Finite Group Databasebeta

← C2×#8778 #190351

Subgroups — C2×#8778 #190351

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2357C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4#8778C218yesmaximal, direct product factor
5C2×#1297C212yesmaximal, complemented
6C2×#1284C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C22×#99—21no—
9C2×#407C2212yes—
10#2357—22no—
11#2357—22no—
12#2357C414yescomplemented
13#2357C2214yes—
14C2×#390—21no—
15C2×#388—21no—
16C2×#388—21no—
17C2×#382C411yescharacteristic, complemented
18C2×#381C411yescharacteristic, complemented
19C2×#381C2211yescharacteristic
20C2×C4×C4⋊C4—21no—
21C2×C4×C4⋊C4C411yescharacteristic, complemented
22C2×C4×C22⋊C4C411yescharacteristic, complemented
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25C4×#99—22no—
26C4×#99C2214yes—
27C2×#342—21no—
28#1465—22no—
29#1465C2214yes—
30C2×#324—21no—
31C2×#323—21no—
32#1297C2218yescomplemented
33#1284C2218yescomplemented
34C23×D4D411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2311yescharacteristic
37C22×M16—22no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4C2×C412yes—
43C2×C4×D4D412yes—
44C2×C4×D4D412yes—
45C2×C4×D4C2312yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D411yescharacteristic
49C22×C22⋊C4—22no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4D411yescharacteristic
55C22×C22⋊C4D411yescharacteristic
56C22×C42—21no—
57C22×C42—21no—
58C22×C42—21no—
59C22×C42—22no—
60C22×C42C2×C411yescharacteristic
61C2×#99—28no—
62C2×#99—24no—
63#407—22no—
64#407—22no—
65#407—22no—
66#407C2314yes—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69#390—24no—
70#390—24no—
71#388—24no—
72#388—24no—
73#388—24no—
74#388—24no—
75#382—22no—
76#382C2×C414yescomplemented
77#381—22no—
78#381—21no—
79#381—21no—
80#381C2×C412yes—
81#381C2×C414yescomplemented
82#381C2312yes—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4C2×C414yescomplemented
87C4×C22⋊C4—22no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4C2×C412yes—
94C4×C22⋊C4C2×C414yescomplemented
95C4×C22⋊C4C2314yes—
96C4×C22⋊C4C2312yes—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105#342—24no—
106#324—24no—
107#323—24no—
108C25C22⋊C411yescharacteristic
109C25C2×D411yescharacteristic
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—24no—
113C22×D4—42no—
114C22×D4C22⋊C412yes—
115C22×D4C2×D412yes—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—42no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C412yes—
131C23×C4C22×C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4#4112yes—
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4#4112yes—
141C23×C4C2411yescharacteristic
142C2×M16—24no—
143C2×M16—42no—
144C2×M16—84no—
145C22×C8—42no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C2×D412yes—
183C2×C22⋊C4C2×D412yes—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—24no—
191C2×C42—22no—
192C2×C42—24no—
193C2×C42—44no—
194C2×C42—42no—
195C2×C42—82no—
196C2×C42—82no—
197C2×C42—84no—
198C2×C42C22⋊C412yes—
199C2×C42C22×C412yes—
200#99—216no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205#95—22no—
206#95—24no—
207#95—24no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—22no—
213#95—24no—
214#95—42no—
215#95C22×C412yes—
216#95C22×C412yes—
217#95C22×C412yes—
218#95C22×C412yes—
219C24—22no—
220C24—22no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24C2×C22⋊C411yescharacteristic
230C24C2×C22⋊C412yes—
231C24C2×C22⋊C411yescharacteristic
232C24#12312yes—
233C24#12312yes—
234C24C22×D411yescharacteristic
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—48no—
242C2×D4—48no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—48no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—81no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4#10414yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4#11711yescharacteristic
299C22×C4#11711yescharacteristic
300C22×C4#11711yescharacteristic
301C22×C4#11711yescharacteristic
302C22×C4#11712yes—
303C22×C4#11712yes—
304C22×C4#12312yes—
305C22×C4#12312yes—
306C22×C4#12312yes—
307C22×C4#12312yes—
308C22×C4#12312yes—
309C22×C4#12312yes—
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4112yes—
313C22×C4C2×#4112yes—
314M16—84no—
315M16—84no—
316C2×C8—44no—
317C2×C8—84no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—48no—
321C4⋊C4—48no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—28no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—48no—
330C22⋊C4—48no—
331C22⋊C4—44no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—84no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—48no—
340C42—82no—
341C42—82no—
342C42—84no—
343C42—84no—
344C42—84no—
345C23—22no—
346C23—22no—
347C23—24no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—84no—
373C23—84no—
374C23—84no—
375C23—84no—
376C23#38112yes—
377C23#38111yescharacteristic
378C23#38111yescharacteristic
379C23C2×#10412yes—
380C23#42112yes—
381C23C22×C22⋊C411yescharacteristic
382C23C2×#12312yes—
383D4—48no—
384D4—48no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—48no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—816no—
439C2×C4#38111yescharacteristic
440C2×C4#38111yescharacteristic
441C2×C4#38112yes—
442C2×C4C2×#10414yes—
443C2×C4C2×#11711yescharacteristic
444C2×C4C2×#11711yescharacteristic
445C2×C4C2×#12312yes—
446C8—84no—
447C22—24no—
448C22—24no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—48no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—84no—
468C22—82no—
469C22—82no—
470C22—84no—
471C22—84no—
472C22#182612yes—
473C22#182612yes—
474C22#182712yes—
475C22#182812yes—
476C22C2×#38111yescharacteristic
477C22C22×#10411yescharacteristic
478C22C2×#42111yescharacteristic
479C4—24no—
480C4—22no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—44no—
487C4—42no—
488C4—42no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—88no—
493C2—21no—
494C2—21no—
495C2—22no—
496C2—42no—
497C2—42no—
498C2—82no—
499C2#877814yesdirect product factor
500C2C2×#182611yescharacteristic
501C2C2×#182711yescharacteristic
502C2C2×#182811yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8778C2direct (N × H)
5. C2×#1297C2semidirect
6. C2×#1284C2semidirect
12. #2357C4semidirect
17. C2×#382C4semidirect
18. C2×#381C4semidirect
21. C2×C4×C4⋊C4C4semidirect
22. C2×C4×C22⋊C4C4semidirect
32. #1297C22semidirect
33. #1284C22semidirect
76. #382C2×C4semidirect
81. #381C2×C4semidirect
86. C4×C4⋊C4C2×C4semidirect
94. C4×C22⋊C4C2×C4semidirect
499. C2#8778direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23572
C2×C4×#992
C2×#14652
#87782
C2×#12972
C2×#12842