Finite Group Databasebeta

← C2×#9493 #201606

Subgroups — C2×#9493 #201606

Order 512.

1236 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2921C211yesmaximal, characteristic
2C2×#2599C211yesmaximal, characteristic
3C2×#1491C212yesmaximal, complemented
4C2×#1464C211yesmaximal, characteristic
5#9493C218yesmaximal, direct product factor
6C2×#1288C212yesmaximal, complemented
7C22×C4×Q8C2211yescharacteristic
8C22×#117C2211yescharacteristic
9#2921—24no—
10#2921—22no—
11#2921C2214yes—
12C2×#433—24no—
13C2×#432C2212yes—
14C2×#423—21no—
15C2×#407—21no—
16C2×C4×M16C2212yes—
17#2599—22no—
18#2599—22no—
19#2599C414yes—
20#2599C2214yes—
21C2×#398C411yescharacteristic
22C2×#397C411yescharacteristic
23C2×#395C411yescharacteristic
24C2×#393C411yescharacteristic
25C2×#388—21no—
26C2×#387—21no—
27C2×#387—21no—
28C2×#385—21no—
29C2×#383C2211yescharacteristic
30#1491—24no—
31#1491C2218yescomplemented
32C2×Q8⋊C8—24no—
33#1464—22no—
34#1464C2214yes—
35C2×#324—21no—
36C2×#323—21no—
37#1288C2218yescomplemented
38C23×Q8C2×C411yescharacteristic
39C24×C4D411yescharacteristic
40C22×M16—22no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8C2312yes—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117C2×C412yes—
54C2×#117D412yes—
55C2×#117D412yes—
56C2×#117C2312yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42D411yescharacteristic
69C22×C42C2311yescharacteristic
70#433—216no—
71#433—216no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432C2314yes—
76#423—24no—
77#423—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—22no—
80#407—24no—
81#407—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—22no—
84C4×M16—22no—
85C4×M16—22no—
86C4×M16D414yes—
87C4×M16C2314yes—
88C2×#97—22no—
89C2×#97—22no—
90C2×C4×C8D414yes—
91#398—22no—
92#398C2×C414yes—
93#397—22no—
94#397C2×C414yes—
95#395—22no—
96#395C2×C414yes—
97#393—22no—
98#393C2×C414yes—
99#388—24no—
100#388—24no—
101#387—24no—
102#387—24no—
103#387—24no—
104#387—24no—
105#385—24no—
106#385—24no—
107#383—21no—
108#383—21no—
109#383C2×C412yes—
110#383C2312yes—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95C2×C411yescharacteristic
118C2×#95C2×C411yescharacteristic
119Q8⋊C8—216no—
120#324—24no—
121#323—24no—
122C25C22⋊C411yescharacteristic
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—44no—
127C22×Q8C22×C412yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C412yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—44no—
149C2×M16—44no—
150C22×C8—24no—
151C22×C8—42no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—42no—
160#117—24no—
161#117—24no—
162#117—24no—
163#117—24no—
164#117—24no—
165#117—24no—
166#117—24no—
167#117—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22×C412yes—
201C2×C22⋊C4C2×D412yes—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C22⋊C412yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42#4112yes—
218C2×C42#4112yes—
219C2×C42C2411yescharacteristic
220C4⋊C8—216no—
221C4⋊C8—24no—
222C4⋊C8—42no—
223C22⋊C8—216no—
224C22⋊C8—24no—
225C22⋊C8—42no—
226#97—24no—
227#97—24no—
228#97—24no—
229#97—42no—
230C4×C8—24no—
231C4×C8C2×D418yes—
232#95—22no—
233#95—22no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95C22×C412yes—
243#95C22×C412yes—
244C24—21no—
245C24—22no—
246C24—21no—
247C24—22no—
248C24—42no—
249C24—42no—
250C24C2×C22⋊C411yescharacteristic
251C24C2×C22⋊C412yes—
252C2×Q8—28no—
253C2×Q8—24no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—48no—
258C2×Q8—42no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4#12011yescharacteristic
312C22×C4#12011yescharacteristic
313C22×C4#12112yes—
314C22×C4#12112yes—
315C22×C4#12311yescharacteristic
316C22×C4#12311yescharacteristic
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320M16—44no—
321M16—44no—
322M16—48no—
323C2×C8—28no—
324C2×C8—48no—
325C2×C8—44no—
326C2×C8—84no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—48no—
333C4⋊C4—48no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—48no—
344C22⋊C4—48no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—44no—
350C42—44no—
351C42C4×D412yes—
352C42C4×D412yes—
353C42#12012yes—
354C42#12112yes—
355C42#12112yes—
356C42#12312yes—
357C42C22×D412yes—
358C42C2×#4112yes—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—42no—
365C23—42no—
366C23—48no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—81no—
372C23—81no—
373C23—84no—
374C23#38611yescharacteristic
375C23#38611yescharacteristic
376C23#42112yes—
377C23#42112yes—
378C23C22×C22⋊C411yescharacteristic
379Q8—28no—
380Q8—44no—
381Q8—48no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#38611yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38612yes—
436C2×C4#38612yes—
437C2×C4C2×C4×D412yes—
438C2×C4C2×#12011yescharacteristic
439C2×C4C2×#12112yes—
440C2×C4C2×#12311yescharacteristic
441C8—48no—
442C8—84no—
443C22—21no—
444C22—21no—
445C22—24no—
446C22—24no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—82no—
454C22—84no—
455C22—84no—
456C22#189112yes—
457C22#189712yes—
458C22#189712yes—
459C22#190912yes—
460C22C2×#38611yescharacteristic
461C22C2×#42111yescharacteristic
462C22C2×#42111yescharacteristic
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—42no—
472C4—44no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—82no—
477C4—82no—
478C2—21no—
479C2—21no—
480C2—22no—
481C2—82no—
482C2#949314yesdirect product factor
483C2C2×#189111yescharacteristic
484C2C2×#189711yescharacteristic
485C2C2×#190911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1491C2semidirect
5. #9493C2direct (N × H)
6. C2×#1288C2semidirect
31. #1491C22semidirect
37. #1288C22semidirect
482. C2#9493direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29212
C2×#25992
C2×#14912
C2×#14642
#94932
C2×#12882