Finite Group Databasebeta

← C2×#7793 #165704

Subgroups — C2×#7793 #165704

Order 512.

1083 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2727C211yesmaximal, characteristic
2C2×#2093C211yesmaximal, characteristic
3C2×#2084C211yesmaximal, characteristic, complemented
4C2×#2083C211yesmaximal, characteristic, complemented
5#7793C218yesmaximal, direct product factor
6C2×#1271C211yesmaximal, characteristic, complemented
7C2×#1271C211yesmaximal, characteristic, complemented
8C2×#1270C211yesmaximal, characteristic
9C22×#126—21no—
10C22×#107—21no—
11C22×#107—21no—
12C22×#106—21no—
13C22×#102—21no—
14#2727—22no—
15#2727—22no—
16#2727—22no—
17#2727C2214yes—
18C2×#399—21no—
19C2×#396—21no—
20C2×#394—21no—
21C2×#390C2211yescharacteristic
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388—21no—
26C2×#384C2211yescharacteristic
27#2093—22no—
28#2093C2214yes—
29#2084—22no—
30#2084C2214yescomplemented
31#2083—22no—
32#2083C2214yescomplemented
33C2×#340—21no—
34C2×#340—21no—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1271C2214yescomplemented
39#1271C2214yescomplemented
40#1270C2214yes—
41C23×D4D411yescharacteristic
42C23×C8—21no—
43C23×C8—21no—
44C23×C8—21no—
45C2×#126—24no—
46C2×#126—24no—
47C2×#126—44no—
48C2×#126—44no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C2×#107—24no—
70C2×#107—24no—
71C2×#107—44no—
72C2×#107—44no—
73C2×#106—24no—
74C2×#106—44no—
75C2×#102—24no—
76C2×#102—44no—
77#399—24no—
78#399—24no—
79#396—24no—
80#396—24no—
81#394—24no—
82#394—24no—
83#390—21no—
84#390—21no—
85#390D412yes—
86#390C2312yes—
87#389—24no—
88#389—24no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389D412yes—
94#389D412yes—
95#389C2312yes—
96#389C2312yes—
97#388—24no—
98#388—24no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384C2312yes—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106#340—24no—
107#340—24no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336C2312yes—
112#336C2312yes—
113#336C2312yes—
114C25—21no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—42no—
119C22×D4—42no—
120C22×D4C2×D412yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C22×C8—24no—
131C22×C8—24no—
132C22×C8—24no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—42no—
142#126—48no—
143#126—416no—
144#126—48no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4D814yes—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4—82no—
202C2×C42—24no—
203C2×C42—42no—
204C2×C42—44no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2411yescharacteristic
209#107—48no—
210#107—48no—
211#106—48no—
212#102—48no—
213#95—24no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218C24—22no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24#12011yescharacteristic
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—48no—
232C2×D4—42no—
233C2×D4—81no—
234C2×D4—81no—
235C2×D4—84no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4C2×D812yes—
270C22×C4#13512yes—
271C22×C4#13512yes—
272C22×C4AGL(1,ℤ/8)12yes—
273C22×C4AGL(1,ℤ/8)12yes—
274C22×C4#13712yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—48no—
282C2×C8—48no—
283C2×C8—48no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—48no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—81no—
312C4⋊C4—81no—
313C4⋊C4—81no—
314C4⋊C4—82no—
315C4⋊C4—84no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4C2×D814yes—
319C22⋊C4—48no—
320C22⋊C4—48no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—84no—
328C22⋊C4—84no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—48no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—84no—
342C23—84no—
343C23—84no—
344C23—84no—
345C23—84no—
346C23—84no—
347C23—84no—
348C23#44712yes—
349C23#44912yes—
350C23#45212yes—
351C23#45312yes—
352C23#45612yes—
353C23#45612yes—
354C23#45812yes—
355C23C2×#12011yescharacteristic
356D4—44no—
357D4—82no—
358D4—84no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—48no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—164no—
401C2×C4#44712yes—
402C2×C4#44912yes—
403C2×C4#45212yes—
404C2×C4C22×D811yescharacteristic
405C2×C4C2×#13511yescharacteristic
406C2×C4C2×#13511yescharacteristic
407C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
408C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
409C2×C4C2×#13711yescharacteristic
410C8—44no—
411C8—44no—
412C8—44no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—168no—
421C22#160914yes—
422C22#161014yes—
423C22#161814yes—
424C22#162114yes—
425C22#164714yes—
426C22#165214yes—
427C22#166014yes—
428C22C2×#44711yescharacteristic
429C22C2×#44911yescharacteristic
430C22C2×#45211yescharacteristic
431C22C2×#45311yescharacteristic
432C22C2×#45611yescharacteristic
433C22C2×#45611yescharacteristic
434C22C2×#45811yescharacteristic
435C4—21no—
436C4—21no—
437C4—22no—
438C4—21no—
439C4—21no—
440C4—22no—
441C4—21no—
442C4—21no—
443C4—22no—
444C4—44no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4—82no—
449C4—162no—
450C2—162no—
451C2#779318yesdirect product factor
452C2C2×#160911yescharacteristic
453C2C2×#161011yescharacteristic
454C2C2×#161811yescharacteristic
455C2C2×#164711yescharacteristic
456C2C2×#165211yescharacteristic
457C2C2×#162111yescharacteristic
458C2C2×#166011yescharacteristic
subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2084C2semidirect
4. C2×#2083C2semidirect
5. #7793C2direct (N × H)
6. C2×#1271C2semidirect
7. C2×#1271C2semidirect
30. #2084C22semidirect
32. #2083C22semidirect
38. #1271C22semidirect
39. #1271C22semidirect
451. C2#7793direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#27272
C2×#20932
C2×#20842
C2×#20832
#77932
C2×#12712
C2×#12702