Finite Group Databasebeta

← C2×#13224 #495170

Subgroups — C2×#13224 #495170

Order 512.

1496 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C22×#354C214yesmaximal
3C2×#1799C211yesmaximal, characteristic
4C2×C4×#99C211yesmaximal, characteristic
5#13224C218yesmaximal, direct product factor
6C22×C4⋊Q8C411yescharacteristic, complemented
7C22×#124C2212yes—
8C22×C4×D4C2211yescharacteristic
9C22×#99C2212yes—
10C22×#99C2212yes—
11#2547—22no—
12#2547—22no—
13#2547C414yescomplemented
14#2547C2214yes—
15C2×#397—21no—
16C2×#397—21no—
17C2×#391C412yescomplemented
18C2×#380—21no—
19C2×#380—21no—
20C2×#380—21no—
21C2×C4×C4⋊C4C411yescharacteristic, complemented
22C2×C4×C22⋊C4—21no—
23C2×#354—216no—
24C2×#354C22116yes—
25#1799—22no—
26#1799C2214yes—
27C4×#99—22no—
28C4×#99C2214yes—
29C2×#342—21no—
30C2×#342—21no—
31C23×Q8C2×C412yes—
32C23×D4C2311yescharacteristic
33C24×C4D411yescharacteristic
34C24×C4D411yescharacteristic
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—42no—
37C2×C4⋊Q8—41no—
38C2×C4⋊Q8—41no—
39C2×C4⋊Q8C2×C414yescomplemented
40C2×#124—22no—
41C2×#124—22no—
42C2×#124—28no—
43C2×#124—28no—
44C2×#124C2×C414yes—
45C2×#124C2314yes—
46C2×C4×D4—41no—
47C2×C4×D4—41no—
48C2×C4×D4—42no—
49C2×C4×D4C2×C412yes—
50C2×C4×D4D412yes—
51C2×C4×D4D412yes—
52C2×C4×D4C2312yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—41no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42—41no—
65C22×C42C2×C412yes—
66C22×C42C2×C411yescharacteristic
67C2×#99—22no—
68C2×#99—24no—
69C2×#99—28no—
70C2×#99—22no—
71C2×#99D418yes—
72C2×#99C2314yes—
73C2×#99C2314yes—
74#397—24no—
75#397—24no—
76#397—24no—
77#397—24no—
78#391—24no—
79#391C2×C418yescomplemented
80#380—24no—
81#380—24no—
82#380—24no—
83#380—24no—
84#380—24no—
85#380—42no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2×C414yescomplemented
88C4×C22⋊C4—24no—
89C4×C22⋊C4—42no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—41no—
94C2×#95—41no—
95#354—232no—
96#342—24no—
97#342—24no—
98C25C2×D411yescharacteristic
99C25C2×D411yescharacteristic
100C22×Q8—22no—
101C22×Q8—28no—
102C22×Q8—42no—
103C22×Q8—42no—
104C22×Q8—44no—
105C22×Q8C22×C414yes—
106C22×D4—41no—
107C22×D4—41no—
108C22×D4—42no—
109C22×D4C22×C411yescharacteristic
110C22×D4C2×D411yescharacteristic
111C22×D4C2×D411yescharacteristic
112C22×D4C2×D414yes—
113C22×D4#4114yes—
114C22×D4C2411yescharacteristic
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C4⋊Q8—44no—
137C4⋊Q8—44no—
138C4⋊Q8—44no—
139C4⋊Q8—44no—
140#124—216no—
141#124—28no—
142#124—28no—
143#124—48no—
144#124—44no—
145#124—44no—
146C4×D4—42no—
147C4×D4—42no—
148C4×D4—42no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—48no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C2×D412yes—
191C2×C22⋊C4C2×D412yes—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—28no—
196C2×C42—22no—
197C2×C42—44no—
198C2×C42—84no—
199C2×C42C22⋊C412yes—
200C2×C42C22×C414yes—
201C2×C42C22×C412yes—
202#99—28no—
203#99—24no—
204#99—28no—
205#99—44no—
206#99C2×D418yes—
207#95—24no—
208#95—28no—
209#95—28no—
210#95—44no—
211#95—44no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—21no—
216C24—22no—
217C24—22no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24C2×C22⋊C411yescharacteristic
223C24C2×C22⋊C411yescharacteristic
224C24C2×C22⋊C411yescharacteristic
225C24#12114yes—
226C24#12412yes—
227C24#12712yes—
228C24C22×D411yescharacteristic
229C24C22×D411yescharacteristic
230C2×Q8—28no—
231C2×Q8—44no—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—88no—
237C2×Q8—88no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—41no—
250C2×D4C4×D414yes—
251C2×D4#12114yes—
252C2×D4#12412yes—
253C2×D4#12712yes—
254C2×D4C22×D412yes—
255C2×D4C2×#4112yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—84no—
296C22×C4—84no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4#9912yes—
305C22×C4#9911yescharacteristic
306C22×C4#9911yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C4×D414yes—
313C22×C4C23×C411yescharacteristic
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—88no—
321C22⋊C4—28no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—48no—
328C22⋊C4—48no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—48no—
332C22⋊C4—48no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—84no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—28no—
341C42—44no—
342C42—88no—
343C23—22no—
344C23—22no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—24no—
350C23—22no—
351C23—22no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—24no—
357C23—22no—
358C23—22no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—48no—
366C23—44no—
367C23—44no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—84no—
379C23—84no—
380C23#39012yes—
381C23#39012yes—
382C23#39012yes—
383C23C2×#9911yescharacteristic
384C23C2×#9911yescharacteristic
385C23#41012yes—
386C23C22×C22⋊C411yescharacteristic
387C23C2×#12112yes—
388C23C2×#12411yescharacteristic
389C23C2×#12711yescharacteristic
390Q8—88no—
391Q8—88no—
392D4—44no—
393D4—44no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—42no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—88no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—816no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—168no—
444C2×C4#35414yescomplemented
445C2×C4#39012yes—
446C2×C4C2×#9912yes—
447C2×C4C2×#9912yes—
448C2×C4C2×C4×D412yes—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—24no—
454C22—24no—
455C22—22no—
456C22—21no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—42no—
464C22—42no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—48no—
474C22—44no—
475C22—44no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—84no—
481C22—82no—
482C22—82no—
483C22#196412yes—
484C22#196412yes—
485C22C2×#35412yes—
486C22#212912yes—
487C22C2×#39011yescharacteristic
488C22C22×#9911yescharacteristic
489C22C2×#41011yescharacteristic
490C4—22no—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—84no—
497C4—88no—
498C4—84no—
499C4—164no—
500C4C2×#35414yescomplemented
501C2—21no—
502C2—21no—
503C2—22no—
504C2—42no—
505C2—41no—
506C2—41no—
507C2—42no—
508C2—42no—
509C2#1322414yesdirect product factor
510C2C2×#196411yescharacteristic
511C2C22×#35411yescharacteristic
512C2C2×#212911yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13224C2direct (N × H)
6. C22×C4⋊Q8C4semidirect
13. #2547C4semidirect
17. C2×#391C4semidirect
21. C2×C4×C4⋊C4C4semidirect
39. C2×C4⋊Q8C2×C4semidirect
79. #391C2×C4semidirect
87. C4×C4⋊C4C2×C4semidirect
444. C2×C4#354semidirect
500. C4C2×#354semidirect
509. C2#13224direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25472
C22×#3542
C2×#17992
C2×C4×#992
#132242