Finite Group Databasebeta

← C2×#10589733 #11902498

Subgroups — C2×#10589733 #11902498

Order 1152.

2448 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
1C22×#3701C211yesmaximal, characteristic, complemented
2C22×#3697C211yesmaximal, characteristic, complemented
3C2×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
4C2×#63506C211yesmaximal, characteristic, complemented
5C2×C4×#611C211yesmaximal, characteristic, complemented
6#10589733C2116yesmaximal, direct product factor
7C2×#63459C211yesmaximal, characteristic, complemented
8C2×#63456C212yesmaximal, complemented
9C2×#63455C211yesmaximal, characteristic, complemented
10C2×C4⋊C4×D9C212yesmaximal, complemented
11C2×#63446C212yesmaximal, complemented
12C2×#63445C212yesmaximal, complemented
13C2×#5631—31nomaximal
14C22×Q8×C9C2211yescharacteristic
15C23×C4×C9C2211yescharacteristic
16C22×#611C2212yes—
17C22×#611C2211yescharacteristic
18C2×#3701—28no—
19C2×#3701C2218yescomplemented
20C2×#3697—28no—
21C2×#3697C2218yescomplemented
22C22×D36C2212yes—
23C22×D36C2211yescharacteristic
24C22×C4×D9C2212yes—
25C22×C4×D9C2212yes—
26C22×C4×D9C2211yescharacteristic
27C22×C4×D9C2212yes—
28C22×Dic18C2211yescharacteristic
29C9×C22⋊Q8C2218yescomplemented
30C2×C9×C4⋊C4C2212yes—
31C2×C9×C4⋊C4C2211yescharacteristic
32C2×C9×C22⋊C4C2212yes—
33C2×#3676C2211yescharacteristic
34#63506C2218yescomplemented
35C4×#611C2218yescomplemented
36C2×(C2×D9)⋊C4C2212yes—
37C2×(C2×D9)⋊C4C2212yes—
38C2×(C2×D9)⋊C4C2211yescharacteristic
39C2×C4⋊Dic9C2212yes—
40C2×Dic9⋊C4C2212yes—
41C2×Dic9⋊C4C2212yes—
42C2×Dic9⋊C4C2211yescharacteristic
43C2×C4×Dic9C2211yescharacteristic
44#63459C2218yescomplemented
45#63456C22116yescomplemented
46#63455C2218yescomplemented
47C4⋊C4×D9C22116yescomplemented
48#63446C22116yescomplemented
49#63445C22116yescomplemented
50C22×#239—31no—
51C22×#235—31no—
52C2×C3×C22⋊Q8D311yescharacteristic
53C2×#944—31no—
54C2×C4×#67—31no—
55#5631—316no—
56C2×#897—31no—
57C2×#894—32no—
58C2×#893—31no—
59C2×D3×C4⋊C4—32no—
60C2×#884—32no—
61C2×#883—32no—
62C24×C9C2311yescharacteristic
63C23×D9C2311yescharacteristic
64C23×D9C2312yes—
65C2×Q8×C9—24no—
66C2×Q8×C9C2314yes—
67C22×C4×C9—24no—
68C22×C4×C9C2312yes—
69C22×C4×C9C2311yescharacteristic
70C22×C4×C9C2314yes—
71C22×C4×C9C2312yes—
72C22×C4×C9C2311yescharacteristic
73C2×#611—28no—
74C2×#611—24no—
75C2×#611C2314yes—
76C2×#611C2318yes—
77C22×Dic9C2311yescharacteristic
78C22×Dic9C2312yes—
79C22×Dic9C2311yescharacteristic
80C22×Dic9C2311yescharacteristic
81#3701—232no—
82#3697—232no—
83C2×D36—28no—
84C2×D36D418yescomplemented
85C2×D36C2318yes—
86C2×D36C2314yes—
87C2×C4×D9—28no—
88C2×C4×D9—28no—
89C2×C4×D9—24no—
90C2×C4×D9—28no—
91C2×C4×D9C2314yes—
92C2×C4×D9C2318yes—
93C2×C4×D9C2318yes—
94C2×C4×D9C2318yes—
95C2×Dic18—24no—
96C2×Dic18C2314yes—
97C9×C4⋊C4C2318yes—
98C9×C4⋊C4C2314yes—
99C9×C22⋊C4C2318yes—
100#3676C2314yes—
101(C2×D9)⋊C4C2314yes—
102(C2×D9)⋊C4C2318yes—
103(C2×D9)⋊C4C2318yes—
104C4⋊Dic9C2318yes—
105Dic9⋊C4C2314yes—
106Dic9⋊C4C2318yes—
107Dic9⋊C4C2318yes—
108C4×Dic9C2314yes—
109C2×#547—91noSylow
110C22×C3×Q8C2×D311yescharacteristic
111C23×C3×C4C2×D311yescharacteristic
112C22×#67—32no—
113C22×#67—31no—
114C2×#239—38no—
115C2×#239—68no—
116C2×#235—38no—
117C2×#235—68no—
118C22×D12—31no—
119C22×D12—32no—
120C22×C4×D3—32no—
121C22×C4×D3—32no—
122C22×C4×D3—31no—
123C22×C4×D3—32no—
124C22×Dic6—31no—
125C3×C22⋊Q8C2×D318yes—
126C2×C3×C4⋊C4C2×D312yes—
127C2×C3×C4⋊C4C2×D311yescharacteristic
128C2×C3×C22⋊C4C2×D312yes—
129C2×#217—31no—
130#944—38no—
131C4×#67—38no—
132C2×(C2×D3)⋊C4—32no—
133C2×(C2×D3)⋊C4—32no—
134C2×(C2×D3)⋊C4—31no—
135C2×C4⋊Dic3—32no—
136C2×Dic3⋊C4—32no—
137C2×Dic3⋊C4—32no—
138C2×Dic3⋊C4—31no—
139C2×C4×Dic3—31no—
140#897—38no—
141#894—316no—
142#893—38no—
143D3×C4⋊C4—316no—
144#884—316no—
145#883—316no—
146C23×C9—22no—
147C23×C9—22no—
148C23×C9—22no—
149C23×C9C2412yes—
150C23×C9C2411yescharacteristic
151C22×D9—24no—
152C22×D9—24no—
153C22×D9—22no—
154C22×D9—22no—
155C22×D9—22no—
156C22×D9C2×D418yes—
157C22×D9C2412yes—
158C22×D9C2414yes—
159Q8×C9—28no—
160C2×C4×C9—22no—
161C2×C4×C9—24no—
162C2×C4×C9—24no—
163C2×C4×C9—28no—
164C2×C4×C9C2×D414yes—
165C2×C4×C9C2414yes—
166C2×C4×C9C2412yes—
167C2×C4×C9C2414yes—
168C2×C4×C9C2412yes—
169#611—216no—
170#611—28no—
171C2×Dic9—24no—
172C2×Dic9—22no—
173C2×Dic9—22no—
174C2×Dic9#4114yescomplemented
175C2×Dic9C2412yes—
176C2×Dic9C2414yes—
177C2×Dic9C2412yes—
178C2×Dic9C2412yes—
179D36—216no—
180D36C2×D4116yescomplemented
181C4×D9—216no—
182C4×D9—216no—
183C4×D9—28no—
184C4×D9—216no—
185Dic18—28no—
186C22×#41—91no—
187C22×#41—91no—
188#547—916no—
189C2×C4⋊Q8—91no—
190C2×#123—92no—
191C2×#123—92no—
192C2×C22⋊Q8—91no—
193C2×C22⋊Q8D911yescharacteristic, complemented
194C2×#121—92no—
195C2×C4×D4—91no—
196C2×C4×D4—91no—
197C22×C4⋊C4—92no—
198C24×C3C22×D311yescharacteristic
199C23×D3—32no—
200C23×D3—31no—
201C2×C3×Q8—24no—
202C2×C3×Q8C22×D314yes—
203C22×C3×C4—24no—
204C22×C3×C4C22×D312yes—
205C22×C3×C4C22×D311yescharacteristic
206C22×C3×C4C22×D312yes—
207C22×C3×C4C22×D314yes—
208C22×C3×C4C22×D311yescharacteristic
209C2×#67—38no—
210C2×#67—34no—
211C2×#67—64no—
212C2×#67—68no—
213C22×Dic3—31no—
214C22×Dic3—31no—
215C22×Dic3—32no—
216C22×Dic3—31no—
217#239—632no—
218#235—632no—
219C2×D12—38no—
220C2×D12—34no—
221C2×D12—38no—
222C2×D12—68no—
223C2×C4×D3—38no—
224C2×C4×D3—38no—
225C2×C4×D3—38no—
226C2×C4×D3—34no—
227C2×C4×D3—68no—
228C2×C4×D3—68no—
229C2×C4×D3—68no—
230C2×C4×D3—64no—
231C2×Dic6—34no—
232C2×Dic6—64no—
233C3×C4⋊C4C22×D318yes—
234C3×C4⋊C4C22×D314yes—
235C3×C22⋊C4C22×D318yes—
236#217—34no—
237(C2×D3)⋊C4—38no—
238(C2×D3)⋊C4—38no—
239(C2×D3)⋊C4—34no—
240C4⋊Dic3—38no—
241Dic3⋊C4—38no—
242Dic3⋊C4—38no—
243Dic3⋊C4—34no—
244C4×Dic3—34no—
245C22×C9—22no—
246C22×C9—22no—
247C22×C9—22no—
248C22×C9—44no—
249C22×C9C22×D412yes—
250C22×C9C2×#4112yes—
251C22×C92−1+412yes—
252C22×C9C2511yescharacteristic
253C2×D9—216no—
254C2×D9—24no—
255C2×D9—24no—
256C2×D9—22no—
257C2×D9—22no—
258C2×D9—22no—
259C2×D9—44no—
260C2×D9C22×D418yes—
261C4×C9—22no—
262C4×C9—24no—
263C4×C9—24no—
264C4×C9C22×D414yes—
265Dic9—24no—
266Dic9—22no—
267Dic9—22no—
268Dic9C2×#4114yescomplemented
269C2×#41—98no—
270C2×#41—98no—
271C2×#41—188no—
272C2×#41—188no—
273C22×Q8—91no—
274C22×Q8C2×D911yescharacteristic
275C22×D4—91no—
276C22×D4—92no—
277C22×D4—91no—
278C22×D4—92no—
279C23×C4—92no—
280C23×C4—92no—
281C23×C4—92no—
282C23×C4—91no—
283C23×C4C2×D911yescharacteristic
284C4⋊Q8—98no—
285#123—916no—
286#123—916no—
287C22⋊Q8—98no—
288C22⋊Q8C2×D918yescomplemented
289#121—916no—
290C4×D4—98no—
291C4×D4—98no—
292C2×C4⋊C4—91no—
293C2×C4⋊C4—92no—
294C2×C4⋊C4—92no—
295C2×C4⋊C4—92no—
296C2×C4⋊C4—916no—
297C2×C4⋊C4C2×D912yes—
298C2×C4⋊C4C2×D911yescharacteristic
299C2×C22⋊C4—91no—
300C2×C22⋊C4—92no—
301C2×C22⋊C4—91no—
302C2×C22⋊C4—92no—
303C2×C22⋊C4C2×D912yes—
304C2×C42—91no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3C23×D312yes—
309C23×C3C23×D311yescharacteristic
310C22×D3—38no—
311C22×D3—34no—
312C22×D3—32no—
313C22×D3—64no—
314C22×D3—64no—
315C22×D3—62no—
316C22×D3—62no—
317C22×D3—62no—
318C3×Q8—28no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—28no—
323C2×C3×C4D3×D414yes—
324C2×C3×C4C23×D314yes—
325C2×C3×C4C23×D312yes—
326C2×C3×C4C23×D314yes—
327C2×C3×C4C23×D312yes—
328#67—68no—
329#67—616no—
330C2×Dic3—34no—
331C2×Dic3—32no—
332C2×Dic3—32no—
333C2×Dic3—34no—
334C2×Dic3—32no—
335C2×Dic3—64no—
336C2×Dic3—62no—
337C2×Dic3—62no—
338D12—316no—
339D12—616no—
340C4×D3—616no—
341C4×D3—616no—
342C4×D3—616no—
343C4×D3—68no—
344Dic6—68no—
345C2×C9—42no—
346C2×C9#54714yescomplemented
347C2×C9C23×D411yescharacteristic
348C2×C9C22×#4111yescharacteristic
349C2×C9C2×2−1+411yescharacteristic
350D9—28no—
351D9—42no—
352C24—92no—
353C24—91no—
354C24C22×D911yescharacteristic
355#41—1832no—
356#41—1832no—
357C2×Q8—24no—
358C2×Q8—94no—
359C2×Q8—184no—
360C2×Q8C22×D914yes—
361C2×D4—94no—
362C2×D4—98no—
363C2×D4—98no—
364C2×D4—94no—
365C2×D4—98no—
366C2×D4—188no—
367C2×D4—184no—
368C2×D4—188no—
369C22×C4—24no—
370C22×C4—91no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—92no—
374C22×C4—98no—
375C22×C4—98no—
376C22×C4—98no—
377C22×C4—94no—
378C22×C4—188no—
379C22×C4—188no—
380C22×C4—188no—
381C22×C4—184no—
382C22×C4C22×D914yes—
383C22×C4C22×D912yes—
384C22×C4C22×D911yescharacteristic
385C22×C4C22×D912yes—
386C22×C4C22×D911yescharacteristic
387C4⋊C4—94no—
388C4⋊C4—98no—
389C4⋊C4—98no—
390C4⋊C4—98no—
391C4⋊C4C22×D918yes—
392C4⋊C4C22×D914yes—
393C22⋊C4—94no—
394C22⋊C4—98no—
395C22⋊C4—94no—
396C22⋊C4—98no—
397C22⋊C4C22×D918yes—
398C42—94no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—44no—
403C22×C3C2×D3×D412yes—
404C22×C3#100712yes—
405C22×C3D3×#4112yes—
406C22×C3C24×D311yescharacteristic
407C2×D3—38no—
408C2×D3—64no—
409C2×D3—64no—
410C2×D3—616no—
411C2×D3—62no—
412C2×D3—62no—
413C2×D3—62no—
414C2×D3—124no—
415C3×C4—22no—
416C3×C4—24no—
417C3×C4—24no—
418C3×C4C2×D3×D414yes—
419Dic3—34no—
420Dic3—64no—
421Dic3—62no—
422Dic3—62no—
423C9C2×#54711yescharacteristic, complemented, Sylow
424C23—22no—
425C23—22no—
426C23—22no—
427C23—94no—
428C23—98no—
429C23—92no—
430C23—184no—
431C23—184no—
432C23—182no—
433C23—182no—
434C23—182no—
435C23C23×D912yes—
436C23C23×D911yescharacteristic
437Q8—28no—
438Q8—188no—
439D4—916no—
440D4—1816no—
441D4—188no—
442D4—1816no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—92no—
448C2×C4—94no—
449C2×C4—92no—
450C2×C4—92no—
451C2×C4—94no—
452C2×C4—1816no—
453C2×C4—1816no—
454C2×C4—1816no—
455C2×C4—182no—
456C2×C4—184no—
457C2×C4—182no—
458C2×C4—188no—
459C2×C4D4×D914yes—
460C2×C4C23×D914yes—
461C2×C4C23×D912yes—
462C2×C4C23×D914yes—
463C2×C4C23×D912yes—
464C2×C3—42no—
465C2×C3#563114yes—
466C2×C3C22×D3×D411yescharacteristic
467C2×C3C2×#100711yescharacteristic
468C2×C3C2×D3×#4111yescharacteristic
469D3—68no—
470D3—122no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—44no—
475C22—98no—
476C22—1816no—
477C22—184no—
478C22—184no—
479C22—182no—
480C22—182no—
481C22—182no—
482C22—364no—
483C22C2×D4×D912yes—
484C22#6371312yes—
485C22#41×D912yes—
486C22C24×D911yescharacteristic
487C4—22no—
488C4—24no—
489C4—24no—
490C4—94no—
491C4—182no—
492C4—184no—
493C4—182no—
494C4C2×D4×D914yes—
495C3C2×#563111yescharacteristic
496C2—42no—
497C2—188no—
498C2—362no—
499C2#1058973314yesdirect product factor
500C2C22×D4×D911yescharacteristic
501C2C2×#6371311yescharacteristic
502C2C2×#41×D911yescharacteristic
subgroupG/H
centerC23C23×D9
commutator, cosocleC22×C9C25
FrattiniC22×C3C24×D3
FittingC2×C9×C22⋊Q8C2
socleC23×C3C23×D3
hypercenterC2×C22⋊Q8D9
hypocenterC9C2×#547
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#3701C2semidirect
2. C22×#3697C2semidirect
3. C2×C9×C22⋊Q8C2semidirect
4. C2×#63506C2semidirect
5. C2×C4×#611C2semidirect
6. #10589733C2direct (N × H)
7. C2×#63459C2semidirect
8. C2×#63456C2semidirect
9. C2×#63455C2semidirect
10. C2×C4⋊C4×D9C2semidirect
11. C2×#63446C2semidirect
12. C2×#63445C2semidirect
19. C2×#3701C22semidirect
21. C2×#3697C22semidirect
29. C9×C22⋊Q8C22semidirect
34. #63506C22semidirect
35. C4×#611C22semidirect
44. #63459C22semidirect
45. #63456C22semidirect
46. #63455C22semidirect
47. C4⋊C4×D9C22semidirect
48. #63446C22semidirect
49. #63445C22semidirect
84. C2×D36D4semidirect
174. C2×Dic9#41semidirect
180. D36C2×D4semidirect
193. C2×C22⋊Q8D9semidirect
268. Dic9C2×#41semidirect
288. C22⋊Q8C2×D9semidirect
346. C2×C9#547semidirect
423. C9C2×#547semidirect
499. C2#10589733direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#37012
C22×#36972
C2×C9×C22⋊Q82
C2×#635062
C2×C4×#6112
#105897332
C2×#634592
C2×#634562
C2×#634552
C2×C4⋊C4×D92
C2×#634462
C2×#634452
C2×#56313