Finite Group Databasebeta

← C2×#7764 #165692

Subgroups — C2×#7764 #165692

Order 512.

1381 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#2043C211yesmaximal, characteristic, complemented
4C2×#2023C211yesmaximal, characteristic
5#7764C218yesmaximal, direct product factor
6C2×#1271C211yesmaximal, characteristic, complemented
7C2×#1271C211yesmaximal, characteristic, complemented
8C2×#1270C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C22⋊Q8—21no—
11C22×C22⋊Q8—21no—
12C22×#107—21no—
13C22×#103—21no—
14C22×#103—21no—
15C22×#102—21no—
16#2683—22no—
17#2683—22no—
18#2683—22no—
19#2683C2214yes—
20C2×#396—21no—
21C2×#396—21no—
22C2×#395—21no—
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#386—21no—
27#2057—22no—
28#2057C2214yescomplemented
29#2043—22no—
30#2043C2214yescomplemented
31#2023—22no—
32#2023C2214yes—
33C2×#340—21no—
34C2×#340—21no—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1271C2214yescomplemented
39#1271C2214yescomplemented
40#1270C2214yes—
41C23×Q8—21no—
42C23×Q8—21no—
43C23×D4D411yescharacteristic
44C24×C4—21no—
45C23×C8—21no—
46C23×C8—21no—
47C23×C8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—44no—
52C2×C4⋊Q8C2312yes—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—44no—
58C2×C22⋊Q8—44no—
59C2×C22⋊Q8—44no—
60C2×C22⋊Q8—44no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42C2311yescharacteristic
78C2×#107—24no—
79C2×#107—44no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—44no—
83C2×#103—44no—
84C2×#102—24no—
85C2×#102—44no—
86#396—24no—
87#396—24no—
88#396—24no—
89#396—24no—
90#395—24no—
91#395—24no—
92#390—21no—
93#390—21no—
94#390D412yes—
95#390C2312yes—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389D412yes—
101#389D412yes—
102#389C2312yes—
103#389C2312yes—
104#386—24no—
105#386—24no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109#340—24no—
110#340—24no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336C2312yes—
115#336C2312yes—
116#336C2312yes—
117C25—21no—
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8—22no—
121C22×Q8—44no—
122C22×Q8—44no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—41no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—82no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4Q1614yes—
135C22×D4C2×D412yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—24no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—41no—
145C23×C4—82no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C22×C8—24no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—42no—
154C22×C8—42no—
155C22×C8—42no—
156C22×C8—42no—
157C22×C8—42no—
158C22×C8—42no—
159C22×C8—42no—
160C22×C8—42no—
161C4⋊Q8—44no—
162C4⋊Q8—44no—
163C4⋊Q8—44no—
164C4⋊Q8—44no—
165C22⋊Q8—48no—
166C22⋊Q8—48no—
167C22⋊Q8—416no—
168C22⋊Q8—416no—
169C22⋊Q8—48no—
170C22⋊Q8—48no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4D814yes—
202C2×C4⋊C4SD1614yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—82no—
222C2×C42—42no—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2411yescharacteristic
227#107—48no—
228#103—48no—
229#103—48no—
230#102—48no—
231#95—24no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236C24—22no—
237C24—24no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24#12011yescharacteristic
245C2×Q8—24no—
246C2×Q8—48no—
247C2×Q8—48no—
248C2×Q8—42no—
249C2×Q8—44no—
250C2×Q8—48no—
251C2×Q8—48no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—84no—
257C2×Q8—84no—
258C2×Q8—82no—
259C2×Q8—84no—
260C2×Q8—816no—
261C2×Q8—82no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—42no—
265C2×D4—48no—
266C2×D4—42no—
267C2×D4—84no—
268C2×D4—81no—
269C2×D4—81no—
270C2×D4C2×Q1614yes—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—84no—
302C22×C4—816no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4C2×D812yes—
309C22×C4C2×SD1612yes—
310C22×C4C2×Q1612yes—
311C22×C4AGL(1,ℤ/8)12yes—
312C22×C4AGL(1,ℤ/8)12yes—
313C22×C4#13712yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C2×C8—24no—
318C2×C8—42no—
319C2×C8—48no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—84no—
328C2×C8—84no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—48no—
339C4⋊C4—48no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4C2×D814yes—
354C4⋊C4C2×SD1614yes—
355C22⋊C4—48no—
356C22⋊C4—48no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—84no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C23—24no—
369C23—48no—
370C23—48no—
371C23—48no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—84no—
379C23—84no—
380C23—84no—
381C23—84no—
382C23#44712yes—
383C23#45012yes—
384C23#45112yes—
385C23#45412yes—
386C23#45512yes—
387C23#45612yes—
388C23B(4,𝔽2)12yes—
389C23C2×#12011yescharacteristic
390Q8—44no—
391Q8—44no—
392Q8—82no—
393Q8—82no—
394Q8—88no—
395Q8—816no—
396D4—44no—
397D4—82no—
398D4—84no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—816no—
436C2×C4—84no—
437C2×C4—88no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4#44712yes—
443C2×C4#45012yes—
444C2×C4#45112yes—
445C2×C4C22×D811yescharacteristic
446C2×C4C22×SD1611yescharacteristic
447C2×C4C22×Q1611yescharacteristic
448C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
449C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
450C2×C4C2×#13711yescharacteristic
451C8—44no—
452C8—82no—
453C8—82no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—82no—
458C22—82no—
459C22—816no—
460C22—82no—
461C22—82no—
462C22#159814yes—
463C22#160214yes—
464C22#160414yes—
465C22#163714yes—
466C22#163914yes—
467C22#164914yes—
468C22#166414yes—
469C22C2×#44711yescharacteristic
470C22C2×#45011yescharacteristic
471C22C2×#45111yescharacteristic
472C22C2×#45511yescharacteristic
473C22C2×#45411yescharacteristic
474C22C2×#45611yescharacteristic
475C22C2×B(4,𝔽2)11yescharacteristic
476C4—21no—
477C4—21no—
478C4—22no—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—21no—
483C4—21no—
484C4—22no—
485C4—44no—
486C4—44no—
487C4—82no—
488C4—82no—
489C4—84no—
490C4—82no—
491C2—84no—
492C2#776418yesdirect product factor
493C2C2×#160211yescharacteristic
494C2C2×#159811yescharacteristic
495C2C2×#160411yescharacteristic
496C2C2×#163911yescharacteristic
497C2C2×#163711yescharacteristic
498C2C2×#164911yescharacteristic
499C2C2×#166411yescharacteristic
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
2. C2×#2057C2semidirect
3. C2×#2043C2semidirect
5. #7764C2direct (N × H)
6. C2×#1271C2semidirect
7. C2×#1271C2semidirect
28. #2057C22semidirect
30. #2043C22semidirect
38. #1271C22semidirect
39. #1271C22semidirect
492. C2#7764direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#20572
C2×#20432
C2×#20232
#77642
C2×#12712
C2×#12702