Finite Group Databasebeta

← C2×#9200 #197495

Subgroups — C2×#9200 #197495

Order 512.

1554 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3C2×#1520C212yesmaximal
4C22×#324C211yesmaximal, characteristic
5#9200C218yesmaximal, direct product factor
6C2×#1289C212yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C22×#117—21no—
9C23×C4⋊C4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2627—22no—
12#2627—22no—
13#2627C414yescomplemented
14#2627C2214yes—
15C2×#400C412yescomplemented
16C2×#398—22no—
17C2×#395C412yescomplemented
18#2316—24no—
19#2316—22no—
20#2316C2214yes—
21C2×#389—24no—
22C2×#385—21no—
23C2×#381—21no—
24C2×#380—21no—
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27#1520—24no—
28#1520C2218yes—
29C2×#328—24no—
30C2×#324—22no—
31C2×#324—22no—
32C2×#324C2214yes—
33#1289C2218yes—
34C23×Q8C2×C411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4D411yescharacteristic
37C24×C4C2311yescharacteristic
38C23×C8—22no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8C2×C412yes—
46C2×C22⋊Q8C2312yes—
47C2×#117—24no—
48C2×#117—24no—
49C2×#117—44no—
50C2×#117—44no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4D412yes—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8C2314yes—
78#400—24no—
79#400C2×C418yescomplemented
80#398—28no—
81#398—28no—
82#395—24no—
83#395C2×C418yescomplemented
84#389—216no—
85#389—216no—
86#385—24no—
87#385—24no—
88#381—24no—
89#381—24no—
90#380—24no—
91#380—22no—
92#380—22no—
93#380—24no—
94#380—21no—
95#380—21no—
96#380C2×C412yes—
97#380D414yes—
98#380C2312yes—
99#380C2314yes—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95C2×C412yes—
105C2×#95D414yes—
106#328—216no—
107#324—28no—
108C25C2×D411yescharacteristic
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—82no—
113C22×Q8C22×C412yes—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—24no—
128C23×C4—82no—
129C23×C4—82no—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4SD1614yes—
135C23×C4Q1614yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4#4112yes—
146C23×C4#4112yes—
147C23×C4C2411yescharacteristic
148C22×C8—24no—
149C22×C8—42no—
150C22×C8—42no—
151C22×C8—84no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8—24no—
155C22⋊Q8—24no—
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158C22⋊Q8—42no—
159C22⋊Q8—42no—
160#117—48no—
161#117—416no—
162#117—48no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—84no—
192C2×C4⋊C4—84no—
193C2×C4⋊C4—84no—
194C2×C4⋊C4C22⋊C414yes—
195C2×C4⋊C4C22⋊C414yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C414yes—
198C2×C4⋊C4C2×D414yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—42no—
213C2×C42—28no—
214C2×C42—48no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—42no—
218C22⋊C8—42no—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—22no—
224#95—24no—
225#95C22×C414yes—
226#95C2×D418yes—
227C24—42no—
228C24—42no—
229C24C2×C22⋊C411yescharacteristic
230C24C2×C22⋊C411yescharacteristic
231C24#12011yescharacteristic
232C24#12011yescharacteristic
233C24#12311yescharacteristic
234C24#12311yescharacteristic
235C24C2×SD1612yes—
236C24C2×Q1612yes—
237C24AGL(1,ℤ/8)12yes—
238C24#13712yes—
239C24C22×D411yescharacteristic
240C2×Q8—24no—
241C2×Q8—42no—
242C2×Q8—84no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—84no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—28no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—48no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—48no—
285C22×C4—44no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—44no—
291C22×C4—48no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—82no—
295C22×C4—81no—
296C22×C4—82no—
297C22×C4—81no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4#10318yes—
304C22×C4#10318yes—
305C22×C4C2×C22⋊C414yes—
306C22×C4C2×C22⋊C414yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4C4×D412yes—
312C22×C4C4×D412yes—
313C22×C4#12012yes—
314C22×C4#12112yes—
315C22×C4#12112yes—
316C22×C4#12112yes—
317C22×C4#12112yes—
318C22×C4#12312yes—
319C22×C4C2×SD1614yes—
320C22×C4C2×Q1614yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D412yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4112yes—
325C2×C8—44no—
326C2×C8—84no—
327C2×C8—84no—
328C2×C8—84no—
329C4⋊C4—28no—
330C4⋊C4—28no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C4⋊C4—24no—
334C4⋊C4—416no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—82no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C4⋊C4—82no—
342C4⋊C4—88no—
343C4⋊C4—88no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—44no—
353C22⋊C4—48no—
354C42—416no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—41no—
373C23#38611yescharacteristic
374C23#38611yescharacteristic
375C23#38611yescharacteristic
376C23#38611yescharacteristic
377C23C2×#10312yes—
378C23C2×#10312yes—
379C23#41712yes—
380C23#41712yes—
381C23#45012yes—
382C23#45012yes—
383C23#45112yes—
384C23#45112yes—
385C23#48312yes—
386C23#48312yes—
387C23#48412yes—
388C23#48412yes—
389C23C22×C22⋊C411yescharacteristic
390C23C2×#12011yescharacteristic
391C23C2×#12311yescharacteristic
392C23C22×SD1611yescharacteristic
393C23C22×Q1611yescharacteristic
394C23C2×AGL(1,ℤ/8)11yescharacteristic
395C23C2×#13711yescharacteristic
396Q8—84no—
397Q8—84no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—48no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—48no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—816no—
443C2×C4—88no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4#38612yes—
449C2×C4#38612yes—
450C2×C4C2×#10318yes—
451C2×C4C2×C4×D412yes—
452C2×C4C2×#12112yes—
453C8—84no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—24no—
470C22—24no—
471C22—24no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22#187212yes—
483C22#187212yes—
484C22#187212yes—
485C22#187212yes—
486C22#191212yes—
487C22#191312yes—
488C22C2×#38611yescharacteristic
489C22C22×#10311yescharacteristic
490C22C2×#41711yescharacteristic
491C22C2×#45011yescharacteristic
492C22C2×#45111yescharacteristic
493C22C2×#48311yescharacteristic
494C22C2×#48411yescharacteristic
495C4—22no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—28no—
500C4—44no—
501C4—88no—
502C4—84no—
503C4—82no—
504C4—82no—
505C2—21no—
506C2—21no—
507C2—21no—
508C2—21no—
509C2—21no—
510C2—22no—
511C2—22no—
512C2—22no—
513C2—21no—
514C2#920014yesdirect product factor
515C2C2×#187211yescharacteristic
516C2C2×#191211yescharacteristic
517C2C2×#191311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9200C2direct (N × H)
13. #2627C4semidirect
15. C2×#400C4semidirect
17. C2×#395C4semidirect
79. #400C2×C4semidirect
83. #395C2×C4semidirect
514. C2#9200direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26272
C2×#23162
C2×#15202
C22×#3242
#92002
C2×#12892