Finite Group Databasebeta

← C2×#7726 #161697

Subgroups — C2×#7726 #161697

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic
2C2×#1839C211yesmaximal, characteristic, complemented
3C2×#1839C211yesmaximal, characteristic, complemented
4#7726C218yesmaximal, direct product factor
5C2×#1271C212yesmaximal, complemented
6C2×#1270C212yesmaximal
7C22×#126—21no—
8#2860—22no—
9#2860—22no—
10#2860C414yes—
11#2860C2214yes—
12C2×#390C411yescharacteristic
13C2×#390C411yescharacteristic
14C2×#390C2211yescharacteristic
15C2×#389C411yescharacteristic
16C2×#389C411yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#388—21no—
19C2×#388—21no—
20C2×#388—21no—
21C2×C43C2211yescharacteristic
22#1839—22no—
23#1839—22no—
24#1839C2214yescomplemented
25#1839C2214yescomplemented
26C2×#336—21no—
27C2×#336—21no—
28C2×#336—21no—
29C2×#336—21no—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1271C2218yescomplemented
33#1270C2218yes—
34C23×D4C2×C411yescharacteristic
35C23×C8—22no—
36C23×C8—22no—
37C2×#126—24no—
38C2×#126—24no—
39C2×#126—48no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C22⋊C4—21no—
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59#390—22no—
60#390—22no—
61#390—21no—
62#390—21no—
63#390C2×C412yes—
64#390C2×C414yes—
65#390C2×C414yes—
66#390C2312yes—
67#389—22no—
68#389—22no—
69#389—21no—
70#389—21no—
71#389C2×C412yes—
72#389C2×C414yes—
73#389C2×C414yes—
74#389C2312yes—
75#388—24no—
76#388—24no—
77#388—24no—
78#388—24no—
79#388—24no—
80#388—24no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C43C2×C412yes—
85C43D412yes—
86C43D412yes—
87C43C2312yes—
88#336—22no—
89#336—22no—
90#336—24no—
91#336—24no—
92#336—24no—
93#336—24no—
94#336C2314yes—
95#336C2314yes—
96C25—21no—
97C22×D4—21no—
98C22×D4—24no—
99C22×D4—42no—
100C22×D4—42no—
101C22×D4C22×C412yes—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C22×C8—24no—
113C22×C8—24no—
114C22×C8—44no—
115C22×C8—44no—
116C22×C8—44no—
117C22×C8—44no—
118C22×C8—44no—
119C22×C8—44no—
120#126—416no—
121#126—416no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4C22×C412yes—
136C2×C4⋊C4C22×C412yes—
137C2×C4⋊C4C22×C412yes—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—44no—
147C2×C22⋊C4—44no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4—44no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—42no—
156C2×C42—42no—
157C2×C42—42no—
158C2×C42—42no—
159C2×C42C22⋊C412yes—
160C2×C42C22⋊C412yes—
161C2×C42C22⋊C412yes—
162C2×C42C22⋊C412yes—
163C2×C42C22⋊C412yes—
164C2×C42C22⋊C412yes—
165C2×C42C22×C412yes—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2411yescharacteristic
175#95—24no—
176#95—24no—
177#95—22no—
178#95—22no—
179C24—22no—
180C24—42no—
181C24—42no—
182C24—42no—
183C24—42no—
184C24—42no—
185C24—42no—
186C24—42no—
187C24C2×C22⋊C411yescharacteristic
188C2×D4—24no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—88no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4#13114yes—
233C22×C4#13514yes—
234C22×C4AGL(1,ℤ/8)14yes—
235C22×C4C23×C411yescharacteristic
236C22×C4C22×D411yescharacteristic
237C22×C4C22×D411yescharacteristic
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—88no—
245C2×C8—88no—
246C4⋊C4—24no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—48no—
252C4⋊C4—48no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C22⋊C4—48no—
257C22⋊C4—48no—
258C22⋊C4—48no—
259C22⋊C4—48no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C42—24no—
265C42—24no—
266C42—24no—
267C42—24no—
268C42—24no—
269C42—24no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42—41no—
274C42—42no—
275C42—41no—
276C42—41no—
277C42—42no—
278C42—42no—
279C42—41no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—84no—
288C23—84no—
289C23—84no—
290C23—84no—
291C23—84no—
292C23—84no—
293C23—84no—
294C23#40812yes—
295C23#41012yes—
296C23#41312yes—
297C23#41612yes—
298C23#41812yes—
299C23#42112yes—
300C23#42112yes—
301C23C22×C22⋊C411yescharacteristic
302D4—84no—
303D4—84no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—28no—
317C2×C4—28no—
318C2×C4—28no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4#40812yes—
348C2×C4#41612yes—
349C2×C4#41812yes—
350C2×C4C2×#13112yes—
351C2×C4C2×#13512yes—
352C2×C4C2×AGL(1,ℤ/8)12yes—
353C8—84no—
354C8—84no—
355C22—82no—
356C22—82no—
357C22—82no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22—168no—
363C22#147314yes—
364C22#148214yes—
365C22#149614yes—
366C22#151414yes—
367C22#152114yes—
368C22#153514yes—
369C22#155314yes—
370C22C2×#40811yescharacteristic
371C22C2×#41011yescharacteristic
372C22C2×#41311yescharacteristic
373C22C2×#41611yescharacteristic
374C22C2×#41811yescharacteristic
375C22C2×#42111yescharacteristic
376C22C2×#42111yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4—82no—
388C4—82no—
389C4—82no—
390C2—162no—
391C2#772618yesdirect product factor
392C2C2×#147311yescharacteristic
393C2C2×#148211yescharacteristic
394C2C2×#149611yescharacteristic
395C2C2×#151411yescharacteristic
396C2C2×#152111yescharacteristic
397C2C2×#153511yescharacteristic
398C2C2×#155311yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1839C2semidirect
3. C2×#1839C2semidirect
4. #7726C2direct (N × H)
5. C2×#1271C2semidirect
24. #1839C22semidirect
25. #1839C22semidirect
32. #1271C22semidirect
391. C2#7726direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28602
C2×#18392
#77262
C2×#12712
C2×#12702