Finite Group Databasebeta

← C2×#9011 #194653

Subgroups — C2×#9011 #194653

Order 512.

1518 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic
2C2×#2395C211yesmaximal, characteristic
3C2×#1826C212yesmaximal, complemented
4C2×C4×#99C211yesmaximal, characteristic
5#9011C218yesmaximal, direct product factor
6C2×#1297C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C22×#117C2211yescharacteristic
9#2973—24no—
10#2973—22no—
11#2973C2214yes—
12C2×#435—24no—
13C2×#432—21no—
14C22×#99—21no—
15C2×#407C2212yes—
16C22×C22⋊C8—21no—
17#2395—22no—
18#2395—22no—
19#2395C414yescomplemented
20#2395C2214yes—
21C2×#400C411yescharacteristic, complemented
22C2×#399C411yescharacteristic, complemented
23C2×#398C411yescharacteristic, complemented
24C2×#397—21no—
25C2×#397C411yescharacteristic, complemented
26C2×#394—21no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C22⋊C4—21no—
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31#1826—24no—
32#1826C2218yescomplemented
33C4×#99—22no—
34C4×#99C2214yes—
35C2×#342—21no—
36C2×#339—24no—
37#1297C2218yescomplemented
38C23×D4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16D414yes—
42C23×C8—22no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D412yes—
49C2×C4×D4D412yes—
50C2×C4×D4C2312yes—
51C2×#117—21no—
52C2×#117—21no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117C2×C412yes—
58C2×#117C2312yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C22⋊C4—22no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42—21no—
73C22×C42—22no—
74C22×C42D411yescharacteristic
75#435—216no—
76#435—216no—
77#432—24no—
78#432—24no—
79C2×C4⋊C8—22no—
80C2×#99—28no—
81C2×#99—24no—
82#407—22no—
83#407—22no—
84#407D414yes—
85#407C2314yes—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—22no—
91C2×#97—22no—
92#400—22no—
93#400C2×C414yescomplemented
94#399—22no—
95#399C2×C414yescomplemented
96#398—22no—
97#398C2×C414yescomplemented
98#397—24no—
99#397—22no—
100#397—24no—
101#397C2×C414yescomplemented
102#394—24no—
103#394—24no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4C2×C412yes—
114C4×C22⋊C4C2314yes—
115C4×C22⋊C4C2312yes—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95C2×C411yescharacteristic
122C2×#95C2×C411yescharacteristic
123#342—24no—
124#339—216no—
125C25C22⋊C411yescharacteristic
126C25C2×D411yescharacteristic
127C22×D4—24no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—42no—
131C22×D4C22⋊C412yes—
132C22×D4C2×D412yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—42no—
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
159C2×M16—24no—
160C2×M16—48no—
161C2×M16C2×D418yes—
162C22×C8—24no—
163C22×C8—28no—
164C22×C8—84no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173#117—24no—
174#117—24no—
175#117—24no—
176#117—24no—
177#117—42no—
178#117—42no—
179#117—42no—
180#117—42no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22×C412yes—
219C2×C22⋊C4C2×D412yes—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—44no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42—48no—
235C2×C42—42no—
236C2×C42C22⋊C412yes—
237C2×C42C2×D412yes—
238C4⋊C8—28no—
239#99—216no—
240C22⋊C8—28no—
241C22⋊C8—24no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—24no—
245#97—28no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—21no—
251#95—21no—
252#95—24no—
253#95—42no—
254#95C22×C412yes—
255#95C22×C412yes—
256C24—21no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24#10414yes—
266C24C2×C22⋊C411yescharacteristic
267C24C2×C22⋊C411yescharacteristic
268C24C2×C22⋊C412yes—
269C24#12112yes—
270C24#12112yes—
271C24C22×D411yescharacteristic
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—48no—
279C2×D4—48no—
280C22×C4—28no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—48no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C4×D412yes—
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4C4×D412yes—
341C22×C4#12011yescharacteristic
342C22×C4#12012yes—
343C22×C4#12011yescharacteristic
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4#12311yescharacteristic
347C22×C4#12311yescharacteristic
348C22×C4#12312yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4112yes—
353M16—48no—
354M16—84no—
355C2×C8—28no—
356C2×C8—48no—
357C2×C8—88no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—28no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—48no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—44no—
384C42—44no—
385C42—48no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—84no—
391C23—22no—
392C23—22no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—22no—
398C23—24no—
399C23—24no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—22no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—41no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—48no—
414C23—42no—
415C23—42no—
416C23—81no—
417C23—84no—
418C23—84no—
419C23—84no—
420C23—81no—
421C23#38612yes—
422C23#38611yescharacteristic
423C23#38611yescharacteristic
424C23C2×#10412yes—
425C23C2×#10414yes—
426C23#42112yes—
427C23C22×C22⋊C411yescharacteristic
428C23C2×#12112yes—
429D4—48no—
430D4—48no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—48no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—48no—
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—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—816no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4#38611yescharacteristic
488C2×C4#38612yes—
489C2×C4#38611yescharacteristic
490C2×C4#45814yes—
491C2×C4C2×C4×D412yes—
492C2×C4C2×#12011yescharacteristic
493C2×C4C2×#12311yescharacteristic
494C8—48no—
495C22—24no—
496C22—24no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—22no—
502C22—22no—
503C22—24no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—44no—
509C22—48no—
510C22—48no—
511C22—42no—
512C22—42no—
513C22—81no—
514C22—81no—
515C22—82no—
516C22—81no—
517C22—81no—
518C22—84no—
519C22—84no—
520C22#187112yes—
521C22#187112yes—
522C22#188412yes—
523C22#189012yes—
524C22C2×#38611yescharacteristic
525C22C22×#10411yescharacteristic
526C22C2×#42111yescharacteristic
527C4—21no—
528C4—21no—
529C4—22no—
530C4—22no—
531C4—44no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—82no—
536C4—88no—
537C4—82no—
538C4—82no—
539C4#189012yes—
540C4C2×#45812yes—
541C2—21no—
542C2—21no—
543C2—22no—
544C2—42no—
545C2—42no—
546C2—44no—
547C2#901114yesdirect product factor
548C2C2×#187111yescharacteristic
549C2C2×#188411yescharacteristic
550C2C2×#189011yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23#386
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1826C2semidirect
5. #9011C2direct (N × H)
6. C2×#1297C2semidirect
19. #2395C4semidirect
21. C2×#400C4semidirect
22. C2×#399C4semidirect
23. C2×#398C4semidirect
25. C2×#397C4semidirect
32. #1826C22semidirect
37. #1297C22semidirect
93. #400C2×C4semidirect
95. #399C2×C4semidirect
97. #398C2×C4semidirect
101. #397C2×C4semidirect
547. C2#9011direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29732
C2×#23952
C2×#18262
C2×C4×#992
#90112
C2×#12972