Finite Group Databasebeta

← C2×#10589667 #11903100

Subgroups — C2×#10589667 #11903100

Order 1152.

1878 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D4×C9C211yesmaximal, characteristic, complemented
2C2×#63501C212yesmaximal, complemented
3C2×#63492C211yesmaximal, characteristic, complemented
4C2×#63492C211yesmaximal, characteristic, complemented
5C2×#63488C211yesmaximal, characteristic, complemented
6C22×C4⋊Dic9C211yesmaximal, characteristic, complemented
7#10589667C2116yesmaximal, direct product factor
8C2×#63460C212yesmaximal, complemented
9C2×#63445C212yesmaximal, complemented
10C2×#611⋊C4C212yesmaximal, complemented
11C2×C4×D36C211yesmaximal, characteristic, complemented
12C2×#63432C211yesmaximal, characteristic, complemented
13C2×#5565—31nomaximal
14C22×D4×C9C2211yescharacteristic
15C23×C4×C9C2211yescharacteristic
16C23×C4×C9C2211yescharacteristic
17C22×#611C2212yes—
18C22×#611C2212yes—
19C23×Dic9C2212yes—
20C22×D36C2211yescharacteristic
21C22×C4×D9C2212yes—
22C4×D4×C9C2218yescomplemented
23C2×C9×C4⋊C4C2211yescharacteristic
24C2×C9×C22⋊C4C2211yescharacteristic
25C2×C9×C22⋊C4C2211yescharacteristic
26C2×C42×C9C2211yescharacteristic
27C2×#3676C2212yes—
28#63501C22116yescomplemented
29#63492C2218yescomplemented
30#63492C2218yescomplemented
31C2×(C2×D9)⋊C4C2212yes—
32C2×(C2×D9)⋊C4C2212yes—
33C2×(C2×D9)⋊C4C2212yes—
34#63488C2218yescomplemented
35C2×C4⋊Dic9C2211yescharacteristic
36C2×C4⋊Dic9C2211yescharacteristic
37C2×C4⋊Dic9C2211yescharacteristic
38C2×C4⋊Dic9C2212yes—
39C2×C4⋊Dic9C2218yescomplemented
40C2×Dic9⋊C4C2212yes—
41C2×Dic9⋊C4C2212yes—
42C2×C4×Dic9C2212yes—
43#63460C22116yescomplemented
44#63445C22116yescomplemented
45#611⋊C4C22116yescomplemented
46C4×D36C2218yescomplemented
47#63432C2218yescomplemented
48C2×C3×C4×D4D311yescharacteristic
49C2×#939—32no—
50C2×#930—31no—
51C2×#930—31no—
52C2×#926—31no—
53C22×C4⋊Dic3—31no—
54#5565—316no—
55C2×#898—32no—
56C2×#883—32no—
57C2×#67⋊C4—32no—
58C2×C4×D12—31no—
59C2×#870—31no—
60C24×C9C2311yescharacteristic
61C24×C9C2311yescharacteristic
62C23×D9C2312yes—
63C2×D4×C9—24no—
64C2×D4×C9C2314yes—
65C22×C4×C9—24no—
66C22×C4×C9—24no—
67C22×C4×C9C2314yes—
68C22×C4×C9C2311yescharacteristic
69C22×C4×C9C2311yescharacteristic
70C22×C4×C9C2311yescharacteristic
71C22×C4×C9C2314yes—
72C22×C4×C9C2311yescharacteristic
73C22×C4×C9C2311yescharacteristic
74C2×#611—28no—
75C2×#611—44no—
76C2×#611C2318yes—
77C2×#611C2318yes—
78C22×Dic9—44no—
79C22×Dic9C2312yes—
80C22×Dic9C2312yes—
81C22×Dic9C2312yes—
82C22×Dic9C2318yes—
83C2×D36—24no—
84C2×D36C2314yes—
85C2×C4×D9—44no—
86C2×C4×D9C2318yes—
87C9×C4⋊C4C2314yes—
88C9×C22⋊C4C2314yes—
89C9×C22⋊C4C2314yes—
90C42×C9C2314yes—
91#3676C2318yes—
92(C2×D9)⋊C4C2318yes—
93(C2×D9)⋊C4C2318yes—
94(C2×D9)⋊C4C2318yes—
95C4⋊Dic9C2314yes—
96C4⋊Dic9C2314yes—
97C4⋊Dic9C2314yes—
98C4⋊Dic9C2318yes—
99Dic9⋊C4C2318yes—
100Dic9⋊C4C2318yes—
101C4×Dic9C2318yes—
102C2×#553—91noSylow
103C22×C3×D4C2×D311yescharacteristic
104C23×C3×C4C2×D311yescharacteristic
105C23×C3×C4C2×D311yescharacteristic
106C22×#67—32no—
107C22×#67—32no—
108C23×Dic3—32no—
109C22×D12—31no—
110C22×C4×D3—32no—
111C3×C4×D4C2×D318yes—
112C2×C3×C4⋊C4C2×D311yescharacteristic
113C2×C3×C22⋊C4C2×D311yescharacteristic
114C2×C3×C22⋊C4C2×D311yescharacteristic
115C2×C3×C42C2×D311yescharacteristic
116C2×#217—32no—
117#939—316no—
118#930—38no—
119#930—38no—
120C2×(C2×D3)⋊C4—32no—
121C2×(C2×D3)⋊C4—32no—
122C2×(C2×D3)⋊C4—32no—
123#926—38no—
124C2×C4⋊Dic3—31no—
125C2×C4⋊Dic3—31no—
126C2×C4⋊Dic3—31no—
127C2×C4⋊Dic3—32no—
128C2×C4⋊Dic3—38no—
129C2×Dic3⋊C4—32no—
130C2×Dic3⋊C4—32no—
131C2×C4×Dic3—32no—
132#898—316no—
133#883—316no—
134#67⋊C4—316no—
135C4×D12—38no—
136#870—38no—
137C23×C9—22no—
138C23×C9—22no—
139C23×C9—22no—
140C23×C9—22no—
141C23×C9—22no—
142C23×C9#4114yes—
143C23×C9C2412yes—
144C23×C9C2412yes—
145C23×C9C2411yescharacteristic
146C22×D9—24no—
147C22×D9—24no—
148C22×D9—24no—
149C22×D9C2414yes—
150D4×C9—28no—
151C2×C4×C9—22no—
152C2×C4×C9—22no—
153C2×C4×C9—22no—
154C2×C4×C9—22no—
155C2×C4×C9—28no—
156C2×C4×C9—28no—
157C2×C4×C9#4114yes—
158C2×C4×C9C2412yes—
159C2×C4×C9C2412yes—
160C2×C4×C9C2412yes—
161C2×C4×C9C2412yes—
162C2×C4×C9C2412yes—
163#611—216no—
164#611—48no—
165C2×Dic9—24no—
166C2×Dic9—24no—
167C2×Dic9—24no—
168C2×Dic9—48no—
169C2×Dic9C2414yes—
170C2×Dic9C2414yes—
171C2×Dic9C2414yes—
172D36—28no—
173C4×D9—48no—
174#553—916no—
175C2×#126—92no—
176C2×#125—91no—
177C2×#123—92no—
178C2×#121—92no—
179C2×#121—91no—
180C2×#121—91no—
181C2×C4×D4—92no—
182C2×C4×D4—91no—
183C2×C4×D4D911yescharacteristic, complemented
184C2×#117—91no—
185C22×C4⋊C4—91no—
186C24×C3C22×D311yescharacteristic
187C24×C3C22×D311yescharacteristic
188C23×D3—32no—
189C2×C3×D4—24no—
190C2×C3×D4C22×D314yes—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4C22×D311yescharacteristic
194C22×C3×C4C22×D311yescharacteristic
195C22×C3×C4C22×D311yescharacteristic
196C22×C3×C4C22×D311yescharacteristic
197C22×C3×C4C22×D314yes—
198C22×C3×C4C22×D314yes—
199C22×C3×C4C22×D311yescharacteristic
200C2×#67—38no—
201C2×#67—38no—
202C2×#67—68no—
203C2×#67—124no—
204C22×Dic3—32no—
205C22×Dic3—32no—
206C22×Dic3—32no—
207C22×Dic3—38no—
208C22×Dic3—124no—
209C2×D12—34no—
210C2×D12—64no—
211C2×C4×D3—38no—
212C2×C4×D3—124no—
213C3×C4⋊C4C22×D314yes—
214C3×C22⋊C4C22×D314yes—
215C3×C22⋊C4C22×D314yes—
216C3×C42C22×D314yes—
217#217—38no—
218(C2×D3)⋊C4—38no—
219(C2×D3)⋊C4—38no—
220(C2×D3)⋊C4—38no—
221C4⋊Dic3—34no—
222C4⋊Dic3—34no—
223C4⋊Dic3—34no—
224C4⋊Dic3—38no—
225Dic3⋊C4—38no—
226Dic3⋊C4—38no—
227C4×Dic3—38no—
228C22×C9—28no—
229C22×C9—22no—
230C22×C9—22no—
231C22×C9—22no—
232C22×C9—22no—
233C22×C9—22no—
234C22×C9—44no—
235C22×C9C2×#4114yes—
236C22×C9C2×#4112yes—
237C22×C9C2×#4112yes—
238C22×C9H(5,𝔽2)12yes—
239C22×C9C2511yescharacteristic
240C2×D9—24no—
241C2×D9—24no—
242C2×D9—24no—
243C2×D9—48no—
244C4×C9—22no—
245C4×C9—22no—
246C4×C9—22no—
247C4×C9—22no—
248C4×C9C2×#4114yes—
249Dic9—24no—
250Dic9—24no—
251Dic9—24no—
252C22×D4—92no—
253C22×D4—92no—
254C22×D4—91no—
255C22×D4C2×D911yescharacteristic
256C23×C4—92no—
257C23×C4—92no—
258C23×C4C2×D911yescharacteristic
259C23×C4C2×D911yescharacteristic
260#126—916no—
261#125—98no—
262#123—916no—
263#121—916no—
264#121—98no—
265#121—98no—
266C4×D4—916no—
267C4×D4—98no—
268C4×D4C2×D918yescomplemented
269#117—98no—
270C2×C4⋊C4—91no—
271C2×C4⋊C4—91no—
272C2×C4⋊C4—92no—
273C2×C4⋊C4—98no—
274C2×C4⋊C4—91no—
275C2×C4⋊C4—92no—
276C2×C4⋊C4—92no—
277C2×C4⋊C4C2×D911yescharacteristic
278C2×C22⋊C4—92no—
279C2×C22⋊C4—92no—
280C2×C22⋊C4—92no—
281C2×C22⋊C4—92no—
282C2×C22⋊C4C2×D911yescharacteristic
283C2×C22⋊C4C2×D911yescharacteristic
284C2×C42—92no—
285C2×C42C2×D911yescharacteristic
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3#23514yes—
292C23×C3C23×D312yes—
293C23×C3C23×D312yes—
294C23×C3C23×D311yescharacteristic
295C22×D3—34no—
296C22×D3—64no—
297C22×D3—64no—
298C22×D3—64no—
299C3×D4—28no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—28no—
305C2×C3×C4—28no—
306C2×C3×C4#23714yes—
307C2×C3×C4C23×D312yes—
308C2×C3×C4C23×D312yes—
309C2×C3×C4C23×D312yes—
310C2×C3×C4C23×D312yes—
311C2×C3×C4C23×D312yes—
312#67—616no—
313#67—128no—
314C2×Dic3—34no—
315C2×Dic3—34no—
316C2×Dic3—34no—
317C2×Dic3—64no—
318C2×Dic3—64no—
319C2×Dic3—64no—
320C2×Dic3—128no—
321D12—68no—
322C4×D3—128no—
323C2×C9—24no—
324C2×C9—42no—
325C2×C9#55314yescomplemented
326C2×C9C22×#4111yescharacteristic
327C2×C9C22×#4111yescharacteristic
328C2×C9C2×H(5,𝔽2)11yescharacteristic
329D9—44no—
330C24—92no—
331C24C22×D911yescharacteristic
332C24C22×D911yescharacteristic
333C2×D4—24no—
334C2×D4—98no—
335C2×D4—98no—
336C2×D4—94no—
337C2×D4—188no—
338C2×D4—184no—
339C2×D4—364no—
340C2×D4C22×D914yes—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—92no—
344C22×C4—92no—
345C22×C4—92no—
346C22×C4—98no—
347C22×C4—98no—
348C22×C4—364no—
349C22×C4—364no—
350C22×C4C22×D914yes—
351C22×C4C22×D914yes—
352C22×C4C22×D911yescharacteristic
353C22×C4C22×D911yescharacteristic
354C22×C4C22×D911yescharacteristic
355C22×C4C22×D911yescharacteristic
356C22×C4C22×D911yescharacteristic
357C4⋊C4—94no—
358C4⋊C4—94no—
359C4⋊C4—98no—
360C4⋊C4—94no—
361C4⋊C4—98no—
362C4⋊C4—98no—
363C4⋊C4C22×D914yes—
364C22⋊C4—98no—
365C22⋊C4—98no—
366C22⋊C4—98no—
367C22⋊C4—98no—
368C22⋊C4C22×D914yes—
369C22⋊C4C22×D914yes—
370C42—98no—
371C42C22×D914yes—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—28no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—44no—
379C22×C3C2×#23514yes—
380C22×C3C2×#23512yes—
381C22×C3C2×#23712yes—
382C22×C3#100912yes—
383C22×C3C24×D311yescharacteristic
384C2×D3—64no—
385C2×D3—64no—
386C2×D3—64no—
387C2×D3—128no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—22no—
392C3×C4C2×#23714yes—
393Dic3—64no—
394Dic3—64no—
395Dic3—64no—
396C9C2×#55311yescharacteristic, complemented, Sylow
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—94no—
403C23—184no—
404C23—184no—
405C23—184no—
406C23#369714yes—
407C23C23×D912yes—
408C23C23×D912yes—
409C23C23×D911yescharacteristic
410D4—28no—
411D4—1816no—
412D4—188no—
413D4—368no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—94no—
421C2×C4—94no—
422C2×C4—94no—
423C2×C4—184no—
424C2×C4—184no—
425C2×C4—184no—
426C2×C4—368no—
427C2×C4—368no—
428C2×C4#369914yes—
429C2×C4C23×D912yes—
430C2×C4C23×D912yes—
431C2×C4C23×D912yes—
432C2×C4C23×D912yes—
433C2×C4C23×D912yes—
434C2×C3—24no—
435C2×C3—42no—
436C2×C3#556514yes—
437C2×C3C22×#23511yescharacteristic
438C2×C3C22×#23711yescharacteristic
439C2×C3C2×#100911yescharacteristic
440D3—124no—
441C22—22no—
442C22—22no—
443C22—28no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—44no—
448C22—184no—
449C22—184no—
450C22—184no—
451C22—368no—
452C22C2×#369714yes—
453C22C2×#369712yes—
454C22C2×#369912yes—
455C22#6371512yes—
456C22C24×D911yescharacteristic
457C4—22no—
458C4—22no—
459C4—22no—
460C4—22no—
461C4—184no—
462C4—184no—
463C4—184no—
464C4C2×#369914yes—
465C3C2×#556511yescharacteristic
466C2—24no—
467C2—42no—
468C2—364no—
469C2#1058966714yesdirect product factor
470C2C22×#369711yescharacteristic
471C2C22×#369911yescharacteristic
472C2C2×#6371511yescharacteristic
subgroupG/H
centerC23C23×D9
commutator, cosocleC22×C9C25
FrattiniC22×C3C24×D3
FittingC2×C4×D4×C9C2
socleC23×C3C23×D3
hypercenterC2×C4×D4D9
hypocenterC9C2×#553
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D4×C9C2semidirect
2. C2×#63501C2semidirect
3. C2×#63492C2semidirect
4. C2×#63492C2semidirect
5. C2×#63488C2semidirect
6. C22×C4⋊Dic9C2semidirect
7. #10589667C2direct (N × H)
8. C2×#63460C2semidirect
9. C2×#63445C2semidirect
10. C2×#611⋊C4C2semidirect
11. C2×C4×D36C2semidirect
12. C2×#63432C2semidirect
22. C4×D4×C9C22semidirect
28. #63501C22semidirect
29. #63492C22semidirect
30. #63492C22semidirect
34. #63488C22semidirect
39. C2×C4⋊Dic9C22semidirect
43. #63460C22semidirect
44. #63445C22semidirect
45. #611⋊C4C22semidirect
46. C4×D36C22semidirect
47. #63432C22semidirect
183. C2×C4×D4D9semidirect
268. C4×D4C2×D9semidirect
325. C2×C9#553semidirect
396. C9C2×#553semidirect
469. C2#10589667direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×D4×C92
C2×#635012
C2×#634922
C2×#634882
C22×C4⋊Dic92
#105896672
C2×#634602
C2×#634452
C2×#611⋊C42
C2×C4×D362
C2×#634322
C2×#55653