Finite Group Databasebeta

← C2×#9453 #201185

Subgroups — C2×#9453 #201185

Order 512.

1364 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic
3C2×#1556C212yesmaximal, complemented
4C2×#1466C211yesmaximal, characteristic
5#9453C218yesmaximal, direct product factor
6C2×#1285C212yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9#2973—24no—
10#2973—22no—
11#2973C2214yes—
12C2×#435—24no—
13C2×#432C2212yes—
14C2×#432C2212yes—
15C2×#407—21no—
16C22×C22⋊C8—21no—
17#2563—22no—
18#2563—22no—
19#2563C414yes—
20#2563C2214yes—
21C2×#393—21no—
22C2×#393—21no—
23C2×#390C412yes—
24C2×#382C412yes—
25C2×#380—21no—
26C2×#380—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1556—24no—
29#1556C2218yescomplemented
30C2×#332—24no—
31#1466—22no—
32#1466C2214yes—
33C2×#323—21no—
34C2×#323—21no—
35#1285C2218yescomplemented
36C23×Q8—21no—
37C23×D4C2×C411yescharacteristic
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C22×M16—22no—
41C23×C8—22no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2312yes—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4D412yes—
56C2×C4×D4D412yes—
57C2×C4×D4C2312yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D411yescharacteristic
65C22×C22⋊C4D411yescharacteristic
66C22×C42C2×C412yes—
67C22×C42C2311yescharacteristic
68#435—216no—
69#435—216no—
70#432—22no—
71#432—22no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432D414yes—
76#432C2314yes—
77#432C2314yes—
78C2×C4⋊C8—22no—
79C2×C4⋊C8D414yes—
80#407—24no—
81#407—24no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—22no—
88C2×#97—22no—
89C2×#97—22no—
90#393—24no—
91#393—24no—
92#393—24no—
93#393—24no—
94#390—24no—
95#390C2×C418yes—
96#382—24no—
97#382C2×C418yes—
98#380—24no—
99#380—24no—
100#380—24no—
101#380—24no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2312yes—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109#332—216no—
110#323—24no—
111#323—24no—
112C25C22⋊C411yescharacteristic
113C25C22⋊C411yescharacteristic
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—42no—
117C22×Q8—42no—
118C22×Q8—84no—
119C22×D4—24no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—42no—
123C22×D4C22⋊C412yes—
124C22×D4C22×C412yes—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C2×M16—28no—
147C2×M16—84no—
148C22×C8—24no—
149C22×C8—28no—
150C22×C8—42no—
151C22×C8—84no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22×C414yes—
184C2×C4⋊C4C22×C412yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—84no—
192C2×C22⋊C4—84no—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C2×D412yes—
196C2×C22⋊C4C2×D412yes—
197C2×C42—22no—
198C2×C42—44no—
199C2×C42C22×C414yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42C2411yescharacteristic
208C4⋊C8—24no—
209C4⋊C8—24no—
210C4⋊C8—42no—
211C4⋊C8C2×D418yes—
212C22⋊C8—28no—
213C22⋊C8—28no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—42no—
218#97—24no—
219#97—24no—
220#97—24no—
221#97—42no—
222#95—22no—
223#95—28no—
224#95—28no—
225#95—22no—
226C24—21no—
227C24—21no—
228C24—22no—
229C24—21no—
230C24—22no—
231C24—21no—
232C24—22no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24#10114yes—
238C24C2×C22⋊C411yescharacteristic
239C24C2×C22⋊C412yes—
240C24C2×C22⋊C412yes—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—88no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×Q8—88no—
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—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C4×D412yes—
300C22×C4C4×D412yes—
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12112yes—
304C22×C4#12112yes—
305C22×C4#12311yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310M16—88no—
311C2×C8—28no—
312C2×C8—48no—
313C2×C8—44no—
314C2×C8—88no—
315C2×C8—84no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—28no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—88no—
331C22⋊C4—88no—
332C42—44no—
333C42—44no—
334C42C4×D412yes—
335C42C4×D412yes—
336C42#12012yes—
337C42#12112yes—
338C42#12112yes—
339C42#12312yes—
340C42C22×D412yes—
341C42C2×#4112yes—
342C23—22no—
343C23—22no—
344C23—24no—
345C23—22no—
346C23—22no—
347C23—24no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—48no—
356C23—48no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—84no—
368C23—84no—
369C23#38611yescharacteristic
370C23#38611yescharacteristic
371C23C2×#10112yes—
372C23C2×#10114yes—
373C23#41312yes—
374C23C22×C22⋊C411yescharacteristic
375Q8—88no—
376Q8—88no—
377D4—48no—
378D4—48no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—88no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4#38612yes—
420C2×C4#38611yescharacteristic
421C2×C4#38612yes—
422C2×C4#38611yescharacteristic
423C2×C4#45514yes—
424C2×C4C2×C4×D412yes—
425C2×C4C2×#12011yescharacteristic
426C2×C4C2×#12112yes—
427C2×C4C2×#12311yescharacteristic
428C8—48no—
429C8—84no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—24no—
435C22—24no—
436C22—24no—
437C22—44no—
438C22—44no—
439C22—42no—
440C22—42no—
441C22—48no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—82no—
446C22—84no—
447C22—82no—
448C22—84no—
449C22—84no—
450C22#188512yes—
451C22#190112yes—
452C22#190112yes—
453C22#191012yes—
454C22C2×#38611yescharacteristic
455C22C22×#10111yescharacteristic
456C22C2×#41311yescharacteristic
457C4—22no—
458C4—22no—
459C4—21no—
460C4—21no—
461C4—22no—
462C4—22no—
463C4—42no—
464C4—42no—
465C4—84no—
466C4—84no—
467C4#188512yes—
468C4C2×#45512yes—
469C2—21no—
470C2—21no—
471C2—22no—
472C2—44no—
473C2—82no—
474C2#945314yesdirect product factor
475C2C2×#188511yescharacteristic
476C2C2×#190111yescharacteristic
477C2C2×#191011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×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×#1556C2semidirect
5. #9453C2direct (N × H)
6. C2×#1285C2semidirect
29. #1556C22semidirect
35. #1285C22semidirect
474. C2#9453direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29732
C2×#25632
C2×#15562
C2×#14662
#94532
C2×#12852