Finite Group Databasebeta

← C2×#7731 #161704

Subgroups — C2×#7731 #161704

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic
4#7731C218yesmaximal, direct product factor
5C2×#1271C212yesmaximal, complemented
6C2×#1271C212yesmaximal, complemented
7C22×#125C411yescharacteristic
8C23×C42C2211yescharacteristic
9C22×C4⋊C8—21no—
10C22×C22⋊C8—21no—
11C22×C22⋊C8—21no—
12#2851—22no—
13#2851—22no—
14#2851C414yes—
15#2851C2214yes—
16C2×#389C2211yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#387—21no—
19C2×#387—21no—
20C2×#385—21no—
21C2×#385—21no—
22C2×#382C411yescharacteristic
23C2×#380C411yescharacteristic
24C2×#380C411yescharacteristic
25#1817—22no—
26#1817C2214yes—
27#1797—22no—
28#1797C2214yes—
29C2×#336—21no—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1271C2218yescomplemented
33#1271C2218yescomplemented
34C24×C4C2×C411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C23×C8—22no—
38C23×C8—22no—
39C2×#125—22no—
40C2×#125—44no—
41C2×#125C2×C414yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C42—44no—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C412yes—
55C22×C42D411yescharacteristic
56C22×C42D412yes—
57C22×C42D412yes—
58C22×C42D411yescharacteristic
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61C2×C4⋊C8—24no—
62C2×C4⋊C8—44no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8—44no—
66C2×C22⋊C8—44no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389—21no—
71#389C2×C412yes—
72#389C2×C412yes—
73#389C2312yes—
74#389C2312yes—
75#387—24no—
76#387—24no—
77#387—24no—
78#387—24no—
79#385—24no—
80#385—24no—
81#385—24no—
82#385—24no—
83#382—22no—
84#382C2×C414yes—
85#380—22no—
86#380—22no—
87#380C2×C414yes—
88#380C2×C414yes—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95#336—22no—
96#336—22no—
97#336—24no—
98#336C2314yes—
99#336C2314yes—
100C25C22⋊C411yescharacteristic
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C412yes—
121C23×C4C22×C412yes—
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C22×C8—24no—
128C22×C8—24no—
129C22×C8—44no—
130C22×C8—44no—
131C22×C8—44no—
132C22×C8—44no—
133C22×C8—44no—
134C22×C8—44no—
135#125—48no—
136#125—48no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C412yes—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—44no—
164C2×C42—24no—
165C2×C42—24no—
166C2×C42—44no—
167C2×C42—44no—
168C2×C42—44no—
169C2×C42—44no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42C22⋊C412yes—
173C2×C42C22⋊C412yes—
174C2×C42C22⋊C412yes—
175C2×C42C22×C412yes—
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2411yescharacteristic
182C4⋊C8—48no—
183C22⋊C8—48no—
184C22⋊C8—48no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193C24—22no—
194C24—22no—
195C24—22no—
196C24—22no—
197C24—22no—
198C24—22no—
199C24#10114yes—
200C24C2×C22⋊C412yes—
201C24C2×C22⋊C411yescharacteristic
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—24no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—416no—
240C22×C4—416no—
241C22×C4—48no—
242C22×C4—48no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4#9914yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4#13114yes—
263C22×C4#13514yes—
264C22×C4AGL(1,ℤ/8)12yes—
265C22×C4#13712yes—
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C2×C8—44no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—88no—
276C2×C8—88no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—48no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—48no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C22⋊C4—48no—
293C22⋊C4—48no—
294C42—24no—
295C42—24no—
296C42—24no—
297C42—24no—
298C42—42no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42—42no—
303C42—41no—
304C23—28no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—28no—
312C23—28no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23#40812yes—
318C23C2×#9912yes—
319C23C2×#10114yes—
320C23C2×#10112yes—
321C23#41612yes—
322C23#41712yes—
323C23#42112yes—
324C23#42112yes—
325C23C22×C22⋊C411yescharacteristic
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—28no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—48no—
352C2×C4—48no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4#40812yes—
377C2×C4C2×#9914yes—
378C2×C4#41612yes—
379C2×C4#41712yes—
380C2×C4C2×#13112yes—
381C2×C4C2×#13512yes—
382C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C2×#13711yescharacteristic
384C8—84no—
385C8—84no—
386C22—24no—
387C22—24no—
388C22—24no—
389C22—42no—
390C22—416no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22#147214yes—
395C22#148214yes—
396C22#149514yes—
397C22#151314yes—
398C22#151614yes—
399C22#153414yes—
400C22#155214yes—
401C22C2×#40811yescharacteristic
402C22C22×#9911yescharacteristic
403C22C22×#10111yescharacteristic
404C22C2×#41611yescharacteristic
405C22C2×#41711yescharacteristic
406C22C2×#42111yescharacteristic
407C22C2×#42111yescharacteristic
408C4—21no—
409C4—21no—
410C4—22no—
411C4—22no—
412C4—24no—
413C4—22no—
414C4—22no—
415C4—22no—
416C4—42no—
417C4—42no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—82no—
422C2—44no—
423C2#773118yesdirect product factor
424C2C2×#147211yescharacteristic
425C2C2×#148211yescharacteristic
426C2C2×#149511yescharacteristic
427C2C2×#151311yescharacteristic
428C2C2×#151611yescharacteristic
429C2C2×#153411yescharacteristic
430C2C2×#155211yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7731C2direct (N × H)
5. C2×#1271C2semidirect
6. C2×#1271C2semidirect
32. #1271C22semidirect
33. #1271C22semidirect
423. C2#7731direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#18172
C2×#17972
#77312
C2×#12712