Finite Group Databasebeta

← C2×#7789 #165686

Subgroups — C2×#7789 #165686

Order 512.

1175 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2732C211yesmaximal, characteristic
2C2×#2093C211yesmaximal, characteristic
3C2×#2083C211yesmaximal, characteristic, complemented
4C2×#2072C211yesmaximal, characteristic, complemented
5#7789C218yesmaximal, direct product factor
6C2×#1272C211yesmaximal, characteristic, complemented
7C2×#1271C211yesmaximal, characteristic, complemented
8C2×#1270C211yesmaximal, characteristic
9C22×C22⋊Q8—21no—
10C22×#107—21no—
11C22×#107—21no—
12C22×#103—21no—
13C22×#102—21no—
14#2732—22no—
15#2732—22no—
16#2732—22no—
17#2732C2214yes—
18C2×#399—21no—
19C2×#398—21no—
20C2×#396—21no—
21C2×#391C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#386—21no—
25C2×#385—21no—
26C2×#384C2211yescharacteristic
27#2093—22no—
28#2093C2214yes—
29#2083—22no—
30#2083C2214yescomplemented
31#2072—22no—
32#2072C2214yescomplemented
33C2×#340—21no—
34C2×#340—21no—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1272C2214yescomplemented
39#1271C2214yescomplemented
40#1270C2214yes—
41C23×Q8D411yescharacteristic
42C23×D4D411yescharacteristic
43C24×C4—21no—
44C23×C8—21no—
45C23×C8—21no—
46C23×C8—21no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—44no—
50C2×C22⋊Q8—44no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42C2311yescharacteristic
68C2×#107—24no—
69C2×#107—24no—
70C2×#107—44no—
71C2×#107—44no—
72C2×#103—24no—
73C2×#103—44no—
74C2×#102—24no—
75C2×#102—44no—
76#399—24no—
77#399—24no—
78#398—24no—
79#398—24no—
80#396—24no—
81#396—24no—
82#391—21no—
83#391—21no—
84#391D412yes—
85#391C2312yes—
86#390—21no—
87#390—21no—
88#390D412yes—
89#390C2312yes—
90#389—21no—
91#389—21no—
92#389D412yes—
93#389C2312yes—
94#386—24no—
95#386—24no—
96#385—24no—
97#385—24no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384C2312yes—
102C2×#95—21no—
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×Q8—21no—
116C22×Q8—22no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8D814yes—
120C22×Q8C2×D412yes—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—42no—
124C22×D4—42no—
125C22×D4Q1614yes—
126C22×D4C2×D412yes—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—24no—
133C23×C4—44no—
134C23×C4—41no—
135C23×C4—82no—
136C23×C4—82no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—42no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8—42no—
149C22×C8—42no—
150C22×C8—42no—
151C22×C8—42no—
152C22⋊Q8—48no—
153C22⋊Q8—416no—
154C22⋊Q8—48no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C42—42no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2411yescharacteristic
209#107—48no—
210#107—48no—
211#103—48no—
212#102—48no—
213#95—24no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220C24—22no—
221C24—24no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24#12011yescharacteristic
229C2×Q8—22no—
230C2×Q8—24no—
231C2×Q8—48no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—84no—
235C2×Q8—81no—
236C2×Q8—81no—
237C2×Q8C2×D814yes—
238C2×D4—22no—
239C2×D4—24no—
240C2×D4—42no—
241C2×D4—48no—
242C2×D4—42no—
243C2×D4—84no—
244C2×D4—81no—
245C2×D4—81no—
246C2×D4C2×Q1614yes—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—48no—
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—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4—816no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4#12011yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4C2×D812yes—
285C22×C4C2×Q1612yes—
286C22×C4#13512yes—
287C22×C4AGL(1,ℤ/8)12yes—
288C22×C4AGL(1,ℤ/8)12yes—
289C22×C4#13712yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—48no—
296C2×C8—48no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—84no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—81no—
323C4⋊C4—81no—
324C4⋊C4—81no—
325C4⋊C4—82no—
326C4⋊C4—84no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C22⋊C4—48no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—84no—
337C22⋊C4—84no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C23—24no—
343C23—48no—
344C23—48no—
345C23—48no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—84no—
353C23—84no—
354C23—84no—
355C23—84no—
356C23#44712yes—
357C23#44912yes—
358C23#45112yes—
359C23#45312yes—
360C23#45512yes—
361C23#45612yes—
362C23#45812yes—
363C23C2×#12011yescharacteristic
364Q8—44no—
365Q8—82no—
366Q8—84no—
367D4—44no—
368D4—82no—
369D4—84no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—24no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—816no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—164no—
413C2×C4#44712yes—
414C2×C4#44912yes—
415C2×C4#45112yes—
416C2×C4C22×D811yescharacteristic
417C2×C4C22×Q1611yescharacteristic
418C2×C4C2×#13511yescharacteristic
419C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
420C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
421C2×C4C2×#13711yescharacteristic
422C8—44no—
423C8—44no—
424C8—82no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22—82no—
429C22—82no—
430C22—816no—
431C22—82no—
432C22—82no—
433C22#160714yes—
434C22#160914yes—
435C22#161414yes—
436C22#162114yes—
437C22#164414yes—
438C22#164914yes—
439C22#166114yes—
440C22C2×#44711yescharacteristic
441C22C2×#45111yescharacteristic
442C22C2×#44911yescharacteristic
443C22C2×#45311yescharacteristic
444C22C2×#45511yescharacteristic
445C22C2×#45611yescharacteristic
446C22C2×#45811yescharacteristic
447C4—21no—
448C4—21no—
449C4—22no—
450C4—21no—
451C4—21no—
452C4—22no—
453C4—21no—
454C4—21no—
455C4—22no—
456C4—44no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4—82no—
461C4—162no—
462C2—84no—
463C2#778918yesdirect product factor
464C2C2×#160711yescharacteristic
465C2C2×#161411yescharacteristic
466C2C2×#160911yescharacteristic
467C2C2×#164411yescharacteristic
468C2C2×#164911yescharacteristic
469C2C2×#162111yescharacteristic
470C2C2×#166111yescharacteristic
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×#2083C2semidirect
4. C2×#2072C2semidirect
5. #7789C2direct (N × H)
6. C2×#1272C2semidirect
7. C2×#1271C2semidirect
30. #2083C22semidirect
32. #2072C22semidirect
38. #1272C22semidirect
39. #1271C22semidirect
463. C2#7789direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#27322
C2×#20932
C2×#20832
C2×#20722
#77892
C2×#12722
C2×#12712
C2×#12702