Finite Group Databasebeta

← C2×#10171 #236903

Subgroups — C2×#10171 #236903

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2C2×#3139C211yesmaximal, characteristic
3C22×#487C211yesmaximal, characteristic
4C2×#1997C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6C2×#1493C211yesmaximal, characteristic
7#10171C218yesmaximal, direct product factor
8C2×#1278C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—22no—
13#3209—22no—
14#3209—22no—
15#3209C2214yes—
16C2×#494—21no—
17#3139—22no—
18#3139—22no—
19#3139—22no—
20#3139C2214yes—
21C2×#491—22no—
22C2×#487—22no—
23C2×#487—22no—
24C2×#487—21no—
25C2×#487—22no—
26C2×#487—22no—
27C2×#487—22no—
28C2×#487—21no—
29C2×#487—21no—
30C2×#487C2214yes—
31C2×#462—22no—
32C2×#451—22no—
33C2×#431C2211yescharacteristic
34C2×#423C2211yescharacteristic
35C2×#419—22no—
36C22×#103—22no—
37C2×C4×M16C2211yescharacteristic
38C22×C4×C8C2211yescharacteristic
39#1997—22no—
40#1997C2214yes—
41C2×#343—22no—
42#1546—22no—
43#1546C2214yes—
44C2×#332—22no—
45#1493—22no—
46#1493C2214yes—
47C2×Q8⋊C8—22no—
48#1278C2214yes—
49C23×Q8—21no—
50C23×Q8D412yes—
51C22×Q16—24no—
52C22×M16—21no—
53C22×M16D412yes—
54C23×C8—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533D412yes—
62#533C2312yes—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8D412yes—
83C2×C4⋊Q8D411yescharacteristic
84C2×C4⋊Q8D412yes—
85C2×C4⋊Q8D412yes—
86C2×C4⋊Q8D411yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#125—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×#117—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C42C2311yescharacteristic
100#494—24no—
101#494—24no—
102#491—28no—
103#491—28no—
104#487—28no—
105#487—28no—
106#487—24no—
107#487—24no—
108#487—24no—
109#487—28no—
110#487—24no—
111#487—24no—
112#487—24no—
113#487—28no—
114#462—28no—
115#462—28no—
116#451—28no—
117#451—28no—
118#431—21no—
119#431—21no—
120#431D412yes—
121#431C2312yes—
122#423—21no—
123#423—21no—
124#423—21no—
125#423C2312yes—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8D412yes—
129#419—28no—
130#419—28no—
131C2×#103—28no—
132C2×#103—28no—
133C2×#103—24no—
134C2×#103—24no—
135C2×#103—24no—
136C2×#103—28no—
137C2×C22⋊C8—21no—
138C4×M16—21no—
139C4×M16—21no—
140C4×M16D412yes—
141C4×M16C2312yes—
142C2×#97—21no—
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C2×C4×C8—21no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8D412yes—
149C2×C4×C8D412yes—
150C2×C4×C8C2312yes—
151#343—28no—
152#332—28no—
153Q8⋊C8—28no—
154C22×Q8—22no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—21no—
163C22×Q8—22no—
164C22×Q8—48no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8—41no—
168C22×Q8—44no—
169C22×Q8—44no—
170C22×Q8C2×D414yes—
171C23×C4—22no—
172C23×C4—41no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4#4111yescharacteristic
176C2×Q16—216no—
177C2×Q16—88no—
178C2×M16—22no—
179C2×M16—24no—
180C2×M16—24no—
181C2×M16—22no—
182C2×M16—41no—
183C2×M16—44no—
184C2×M16C2×D414yes—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—44no—
191C22×C8—41no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—21no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C4⋊Q8—22no—
211C4⋊Q8—22no—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—41no—
216C4⋊Q8—41no—
217C4⋊Q8—41no—
218C4⋊Q8—41no—
219C4⋊Q8—41no—
220C4⋊Q8—41no—
221C4⋊Q8—41no—
222C4⋊Q8—41no—
223C4⋊Q8—41no—
224C4⋊Q8—41no—
225C4⋊Q8—41no—
226C4⋊Q8—41no—
227C4⋊Q8—41no—
228C4⋊Q8—41no—
229C4⋊Q8C2×D414yes—
230C4⋊Q8C2×D412yes—
231C4⋊Q8C2×D412yes—
232#125—22no—
233#125—22no—
234#125—41no—
235#125—41no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240C22⋊Q8—41no—
241C22⋊Q8—41no—
242C22⋊Q8—41no—
243C22⋊Q8—41no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—41no—
249C4×Q8—41no—
250C4×Q8—41no—
251C4×Q8—41no—
252#117—22no—
253#117—22no—
254#117—41no—
255#117—41no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C42—41no—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42#4111yescharacteristic
299C2×C42C2411yescharacteristic
300C4⋊C8—22no—
301C4⋊C8—22no—
302C4⋊C8—22no—
303C4⋊C8—41no—
304C4⋊C8—41no—
305C4⋊C8C2×D414yes—
306#103—232no—
307#103—216no—
308#103—216no—
309#103—216no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312#97—22no—
313#97—22no—
314C4×C8—22no—
315C4×C8—22no—
316C4×C8—22no—
317C4×C8—22no—
318C4×C8—22no—
319C4×C8—22no—
320C4×C8—22no—
321C4×C8C2×D414yes—
322C24#12111yescharacteristic
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—42no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—44no—
343C2×Q8—48no—
344C2×Q8—44no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×Q8—82no—
349C2×Q8—82no—
350C2×Q8—82no—
351C2×Q8—82no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4#12011yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12011yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4#12011yescharacteristic
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12711yescharacteristic
378C22×C4#12711yescharacteristic
379C22×C4#13514yes—
380C22×C4#13714yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C2×#4111yescharacteristic
384Q16—816no—
385M16—24no—
386M16—24no—
387M16—44no—
388M16—44no—
389M16—82no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—44no—
394C2×C8—44no—
395C2×C8—44no—
396C2×C8—44no—
397C2×C8—44no—
398C2×C8—42no—
399C2×C8—82no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C42—42no—
431C42—42no—
432C42#12011yescharacteristic
433C42#12011yescharacteristic
434C42#12011yescharacteristic
435C42#12011yescharacteristic
436C42#12011yescharacteristic
437C42#12011yescharacteristic
438C42#12111yescharacteristic
439C42#12111yescharacteristic
440C42#12111yescharacteristic
441C42#12111yescharacteristic
442C42#12711yescharacteristic
443C42#12711yescharacteristic
444C42C22×D411yescharacteristic
445C42C22×D411yescharacteristic
446C42C22×D411yescharacteristic
447C42C22×D411yescharacteristic
448C23—22no—
449C23—22no—
450C23#39211yescharacteristic
451C23#39211yescharacteristic
452C23#46414yes—
453C23C2×#12111yescharacteristic
454Q8—48no—
455Q8—48no—
456Q8—44no—
457Q8—48no—
458Q8—44no—
459Q8—84no—
460Q8—82no—
461Q8—82no—
462Q8—82no—
463Q8—84no—
464Q8—84no—
465Q8—84no—
466Q8—82no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—82no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4#39211yescharacteristic
490C2×C4#39211yescharacteristic
491C2×C4#39211yescharacteristic
492C2×C4#39211yescharacteristic
493C2×C4#39211yescharacteristic
494C2×C4#39211yescharacteristic
495C2×C4#45214yes—
496C2×C4#45214yes—
497C2×C4#45514yes—
498C2×C4#46414yes—
499C2×C4#47314yes—
500C2×C4#49512yes—
501C2×C4#49512yes—
502C2×C4#49712yes—
503C2×C4#49712yes—
504C2×C4C2×#12011yescharacteristic
505C2×C4C2×#12011yescharacteristic
506C2×C4C2×#12011yescharacteristic
507C2×C4C2×#12111yescharacteristic
508C2×C4C2×#12111yescharacteristic
509C2×C4C2×#12711yescharacteristic
510C2×C4C2×#13512yes—
511C2×C4C2×#13712yes—
512C8—44no—
513C8—44no—
514C8—44no—
515C8—82no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—28no—
521C22#200412yes—
522C22#201512yes—
523C22#201512yes—
524C22#202012yes—
525C22C2×#39211yescharacteristic
526C22C2×#46412yes—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—84no—
532C4—82no—
533C4—84no—
534C4—82no—
535C4#200412yes—
536C4#201512yes—
537C4#201512yes—
538C4#202012yes—
539C4C2×#45212yes—
540C4C2×#45512yes—
541C4C2×#47312yes—
542C4C2×#49511yescharacteristic
543C4C2×#49711yescharacteristic
544C2—22no—
545C2—22no—
546C2#1017114yesdirect product factor
547C2C2×#200411yescharacteristic
548C2C2×#201511yescharacteristic
549C2C2×#202011yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ 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
7. #10171C2direct (N × H)
546. C2#10171direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32092
C2×#31392
C22×#4872
C2×#19972
C2×#15462
C2×#14932
#101712
C2×#12782