Finite Group Databasebeta

← C2×#11724 #481497

Subgroups — C2×#11724 #481497

Order 512.

1176 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#1742C211yesmaximal, characteristic
3C2×#1458C211yesmaximal, characteristic
4#11724C218yesmaximal, direct product factor
5C2×#1354C214yesmaximal, complemented
6C23×M16—21no—
7C22×C4⋊Q8C411yescharacteristic
8C22×#117C2211yescharacteristic
9C2×#431—21no—
10#2628—22no—
11#2628—22no—
12#2628C414yes—
13#2628C2214yes—
14C2×#400—21no—
15C2×#399—21no—
16C2×#395C2212yes—
17C2×#393C412yes—
18C2×#389—21no—
19C2×#382C411yescharacteristic
20C2×#380—21no—
21#1742—22no—
22#1742C2214yes—
23C2×#343—21no—
24C2×#343C2212yes—
25#1458—22no—
26#1458C2214yes—
27C2×#324C2212yes—
28C2×#323—21no—
29#1354C22116yescomplemented
30C23×Q8C2×C412yes—
31C24×C4D411yescharacteristic
32C22×M16—28no—
33C22×M16—22no—
34C22×M16—24no—
35C22×M16—41no—
36C22×M16—42no—
37C22×M16—44no—
38C23×C8—41no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—42no—
41C2×C4⋊Q8—41no—
42C2×C4⋊Q8—41no—
43C2×C4⋊Q8C2×C414yes—
44C2×#117—41no—
45C2×#117—41no—
46C2×#117—42no—
47C2×#117C2×C412yes—
48C2×#117D412yes—
49C2×#117D412yes—
50C2×#117C2312yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—22no—
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42D411yescharacteristic
62#431—24no—
63#431—42no—
64C2×C4⋊C8—41no—
65C2×C22⋊C8—22no—
66C2×C4×C8—41no—
67#400—24no—
68#400—24no—
69#399—24no—
70#399—24no—
71#395—22no—
72#395—22no—
73#395C2×C414yes—
74#395C2314yes—
75#393—24no—
76#393C2×C418yes—
77#389—24no—
78#389—24no—
79#382—22no—
80#382C2×C414yes—
81#380—24no—
82#380—24no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95C2×C412yes—
87#343—24no—
88#343—22no—
89#343C2314yes—
90#324—22no—
91#324C2314yes—
92#323—24no—
93C25C22⋊C411yescharacteristic
94C22×Q8—22no—
95C22×Q8—42no—
96C22×Q8—42no—
97C22×Q8—84no—
98C22×Q8C22×C414yes—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—24no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4C22⋊C412yes—
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4SD1614yes—
110C23×C4Q1614yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C2×M16—28no—
116C2×M16—216no—
117C2×M16—24no—
118C2×M16—42no—
119C2×M16—48no—
120C2×M16—416no—
121C2×M16—44no—
122C2×M16—44no—
123C2×M16—48no—
124C22×C8—42no—
125C22×C8—48no—
126C22×C8—44no—
127C22×C8—42no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131C4⋊Q8—44no—
132#117—42no—
133#117—42no—
134#117—42no—
135#117—42no—
136#117—42no—
137#117—42no—
138#117—42no—
139#117—42no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—84no—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22×C411yescharacteristic
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C4⋊C4C2411yescharacteristic
162C2×C22⋊C4—28no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—84no—
168C2×C22⋊C4C22⋊C412yes—
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C22×C412yes—
171C2×C22⋊C4C2×D412yes—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42C22⋊C412yes—
179C2×C42C22⋊C412yes—
180C2×C42C22×C412yes—
181C2×C42C2×D412yes—
182C4⋊C8—44no—
183C22⋊C8—24no—
184C22⋊C8—42no—
185C4×C8—44no—
186#95—24no—
187#95—24no—
188#95—28no—
189#95—22no—
190#95—24no—
191#95C22×C414yes—
192C24—21no—
193C24—21no—
194C24—22no—
195C24—42no—
196C24—42no—
197C24#10314yes—
198C24C2×C22⋊C411yescharacteristic
199C24C2×C22⋊C412yes—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—84no—
203C2×Q8—84no—
204C2×Q8—88no—
205C2×Q8—88no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—28no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4#9912yes—
243C22×C4#9911yescharacteristic
244C22×C4#9911yescharacteristic
245C22×C4#10318yes—
246C22×C4#10314yes—
247C22×C4#10414yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×SD1612yes—
255C22×C4C2×SD1614yes—
256C22×C4C2×Q1612yes—
257C22×C4C2×Q1614yes—
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261M16—28no—
262M16—48no—
263M16—416no—
264M16—44no—
265C2×C8—44no—
266C2×C8—48no—
267C2×C8—416no—
268C2×C8—44no—
269C2×C8—84no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—48no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—48no—
280C4⋊C4—84no—
281C4⋊C4—84no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—88no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—24no—
299C23—22no—
300C23—24no—
301C23—42no—
302C23—48no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—81no—
307C23—81no—
308C23—84no—
309C23C2×#9911yescharacteristic
310C23C2×#9911yescharacteristic
311C23C2×#10312yes—
312C23C2×#10314yes—
313C23C2×#10412yes—
314C23C22×C22⋊C411yescharacteristic
315Q8—88no—
316Q8—88no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—28no—
328C2×C4—28no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—88no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4#32818yes—
361C2×C4#32814yes—
362C2×C4#32814yes—
363C2×C4C2×#9912yes—
364C2×C4C2×#9912yes—
365C2×C4C2×#10312yes—
366C2×C4C2×#10314yes—
367C2×C4C2×#10414yes—
368C2×C4C22×SD1611yescharacteristic
369C2×C4C22×Q1611yescharacteristic
370C8—48no—
371C8—84no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—48no—
383C22—82no—
384C22—84no—
385C22C2×#32812yes—
386C22C2×#32812yes—
387C22#212212yes—
388C22#213512yes—
389C22C22×#9911yescharacteristic
390C22C22×#10311yescharacteristic
391C22C22×#10411yescharacteristic
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—42no—
400C4—42no—
401C4—84no—
402C4—84no—
403C4C2×#32814yes—
404C4C2×#32814yes—
405C2—21no—
406C2—21no—
407C2—22no—
408C2—44no—
409C2#1172414yesdirect product factor
410C2C22×#32811yescharacteristic
411C2C2×#212211yescharacteristic
412C2C2×#213511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11724C2direct (N × H)
5. C2×#1354C2semidirect
29. #1354C22semidirect
409. C2#11724direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#17422
C2×#14582
#117242
C2×#13542