Finite Group Databasebeta

← C2×#9202 #197501

Subgroups — C2×#9202 #197501

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic
3C2×#1520C212yesmaximal
4C22×#324C211yesmaximal, characteristic
5#9202C218yesmaximal, direct product factor
6C2×#1289C212yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C22×C22⋊C8C2212yes—
10#2627—22no—
11#2627—22no—
12#2627C414yescomplemented
13#2627C2214yes—
14C2×#400C412yescomplemented
15C2×#398—22no—
16C2×#395C412yescomplemented
17#2313—24no—
18#2313—22no—
19#2313C2214yes—
20C2×#387—24no—
21C2×#385—21no—
22C2×#381—21no—
23C2×#381—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2212yes—
26C2×C4×C22⋊C4—21no—
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—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C42—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8C2314yes—
72#400—24no—
73#400C2×C418yescomplemented
74#398—28no—
75#398—28no—
76#395—24no—
77#395C2×C418yescomplemented
78#387—216no—
79#387—216no—
80#385—24no—
81#385—24no—
82#381—24no—
83#381—24no—
84#381—24no—
85#381—24no—
86#380—22no—
87#380—22no—
88#380—21no—
89#380—21no—
90#380C2×C412yes—
91#380D414yes—
92#380C2312yes—
93#380C2314yes—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96C2×#95—22no—
97C2×#95—24no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95C2×C412yes—
102C2×#95D414yes—
103#328—216no—
104#324—28no—
105C25C2×D411yescharacteristic
106C22×Q8—21no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—82no—
110C22×Q8C22×C412yes—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—24no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4#4112yes—
141C23×C4#4112yes—
142C23×C4C2411yescharacteristic
143C22×C8—24no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—84no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C22⋊Q8—24no—
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153C22⋊Q8—42no—
154C22⋊Q8—42no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—84no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—84no—
182C2×C4⋊C4C22⋊C414yes—
183C2×C4⋊C4C22⋊C414yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C414yes—
186C2×C4⋊C4C2×D414yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C42—28no—
201C2×C42—84no—
202C22⋊C8—24no—
203C22⋊C8—24no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206#95—28no—
207#95—216no—
208#95—28no—
209#95—28no—
210#95—28no—
211#95—22no—
212#95—24no—
213#95C22×C414yes—
214#95C2×D418yes—
215C24—42no—
216C24—42no—
217C24C2×C22⋊C411yescharacteristic
218C24C2×C22⋊C411yescharacteristic
219C24#12011yescharacteristic
220C24#12011yescharacteristic
221C24#12311yescharacteristic
222C24#12311yescharacteristic
223C24#13514yes—
224C24#13714yes—
225C24C22×D411yescharacteristic
226C2×Q8—24no—
227C2×Q8—42no—
228C2×Q8—84no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—84no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—44no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—44no—
275C22×C4—48no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—82no—
279C22×C4—81no—
280C22×C4—82no—
281C22×C4—81no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4C2×C22⋊C414yes—
288C22×C4C2×C22⋊C414yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C4×D412yes—
292C22×C4C4×D412yes—
293C22×C4C4×D412yes—
294C22×C4C4×D412yes—
295C22×C4#12012yes—
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4#12312yes—
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D412yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4112yes—
305C2×C8—44no—
306C2×C8—84no—
307C2×C8—84no—
308C2×C8—84no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4—28no—
313C4⋊C4—24no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—82no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4—82no—
321C4⋊C4—88no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—48no—
331C42—88no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—48no—
346C23—48no—
347C23—41no—
348C23#38611yescharacteristic
349C23#38611yescharacteristic
350C23#38611yescharacteristic
351C23#38611yescharacteristic
352C23#41612yes—
353C23#41612yes—
354C23#41912yes—
355C23#41912yes—
356C23#45214yes—
357C23#45214yes—
358C23#48514yes—
359C23#48514yes—
360C23C22×C22⋊C411yescharacteristic
361C23C2×#12011yescharacteristic
362C23C2×#12311yescharacteristic
363C23C2×#13512yes—
364C23C2×#13712yes—
365Q8—84no—
366Q8—84no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—48no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—48no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—48no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—88no—
411C2×C4—816no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4#38612yes—
417C2×C4#38612yes—
418C2×C4C2×C4×D412yes—
419C2×C4C2×#12112yes—
420C8—84no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—24no—
434C22—28no—
435C22—24no—
436C22—28no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—44no—
442C22—44no—
443C22#187312yes—
444C22#187312yes—
445C22#187312yes—
446C22#187312yes—
447C22#191212yes—
448C22#191312yes—
449C22C2×#38611yescharacteristic
450C22C2×#41611yescharacteristic
451C22C2×#41911yescharacteristic
452C22C2×#45212yes—
453C22C2×#48512yes—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—22no—
458C4—44no—
459C4—44no—
460C4—84no—
461C4—88no—
462C4—82no—
463C4—82no—
464C2—21no—
465C2—22no—
466C2—22no—
467C2—21no—
468C2—22no—
469C2—22no—
470C2—22no—
471C2#920214yesdirect product factor
472C2C2×#187311yescharacteristic
473C2C2×#191211yescharacteristic
474C2C2×#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. #9202C2direct (N × H)
12. #2627C4semidirect
14. C2×#400C4semidirect
16. C2×#395C4semidirect
73. #400C2×C4semidirect
77. #395C2×C4semidirect
471. C2#9202direct (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×#23132
C2×#15202
C22×#3242
#92022
C2×#12892