Finite Group Databasebeta

← C2×#7787 #165685

Subgroups — C2×#7787 #165685

Order 512.

1131 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2732C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#2072C211yesmaximal, characteristic, complemented
4C2×#2071C211yesmaximal, characteristic
5#7787C218yesmaximal, direct product factor
6C2×#1272C211yesmaximal, characteristic, complemented
7C2×#1271C211yesmaximal, characteristic, complemented
8C2×#1270C211yesmaximal, characteristic
9C22×C22⋊Q8—21no—
10C22×#103—21no—
11C22×#102—21no—
12#2732—22no—
13#2732—22no—
14#2732—22no—
15#2732C2214yes—
16C2×#399—21no—
17C2×#398—21no—
18C2×#396—21no—
19C2×#391C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#386—21no—
23C2×#385—21no—
24C2×#384C2211yescharacteristic
25#2099—22no—
26#2099C2214yescomplemented
27#2072—22no—
28#2072C2214yescomplemented
29#2071—22no—
30#2071C2214yes—
31C2×#340—21no—
32C2×#340—21no—
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×#103—24no—
69C2×#103—44no—
70C2×#102—24no—
71C2×#102—44no—
72#399—24no—
73#399—24no—
74#398—24no—
75#398—24no—
76#396—24no—
77#396—24no—
78#391—21no—
79#391—21no—
80#391D412yes—
81#391C2312yes—
82#390—21no—
83#390—21no—
84#390D412yes—
85#390C2312yes—
86#389—21no—
87#389—21no—
88#389D412yes—
89#389C2312yes—
90#386—24no—
91#386—24no—
92#385—24no—
93#385—24no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384C2312yes—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336C2312yes—
110#336C2312yes—
111#336C2312yes—
112C25—21no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8D814yes—
118C22×Q8C2×D412yes—
119C22×D4—21no—
120C22×D4—22no—
121C22×D4—42no—
122C22×D4—42no—
123C22×D4SD1614yes—
124C22×D4C2×D412yes—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—44no—
132C23×C4—41no—
133C23×C4—82no—
134C23×C4—82no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8—42no—
149C22×C8—42no—
150C22⋊Q8—48no—
151C22⋊Q8—416no—
152C22⋊Q8—48no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
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—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C42—42no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2411yescharacteristic
206#103—48no—
207#102—48no—
208#95—24no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215C24—22no—
216C24—24no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24#12011yescharacteristic
224C2×Q8—22no—
225C2×Q8—24no—
226C2×Q8—48no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—84no—
230C2×Q8—81no—
231C2×Q8—81no—
232C2×Q8C2×D814yes—
233C2×D4—22no—
234C2×D4—24no—
235C2×D4—48no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—84no—
239C2×D4—81no—
240C2×D4—81no—
241C2×D4C2×SD1614yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—48no—
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—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—84no—
271C22×C4—84no—
272C22×C4—816no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4#12011yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12011yescharacteristic
279C22×C4C2×D812yes—
280C22×C4C2×SD1612yes—
281C22×C4#13512yes—
282C22×C4AGL(1,ℤ/8)12yes—
283C22×C4#13712yes—
284C22×C4#13712yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—84no—
298C2×C8—84no—
299C2×C8—84no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—81no—
317C4⋊C4—81no—
318C4⋊C4—81no—
319C4⋊C4—82no—
320C4⋊C4—84no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C22⋊C4—48no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—84no—
331C22⋊C4—84no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—41no—
336C23—24no—
337C23—48no—
338C23—48no—
339C23—48no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—84no—
347C23—84no—
348C23—84no—
349C23—84no—
350C23#44712yes—
351C23#44812yes—
352C23#45212yes—
353C23#45412yes—
354C23#45512yes—
355C23#45512yes—
356C23#45812yes—
357C23C2×#12011yescharacteristic
358Q8—44no—
359Q8—82no—
360Q8—84no—
361D4—44no—
362D4—82no—
363D4—84no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—24no—
383C2×C4—48no—
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—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—816no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—164no—
407C2×C4#44712yes—
408C2×C4#44812yes—
409C2×C4#45212yes—
410C2×C4C22×D811yescharacteristic
411C2×C4C22×SD1611yescharacteristic
412C2×C4C2×#13511yescharacteristic
413C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
414C2×C4C2×#13711yescharacteristic
415C2×C4C2×#13711yescharacteristic
416C8—82no—
417C8—82no—
418C8—82no—
419C22—44no—
420C22—44no—
421C22—44no—
422C22—82no—
423C22—82no—
424C22—816no—
425C22—82no—
426C22—82no—
427C22#160714yes—
428C22#160814yes—
429C22#161514yes—
430C22#162714yes—
431C22#164114yes—
432C22#164314yes—
433C22#166214yes—
434C22C2×#44711yescharacteristic
435C22C2×#44811yescharacteristic
436C22C2×#45211yescharacteristic
437C22C2×#45511yescharacteristic
438C22C2×#45511yescharacteristic
439C22C2×#45411yescharacteristic
440C22C2×#45811yescharacteristic
441C4—21no—
442C4—21no—
443C4—22no—
444C4—21no—
445C4—21no—
446C4—22no—
447C4—21no—
448C4—21no—
449C4—22no—
450C4—44no—
451C4—82no—
452C4—82no—
453C4—82no—
454C4—82no—
455C4—162no—
456C2—84no—
457C2#778718yesdirect product factor
458C2C2×#160711yescharacteristic
459C2C2×#160811yescharacteristic
460C2C2×#161511yescharacteristic
461C2C2×#164311yescharacteristic
462C2C2×#162711yescharacteristic
463C2C2×#164111yescharacteristic
464C2C2×#166211yescharacteristic
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×#2099C2semidirect
3. C2×#2072C2semidirect
5. #7787C2direct (N × H)
6. C2×#1272C2semidirect
7. C2×#1271C2semidirect
26. #2099C22semidirect
28. #2072C22semidirect
38. #1272C22semidirect
39. #1271C22semidirect
457. C2#7787direct (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×#20992
C2×#20722
C2×#20712
#77872
C2×#12722
C2×#12712
C2×#12702