Finite Group Databasebeta

← C2×#9491 #201600

Subgroups — C2×#9491 #201600

Order 512.

1522 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×#2873C211yesmaximal, characteristic
2C2×#2599C211yesmaximal, characteristic
3C2×#1491C212yesmaximal, complemented
4C2×#1464C211yesmaximal, characteristic
5#9491C218yesmaximal, direct product factor
6C2×#1288C212yesmaximal, complemented
7C23×M16—21no—
8C22×C4×Q8C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C2×#432C2212yes—
11#2873—24no—
12#2873—22no—
13#2873C2214yes—
14C2×#405—24no—
15C2×C4×M16—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—
41C22×M16—24no—
42C22×M16—28no—
43C22×M16—28no—
44C22×M16—22no—
45C22×M16—24no—
46C23×C8—22no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2312yes—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117C2×C412yes—
60C2×#117D412yes—
61C2×#117D412yes—
62C2×#117C2312yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76#432—22no—
77#432—22no—
78#432—22no—
79#432C2314yes—
80C2×C4⋊C8—22no—
81C2×C22⋊C8—22no—
82#405—216no—
83#405—216no—
84C4×M16—24no—
85C4×M16—24no—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16D414yes—
89C4×M16C2314yes—
90C2×#97—22no—
91C2×#97—22no—
92C2×#97—22no—
93C2×C4×C8—22no—
94C2×C4×C8D414yes—
95#398—22no—
96#398C2×C414yes—
97#397—22no—
98#397C2×C414yes—
99#395—22no—
100#395C2×C414yes—
101#393—22no—
102#393C2×C414yes—
103#388—24no—
104#388—24no—
105#387—24no—
106#387—24no—
107#387—24no—
108#387—24no—
109#385—24no—
110#385—24no—
111#383—21no—
112#383—21no—
113#383C2×C412yes—
114#383C2312yes—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95C2×C411yescharacteristic
122C2×#95C2×C411yescharacteristic
123Q8⋊C8—216no—
124#324—24no—
125#323—24no—
126C25C22⋊C411yescharacteristic
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—44no—
131C22×Q8C22×C412yes—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—216no—
151C2×M16—232no—
152C2×M16—28no—
153C2×M16—28no—
154C2×M16—216no—
155C2×M16—216no—
156C2×M16—28no—
157C2×M16—24no—
158C2×M16—24no—
159C22×C8—216no—
160C22×C8—28no—
161C22×C8—24no—
162C22×C8—42no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8—42no—
171#117—24no—
172#117—24no—
173#117—24no—
174#117—24no—
175#117—24no—
176#117—24no—
177#117—24no—
178#117—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22×C412yes—
212C2×C22⋊C4C2×D412yes—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42C22⋊C412yes—
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42#4112yes—
229C2×C42#4112yes—
230C2×C42C2411yescharacteristic
231C4⋊C8—24no—
232C4⋊C8—42no—
233C22⋊C8—24no—
234C22⋊C8—42no—
235#97—28no—
236#97—24no—
237#97—24no—
238#97—24no—
239#97—42no—
240C4×C8—28no—
241C4×C8—24no—
242C4×C8C2×D418yes—
243#95—22no—
244#95—22no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—21no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95C22×C412yes—
254#95C22×C412yes—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—42no—
259C24—42no—
260C24#10414yes—
261C24C2×C22⋊C411yescharacteristic
262C24C2×C22⋊C412yes—
263C2×Q8—28no—
264C2×Q8—24no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—48no—
269C2×Q8—42no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4#10414yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C4×D412yes—
322C22×C4C4×D412yes—
323C22×C4#12011yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4#12112yes—
326C22×C4#12112yes—
327C22×C4#12311yescharacteristic
328C22×C4#12311yescharacteristic
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332M16—232no—
333M16—28no—
334M16—28no—
335M16—216no—
336C2×C8—232no—
337C2×C8—216no—
338C2×C8—28no—
339C2×C8—44no—
340C2×C8—84no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—48no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—48no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—48no—
358C22⋊C4—48no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—44no—
364C42—44no—
365C42C4×D412yes—
366C42C4×D412yes—
367C42#12012yes—
368C42#12112yes—
369C42#12112yes—
370C42#12312yes—
371C42C22×D412yes—
372C42C2×#4112yes—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—22no—
377C23—22no—
378C23—24no—
379C23—42no—
380C23—48no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—81no—
385C23—84no—
386C23—81no—
387C23#38611yescharacteristic
388C23#38611yescharacteristic
389C23C2×#10412yes—
390C23C2×#10412yes—
391C23C2×#10414yes—
392C23C22×C22⋊C411yescharacteristic
393Q8—28no—
394Q8—44no—
395Q8—48no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4#38611yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4#38612yes—
450C2×C4#38612yes—
451C2×C4C2×#10414yes—
452C2×C4C2×C4×D412yes—
453C2×C4C2×#12011yescharacteristic
454C2×C4C2×#12112yes—
455C2×C4C2×#12311yescharacteristic
456C8—216no—
457C8—84no—
458C22—21no—
459C22—21no—
460C22—24no—
461C22—24no—
462C22—21no—
463C22—21no—
464C22—24no—
465C22—44no—
466C22—42no—
467C22—48no—
468C22—42no—
469C22—82no—
470C22—84no—
471C22#189112yes—
472C22#189612yes—
473C22#189612yes—
474C22#190912yes—
475C22C2×#38611yescharacteristic
476C22C22×#10411yescharacteristic
477C22C22×#10411yescharacteristic
478C4—22no—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—24no—
487C4—42no—
488C4—44no—
489C4—42no—
490C4—82no—
491C4—82no—
492C4—82no—
493C2—21no—
494C2—21no—
495C2—22no—
496C2—44no—
497C2#949114yesdirect product factor
498C2C2×#189111yescharacteristic
499C2C2×#189611yescharacteristic
500C2C2×#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. #9491C2direct (N × H)
6. C2×#1288C2semidirect
31. #1491C22semidirect
37. #1288C22semidirect
497. C2#9491direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28732
C2×#25992
C2×#14912
C2×#14642
#94912
C2×#12882