Finite Group Databasebeta

← C2×#7765 #165696

Subgroups — C2×#7765 #165696

Order 512.

1175 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2727C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#2043C211yesmaximal, characteristic, complemented
4C2×#2023C211yesmaximal, characteristic
5#7765C218yesmaximal, direct product factor
6C2×#1271C211yesmaximal, characteristic, complemented
7C2×#1271C211yesmaximal, characteristic, complemented
8C2×#1270C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×#126—21no—
11C22×#107—21no—
12C22×#103—21no—
13C22×#103—21no—
14C22×#102—21no—
15#2727—22no—
16#2727—22no—
17#2727—22no—
18#2727C2214yes—
19C2×#399—21no—
20C2×#396—21no—
21C2×#394—21no—
22C2×#390C2211yescharacteristic
23C2×#389—21no—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388—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×D4D411yescharacteristic
43C23×C8—21no—
44C23×C8—21no—
45C23×C8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—44no—
50C2×C4⋊Q8C2312yes—
51C2×#126—24no—
52C2×#126—24no—
53C2×#126—44no—
54C2×#126—44no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74C2×#107—24no—
75C2×#107—44no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—44no—
79C2×#103—44no—
80C2×#102—24no—
81C2×#102—44no—
82#399—24no—
83#399—24no—
84#396—24no—
85#396—24no—
86#394—24no—
87#394—24no—
88#390—21no—
89#390—21no—
90#390D412yes—
91#390C2312yes—
92#389—24no—
93#389—24no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389D412yes—
99#389D412yes—
100#389C2312yes—
101#389C2312yes—
102#388—24no—
103#388—24no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107#340—24no—
108#340—24no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336C2312yes—
113#336C2312yes—
114#336C2312yes—
115C25—21no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8—41no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—42no—
128C22×D4—42no—
129C22×D4C2×D412yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C22×C8—24no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8—42no—
149C22×C8—42no—
150C22×C8—42no—
151C4⋊Q8—44no—
152C4⋊Q8—44no—
153C4⋊Q8—44no—
154C4⋊Q8—44no—
155#126—48no—
156#126—416no—
157#126—48no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4D814yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4—82no—
211C2×C42—24no—
212C2×C42—42no—
213C2×C42—44no—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2411yescharacteristic
218#107—48no—
219#103—48no—
220#103—48no—
221#102—48no—
222#95—24no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227C24—22no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24#12011yescharacteristic
236C2×Q8—24no—
237C2×Q8—42no—
238C2×Q8—44no—
239C2×Q8—48no—
240C2×Q8—48no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—84no—
246C2×Q8—84no—
247C2×Q8—82no—
248C2×Q8—82no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—48no—
254C2×D4—42no—
255C2×D4—81no—
256C2×D4—81no—
257C2×D4—84no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4C2×D812yes—
292C22×C4#13512yes—
293C22×C4#13512yes—
294C22×C4AGL(1,ℤ/8)12yes—
295C22×C4#13712yes—
296C22×C4#13712yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C2×C8—24no—
301C2×C8—42no—
302C2×C8—48no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—84no—
311C2×C8—84no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—48no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C4⋊C4—84no—
337C4⋊C4C2×D814yes—
338C22⋊C4—48no—
339C22⋊C4—48no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—84no—
347C22⋊C4—84no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—48no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—84no—
361C23—84no—
362C23—84no—
363C23—84no—
364C23—84no—
365C23—84no—
366C23—84no—
367C23#44712yes—
368C23#45212yes—
369C23#45212yes—
370C23#45412yes—
371C23#45512yes—
372C23#45612yes—
373C23B(4,𝔽2)12yes—
374C23C2×#12011yescharacteristic
375Q8—44no—
376Q8—44no—
377Q8—82no—
378Q8—82no—
379Q8—88no—
380D4—44no—
381D4—82no—
382D4—84no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—48no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—84no—
419C2×C4—88no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4#44712yes—
426C2×C4#45212yes—
427C2×C4#45212yes—
428C2×C4C22×D811yescharacteristic
429C2×C4C2×#13511yescharacteristic
430C2×C4C2×#13511yescharacteristic
431C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
432C2×C4C2×#13711yescharacteristic
433C2×C4C2×#13711yescharacteristic
434C8—44no—
435C8—82no—
436C8—82no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—168no—
445C22#159814yes—
446C22#160314yes—
447C22#160614yes—
448C22#163814yes—
449C22#164114yes—
450C22#164714yes—
451C22#166414yes—
452C22C2×#44711yescharacteristic
453C22C2×#45211yescharacteristic
454C22C2×#45211yescharacteristic
455C22C2×#45511yescharacteristic
456C22C2×#45411yescharacteristic
457C22C2×#45611yescharacteristic
458C22C2×B(4,𝔽2)11yescharacteristic
459C4—21no—
460C4—21no—
461C4—22no—
462C4—21no—
463C4—21no—
464C4—22no—
465C4—21no—
466C4—21no—
467C4—22no—
468C4—44no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4—84no—
473C4—82no—
474C2—162no—
475C2#776518yesdirect product factor
476C2C2×#160311yescharacteristic
477C2C2×#159811yescharacteristic
478C2C2×#160611yescharacteristic
479C2C2×#163811yescharacteristic
480C2C2×#164711yescharacteristic
481C2C2×#164111yescharacteristic
482C2C2×#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. #7765C2direct (N × H)
6. C2×#1271C2semidirect
7. C2×#1271C2semidirect
28. #2057C22semidirect
30. #2043C22semidirect
38. #1271C22semidirect
39. #1271C22semidirect
475. C2#7765direct (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×#20572
C2×#20432
C2×#20232
#77652
C2×#12712
C2×#12702