Finite Group Databasebeta

← C2×#7613 #160679

Subgroups — C2×#7613 #160679

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic
3C2×#1749C211yesmaximal, characteristic
4#7613C218yesmaximal, direct product factor
5C2×#1271C212yesmaximal, complemented
6C2×#1271C212yesmaximal, complemented
7C22×#117C411yescharacteristic
8C22×#117C411yescharacteristic
9C23×C42C2211yescharacteristic
10C22×C4⋊C8—21no—
11C22×C22⋊C8—21no—
12C22×C22⋊C8—21no—
13C22×#97—21no—
14#2296—22no—
15#2296—22no—
16#2296C414yes—
17#2296C2214yes—
18C2×#389C411yescharacteristic
19C2×#389C411yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#385—21no—
23C2×#385—21no—
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4—21no—
26#1797—22no—
27#1797C2214yes—
28#1749—22no—
29#1749C2214yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1271C2218yescomplemented
33#1271C2218yescomplemented
34C24×C4C2×C411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C23×C8—22no—
38C23×C8—22no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—24no—
42C2×#117—24no—
43C2×#117—44no—
44C2×#117C2×C414yes—
45C2×#117C2×C414yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C42—21no—
57C22×C42—21no—
58C22×C42—44no—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C412yes—
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D412yes—
64C22×C42D412yes—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—24no—
68C2×C4⋊C8—44no—
69C2×C22⋊C8—28no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—44no—
73C2×#97—24no—
74C2×#97—44no—
75#389—22no—
76#389—22no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389C2×C414yes—
82#389C2×C414yes—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2312yes—
86#389C2312yes—
87#385—24no—
88#385—24no—
89#385—24no—
90#385—24no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—24no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99#336—22no—
100#336—22no—
101#336C2314yes—
102#336C2314yes—
103C25C22⋊C411yescharacteristic
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C412yes—
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22⋊C411yescharacteristic
124C23×C4C22⋊C412yes—
125C23×C4C22×C412yes—
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C22×C8—28no—
132C22×C8—24no—
133C22×C8—24no—
134C22×C8—44no—
135C22×C8—44no—
136C22×C8—44no—
137C22×C8—44no—
138C22×C8—44no—
139#117—216no—
140#117—28no—
141#117—28no—
142#117—48no—
143#117—48no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4M1614yes—
164C2×C4⋊C4M1614yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—44no—
173C2×C42—28no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—44no—
179C2×C42—44no—
180C2×C42—44no—
181C2×C42—44no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—44no—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22⋊C412yes—
188C2×C42C22×C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2411yescharacteristic
195C4⋊C8—48no—
196C22⋊C8—216no—
197C22⋊C8—48no—
198#97—48no—
199#95—24no—
200#95—22no—
201#95—22no—
202#95—24no—
203#95—22no—
204#95—22no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—22no—
211C24#10114yes—
212C24C2×C22⋊C412yes—
213C24C2×C22⋊C411yescharacteristic
214C22×C4—216no—
215C22×C4—216no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—416no—
256C22×C4—48no—
257C22×C4—48no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4#9914yes—
269C22×C4#10414yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×M1612yes—
277C22×C4C2×M1612yes—
278C22×C4#13514yes—
279C22×C4AGL(1,ℤ/8)12yes—
280C22×C4#13712yes—
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C2×C8—28no—
285C2×C8—44no—
286C2×C8—416no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—88no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—48no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—48no—
307C4⋊C4C2×M1614yes—
308C4⋊C4C2×M1614yes—
309C22⋊C4—216no—
310C22⋊C4—48no—
311C42—216no—
312C42—24no—
313C42—24no—
314C42—24no—
315C42—24no—
316C42—42no—
317C42—41no—
318C42—42no—
319C42—41no—
320C42—41no—
321C42—41no—
322C42—48no—
323C23—28no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—28no—
331C23—28no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23#40712yes—
337C23C2×#9912yes—
338C23C2×#10114yes—
339C23C2×#10112yes—
340C23#41612yes—
341C23#41712yes—
342C23C2×#10412yes—
343C23#42112yes—
344C23C22×C22⋊C411yescharacteristic
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—216no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—28no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—48no—
377C2×C4—48no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—48no—
386C2×C4—48no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4#40712yes—
397C2×C4C2×#9914yes—
398C2×C4#41612yes—
399C2×C4#41712yes—
400C2×C4C2×#10414yes—
401C2×C4C22×M1611yescharacteristic
402C2×C4C22×M1611yescharacteristic
403C2×C4C2×#13512yes—
404C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
405C2×C4C2×#13711yescharacteristic
406C8—48no—
407C8—84no—
408C22—24no—
409C22—24no—
410C22—24no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—416no—
416C22#146314yes—
417C22#148814yes—
418C22#148914yes—
419C22#150114yes—
420C22#151314yes—
421C22#153114yes—
422C22#155214yes—
423C22C2×#40711yescharacteristic
424C22C22×#9911yescharacteristic
425C22C22×#10111yescharacteristic
426C22C2×#41611yescharacteristic
427C22C2×#41711yescharacteristic
428C22C22×#10411yescharacteristic
429C22C2×#42111yescharacteristic
430C4—22no—
431C4—22no—
432C4—24no—
433C4—21no—
434C4—21no—
435C4—22no—
436C4—21no—
437C4—21no—
438C4—22no—
439C4—24no—
440C4—44no—
441C4—44no—
442C4—42no—
443C4—82no—
444C4—82no—
445C2—44no—
446C2#761318yesdirect product factor
447C2C2×#146311yescharacteristic
448C2C2×#148811yescharacteristic
449C2C2×#148911yescharacteristic
450C2C2×#150111yescharacteristic
451C2C2×#151311yescharacteristic
452C2C2×#153111yescharacteristic
453C2C2×#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. #7613C2direct (N × H)
5. C2×#1271C2semidirect
6. C2×#1271C2semidirect
32. #1271C22semidirect
33. #1271C22semidirect
446. C2#7613direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#22962
C2×#17972
C2×#17492
#76132
C2×#12712