Finite Group Databasebeta

← C2×#7247 #127459

Subgroups — C2×#7247 #127459

Order 512.

846 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1869C211yesmaximal, characteristic
2C2×#1466C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4#7247C214yesmaximal, direct product factor
5C24×C8C411yescharacteristic
6C22×C4×D4C2211yescharacteristic
7C2×#432—21no—
8C2×#407—21no—
9C22×C22⋊C8—21no—
10C22×C22⋊C8C411yescharacteristic
11#1869—21no—
12#1869—21no—
13#1869C412yes—
14#1869C2212yes—
15C2×#336—21no—
16#1466—21no—
17#1466—21no—
18#1466C412yes—
19#1466C2212yes—
20#1465—21no—
21#1465—21no—
22#1465C412yes—
23#1465C2212yes—
24C2×#324C412yes—
25C2×#323—21no—
26C2×#323—21no—
27C2×#323C412yes—
28C23×D4Q811yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—41no—
32C23×C8—21no—
33C23×C8—21no—
34C23×C8—21no—
35C23×C8—44no—
36C23×C8—41no—
37C23×C8—41no—
38C23×C8C2×C412yes—
39C2×C4×D4—41no—
40C2×C4×D4—41no—
41C2×C4×D4—41no—
42C2×C4×D4—41no—
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4C2×C411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×C4×D4D411yescharacteristic
49C2×C4×D4Q811yescharacteristic
50C2×C4×D4C2311yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42D411yescharacteristic
55#432—22no—
56#432—22no—
57#432—41no—
58#432—41no—
59C2×C4⋊C8—41no—
60#407—22no—
61#407—22no—
62#407—41no—
63#407—41no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—44no—
70C2×C22⋊C8—41no—
71C2×C22⋊C8—41no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8C2×C412yes—
74C2×#97—41no—
75#336—22no—
76#336—22no—
77#324—22no—
78#324C2×C414yes—
79#323—22no—
80#323—22no—
81#323—22no—
82#323—22no—
83#323—22no—
84#323C2×C414yes—
85C25C4⋊C411yescharacteristic
86C25C4⋊C411yescharacteristic
87C22×D4—24no—
88C22×D4—41no—
89C22×D4—41no—
90C22×D4—42no—
91C22×D4C4⋊C411yescharacteristic
92C22×D4C4⋊C411yescharacteristic
93C22×D4C4⋊C411yescharacteristic
94C22×D4C2×Q811yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4C4211yescharacteristic
108C23×C4C4211yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—42no—
121C2×M16—42no—
122C2×M16—82no—
123C2×M16—82no—
124C22×C8—24no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—44no—
128C22×C8—44no—
129C22×C8—42no—
130C22×C8—42no—
131C22×C8—48no—
132C22×C8—44no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8—42no—
136C22×C8—44no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—41no—
142C22×C8—41no—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—41no—
153C4×D4—41no—
154C4×D4—41no—
155C4×D4—41no—
156C4×D4—41no—
157C4×D4—41no—
158C4×D4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4C4211yescharacteristic
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C4⋊C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4C4211yescharacteristic
170C2×C22⋊C4C22⋊C411yescharacteristic
171C2×C22⋊C4C22⋊C411yescharacteristic
172C2×C22⋊C4C22⋊C411yescharacteristic
173C2×C22⋊C4C4⋊C411yescharacteristic
174C2×C22⋊C4C4⋊C411yescharacteristic
175C2×C22⋊C4C22×C411yescharacteristic
176C2×C22⋊C4C2×D411yescharacteristic
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C4⋊C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C4⋊C8—42no—
186C4⋊C8—42no—
187C22⋊C8—24no—
188C22⋊C8—24no—
189C22⋊C8—44no—
190C22⋊C8—44no—
191C22⋊C8—42no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C22⋊C8—42no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197#97—42no—
198#97—42no—
199C24—22no—
200C24—21no—
201C24—21no—
202C24—22no—
203C24—21no—
204C24—22no—
205C24—21no—
206C24—42no—
207C24—42no—
208C24—42no—
209C24—42no—
210C24#9511yescharacteristic
211C24#9511yescharacteristic
212C24#9511yescharacteristic
213C24#10814yes—
214C24C2×C4⋊C411yescharacteristic
215C24C2×C4⋊C411yescharacteristic
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—48no—
229C2×D4—48no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4#9511yescharacteristic
286C22×C4#9511yescharacteristic
287C22×C4#9511yescharacteristic
288C22×C4#9511yescharacteristic
289C22×C4#9511yescharacteristic
290C22×C4#9911yescharacteristic
291C22×C4#9911yescharacteristic
292C22×C4#9911yescharacteristic
293C22×C4#9911yescharacteristic
294C22×C4C2×C4211yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C4⋊C411yescharacteristic
299M16—82no—
300M16—82no—
301M16—84no—
302C2×C8—24no—
303C2×C8—42no—
304C2×C8—44no—
305C2×C8—48no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—42no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—82no—
316C2×C8—82no—
317C2×C8—84no—
318C2×C8—84no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—24no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—48no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—48no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—48no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—81no—
373C23—81no—
374C23—84no—
375C23#33812yes—
376C23#34112yes—
377C23#34112yes—
378C23#34211yescharacteristic
379C23#34211yescharacteristic
380C23C2×#9511yescharacteristic
381C23C2×#10812yes—
382D4—44no—
383D4—44no—
384D4—44no—
385D4—44no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4#34211yescharacteristic
440C2×C4#34211yescharacteristic
441C2×C4AΣL(1,𝔽16)12yes—
442C2×C4#35212yes—
443C2×C4C2×#9911yescharacteristic
444C2×C4C2×#9911yescharacteristic
445C8—44no—
446C8—82no—
447C8—84no—
448C8—84no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—24no—
454C22—24no—
455C22—22no—
456C22—22no—
457C22—42no—
458C22—42no—
459C22—48no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—48no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—81no—
476C22—81no—
477C22—84no—
478C22—84no—
479C22#129312yes—
480C22#129312yes—
481C22#139012yes—
482C22#139512yes—
483C22C2×#33811yescharacteristic
484C22C2×#34111yescharacteristic
485C22C2×#34211yescharacteristic
486C4—21no—
487C4—21no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4#139012yes—
497C4C2×AΣL(1,𝔽16)11yescharacteristic
498C4C2×#35211yescharacteristic
499C2—21no—
500C2—22no—
501C2—21no—
502C2—42no—
503C2—42no—
504C2—81no—
505C2—81no—
506C2#724714yesdirect product factor
507C2C2×#129311yescharacteristic
508C2C2×#139011yescharacteristic
509C2C2×#139511yescharacteristic
subgroupG/H
center, socleC23#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7247C2direct (N × H)
506. C2#7247direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#18692
C2×#14662
C2×#14652
#72472