Finite Group Databasebeta

← C2×#10591156 #11859286

Subgroups — C2×#10591156 #11859286

Order 1152.

1205 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#67C211yesmaximal, characteristic
2C22×C3×#208C211yesmaximal, characteristic, complemented
3C22×#3711C211yesmaximal, characteristic, complemented
4C2×#63657C211yesmaximal, characteristic, complemented
5#10591156C218yesmaximal, direct product factor
6C2×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C2×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
9C2×#5128—31nomaximal
10C2×#5012—31nomaximal
11C23×C32×C4C2211yescharacteristic
12C22×C3×#67C411yescharacteristic
13C22×C3×C4×D3C2211yescharacteristic
14C22×C3⋊(C3⋊C8)C2211yescharacteristic
15C22×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×#217C411yescharacteristic
17C3×C4×#67C414yes—
18C3×C4×#67C2214yes—
19C2×C3×(C2×D3)⋊C4C411yescharacteristic
20C2×C3×Dic3⋊C4C411yescharacteristic
21C2×C3×C4×Dic3C2211yescharacteristic
22C2×C3×#208—24no—
23C2×C3×#208C2214yescomplemented
24C22×C3×C3⋊C8C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C2×#3711—24no—
27C2×#3711C2214yescomplemented
28#63657C2214yescomplemented
29Dic3⋊(C3⋊C8)C2214yescomplemented
30(C2×D3)⋊(C3⋊C8)C2214yescomplemented
31Dic3×C3⋊C8C2214yescomplemented
32C22×C3×M16—31no—
33C2×C3×C4×D4—31no—
34C2×C4×#67D311yescharacteristic
35C22×#208—31no—
36C22×#208D311yescharacteristic, complemented
37C22×ΓL(1,𝔽25)—31no—
38C2×#830—31no—
39C2×#830—31no—
40C2×#830—31no—
41C2×#830—61no—
42#5128—38no—
43C2×(C2×D3)⋊C8—31no—
44C2×Dic3⋊C8—31no—
45C2×C8×Dic3—31no—
46#5012—38no—
47C2×C4⋊(C3⋊C8)—31no—
48C2×C4×C3⋊C8—31no—
49C24×C32C2×C411yescharacteristic
50C23×C3×D3C2×C411yescharacteristic
51C22×C32×C4—22no—
52C22×C32×C4—22no—
53C22×C32×C4C2×C411yescharacteristic
54C22×C32×C4C2×C412yes—
55C22×C32×C4C2312yes—
56C22×C32×C4C2311yescharacteristic
57C2×C3×#67—24no—
58C2×C3×#67C2×C414yes—
59C22×C3×Dic3C2×C411yescharacteristic
60C22×C3×Dic3C2×C411yescharacteristic
61C22×C3×Dic3C2×C411yescharacteristic
62C2×C3×C4×D3—22no—
63C2×C3×C4×D3—22no—
64C2×C3×C4×D3C2×C412yes—
65C2×C3×C4×D3C2312yes—
66C2×C3⋊(C3⋊C8)—22no—
67C2×C3⋊(C3⋊C8)—22no—
68C2×C3⋊(C3⋊C8)C2312yes—
69C2×C3⋊(C3⋊C8)C2312yes—
70C3×#217C2×C414yes—
71C3×(C2×D3)⋊C4C2×C414yes—
72C3×Dic3⋊C4C2×C414yes—
73C3×C4×Dic3C2×C412yes—
74C3×C4×Dic3C2312yes—
75C3×#208—28no—
76C2×C3×C3⋊C8—22no—
77C2×C3×C3⋊C8D414yescomplemented
78C2×C3×C3⋊C8C2312yes—
79C2×C3×C3⋊C8C2312yes—
80#3711—28no—
81C2×#436—91noSylow
82C22×C3×D4—31no—
83C23×C3×C4—21no—
84C23×C3×C4—31no—
85C23×C3×C4C2×D311yescharacteristic
86C23×C3×C4C2×D311yescharacteristic
87C22×#67Dic311yescharacteristic
88C22×C4×D3C2×D311yescharacteristic
89C2×C3×M16—34no—
90C2×C3×M16—64no—
91C22×C3×C8—31no—
92C22×C3×C8—31no—
93C3×C4×D4—34no—
94C3×C4×D4—34no—
95C2×C3×C4⋊C4—31no—
96C2×C3×C22⋊C4—31no—
97C2×C3×C22⋊C4—31no—
98C2×C3×C42—31no—
99C2×#217Dic311yescharacteristic
100C4×#67Dic314yes—
101C4×#67C2×D314yes—
102C2×(C2×D3)⋊C4Dic311yescharacteristic
103C2×Dic3⋊C4Dic311yescharacteristic
104C2×C4×Dic3C2×D311yescharacteristic
105C2×#208—24no—
106C2×#208—34no—
107C2×#208—64no—
108C2×#208C2×D314yescomplemented
109C22×C3⋊C8—31no—
110C22×C3⋊C8—31no—
111C22×C3⋊C8—31no—
112C22×C3⋊C8—31no—
113C22×C3⋊C8—61no—
114C22×C3⋊C8—61no—
115C22×C3⋊C8C2×D311yescharacteristic
116C22×C3⋊C8C2×D311yescharacteristic
117C2×ΓL(1,𝔽25)—34no—
118C2×ΓL(1,𝔽25)—64no—
119#830—34no—
120#830—34no—
121#830—34no—
122#830—64no—
123(C2×D3)⋊C8—34no—
124Dic3⋊C8—34no—
125C8×Dic3—34no—
126C4⋊(C3⋊C8)—34no—
127C4×C3⋊C8—34no—
128C23×C32—22no—
129C23×C32—22no—
130C23×C32—22no—
131C23×C32C22×C412yes—
132C23×C32C22×C411yescharacteristic
133C22×C3×D3—22no—
134C22×C3×D3—22no—
135C22×C3×D3—22no—
136C22×C3×D3C22×C412yes—
137C2×C32×C4—22no—
138C2×C32×C4—24no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4C22×C412yes—
142C2×C32×C4C22×C411yescharacteristic
143C2×C32×C4C2×D412yes—
144C2×C32×C4#4112yes—
145C2×C32×C4C2411yescharacteristic
146C3×#67—28no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3M1614yescomplemented
150C2×C3×Dic3C22×C412yes—
151C2×C3×Dic3C22×C412yes—
152C2×C3×Dic3C22×C412yes—
153C3×C4×D3—24no—
154C3×C4×D3—22no—
155C3×C4×D3—22no—
156C3⋊(C3⋊C8)—22no—
157C3⋊(C3⋊C8)—22no—
158C3×C3⋊C8—22no—
159C3×C3⋊C8C2×D414yescomplemented
160C22×M16—31no—
161C22×M16—91no—
162C2×C4×D4—31no—
163#436—98no—
164C2×C4⋊C8—91no—
165C2×C22⋊C8—91no—
166C2×C22⋊C8—91no—
167C2×C4×C8—91no—
168C24×C3—21no—
169C24×C3—31no—
170C24×C3C4×D311yescharacteristic
171C24×C3C2×Dic311yescharacteristic
172C23×D3C2×Dic311yescharacteristic
173C2×C3×D4—34no—
174C2×C3×D4—64no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—31no—
184C22×C3×C4—31no—
185C22×C3×C4—31no—
186C22×C3×C4—32no—
187C22×C3×C4—32no—
188C22×C3×C4—42no—
189C22×C3×C4—42no—
190C22×C3×C4—62no—
191C22×C3×C4—62no—
192C22×C3×C4C4×D311yescharacteristic
193C22×C3×C4C4×D312yes—
194C22×C3×C4C2×Dic311yescharacteristic
195C22×C3×C4C2×Dic312yes—
196C22×C3×C4C22×D312yes—
197C22×C3×C4C22×D312yes—
198C22×C3×C4C22×D311yescharacteristic
199C22×C3×C4C22×D311yescharacteristic
200C2×#67—24no—
201C2×#67C2×Dic314yes—
202C22×Dic3C2×Dic311yescharacteristic
203C22×Dic3C2×Dic311yescharacteristic
204C22×Dic3C2×Dic311yescharacteristic
205C2×C4×D3—22no—
206C2×C4×D3—22no—
207C2×C4×D3C2×Dic312yes—
208C2×C4×D3C22×D312yes—
209C3×M16—68no—
210C2×C3×C8—34no—
211C2×C3×C8—32no—
212C2×C3×C8—32no—
213C2×C3×C8—62no—
214C3×C4⋊C4—34no—
215C3×C22⋊C4—34no—
216C3×C22⋊C4—34no—
217C3×C42—32no—
218C3×C42—32no—
219#217C2×Dic314yes—
220(C2×D3)⋊C4C2×Dic314yes—
221Dic3⋊C4C2×Dic314yes—
222C4×Dic3C2×Dic312yes—
223C4×Dic3C22×D312yes—
224#208—28no—
225#208—68no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—32no—
228C2×C3⋊C8—32no—
229C2×C3⋊C8—32no—
230C2×C3⋊C8—32no—
231C2×C3⋊C8—62no—
232C2×C3⋊C8—62no—
233C2×C3⋊C8—62no—
234C2×C3⋊C8—62no—
235C2×C3⋊C8—62no—
236C2×C3⋊C8—62no—
237C2×C3⋊C8—122no—
238C2×C3⋊C8—122no—
239C2×C3⋊C8#6714yescomplemented
240C2×C3⋊C8C22×D312yes—
241C2×C3⋊C8C22×D312yes—
242ΓL(1,𝔽25)—68no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—44no—
247C22×C32C4×D412yes—
248C22×C32C2×M1612yes—
249C22×C32#13112yes—
250C22×C32C23×C411yescharacteristic
251C2×C3×D3—22no—
252C2×C3×D3—22no—
253C2×C3×D3—22no—
254C2×C3×D3—44no—
255C32×C4—22no—
256C32×C4C4×D412yes—
257C32×C4C22×D411yescharacteristic
258C32×C4C2×#4111yescharacteristic
259C3×Dic3—22no—
260C3×Dic3—22no—
261C3×Dic3C2×M1614yescomplemented
262C22×D4—31no—
263C23×C4—31no—
264C23×C4D3211yescharacteristic
265C2×M16—34no—
266C2×M16—64no—
267C2×M16—94no—
268C2×M16—184no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8—91no—
272C22×C8—91no—
273C4×D4—34no—
274C4×D4—34no—
275C2×C4⋊C4—31no—
276C2×C22⋊C4—31no—
277C2×C22⋊C4—31no—
278C2×C42—31no—
279C4⋊C8—94no—
280C22⋊C8—94no—
281C22⋊C8—94no—
282C4×C8—94no—
283C23×C3—24no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—21no—
291C23×C3—22no—
292C23×C3—32no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—62no—
296C23×C3—62no—
297C23×C3—62no—
298C23×C3C2×C4×D312yes—
299C23×C3C2×C4×D311yescharacteristic
300C23×C3C22×Dic312yes—
301C23×C3C22×Dic311yescharacteristic
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3C22×Dic312yes—
306C3×D4—68no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—34no—
321C2×C3×C4—32no—
322C2×C3×C4—32no—
323C2×C3×C4—32no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—44no—
327C2×C3×C4—42no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—64no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4C2×C4×D312yes—
334C2×C3×C4C2×C4×D311yescharacteristic
335C2×C3×C4#23512yes—
336C2×C3×C4D3×D412yes—
337C2×C3×C4#23712yes—
338C2×C3×C4C22×Dic312yes—
339C2×C3×C4C22×Dic311yescharacteristic
340C2×C3×C4C2×#6712yes—
341C2×C3×C4C23×D311yescharacteristic
342C2×C3×C4C23×D311yescharacteristic
343#67—28no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3#20814yescomplemented
347C2×Dic3C22×Dic312yes—
348C2×Dic3C22×Dic312yes—
349C2×Dic3C22×Dic312yes—
350C4×D3—24no—
351C4×D3—22no—
352C4×D3—22no—
353C3×C8—34no—
354C3×C8—62no—
355C3⋊C8—22no—
356C3⋊C8—62no—
357C3⋊C8—62no—
358C3⋊C8—62no—
359C3⋊C8—62no—
360C3⋊C8—122no—
361C3⋊C8—122no—
362C3⋊C8C2×#6714yescomplemented
363C2×C32—42no—
364C2×C32#43614yescomplemented
365C2×C32C2×C4×D411yescharacteristic
366C2×C32C22×M1611yescharacteristic
367C2×C32C2×#13111yescharacteristic
368C3×D3—42no—
369C24—31no—
370C24D3×Dic311yescharacteristic
371C2×D4—34no—
372C2×D4—64no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—31no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—32no—
379C22×C4—32no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4D3×Dic311yescharacteristic
383C22×C4D3×Dic312yes—
384C22×C4C2×D3212yes—
385C22×C4C2×D3211yescharacteristic
386M16—68no—
387M16—188no—
388C2×C8—32no—
389C2×C8—34no—
390C2×C8—32no—
391C2×C8—62no—
392C2×C8—92no—
393C2×C8—92no—
394C2×C8—182no—
395C2×C8—182no—
396C4⋊C4—34no—
397C22⋊C4—34no—
398C22⋊C4—34no—
399C42—32no—
400C42—32no—
401C22×C3—24no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—21no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—48no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—62no—
420C22×C3—124no—
421C22×C3D3×M1612yes—
422C22×C3#90712yes—
423C22×C3C2×#20812yes—
424C22×C3C4×#6712yes—
425C22×C3D4×Dic312yes—
426C22×C3#94812yes—
427C22×C3C22×C4×D311yescharacteristic
428C22×C3C23×Dic311yescharacteristic
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—22no—
432C2×D3—44no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—22no—
438C3×C4—34no—
439C3×C4—42no—
440C3×C4—62no—
441C3×C4—62no—
442C3×C4C4×#6712yes—
443C3×C4D4×Dic312yes—
444C3×C4C2×#23511yescharacteristic
445C3×C4C2×D3×D411yescharacteristic
446C3×C4C2×#23711yescharacteristic
447C3×C4C22×#6711yescharacteristic
448Dic3—22no—
449Dic3—22no—
450Dic3C2×#20814yescomplemented
451C32C2×#43611yescharacteristic, complemented, Sylow
452C23—22no—
453C23—22no—
454C23—22no—
455C23—32no—
456C23—62no—
457C23—62no—
458C23—62no—
459C23C2×D3×Dic312yes—
460C23C2×D3×Dic311yescharacteristic
461D4—68no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—32no—
467C2×C4—34no—
468C2×C4—32no—
469C2×C4—32no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—64no—
475C2×C4C2×D3×Dic312yes—
476C2×C4C2×D3×Dic311yescharacteristic
477C2×C4#380412yes—
478C2×C4D3×#6712yes—
479C2×C4C22×D3211yescharacteristic
480C8—34no—
481C8—62no—
482C8—182no—
483C8—182no—
484C2×C3—21no—
485C2×C3—24no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—44no—
489C2×C3—42no—
490C2×C3—42no—
491C2×C3—122no—
492C2×C3#501214yescomplemented
493C2×C3#512814yescomplemented
494C2×C3C2×D3×M1611yescharacteristic
495C2×C3C2×#90711yescharacteristic
496C2×C3C22×#20811yescharacteristic
497C2×C3C2×C4×#6711yescharacteristic
498C2×C3C2×D4×Dic311yescharacteristic
499C2×C3C2×#94811yescharacteristic
500D3—42no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—44no—
505C22—62no—
506C22—62no—
507C22—62no—
508C22—124no—
509C22D3×#20812yes—
510C22#6381512yes—
511C22Dic3×#6712yes—
512C22C22×D3×Dic311yescharacteristic
513C4—22no—
514C4—34no—
515C4—62no—
516C4—62no—
517C4Dic3×#6712yes—
518C4C2×#380411yescharacteristic
519C4C2×D3×#6711yescharacteristic
520C3—21no—
521C3C2×#501211yescharacteristic, complemented
522C3C2×#512811yescharacteristic, complemented
523C2—42no—
524C2—122no—
525C2#1059115614yesdirect product factor
526C2C2×D3×#20811yescharacteristic
527C2C2×#6381511yescharacteristic
528C2C2×Dic3×#6711yescharacteristic
subgroupG/H
centerC22×C4C2×D32
commutatorC22×C32C23×C4
FrattiniC2×C4C22×D32
FittingC23×C32×C4C22
socleC23×C32C22×C4
prime radicalC22×C3×#67C4
cosocleC2×C32×C4C24
hypercenterC23×C4D32
hypocenterC32C2×#436
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×#208C2semidirect
3. C22×#3711C2semidirect
4. C2×#63657C2semidirect
5. #10591156C2direct (N × H)
6. C2×Dic3⋊(C3⋊C8)C2semidirect
7. C2×(C2×D3)⋊(C3⋊C8)C2semidirect
8. C2×Dic3×C3⋊C8C2semidirect
23. C2×C3×#208C22semidirect
27. C2×#3711C22semidirect
28. #63657C22semidirect
29. Dic3⋊(C3⋊C8)C22semidirect
30. (C2×D3)⋊(C3⋊C8)C22semidirect
31. Dic3×C3⋊C8C22semidirect
36. C22×#208D3semidirect
77. C2×C3×C3⋊C8D4semidirect
108. C2×#208C2×D3semidirect
149. C2×C3×Dic3M16semidirect
159. C3×C3⋊C8C2×D4semidirect
239. C2×C3⋊C8#67semidirect
261. C3×Dic3C2×M16semidirect
346. C2×Dic3#208semidirect
362. C3⋊C8C2×#67semidirect
364. C2×C32#436semidirect
450. Dic3C2×#208semidirect
451. C32C2×#436semidirect
492. C2×C3#5012semidirect
493. C2×C3#5128semidirect
521. C3C2×#5012semidirect
522. C3C2×#5128semidirect
525. C2#10591156direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#672
C22×C3×#2082
C22×#37112
C2×#636572
#105911562
C2×Dic3⋊(C3⋊C8)2
C2×(C2×D3)⋊(C3⋊C8)2
C2×Dic3×C3⋊C82
C2×#51283
C2×#50123