Finite Group Databasebeta

← C2×#9487 #201531

Subgroups — C2×#9487 #201531

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2975C211yesmaximal, characteristic
2C2×#2580C211yesmaximal, characteristic
3C2×#1800C211yesmaximal, characteristic
4C2×#1487C212yesmaximal, complemented
5#9487C218yesmaximal, direct product factor
6C2×#1286C212yesmaximal, complemented
7C22×C4×Q8C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9#2975—24no—
10#2975—22no—
11#2975C2214yes—
12C2×#436—24no—
13C2×#431C2212yes—
14C2×#423C2212yes—
15C22×#100—21no—
16C2×#407—22no—
17#2580—22no—
18#2580—22no—
19#2580C414yes—
20#2580C2214yes—
21C2×#399C412yes—
22C2×#393C412yes—
23C2×#390—21no—
24C2×#389—21no—
25C2×#385—22no—
26C2×#382C2211yescharacteristic
27#1800—22no—
28#1800C2214yes—
29C2×#343—21no—
30#1487—24no—
31#1487C2218yescomplemented
32C2×Q8⋊C8—24no—
33#1286C2218yescomplemented
34C23×Q8C2×C411yescharacteristic
35C23×D4D411yescharacteristic
36C24×C4D412yes—
37C22×M16—24no—
38C22×M16—22no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—22no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8C2312yes—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4C2×C412yes—
52C2×C4×D4D412yes—
53C2×C4×D4D412yes—
54C2×C4×D4C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4C2×C412yes—
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65#436—216no—
66#436—216no—
67#431—22no—
68#431—22no—
69#431D414yes—
70#431C2314yes—
71#423—22no—
72#423—22no—
73#423—22no—
74#423C2314yes—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×#100—24no—
79C2×#100—44no—
80#407—28no—
81#407—28no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—22no—
84C2×C4×C8D414yes—
85#399—24no—
86#399C2×C418yes—
87#393—24no—
88#393C2×C418yes—
89#390—24no—
90#390—24no—
91#389—24no—
92#389—24no—
93#385—28no—
94#385—28no—
95#382—21no—
96#382—21no—
97#382C2×C412yes—
98#382C2312yes—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95C2×C412yes—
102#343—24no—
103Q8⋊C8—216no—
104C25C22⋊C412yes—
105C22×Q8—21no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—44no—
109C22×Q8C22×C412yes—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—24no—
115C22×D4C22⋊C412yes—
116C22×D4C2×D412yes—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—84no—
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C414yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D414yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C2×M16—216no—
132C2×M16—24no—
133C2×M16—42no—
134C2×M16—44no—
135C2×M16—44no—
136C2×M16—88no—
137C22×C8—24no—
138C22×C8—42no—
139C4×Q8—24no—
140C4×Q8—24no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—84no—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—84no—
181C2×C22⋊C4C22⋊C414yes—
182C2×C22⋊C4C22×C414yes—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42#4112yes—
193C2×C42#4112yes—
194C2×C42C2411yescharacteristic
195C4⋊C8—24no—
196C4⋊C8—24no—
197C4⋊C8—24no—
198C4⋊C8—24no—
199C4⋊C8—42no—
200C4⋊C8—42no—
201#100—48no—
202C22⋊C8—216no—
203C22⋊C8—24no—
204C22⋊C8—24no—
205C4×C8—24no—
206C4×C8C2×D418yes—
207#95—22no—
208#95—28no—
209#95—22no—
210#95—22no—
211#95C22×C414yes—
212C24—22no—
213C24—24no—
214C24—22no—
215C24—24no—
216C24—44no—
217C24—44no—
218C24C2×C22⋊C411yescharacteristic
219C24C2×C22⋊C414yes—
220C2×Q8—28no—
221C2×Q8—24no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×Q8—48no—
226C2×Q8—42no—
227C2×D4—24no—
228C2×D4—22no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—48no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—44no—
262C22×C4—84no—
263C22×C4—84no—
264C22×C4—84no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4C2×C22⋊C414yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C4×D412yes—
272C22×C4C4×D412yes—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12112yes—
276C22×C4#12112yes—
277C22×C4#12311yescharacteristic
278C22×C4#12311yescharacteristic
279C22×C4AGL(1,ℤ/8)12yes—
280C22×C4#13712yes—
281C22×C4#13714yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285M16—44no—
286M16—44no—
287M16—816no—
288M16—84no—
289C2×C8—28no—
290C2×C8—48no—
291C2×C8—44no—
292C2×C8—84no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—88no—
305C22⋊C4—28no—
306C22⋊C4—28no—
307C22⋊C4—416no—
308C22⋊C4—88no—
309C42—44no—
310C42—44no—
311C42C4×D412yes—
312C42C4×D412yes—
313C42#12012yes—
314C42#12112yes—
315C42#12112yes—
316C42#12312yes—
317C42C22×D412yes—
318C42C2×#4112yes—
319C23—22no—
320C23—22no—
321C23—24no—
322C23—24no—
323C23—24no—
324C23—44no—
325C23—44no—
326C23—416no—
327C23—44no—
328C23—44no—
329C23—44no—
330C23—44no—
331C23—82no—
332C23—82no—
333C23—88no—
334C23#38611yescharacteristic
335C23#38611yescharacteristic
336C23#41712yes—
337C23#41912yes—
338C23C22×C22⋊C411yescharacteristic
339Q8—28no—
340Q8—44no—
341Q8—48no—
342D4—48no—
343D4—44no—
344D4—44no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4—88no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4#38611yescharacteristic
378C2×C4#38611yescharacteristic
379C2×C4#38612yes—
380C2×C4#38612yes—
381C2×C4#41712yes—
382C2×C4#41912yes—
383C2×C4#46814yes—
384C2×C4#46814yes—
385C2×C4#47014yes—
386C2×C4#47014yes—
387C2×C4C2×C4×D412yes—
388C2×C4C2×#12011yescharacteristic
389C2×C4C2×#12112yes—
390C2×C4C2×#12311yescharacteristic
391C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
392C2×C4C2×#13712yes—
393C2×C4C2×#13711yescharacteristic
394C8—48no—
395C8—84no—
396C22—24no—
397C22—24no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—48no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—84no—
407C22—88no—
408C22—88no—
409C22#189112yes—
410C22#189412yes—
411C22#189412yes—
412C22#191112yes—
413C22C2×#38611yescharacteristic
414C22C2×#41911yescharacteristic
415C22C2×#41711yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—22no—
421C4—22no—
422C4—44no—
423C4—44no—
424C4—84no—
425C4—82no—
426C4#189414yes—
427C4C2×#46812yes—
428C4C2×#47012yes—
429C2—21no—
430C2—21no—
431C2—22no—
432C2—84no—
433C2#948714yesdirect product factor
434C2C2×#189111yescharacteristic
435C2C2×#189411yescharacteristic
436C2C2×#191111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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
4. C2×#1487C2semidirect
5. #9487C2direct (N × H)
6. C2×#1286C2semidirect
31. #1487C22semidirect
33. #1286C22semidirect
433. C2#9487direct (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×#25802
C2×#18002
C2×#14872
#94872
C2×#12862