Finite Group Databasebeta

← C2×#8762 #190200

Subgroups — C2×#8762 #190200

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2353C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#8762C218yesmaximal, direct product factor
5C2×#1294C212yesmaximal, complemented
6C2×#1289C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C22×#99—21no—
9C22×C22⋊C8C2212yes—
10#2353—22no—
11#2353—22no—
12#2353C414yescomplemented
13#2353C2214yes—
14C2×#388—21no—
15C2×#387—21no—
16C2×#386—21no—
17C2×#383C411yescharacteristic, complemented
18C2×#381C411yescharacteristic, complemented
19C2×#380C2212yes—
20C2×#380C2211yescharacteristic
21C2×C4×C22⋊C4—21no—
22C2×C4×C22⋊C4C411yescharacteristic, complemented
23C2×#376C411yescharacteristic, complemented
24C22×#95C2211yescharacteristic
25#1799—22no—
26#1799C2214yes—
27C2×#342—21no—
28#1459—22no—
29#1459C2214yes—
30C2×#324—21no—
31C2×#323—21no—
32#1294C2218yescomplemented
33#1289C2218yescomplemented
34C23×D4D411yescharacteristic
35C24×C4—21no—
36C24×C4C2×C411yescharacteristic
37C24×C4C2311yescharacteristic
38C23×C8—22no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4C2×C412yes—
44C2×C4×D4D412yes—
45C2×C4×D4D412yes—
46C2×C4×D4C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D411yescharacteristic
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4D411yescharacteristic
55C22×C22⋊C4D411yescharacteristic
56C22×C42—21no—
57C22×C42—21no—
58C22×C42C2×C411yescharacteristic
59C2×#99—24no—
60C2×#99—44no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8C2314yes—
67#388—24no—
68#388—24no—
69#387—24no—
70#387—24no—
71#386—24no—
72#386—24no—
73#383—22no—
74#383C2×C414yescomplemented
75#381—22no—
76#381C2×C414yescomplemented
77#380—21no—
78#380—21no—
79#380—22no—
80#380—22no—
81#380—22no—
82#380C2×C412yes—
83#380C2314yes—
84#380C2312yes—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—24no—
87C4×C22⋊C4—24no—
88C4×C22⋊C4C2×C414yescomplemented
89#376—22no—
90#376C2×C414yescomplemented
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95C2×C412yes—
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2312yes—
105#342—24no—
106#324—24no—
107#323—24no—
108C25C22⋊C411yescharacteristic
109C25C2×D411yescharacteristic
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—24no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4C22⋊C412yes—
116C22×D4C2×D412yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—44no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—82no—
132C23×C4—82no—
133C23×C4—82no—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C412yes—
138C23×C4C22×C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4#4112yes—
145C23×C4#4112yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C22×C8—24no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8—84no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—84no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C2×D412yes—
189C2×C22⋊C4C2×D412yes—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—44no—
197C2×C42—82no—
198C2×C42C22⋊C412yes—
199C2×C42C22×C412yes—
200#99—48no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205#95—21no—
206#95—21no—
207#95—22no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—21no—
213#95—21no—
214#95—22no—
215#95—24no—
216#95—42no—
217#95C22×C412yes—
218#95C22×C412yes—
219#95C22×C412yes—
220#95C22×C412yes—
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—
234C24#13514yes—
235C24AGL(1,ℤ/8)12yes—
236C24#13712yes—
237C24C22×D411yescharacteristic
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—44no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—81no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—88no—
304C22×C4—88no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4—84no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4#11711yescharacteristic
314C22×C4#11711yescharacteristic
315C22×C4#11711yescharacteristic
316C22×C4#11711yescharacteristic
317C22×C4#11712yes—
318C22×C4#11712yes—
319C22×C4#12312yes—
320C22×C4#12312yes—
321C22×C4#12312yes—
322C22×C4#12312yes—
323C22×C4#12312yes—
324C22×C4#12312yes—
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4C2×#4112yes—
329C2×C8—44no—
330C2×C8—84no—
331C2×C8—84no—
332C2×C8—84no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—84no—
338C4⋊C4—84no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—48no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—84no—
349C22⋊C4—84no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—48no—
355C42—84no—
356C23—22no—
357C23—24no—
358C23—22no—
359C23—24no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—48no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—84no—
381C23—84no—
382C23#38112yes—
383C23#38111yescharacteristic
384C23#38111yescharacteristic
385C23#41612yes—
386C23#41612yes—
387C23#41712yes—
388C23#41712yes—
389C23#48214yes—
390C23#48214yes—
391C23#48514yes—
392C23#48514yes—
393C23C22×C22⋊C411yescharacteristic
394C23C2×#12312yes—
395C23C2×#13512yes—
396C23C2×AGL(1,ℤ/8)11yescharacteristic
397C23C2×#13711yescharacteristic
398D4—44no—
399D4—44no—
400D4—48no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—48no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—88no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—168no—
459C2×C4#38111yescharacteristic
460C2×C4#38111yescharacteristic
461C2×C4#38112yes—
462C2×C4C2×#11711yescharacteristic
463C2×C4C2×#11711yescharacteristic
464C2×C4C2×#12312yes—
465C8—84no—
466C22—24no—
467C22—24no—
468C22—28no—
469C22—28no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—22no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—22no—
478C22—41no—
479C22—41no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—44no—
484C22—44no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22—82no—
490C22—84no—
491C22—84no—
492C22#182512yes—
493C22#182514yes—
494C22#182512yes—
495C22#182712yes—
496C22#182812yes—
497C22C2×#38111yescharacteristic
498C22C2×#41611yescharacteristic
499C22C2×#41711yescharacteristic
500C22C2×#48212yes—
501C22C2×#48512yes—
502C4—22no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—44no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4—164no—
516C2—21no—
517C2—21no—
518C2—22no—
519C2—22no—
520C2—22no—
521C2—24no—
522C2—82no—
523C2#876214yesdirect product factor
524C2C2×#182511yescharacteristic
525C2C2×#182711yescharacteristic
526C2C2×#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. #8762C2direct (N × H)
5. C2×#1294C2semidirect
6. C2×#1289C2semidirect
12. #2353C4semidirect
17. C2×#383C4semidirect
18. C2×#381C4semidirect
22. C2×C4×C22⋊C4C4semidirect
23. C2×#376C4semidirect
32. #1294C22semidirect
33. #1289C22semidirect
74. #383C2×C4semidirect
76. #381C2×C4semidirect
88. C4×C22⋊C4C2×C4semidirect
90. #376C2×C4semidirect
523. C2#8762direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23532
C2×#17992
C2×#14592
#87622
C2×#12942
C2×#12892