Finite Group Databasebeta

← C2×#9508 #201735

Subgroups — C2×#9508 #201735

Order 512.

1510 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2975C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic
3C2×#1821C212yesmaximal, complemented
4C2×#1465C211yesmaximal, characteristic
5#9508C218yesmaximal, direct product factor
6C2×#1305C212yesmaximal, complemented
7C22×#123C412yes—
8C22×C4×D4C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10#2975—24no—
11#2975—22no—
12#2975C2214yes—
13C2×#436—24no—
14C2×#431—21no—
15C2×#407—21no—
16C2×#407C2212yes—
17C2×#407C2212yes—
18#2486—22no—
19#2486—22no—
20#2486C414yes—
21#2486C2214yes—
22C2×#398—21no—
23C2×#397—21no—
24C2×#389—21no—
25C2×#381C412yes—
26C2×#380—21no—
27C2×#380C2211yescharacteristic
28#1821—24no—
29#1821C2218yescomplemented
30C2×#343—24no—
31#1465—22no—
32#1465C2214yes—
33C2×#324—21no—
34C2×#323—21no—
35#1305C2218yescomplemented
36C23×D4C2×C411yescharacteristic
37C24×C4C2×C412yes—
38C24×C4D411yescharacteristic
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41C22×M16—22no—
42C22×M16D414yes—
43C2×#123—24no—
44C2×#123—48no—
45C2×#123C2×C418yes—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×C4×D4C2312yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4D411yescharacteristic
71C22×C42D411yescharacteristic
72#436—216no—
73#436—216no—
74#431—24no—
75#431—24no—
76C2×C4⋊C8—22no—
77#407—24no—
78#407—24no—
79#407—22no—
80#407—22no—
81#407—22no—
82#407—22no—
83#407—22no—
84#407D414yes—
85#407C2314yes—
86#407C2314yes—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C4×C8—22no—
93#398—24no—
94#398—24no—
95#397—24no—
96#397—24no—
97#389—24no—
98#389—24no—
99#381—24no—
100#381C2×C418yes—
101#380—24no—
102#380—21no—
103#380—21no—
104#380—24no—
105#380C2×C412yes—
106#380C2312yes—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110#343—216no—
111#324—24no—
112#323—24no—
113C25C22⋊C411yescharacteristic
114C25C22×C411yescharacteristic
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—24no—
118C22×D4—42no—
119C22×D4C22⋊C412yes—
120C22×D4C22×C412yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—44no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4—44no—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C412yes—
138C23×C4C22×C414yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C2×M16—28no—
151C2×M16—24no—
152C2×M16—24no—
153C2×M16—42no—
154C2×M16—48no—
155C2×M16—48no—
156C2×M16—84no—
157C2×M16C2×D418yes—
158C22×C8—24no—
159C22×C8—42no—
160#123—416no—
161#123—416no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22×C412yes—
208C2×C22⋊C4C22×C414yes—
209C2×C22⋊C4C2×D412yes—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C22⋊C412yes—
215C2×C42C2×D412yes—
216C4⋊C8—28no—
217C22⋊C8—28no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222C22⋊C8—24no—
223C22⋊C8—24no—
224C22⋊C8—42no—
225C22⋊C8—42no—
226C4×C8—28no—
227#95—28no—
228#95—22no—
229#95—28no—
230#95—22no—
231C24—22no—
232C24—22no—
233C24—21no—
234C24—22no—
235C24—21no—
236C24—22no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24C2×C22⋊C411yescharacteristic
242C24C2×C22⋊C412yes—
243C24C2×C22⋊C411yescharacteristic
244C24C4×D412yes—
245C24C4×D412yes—
246C24C23×C411yescharacteristic
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—48no—
254C2×D4—48no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—28no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—88no—
300C22×C4—88no—
301C22×C4#10014yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C4×D412yes—
308C22×C4C4×D412yes—
309C22×C4#12011yescharacteristic
310C22×C4#12012yes—
311C22×C4#12011yescharacteristic
312C22×C4#12112yes—
313C22×C4#12112yes—
314C22×C4#12112yes—
315C22×C4#12112yes—
316C22×C4#12311yescharacteristic
317C22×C4#12311yescharacteristic
318C22×C4#12312yes—
319C22×C4C22×D412yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4112yes—
323M16—48no—
324M16—48no—
325M16—84no—
326M16—88no—
327M16—84no—
328C2×C8—28no—
329C2×C8—48no—
330C2×C8—44no—
331C2×C8—84no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—416no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—44no—
348C22⋊C4—416no—
349C22⋊C4—44no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C23—22no—
355C23—22no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—41no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—41no—
373C23—48no—
374C23—48no—
375C23—42no—
376C23—42no—
377C23—48no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—81no—
383C23—81no—
384C23—84no—
385C23#38612yes—
386C23#38611yescharacteristic
387C23#38611yescharacteristic
388C23C2×#10012yes—
389C23#41312yes—
390C23C22×C22⋊C411yescharacteristic
391C23C2×C4×D412yes—
392D4—48no—
393D4—48no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—88no—
434C2×C4—88no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—88no—
438C2×C4#38611yescharacteristic
439C2×C4#38612yes—
440C2×C4#38611yescharacteristic
441C2×C4C2×#10014yes—
442C2×C4#45814yes—
443C2×C4C2×#12011yescharacteristic
444C2×C4C2×#12112yes—
445C2×C4C2×#12311yescharacteristic
446C8—48no—
447C8—84no—
448C22—21no—
449C22—21no—
450C22—24no—
451C22—24no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—48no—
465C22—44no—
466C22—48no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—82no—
471C22—84no—
472C22—84no—
473C22#189012yes—
474C22#190012yes—
475C22#190012yes—
476C22#191512yes—
477C22C2×#38611yescharacteristic
478C22C22×#10011yescharacteristic
479C22C2×#41311yescharacteristic
480C4—22no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—24no—
485C4—42no—
486C4—42no—
487C4—84no—
488C4—84no—
489C4#189012yes—
490C4C2×#45812yes—
491C2—21no—
492C2—21no—
493C2—22no—
494C2—42no—
495C2—42no—
496C2—82no—
497C2#950814yesdirect product factor
498C2C2×#189011yescharacteristic
499C2C2×#190011yescharacteristic
500C2C2×#191511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ 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
3. C2×#1821C2semidirect
5. #9508C2direct (N × H)
6. C2×#1305C2semidirect
29. #1821C22semidirect
35. #1305C22semidirect
497. C2#9508direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29752
C2×#24862
C2×#18212
C2×#14652
#95082
C2×#13052