Finite Group Databasebeta

← C2×#17614 #6343449

Subgroups — C2×#17614 #6343449

Order 512.

3484 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C2×#2972C211yesmaximal, characteristic
3C2×#2349C211yesmaximal, characteristic
4#17614C2116yesmaximal, direct product factor
5C2×#1518C212yesmaximal, complemented
6C2×#1520C214yesmaximal, complemented
7C2×#1503C212yesmaximal, complemented
8C22×#328C214yesmaximal, complemented
9#3501—24no—
10#3501—24no—
11#3501C418yescomplemented
12#3501C2218yes—
13C2×#554—24no—
14C2×#554C414yescomplemented
15C2×D4×Q8C414yescomplemented
16C2×#544—21no—
17C2×#543—22no—
18C2×#533C2212yes—
19C22×C22⋊Q8C2214yes—
20C2×#518—21no—
21C22×C4×D4—21no—
22C22×C4×D4C2211yescharacteristic
23#2972—24no—
24#2972C2218yes—
25C2×C8×D4—24no—
26C2×#431C2212yes—
27C22×C22⋊C8C2214yes—
28#2349—24no—
29#2349C2218yes—
30C2×#389—21no—
31C2×#385—22no—
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C22×#95C2212yes—
36#1520C22132yescomplemented
37#1518C22116yescomplemented
38#1503C22116yescomplemented
39C2×#328C2214yes—
40C2×#328C2218yes—
41C2×#328C2214yes—
42C2×#328C22132yescomplemented
43C23×Q8C2×C414yescomplemented
44C23×D4D411yescharacteristic
45C24×C4—22no—
46C24×C4C2312yes—
47C23×C8—24no—
48#554—232no—
49#554—216no—
50#554—232no—
51#554C2×C4132yescomplemented
52D4×Q8—216no—
53D4×Q8C2×C4132yescomplemented
54#544—28no—
55#544—28no—
56#543—216no—
57#543—216no—
58#533—24no—
59#533—24no—
60#533C2×C418yescomplemented
61#533C2318yes—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8C2×C412yescomplemented
64C2×C4⋊Q8C2×C412yescomplemented
65C2×#125—24no—
66C2×#125—22no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8C2×C418yescomplemented
75C2×C22⋊Q8C2×C4116yescomplemented
76C2×C22⋊Q8C2314yes—
77C2×C22⋊Q8C23116yes—
78#518—28no—
79#518—28no—
80C2×C4×Q8—22no—
81C2×C4×Q8C2×C414yescomplemented
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4—28no—
85C2×C4×D4—48no—
86C2×C4×D4C2×C414yes—
87C2×C4×D4C2×C414yes—
88C2×C4×D4D414yes—
89C2×C4×D4D414yes—
90C2×C4×D4D414yes—
91C2×C4×D4C2314yes—
92C2×#117—22no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4C2×C414yescomplemented
98C22×C4⋊C4D411yescharacteristic
99C22×C22⋊C4—24no—
100C22×C22⋊C4—22no—
101C22×C22⋊C4D412yes—
102C22×C42—21no—
103C22×C42C2311yescharacteristic
104C8×D4—232no—
105#431—24no—
106#431C2318yes—
107C2×C4⋊C8—22no—
108C2×C22⋊C8—24no—
109C2×C22⋊C8—28no—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C23116yes—
112C2×C4×C8—22no—
113#389—28no—
114#385—216no—
115#382—22no—
116#382C2314yes—
117#380—24no—
118#380C2318yes—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2314yes—
121C2×#95—24no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95C2318yes—
125C2×#95C2311yescharacteristic
126C2×#95C2312yes—
127C2×#95C2311yescharacteristic
128#328C23116yes—
129#328C23132yes—
130#328C23116yes—
131C25C2×D412yes—
132C22×Q8—22no—
133C22×Q8—24no—
134C22×Q8—28no—
135C22×Q8—88no—
136C22×Q8C22×C414yes—
137C22×Q8C22×C4116yescomplemented
138C22×D4C22⋊C414yes—
139C22×D4C22⋊C414yes—
140C22×D4SD1618yes—
141C22×D4Q1618yes—
142C22×D4C2×D414yes—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—28no—
149C23×C4—48no—
150C23×C4—48no—
151C23×C4—84no—
152C23×C4—84no—
153C23×C4C22×C414yes—
154C23×C4C22×C414yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D414yes—
158C23×C4C2×D414yes—
159C23×C4C2×D414yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C23×C4C2414yes—
163C22×C8—24no—
164C22×C8—216no—
165C22×C8—88no—
166C4⋊Q8—232no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8C22×C418yescomplemented
170C4⋊Q8C22×C418yescomplemented
171#125—232no—
172#125—28no—
173#125—28no—
174C22⋊Q8—216no—
175C22⋊Q8—216no—
176C22⋊Q8—216no—
177C22⋊Q8—28no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—216no—
181C22⋊Q8—216no—
182C22⋊Q8—28no—
183C22⋊Q8—264no—
184C22⋊Q8—264no—
185C22⋊Q8C22×C418yes—
186C22⋊Q8C22×C4132yescomplemented
187C22⋊Q8C2418yes—
188C4×Q8—28no—
189C4×Q8—28no—
190C4×Q8—28no—
191C4×Q8C22×C4116yescomplemented
192C4×D4—232no—
193C4×D4—232no—
194C4×D4—432no—
195C4×D4C22⋊C4116yes—
196C4×D4C22⋊C4116yes—
197C4×D4C22×C4116yes—
198C4×D4C2×D4116yes—
199#117—28no—
200#117—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—216no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—216no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—84no—
217C2×C4⋊C4C22⋊C414yes—
218C2×C4⋊C4C22⋊C414yes—
219C2×C4⋊C4C22×C414yes—
220C2×C4⋊C4C22×C4116yescomplemented
221C2×C4⋊C4C2×D414yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—216no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—216no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—48no—
228C2×C22⋊C4C22⋊C418yes—
229C2×C22⋊C4C22⋊C418yes—
230C2×C22⋊C4C2×D418yes—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—44no—
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2412yes—
240C4⋊C8—28no—
241C22⋊C8—24no—
242C22⋊C8—216no—
243C22⋊C8C2418yes—
244C4×C8—28no—
245#95—21no—
246#95—22no—
247#95—28no—
248#95—24no—
249#95—21no—
250#95C2412yes—
251#95C2414yes—
252#95C2412yes—
253C24—44no—
254C24—44no—
255C24C2×C22⋊C414yes—
256C24C2×C22⋊C414yes—
257C24C2×SD1618yes—
258C24C2×Q1618yes—
259C24C22×D411yescharacteristic
260C24C22×D414yes—
261C2×Q8—24no—
262C2×Q8—28no—
263C2×Q8—216no—
264C2×Q8—48no—
265C2×Q8—84no—
266C2×Q8—88no—
267C2×Q8—816no—
268C2×Q8—816no—
269C2×Q8C23×C418yes—
270C2×D4#103132yes—
271C2×D4#103132yes—
272C2×D4C2×C22⋊C4116yes—
273C2×D4C2×SD16116yes—
274C2×D4C2×Q16116yes—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—216no—
280C22×C4—216no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—48no—
286C22×C4—42no—
287C22×C4—48no—
288C22×C4—416no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—416no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—88no—
303C22×C4—88no—
304C22×C4—88no—
305C22×C4—832no—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C414yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C414yes—
310C22×C4C2×C22⋊C418yes—
311C22×C4C2×C22⋊C418yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×SD1614yes—
315C22×C4C2×Q1614yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C23×C411yescharacteristic
318C22×C4C23×C418yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D412yes—
322C22×C4C22×D414yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D418yes—
325C22×C4C22×D412yes—
326C22×C4C2511yescharacteristic
327C2×C8—28no—
328C2×C8—48no—
329C2×C8—816no—
330C4⋊C4—28no—
331C4⋊C4—24no—
332C4⋊C4—232no—
333C4⋊C4—216no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—28no—
337C4⋊C4—216no—
338C4⋊C4—28no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—88no—
342C4⋊C4C2×C22⋊C4116yes—
343C4⋊C4C23×C418yes—
344C22⋊C4—28no—
345C22⋊C4—232no—
346C22⋊C4—28no—
347C22⋊C4—416no—
348C22⋊C4C2×C22⋊C4132yes—
349C42—216no—
350C42—48no—
351C42C2×C22⋊C414yes—
352C42C2×C22⋊C414yes—
353C42C23×C414yes—
354C42C22×D414yes—
355C23—22no—
356C23—28no—
357C23—22no—
358C23—28no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23—416no—
363C23—44no—
364C23—416no—
365C23C2×#103116yes—
366C23C2×#103116yes—
367C23C22×C22⋊C411yescharacteristic
368C23C22×C22⋊C418yes—
369C23C22×C22⋊C411yescharacteristic
370C23C22×SD1612yes—
371C23C22×SD1618yes—
372C23C22×Q1612yes—
373C23C22×Q1618yes—
374C23C23×D411yescharacteristic
375Q8—48no—
376Q8—48no—
377Q8—88no—
378Q8—816no—
379D4C2×#103164yes—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—28no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—28no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—48no—
395C2×C4—44no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—48no—
401C2×C4—48no—
402C2×C4—48no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—816no—
411C2×C4—816no—
412C2×C4—832no—
413C2×C4—84no—
414C2×C4—88no—
415C2×C4C2×#10318yes—
416C2×C4C2×#10318yes—
417C2×C4C22×C22⋊C411yescharacteristic
418C2×C4C22×C22⋊C411yescharacteristic
419C2×C4C22×C22⋊C414yes—
420C2×C4C22×C22⋊C418yes—
421C2×C4C22×C22⋊C414yes—
422C2×C4C22×SD1614yes—
423C2×C4C22×Q1614yes—
424C2×C4C24×C411yescharacteristic
425C2×C4C23×D411yescharacteristic
426C8—48no—
427C22—21no—
428C22—21no—
429C22—28no—
430C22—21no—
431C22—21no—
432C22—28no—
433C22—24no—
434C22—24no—
435C22—232no—
436C22—48no—
437C22—48no—
438C22—416no—
439C22#288312yes—
440C22C22×#10312yes—
441C22C22×#103116yes—
442C22C22×#10312yes—
443C22#290112yes—
444C22C23×C22⋊C411yescharacteristic
445C22C23×SD1611yescharacteristic
446C22C23×Q1611yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—28no—
452C4—24no—
453C4—44no—
454C4—48no—
455C4—48no—
456C4—82no—
457C4—84no—
458C4C22×#10318yes—
459C2—21no—
460C2—22no—
461C2—21no—
462C2—216no—
463C2#1761414yesdirect product factor
464C2C2×#288311yescharacteristic
465C2C23×#10311yescharacteristic
466C2C2×#290111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17614C2direct (N × H)
5. C2×#1518C2semidirect
6. C2×#1520C2semidirect
7. C2×#1503C2semidirect
8. C22×#328C2semidirect
11. #3501C4semidirect
14. C2×#554C4semidirect
15. C2×D4×Q8C4semidirect
36. #1520C22semidirect
37. #1518C22semidirect
38. #1503C22semidirect
42. C2×#328C22semidirect
43. C23×Q8C2×C4semidirect
51. #554C2×C4semidirect
53. D4×Q8C2×C4semidirect
60. #533C2×C4semidirect
63. C2×C4⋊Q8C2×C4semidirect
64. C2×C4⋊Q8C2×C4semidirect
74. C2×C22⋊Q8C2×C4semidirect
75. C2×C22⋊Q8C2×C4semidirect
81. C2×C4×Q8C2×C4semidirect
97. C22×C4⋊C4C2×C4semidirect
137. C22×Q8C22×C4semidirect
169. C4⋊Q8C22×C4semidirect
170. C4⋊Q8C22×C4semidirect
186. C22⋊Q8C22×C4semidirect
191. C4×Q8C22×C4semidirect
220. C2×C4⋊C4C22×C4semidirect
463. C2#17614direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#29722
C2×#23492
#176142
C2×#15182
C2×#15202
C2×#15032
C22×#3282