Finite Group Databasebeta

← C2×#11723 #481496

Subgroups — C2×#11723 #481496

Order 512.

1176 conjugacy classes of subgroups.

The subgroup lattice has 411 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#11723C218yesmaximal, direct product factor
5C2×#1353C214yesmaximal, complemented
6C23×M16—21no—
7C22×C4⋊Q8C411yescharacteristic
8C22×#117C2211yescharacteristic
9C2×#431—21no—
10#2628—22no—
11#2628—22no—
12#2628C414yes—
13#2628C2214yes—
14C2×#400C2212yes—
15C2×#399C412yes—
16C2×#395—21no—
17C2×#393—21no—
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#1353C22116yescomplemented
30C23×Q8—21no—
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—22no—
54C22×C4⋊C4C2×C412yes—
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—22no—
68#400—22no—
69#400C2×C414yes—
70#400C2314yes—
71#399—24no—
72#399C2×C418yes—
73#395—24no—
74#395—24no—
75#393—24no—
76#393—24no—
77#389—24no—
78#389—24no—
79#382—22no—
80#382C2×C414yes—
81#380—24no—
82#380—24no—
83C2×#95—21no—
84C2×#95—22no—
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—24no—
95C22×Q8—42no—
96C22×Q8—42no—
97C22×Q8—84no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4C22⋊C412yes—
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4SD1614yes—
109C23×C4Q1614yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C2×M16—28no—
115C2×M16—216no—
116C2×M16—24no—
117C2×M16—42no—
118C2×M16—48no—
119C2×M16—416no—
120C2×M16—44no—
121C2×M16—44no—
122C2×M16—48no—
123C22×C8—42no—
124C22×C8—48no—
125C22×C8—44no—
126C22×C8—42no—
127C4⋊Q8—44no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131#117—42no—
132#117—42no—
133#117—42no—
134#117—42no—
135#117—42no—
136#117—42no—
137#117—42no—
138#117—42no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—84no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22×C414yes—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C4⋊C4C2411yescharacteristic
161C2×C22⋊C4—28no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—84no—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C22⋊C412yes—
169C2×C22⋊C4C22×C412yes—
170C2×C22⋊C4C2×D412yes—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42C22⋊C412yes—
178C2×C42C22⋊C412yes—
179C2×C42C22×C412yes—
180C2×C42C2×D412yes—
181C4⋊C8—44no—
182C22⋊C8—24no—
183C22⋊C8—42no—
184C4×C8—44no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—22no—
189#95—24no—
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—48no—
201C2×Q8—88no—
202C2×Q8—88no—
203C2×Q8—88no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—28no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4#9912yes—
241C22×C4#9911yescharacteristic
242C22×C4#9911yescharacteristic
243C22×C4#10318yes—
244C22×C4#10314yes—
245C22×C4#10414yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×SD1612yes—
253C22×C4C2×SD1614yes—
254C22×C4C2×Q1612yes—
255C22×C4C2×Q1614yes—
256C22×C4C23×C411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259M16—28no—
260M16—48no—
261M16—416no—
262M16—44no—
263C2×C8—44no—
264C2×C8—48no—
265C2×C8—416no—
266C2×C8—44no—
267C2×C8—84no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—48no—
279C4⋊C4—88no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—88no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—24no—
297C23—22no—
298C23—24no—
299C23—42no—
300C23—48no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—81no—
305C23—81no—
306C23—84no—
307C23C2×#9911yescharacteristic
308C23C2×#9911yescharacteristic
309C23C2×#10312yes—
310C23C2×#10314yes—
311C23C2×#10412yes—
312C23C22×C22⋊C411yescharacteristic
313Q8—816no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—88no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4#32818yes—
358C2×C4#32814yes—
359C2×C4#32814yes—
360C2×C4C2×#9912yes—
361C2×C4C2×#9912yes—
362C2×C4C2×#10312yes—
363C2×C4C2×#10314yes—
364C2×C4C2×#10414yes—
365C2×C4C22×SD1611yescharacteristic
366C2×C4C22×Q1611yescharacteristic
367C8—48no—
368C8—84no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—24no—
374C22—24no—
375C22—24no—
376C22—44no—
377C22—42no—
378C22—42no—
379C22—48no—
380C22—82no—
381C22—84no—
382C22C2×#32812yes—
383C22C2×#32812yes—
384C22#212312yes—
385C22#213412yes—
386C22C22×#9911yescharacteristic
387C22C22×#10311yescharacteristic
388C22C22×#10411yescharacteristic
389C4—21no—
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—42no—
397C4—42no—
398C4—84no—
399C4—84no—
400C4C2×#32814yes—
401C4C2×#32814yes—
402C2—21no—
403C2—21no—
404C2—22no—
405C2—44no—
406C2#1172314yesdirect product factor
407C2C22×#32811yescharacteristic
408C2C2×#212311yescharacteristic
409C2C2×#213411yescharacteristic
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. #11723C2direct (N × H)
5. C2×#1353C2semidirect
29. #1353C22semidirect
406. C2#11723direct (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
#117232
C2×#13532