Finite Group Databasebeta

← C2×#13228 #495183

Subgroups — C2×#13228 #495183

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2545C211yesmaximal, characteristic
2C22×#356C212yesmaximal, complemented
3C22×#355C212yesmaximal
4C2×#1801C211yesmaximal, characteristic, complemented
5C2×C4×#101C211yesmaximal, characteristic, complemented
6#13228C218yesmaximal, direct product factor
7C22×C4⋊Q8C2211yescharacteristic
8C22×#124C411yescharacteristic
9C22×#124C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#423—21no—
12C22×#101C2212yes—
13C22×#101C2212yes—
14C2×C4×M16—21no—
15#2545—22no—
16#2545—22no—
17#2545C414yes—
18#2545C2214yes—
19C2×#396—21no—
20C2×#396—21no—
21C2×#391C411yescharacteristic
22C2×#390C411yescharacteristic
23C2×#388—21no—
24C2×#388—21no—
25C2×C4×C4⋊C4C411yescharacteristic
26C2×#356—28no—
27C2×#356C2218yescomplemented
28C2×#355—28no—
29C2×#355C2218yes—
30#1801—22no—
31#1801C2214yescomplemented
32C4×#101—22no—
33C4×#101C2214yescomplemented
34C2×#343—21no—
35C2×#343—21no—
36C23×Q8C2×C411yescharacteristic
37C23×Q8C2311yescharacteristic
38C23×D4C2×C411yescharacteristic
39C22×M16—22no—
40C22×M16—22no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2312yes—
47C2×#124—21no—
48C2×#124—21no—
49C2×#124—24no—
50C2×#124—24no—
51C2×#124—22no—
52C2×#124—42no—
53C2×#124—41no—
54C2×#124—41no—
55C2×#124C2×C412yes—
56C2×#124C2×C414yes—
57C2×#124C2312yes—
58C2×C4×Q8—41no—
59C2×C4×Q8—41no—
60C2×C4×Q8—42no—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8D412yes—
63C2×C4×Q8D412yes—
64C2×C4×Q8C2312yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82#423—24no—
83#423—42no—
84C2×C4⋊C8—41no—
85C2×C4⋊C8—41no—
86C2×#101—22no—
87C2×#101—24no—
88C2×#101—24no—
89C2×#101—24no—
90C2×#101—22no—
91C2×#101D418yes—
92C2×#101C2314yes—
93C2×#101C2314yes—
94C4×M16—24no—
95C4×M16—42no—
96C2×#97—41no—
97C2×C4×C8—41no—
98#396—24no—
99#396—24no—
100#396—24no—
101#396—24no—
102#391—22no—
103#391C2×C414yes—
104#390—22no—
105#390C2×C414yes—
106#388—24no—
107#388—24no—
108#388—24no—
109#388—24no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C414yes—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115#356—216no—
116#355—216no—
117#343—24no—
118#343—24no—
119C25—21no—
120C22×Q8—21no—
121C22×Q8—24no—
122C22×Q8—41no—
123C22×Q8—41no—
124C22×Q8—42no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8—42no—
128C22×Q8C22×C412yes—
129C22×Q8C22×C411yescharacteristic
130C22×Q8C2×D411yescharacteristic
131C22×Q8C2×D411yescharacteristic
132C22×Q8C2×D414yes—
133C22×Q8#4114yes—
134C22×Q8C2411yescharacteristic
135C22×D4—21no—
136C22×D4—24no—
137C22×D4—41no—
138C22×D4—41no—
139C22×D4—42no—
140C22×D4C22×C412yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—28no—
152C2×M16—24no—
153C2×M16—24no—
154C2×M16—44no—
155C2×M16—42no—
156C2×M16—42no—
157C2×M16—44no—
158C2×M16—44no—
159C22×C8—42no—
160C22×C8—42no—
161C4⋊Q8—28no—
162C4⋊Q8—24no—
163C4⋊Q8—24no—
164C4⋊Q8—44no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167#124—28no—
168#124—24no—
169#124—24no—
170#124—44no—
171#124—44no—
172#124—44no—
173#124—44no—
174#124—44no—
175#124—42no—
176#124—42no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42C22⋊C412yes—
238C2×C42C22⋊C412yes—
239C2×C42C22⋊C412yes—
240C2×C42C22×C412yes—
241C2×C42C22×C412yes—
242C2×C42C22×C412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C4⋊C8—44no—
246C4⋊C8—44no—
247#101—28no—
248#101—24no—
249#101—24no—
250#101—24no—
251#101—44no—
252#101C2×D418yes—
253#97—44no—
254C4×C8—44no—
255#95—24no—
256#95—24no—
257#95—24no—
258C24—22no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—41no—
265C24—41no—
266C24—42no—
267C24C2×C22⋊C411yescharacteristic
268C2×Q8—24no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—41no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—84no—
287C2×Q8—84no—
288C2×Q8C4×D414yes—
289C2×Q8#12114yes—
290C2×Q8#12412yes—
291C2×Q8#12712yes—
292C2×Q8C22×D412yes—
293C2×Q8C2×#4112yes—
294C2×D4—24no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—84no—
301C2×D4—84no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4#9911yescharacteristic
335C22×C4#9911yescharacteristic
336C22×C4#9912yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C4×D414yes—
345C22×C4#12114yes—
346C22×C4#12412yes—
347C22×C4#12712yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351M16—28no—
352M16—44no—
353M16—48no—
354M16—44no—
355M16—44no—
356M16—84no—
357C2×C8—44no—
358C2×C8—48no—
359C2×C8—44no—
360C2×C8—84no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—48no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—48no—
376C22⋊C4—44no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—24no—
396C42—24no—
397C42—42no—
398C42—42no—
399C23—22no—
400C23—22no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—82no—
409C23—82no—
410C23—84no—
411C23—84no—
412C23—84no—
413C23—84no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23#39012yes—
419C23C2×#9911yescharacteristic
420C23C2×#9911yescharacteristic
421C23#41012yes—
422C23C22×C22⋊C411yescharacteristic
423Q8—44no—
424Q8—44no—
425Q8—42no—
426Q8—42no—
427Q8—42no—
428Q8—42no—
429Q8—84no—
430Q8—84no—
431D4—84no—
432D4—84no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—48no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—84no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4#39012yes—
483C2×C4#39012yes—
484C2×C4#39012yes—
485C2×C4C2×#9912yes—
486C2×C4C2×#9912yes—
487C2×C4C2×C4×D412yes—
488C2×C4C2×#12112yes—
489C2×C4C2×#12411yescharacteristic
490C2×C4C2×#12711yescharacteristic
491C8—48no—
492C8—84no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—24no—
498C22—24no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—164no—
507C22—164no—
508C22#196412yes—
509C22#196412yes—
510C22#213212yes—
511C22#213312yes—
512C22C2×#39011yescharacteristic
513C22C22×#9911yescharacteristic
514C22C2×#41011yescharacteristic
515C4—22no—
516C4—22no—
517C4—22no—
518C4—21no—
519C4—22no—
520C4—21no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—44no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—82no—
529C4—82no—
530C2—21no—
531C2—21no—
532C2—22no—
533C2—162no—
534C2#1322814yesdirect product factor
535C2C2×#196411yescharacteristic
536C2C2×#213211yescharacteristic
537C2C2×#213311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#356C2semidirect
4. C2×#1801C2semidirect
5. C2×C4×#101C2semidirect
6. #13228C2direct (N × H)
27. C2×#356C22semidirect
31. #1801C22semidirect
33. C4×#101C22semidirect
534. C2#13228direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25452
C22×#3562
C22×#3552
C2×#18012
C2×C4×#1012
#132282