Finite Group Databasebeta

← C2×#7251 #127495

Subgroups — C2×#7251 #127495

Order 512.

1286 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#343C211yesmaximal, characteristic
2C22×#324C211yesmaximal, characteristic
3C2×#1441C211yesmaximal, characteristic
4#7251C214yesmaximal, direct product factor
5C23×M16—21no—
6C23×C4⋊C4C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C22×C22⋊C8—21no—
9C22×#95—21no—
10C22×#95—21no—
11C22×#95C411yescharacteristic
12C2×#343—21no—
13C2×#343—22no—
14C2×#343C414yescomplemented
15C2×#343C2212yes—
16C2×#324—21no—
17C2×#324—22no—
18C2×#324C414yescomplemented
19C2×#324C2212yes—
20#1441—21no—
21#1441—21no—
22#1441C412yes—
23#1441C2212yes—
24C24×C4—21no—
25C24×C4—21no—
26C24×C4C2×C411yescharacteristic
27C24×C4D411yescharacteristic
28C24×C4D411yescharacteristic
29C22×M16—24no—
30C22×M16—24no—
31C22×M16—24no—
32C22×M16—22no—
33C22×M16—42no—
34C22×M16—41no—
35C22×M16—42no—
36C22×M16—42no—
37C23×C8—41no—
38C23×C8—41no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—41no—
41C22×C4⋊C4—42no—
42C22×C4⋊C4—41no—
43C22×C4⋊C4—41no—
44C22×C4⋊C4—41no—
45C22×C4⋊C4—44no—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D412yes—
53C22×C4⋊C4Q812yes—
54C22×C4⋊C4C2311yescharacteristic
55C2×C22⋊C8—24no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—41no—
58C2×C22⋊C8—42no—
59C2×#95—21no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—42no—
65C2×#95—44no—
66C2×#95—42no—
67C2×#95—42no—
68C2×#95C2×C412yes—
69#343—24no—
70#343—22no—
71#343C2×C414yescomplemented
72#324—24no—
73#324—22no—
74#324C2×C414yescomplemented
75C25C22⋊C411yescharacteristic
76C23×C4—22no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—21no—
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—42no—
101C23×C4—41no—
102C23×C4—42no—
103C23×C4—41no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4C4212yes—
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C412yes—
114C23×C4C4⋊C412yes—
115C23×C4C4⋊C412yes—
116C23×C4C4⋊C412yes—
117C23×C4D812yes—
118C23×C4SD1612yes—
119C23×C4SD1612yes—
120C23×C4Q1612yes—
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×M16—216no—
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—48no—
129C2×M16—44no—
130C2×M16—48no—
131C2×M16—44no—
132C2×M16—44no—
133C2×M16—42no—
134C2×M16—48no—
135C2×M16—42no—
136C2×M16—42no—
137C22×C8—44no—
138C22×C8—44no—
139C22×C8—44no—
140C22×C8—42no—
141C22×C8—44no—
142C22×C8—42no—
143C22×C8—82no—
144C22×C8—82no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4C4212yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C4⋊C412yes—
172C2×C4⋊C4C4⋊C412yes—
173C2×C4⋊C4C4⋊C412yes—
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C2×D411yescharacteristic
177C2×C4⋊C4C2×Q811yescharacteristic
178C22⋊C8—24no—
179C22⋊C8—44no—
180C22⋊C8—42no—
181#95—42no—
182#95—42no—
183#95—44no—
184#95—44no—
185#95—44no—
186#95—42no—
187#95—42no—
188C24—22no—
189C24—22no—
190C24—22no—
191C24—22no—
192C24#9512yes—
193C24#9911yescharacteristic
194C24#9912yes—
195C24#9911yescharacteristic
196C24#10212yes—
197C24#10312yes—
198C24#10412yes—
199C24#10412yes—
200C24C2×C22⋊C411yescharacteristic
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—48no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—44no—
255C22×C4—41no—
256C22×C4—44no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—48no—
268C22×C4—42no—
269C22×C4—48no—
270C22×C4—42no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—84no—
286C22×C4—84no—
287C22×C4—84no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—84no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4#9512yes—
297C22×C4#9512yes—
298C22×C4#9512yes—
299C22×C4#10212yes—
300C22×C4#10214yes—
301C22×C4#10312yes—
302C22×C4#10314yes—
303C22×C4#10614yes—
304C22×C4#10714yes—
305C22×C4C2×C4211yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C4⋊C411yescharacteristic
309C22×C4C2×C4⋊C411yescharacteristic
310C22×C4C2×C4⋊C411yescharacteristic
311C22×C4C2×D811yescharacteristic
312C22×C4C2×SD1611yescharacteristic
313C22×C4C2×SD1611yescharacteristic
314C22×C4C2×Q1611yescharacteristic
315M16—28no—
316M16—416no—
317M16—44no—
318M16—44no—
319M16—44no—
320C2×C8—416no—
321C2×C8—44no—
322C2×C8—44no—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—82no—
326C2×C8—82no—
327C2×C8—88no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—48no—
341C23—24no—
342C23—24no—
343C23—24no—
344C23—24no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—24no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—24no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23#32714yes—
362C23#32712yes—
363C23#32712yes—
364C23#32814yes—
365C23#32812yes—
366C23#32812yes—
367C23#33914yes—
368C23#34014yes—
369C23#34212yes—
370C23#34212yes—
371C23AΣL(1,𝔽16)12yes—
372C23#35212yes—
373C23#35512yes—
374C23#35612yes—
375C23C2×#9511yescharacteristic
376C23C2×#9911yescharacteristic
377C23C2×#9911yescharacteristic
378C23C2×#10211yescharacteristic
379C23C2×#10311yescharacteristic
380C23C2×#10411yescharacteristic
381C23C2×#10411yescharacteristic
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—816no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—88no—
453C2×C4—88no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4#34014yes—
458C2×C4C2×#10211yescharacteristic
459C2×C4C2×#10311yescharacteristic
460C2×C4C2×#10611yescharacteristic
461C2×C4C2×#10711yescharacteristic
462C8—48no—
463C8—84no—
464C22—21no—
465C22—21no—
466C22—22no—
467C22—21no—
468C22—22no—
469C22—21no—
470C22—28no—
471C22—28no—
472C22—28no—
473C22—28no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—21no—
478C22—22no—
479C22—21no—
480C22—21no—
481C22—22no—
482C22—21no—
483C22—21no—
484C22—22no—
485C22—21no—
486C22#129414yes—
487C22#129414yes—
488C22#134514yes—
489C22#134614yes—
490C22#135314yes—
491C22#135414yes—
492C22#139014yes—
493C22#139414yes—
494C22C2×#32711yescharacteristic
495C22C2×#32711yescharacteristic
496C22C2×#32811yescharacteristic
497C22C2×#32811yescharacteristic
498C22C2×#33911yescharacteristic
499C22C2×#34011yescharacteristic
500C22C2×#34211yescharacteristic
501C22C2×AΣL(1,𝔽16)11yescharacteristic
502C22C2×#35211yescharacteristic
503C22C2×#35511yescharacteristic
504C22C2×#35611yescharacteristic
505C4—21no—
506C4—24no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—41no—
511C4—42no—
512C4—41no—
513C4—41no—
514C4—42no—
515C4—41no—
516C4—84no—
517C4—84no—
518C2—21no—
519C2—24no—
520C2—21no—
521C2—21no—
522C2—21no—
523C2#725118yesdirect product factor
524C2C2×#129411yescharacteristic
525C2C2×#134511yescharacteristic
526C2C2×#134611yescharacteristic
527C2C2×#135311yescharacteristic
528C2C2×#135411yescharacteristic
529C2C2×#139011yescharacteristic
530C2C2×#139411yescharacteristic
subgroupG/H
center, socleC24#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7251C2direct (N × H)
14. C2×#343C4semidirect
18. C2×#324C4semidirect
71. #343C2×C4semidirect
74. #324C2×C4semidirect
523. C2#7251direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3432
C22×#3242
C2×#14412
#72512