Finite Group Databasebeta

← C2×#10415 #239590

Subgroups — C2×#10415 #239590

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#1956C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C22×#333C211yesmaximal, characteristic
6C22×Q8⋊C8C211yesmaximal, characteristic
7#10415C218yesmaximal, direct product factor
8C2×#1279C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C22×#106—21no—
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103—21no—
15C22×#103—22no—
16C22×C4×C8C2211yescharacteristic
17#2567—22no—
18#2567—22no—
19#2567—22no—
20#2567C2214yes—
21C2×#395—22no—
22C2×#391—21no—
23C2×#389—21no—
24C2×#389—21no—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1956—22no—
29#1956C2214yes—
30#1948—22no—
31#1948C2214yes—
32#1803—22no—
33#1803C2214yes—
34C2×#340—21no—
35C2×#340—21no—
36C2×#333—24no—
37C2×#333C2214yes—
38C2×Q8⋊C8—24no—
39C2×Q8⋊C8C2214yes—
40#1279C2214yes—
41C23×Q8D411yescharacteristic
42C23×Q8D412yes—
43C23×C8—21no—
44C23×C8—21no—
45C23×C8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8D412yes—
53C2×C4⋊Q8C2312yes—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8D412yes—
61C2×C4×Q8C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C2×#106—24no—
77C2×#106—44no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8D414yes—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86C2×#103—24no—
87C2×#103—28no—
88C2×#103—48no—
89C2×#103—44no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—21no—
93C2×C4×C8C2312yes—
94#395—28no—
95#395—28no—
96#391—24no—
97#391—24no—
98#389—24no—
99#389—24no—
100#389—24no—
101#389—24no—
102#384—24no—
103#384—24no—
104#384—21no—
105#384—21no—
106#384—21no—
107#384C2312yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4C2312yes—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115#340—24no—
116#340—24no—
117#333—28no—
118Q8⋊C8—28no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—24no—
124C22×Q8—22no—
125C22×Q8—24no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—44no—
129C22×Q8—44no—
130C22×Q8SD1614yes—
131C22×Q8C2×D412yes—
132C22×Q8C2×D414yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—41no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C22×C8—24no—
143C22×C8—24no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—42no—
148C22×C8—42no—
149C22×C8—42no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8—42no—
153C22×C8—42no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—84no—
207C2×C4⋊C4—84no—
208C2×C4⋊C4SD1614yes—
209C2×C4⋊C4Q1614yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42C2411yescharacteristic
229#106—48no—
230C4⋊C8—24no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—42no—
235C4⋊C8C2×D414yes—
236#103—416no—
237#103—48no—
238C4×C8—22no—
239C4×C8—22no—
240C4×C8—42no—
241#95—28no—
242#95—28no—
243#95—22no—
244#95—22no—
245C24#12011yescharacteristic
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—28no—
255C2×Q8—28no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—416no—
259C2×Q8—42no—
260C2×Q8—44no—
261C2×Q8—42no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—82no—
265C2×Q8—82no—
266C2×Q8—88no—
267C2×Q8C2×SD1614yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4#12312yes—
301C22×C4#12312yes—
302C22×C4#12412yes—
303C22×C4C2×SD1612yes—
304C22×C4C2×SD1612yes—
305C22×C4C2×Q1612yes—
306C22×C4#13512yes—
307C22×C4#13712yes—
308C22×C4#13712yes—
309C22×C4#13712yes—
310C22×C4#13712yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—48no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—48no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—84no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—82no—
348C4⋊C4—88no—
349C4⋊C4—88no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4C2×SD1614yes—
353C4⋊C4C2×Q1614yes—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—44no—
359C42—44no—
360C42#12112yes—
361C42#12112yes—
362C42#12112yes—
363C42#12312yes—
364C42#12312yes—
365C42#12412yes—
366C42C22×D411yescharacteristic
367C42C2×#4111yescharacteristic
368C42C2×#4111yescharacteristic
369C42C2×#4111yescharacteristic
370C23#39412yes—
371C23#44812yes—
372C23#44812yes—
373C23#45112yes—
374C23#45212yes—
375C23#45312yes—
376C23#45512yes—
377C23C2×#12011yescharacteristic
378Q8—24no—
379Q8—24no—
380Q8—48no—
381Q8—42no—
382Q8—42no—
383Q8—48no—
384Q8—84no—
385Q8—88no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—48no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—84no—
411C2×C4—88no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—164no—
416C2×C4#39412yes—
417C2×C4#39412yes—
418C2×C4#39412yes—
419C2×C4#44812yes—
420C2×C4#44812yes—
421C2×C4#45112yes—
422C2×C4#45212yes—
423C2×C4#46114yes—
424C2×C4#46214yes—
425C2×C4#46514yes—
426C2×C4#47014yes—
427C2×C4#48314yes—
428C2×C4#48514yes—
429C2×C4#48714yes—
430C2×C4#49114yes—
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C2×C4C2×#12111yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4C2×#12311yescharacteristic
436C2×C4C2×#12411yescharacteristic
437C2×C4C22×SD1611yescharacteristic
438C2×C4C22×SD1611yescharacteristic
439C2×C4C22×Q1611yescharacteristic
440C2×C4C2×#13511yescharacteristic
441C2×C4C2×#13711yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4C2×#13711yescharacteristic
444C2×C4C2×#13711yescharacteristic
445C8—44no—
446C8—44no—
447C8—82no—
448C22#162314yes—
449C22#163214yes—
450C22#164314yes—
451C22#164514yes—
452C22#203514yes—
453C22#203714yes—
454C22#204314yes—
455C22C2×#39411yescharacteristic
456C22C2×#44811yescharacteristic
457C22C2×#44811yescharacteristic
458C22C2×#45111yescharacteristic
459C22C2×#45211yescharacteristic
460C22C2×#45311yescharacteristic
461C22C2×#45511yescharacteristic
462C4—21no—
463C4—21no—
464C4—22no—
465C4—44no—
466C4—44no—
467C4—44no—
468C4—82no—
469C4—84no—
470C4—162no—
471C4#203514yes—
472C4#204314yes—
473C4C2×#46111yescharacteristic
474C4C2×#46211yescharacteristic
475C4C2×#46511yescharacteristic
476C4C2×#47011yescharacteristic
477C4C2×#48311yescharacteristic
478C4C2×#48511yescharacteristic
479C4C2×#48711yescharacteristic
480C4C2×#49111yescharacteristic
481C2#1041518yesdirect product factor
482C2C2×#163211yescharacteristic
483C2C2×#164311yescharacteristic
484C2C2×#162311yescharacteristic
485C2C2×#164511yescharacteristic
486C2C2×#203511yescharacteristic
487C2C2×#203711yescharacteristic
488C2C2×#204311yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10415C2direct (N × H)
481. C2#10415direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#19562
C2×#19482
C2×#18032
C22×#3332
C22×Q8⋊C82
#104152
C2×#12792