yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—42no—
44C22×C4⋊C4—42no—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4D414yes—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C22⋊C4—21no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C2×C22⋊C8—24no—
57C2×C22⋊C8—28no—
58C2×C22⋊C8—44no—
59C2×C22⋊C8—42no—
60#391—22no—
61#391—22no—
62#391C2×C412yes—
63#391C2×C414yes—
64#391C2312yes—
65#390—22no—
66#390—22no—
67#390C2×C412yes—
68#390C2×C414yes—
69#390C2312yes—
70#386—28no—
71#386—28no—
72#380—28no—
73#380—28no—
74C2×#95—22no—
75C2×#95—22no—
76#324—24no—
77#324—28no—
78#324C2318yes—
79C25—21no—
80C25C22⋊C411yescharacteristic
81C2×#41—216no—
82C2×#41—28no—
83C2×#41—416no—
84C2×#41—416no—
85C2×#41—48no—
86C2×#41—48no—
87C2×#41—416no—
88C2×#41—416no—
89C2×#41—48no—
90C2×#41—48no—
91C2×#41C2×C8116yescomplemented
92C22×Q8—21no—
93C22×Q8—22no—
94C22×Q8—41no—
95C22×Q8—41no—
96C22×Q8—42no—
97C22×Q8C2×C814yescomplemented
98C22×Q8C22×C412yes—
99C22×D4—28no—
100C22×D4—24no—
101C22×D4—21no—
102C22×D4—22no—
103C22×D4—41no—
104C22×D4—41no—
105C22×D4—42no—
106C22×D4—42no—
107C22×D4—42no—
108C22×D4—44no—
109C22×D4C2×C814yescomplemented
110C22×D4C22×C412yes—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—28no—
119C23×C4—24no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—44no—
125C23×C4—84no—
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C414yes—
130C23×C4C2×C814yes—
131C23×C4M1614yes—
132C23×C4C22×C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D414yes—
136C22×C8—48no—
137C22×C8—44no—
138C22×C8—88no—
139#117—48no—
140#117—48no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4C22⋊C414yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C2×D414yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2411yescharacteristic
160C2×C22⋊C4—24no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—84no—
166C2×C22⋊C4—84no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42C22×C412yes—
172C2×C42C22×C412yes—
173C22⋊C8—28no—
174C22⋊C8—44no—
175C22⋊C8—48no—
176#95—28no—
177#95—24no—
178#95—24no—
179C24—22no—
180C24—24no—
181C24—42no—
182C24—42no—
183C24—44no—
184C24—44no—
185C24—41no—
186C24—41no—
187C24—42no—
188C24C22⋊C814yes—
189C24#9912yes—
190C24#9912yes—
191C24#10014yes—
192C24C2×C22⋊C411yescharacteristic
193C24C2×C22⋊C412yes—
194#41—464no—
195#41—432no—
196#41—432no—
197C2×Q8—22no—
198C2×Q8—24no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8C22×C814yescomplemented
208C2×D4—216no—
209C2×D4—22no—
210C2×D4—24no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—416no—
220C2×D4—416no—
221C2×D4—48no—
222C2×D4—48no—
223C2×D4C22×C814yescomplemented
224C22×C4—28no—
225C22×C4—28no—
226C22×C4—28no—
227C22×C4—28no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—216no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—416no—
254C22×C4—416no—
255C22×C4—48no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4—88no—
271C22×C4C22⋊C814yes—
272C22×C4C22⋊C818yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C414yes—
277C22×C4C22×C812yes—
278C22×C4C22×C814yes—
279C22×C4C2×M1612yes—
280C22×C4C2×M1614yes—
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D412yes—
283C2×C8—48no—
284C2×C8—816no—
285C2×C8—88no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—48no—
294C22⋊C4—48no—
295C22⋊C4—88no—
296C22⋊C4—84no—
297C22⋊C4—84no—
298C42—42no—
299C42—42no—
300C42—42no—
301C42—42no—
302C23—22no—
303C23—24no—
304C23—22no—
305C23—24no—
306C23—24no—
307C23—42no—
308C23—42no—
309C23—48no—
310C23—42no—
311C23—48no—
312C23—42no—
313C23—44no—
314C23—44no—
315C23—416no—
316C23—42no—
317C23—42no—
318C23—44no—
319C23—44no—
320C23—82no—
321C23—82no—
322C23—88no—
323C23—82no—
324C23—82no—
325C23#32318yes—
326C23#32318yes—
327C23C2×C22⋊C812yes—
328C23C2×C22⋊C814yes—
329C23C2×#9911yescharacteristic
330C23C2×#9914yes—
331C23C2×#9911yescharacteristic
332C23C2×#10012yes—
333C23C2×#10014yes—
334C23C22×C22⋊C411yescharacteristic
335Q8—44no—
336Q8—44no—
337Q8—48no—
338D4—44no—
339D4—44no—
340D4—48no—
341D4—432no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—216no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—28no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—28no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—432no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4C2×C22⋊C814yes—
386C2×C4C2×C22⋊C812yes—
387C2×C4C23×C811yescharacteristic
388C2×C4C22×M1611yescharacteristic
389C8—88no—
390C22—21no—
391C22—24no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—24no—
396C22—24no—
397C22—24no—
398C22—216no—
399C22—48no—
400C22—44no—
401C22—44no—
402C22—48no—
403C22—44no—
404C22—84no—
405C22—88no—
406C22—82no—
407C22—82no—
408C22—88no—
409C22C2×#32312yes—
410C22C2×#32318yes—
411C22C2×#32312yes—
412C22#212712yes—
413C22#213312yes—
414C22C22×C22⋊C811yescharacteristic
415C22C22×#9911yescharacteristic
416C22C22×#10011yescharacteristic
417C4—24no—
418C4—21no—
419C4—21no—
420C4—22no—
421C4—42no—
422C4—44no—
423C4—42no—
424C4—44no—
425C4—82no—
426C4—82no—
427C2—21no—
428C2—21no—
429C2—22no—
430C2—28no—
431C2—84no—
432C2#1143214yesdirect product factor
433C2C22×#32311yescharacteristic
434C2C2×#212711yescharacteristic
435C2C2×#213311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC2×C4C23×C8
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×#41C4
LCSG ▷ C2×C4 ▷ C22 ▷ 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
2. C22×#324C2semidirect
3. #11432C2direct (N × H)
4. C2×#1319C2semidirect
5. C2×#1318C2semidirect
22. C2×#324C22semidirect
23. #1319C22semidirect
24. #1318C22semidirect
30. C22×#41C8semidirect
91. C2×#41C2×C8semidirect
97. C22×Q8C2×C8semidirect
109. C22×D4C2×C8semidirect
207. C2×Q8C22×C8semidirect
223. C2×D4C22×C8semidirect
432. C2#11432direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23112
C22×#3242
#114322
C2×#13192
C2×#13182