Finite Group Databasebeta

← C2×#10590976 #11861060

Subgroups — C2×#10590976 #11861060

Order 1152.

981 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#907C211yesmaximal, characteristic, complemented
2C2×#63822C211yesmaximal, characteristic
3C2×#63814C211yesmaximal, characteristic, complemented
4C2×#63649C211yesmaximal, characteristic, complemented
5C2×C3×#825C211yesmaximal, characteristic, complemented
6C2×#63569C211yesmaximal, characteristic, complemented
7C2×#63554C211yesmaximal, characteristic, complemented
8#10590976C218yesmaximal, direct product factor
9C2×#5145—31nomaximal
10C2×#4823—31nomaximal
11C2×C3×#235C2211yescharacteristic
12C2×C3×#235C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C22×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C4×Dic3C2211yescharacteristic
17C22×C3×C3⋊C8C2211yescharacteristic
18C3×#907—22no—
19C3×#907C2214yescomplemented
20C2×C3×ΓL(1,𝔽25)—21no—
21C2×C3×D3×C8—21no—
22C2×#3718C411yescharacteristic, complemented
23C2×#3715C411yescharacteristic, complemented
24C2×#3715C411yescharacteristic, complemented
25#63822C414yescomplemented
26#63822C2214yes—
27C2×#3713C411yescharacteristic, complemented
28C2×#3711—21no—
29#63814—22no—
30#63814C2214yescomplemented
31C2×D3×C3⋊C8—21no—
32#63649C2214yescomplemented
33C3×#825C2214yescomplemented
34#63569C2214yescomplemented
35#63554C2214yescomplemented
36C2×C3×#131—31no—
37C2×#950—31no—
38C2×#950—31no—
39C2×#948—31no—
40C2×#907D311yescharacteristic, complemented
41C2×#901—31no—
42C2×C3×#104—31no—
43C2×#837—31no—
44#5145—38no—
45C2×#825D311yescharacteristic, complemented
46C2×#822—31no—
47C2×#822—31no—
48C2×#822—61no—
49C2×#815—31no—
50C2×#805—31no—
51C2×#803—31no—
52#4823—38no—
53C22×C32×C4C2311yescharacteristic
54C2×C3×#67—21no—
55C2×C3×#67—21no—
56C22×C3×Dic3—21no—
57C3×#235—21no—
58C3×#235—21no—
59C3×#235—21no—
60C3×#235—21no—
61C3×#235C2×C412yes—
62C3×#235C2×C412yes—
63C3×#235C2312yes—
64C3×#235C2312yes—
65C2×C3×D12C2×C411yescharacteristic
66C2×C3×D12C2×C411yescharacteristic
67C2×C3×C4×D3—21no—
68C2×C3×C4×D3—21no—
69C2×C3×Dic6C2×C411yescharacteristic
70C2×C3×Dic6C2×C411yescharacteristic
71C32×M16—21no—
72C32×M16C2312yes—
73C2×C32×C8—21no—
74#3748—21no—
75#3748C2312yes—
76C2×C3⋊(C3⋊C8)—21no—
77C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
78C2×C3⋊(C3⋊C8)C2×C412yes—
79C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
80C2×C3⋊(C3⋊C8)C2312yes—
81C3×C4×Dic3—21no—
82C3×C4×Dic3C2312yes—
83C2×C3×C3⋊C8D411yescharacteristic
84C2×C3×C3⋊C8D411yescharacteristic
85C2×C3×C3⋊C8D412yes—
86C2×C3×C3⋊C8C2312yes—
87C3×ΓL(1,𝔽25)—24no—
88C3×D3×C8—24no—
89#3718C2×C414yescomplemented
90#3715C2×C414yescomplemented
91#3715C2×C414yescomplemented
92#3713C2×C414yescomplemented
93#3711—24no—
94D3×C3⋊C8—24no—
95C2×#444—91noSylow
96C2×C3×#41—31no—
97C2×C3×#41—31no—
98C2×#235C2×D311yescharacteristic
99C2×#235C2×D311yescharacteristic
100C3×#131—34no—
101C3×#131—62no—
102C2×C3×M16—21no—
103C2×C3×M16—61no—
104C2×C3×M16C2×D311yescharacteristic
105C2×C3×M16C2×D311yescharacteristic
106C22×C3×C8—31no—
107C22×C3×C8—61no—
108C2×C3×C42—31no—
109#950—34no—
110#950—34no—
111#950—34no—
112#950—34no—
113#948—34no—
114#948—62no—
115C2×#216—31no—
116C2×#216—31no—
117C2×#215—31no—
118C2×#215—31no—
119C2×#214—31no—
120C2×#214—31no—
121C2×#213—31no—
122C2×#213—31no—
123C2×C4×Dic3C2×D311yescharacteristic
124C2×#208—31no—
125C2×#208—31no—
126C2×#208—61no—
127C2×#208—61no—
128C22×C3⋊C8—31no—
129C22×C3⋊C8—31no—
130C22×C3⋊C8—61no—
131C22×C3⋊C8—61no—
132C22×C3⋊C8C2×D311yescharacteristic
133#907—22no—
134#907C2×D314yescomplemented
135#901—34no—
136#901—62no—
137C2×ΓL(1,𝔽25)—21no—
138C2×ΓL(1,𝔽25)—61no—
139C2×D3×C8—21no—
140C2×D3×C8—61no—
141C3×#104—34no—
142#837—34no—
143#825C2×D314yescomplemented
144#822—34no—
145#822—34no—
146#822—64no—
147#815—34no—
148#805—34no—
149#803—34no—
150C23×C32#4111yescharacteristic
151C22×C3×D3—21no—
152C22×C3×D3—21no—
153C2×C32×C4C22×C411yescharacteristic
154C2×C32×C4C22×C411yescharacteristic
155C2×C32×C4C2×D411yescharacteristic
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4#4111yescharacteristic
158C2×C32×C4C2411yescharacteristic
159C3×#67—22no—
160C3×#67—22no—
161C3×#67—22no—
162C3×#67—22no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—42no—
167C3×D12—21no—
168C3×D12—21no—
169C3×D12C22×C412yes—
170C3×D12C22×C412yes—
171C3×C4×D3—22no—
172C3×C4×D3—22no—
173C3×C4×D3—22no—
174C3×C4×D3—22no—
175C3×Dic6—21no—
176C3×Dic6—21no—
177C3×Dic6C22×C412yes—
178C3×Dic6C22×C412yes—
179C32×C8—22no—
180C3⋊(C3⋊C8)—22no—
181C3⋊(C3⋊C8)C22×C412yes—
182C3⋊(C3⋊C8)C22×C412yes—
183C3×C3⋊C8C2×D412yes—
184C3×C3⋊C8C2×D412yes—
185C2×#135—91no—
186C2×#131—31no—
187C2×#131—91no—
188#444—98no—
189C2×#108—91no—
190C2×#104—31no—
191C2×#104—91no—
192C2×#97—91no—
193C3×#41—32no—
194C3×#41—32no—
195C3×#41—32no—
196C3×#41—32no—
197C3×#41—61no—
198C3×#41—61no—
199C3×#41—61no—
200C3×#41—61no—
201C2×C3×Q8—31no—
202C2×C3×Q8—31no—
203C2×C3×D4—31no—
204C2×C3×D4—31no—
205C2×C3×D4—61no—
206C2×C3×D4—61no—
207C22×C3×C4—21no—
208C22×C3×C4—61no—
209C22×C3×C4—61no—
210C22×C3×C4—61no—
211C22×C3×C4C22×D311yescharacteristic
212C22×C3×C4C22×D311yescharacteristic
213C2×#67—21no—
214C2×#67—21no—
215C22×Dic3—21no—
216#235—21no—
217#235—21no—
218#235—21no—
219#235—21no—
220#235C4×D312yes—
221#235C4×D312yes—
222#235C22×D312yes—
223#235C22×D312yes—
224C2×D12C4×D311yescharacteristic
225C2×D12C4×D311yescharacteristic
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×Dic6C4×D311yescharacteristic
229C2×Dic6C4×D311yescharacteristic
230C3×M16—21no—
231C3×M16—21no—
232C3×M16—22no—
233C3×M16—41no—
234C3×M16—64no—
235C3×M16C22×D312yes—
236C3×M16C22×D312yes—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—32no—
242C2×C3×C8—32no—
243C2×C3×C8—41no—
244C2×C3×C8—64no—
245C3×C42—32no—
246C3×C42—61no—
247#216—34no—
248#216—34no—
249#215—34no—
250#215—34no—
251#214—34no—
252#214—34no—
253#213—34no—
254#213—34no—
255C4×Dic3—21no—
256C4×Dic3C22×D312yes—
257#208—32no—
258#208—32no—
259#208—62no—
260#208—61no—
261#208—61no—
262#208—64no—
263#208—121no—
264C2×C3⋊C8—31no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—31no—
268C2×C3⋊C8—32no—
269C2×C3⋊C8—32no—
270C2×C3⋊C8—32no—
271C2×C3⋊C8—32no—
272C2×C3⋊C8—62no—
273C2×C3⋊C8—61no—
274C2×C3⋊C8—61no—
275C2×C3⋊C8—62no—
276C2×C3⋊C8—61no—
277C2×C3⋊C8—61no—
278C2×C3⋊C8—64no—
279C2×C3⋊C8—121no—
280C2×C3⋊C8#6711yescharacteristic
281C2×C3⋊C8#6711yescharacteristic
282C2×C3⋊C8#6712yes—
283C2×C3⋊C8C22×D312yes—
284ΓL(1,𝔽25)—24no—
285ΓL(1,𝔽25)—64no—
286D3×C8—24no—
287D3×C8—64no—
288C22×C32—22no—
289C22×C32C4×D411yescharacteristic
290C22×C32C4×D411yescharacteristic
291C22×C32C2×#4111yescharacteristic
292C2×C3×D3—22no—
293C2×C3×D3—22no—
294C2×C3×D3—42no—
295C2×C3×D3—42no—
296C32×C4C4×D411yescharacteristic
297C32×C4C4×D411yescharacteristic
298C32×C4C23×C411yescharacteristic
299C32×C4C22×D411yescharacteristic
300C3×Dic3—22no—
301C3×Dic3—22no—
302C3×Dic3—42no—
303C2×#41—31no—
304C2×#41—31no—
305#135—94no—
306#135—94no—
307C2×Q16—91no—
308C2×SD16—91no—
309C2×SD16—91no—
310C2×D8—91no—
311#131—34no—
312#131—62no—
313#131—94no—
314#131—182no—
315C2×M16—61no—
316C2×M16—91no—
317C2×M16—181no—
318C2×M16D3211yescharacteristic
319C22×C8—31no—
320C22×C8—61no—
321C22×C8—91no—
322C22×C8—181no—
323C2×C42—31no—
324#108—94no—
325#104—34no—
326#104—94no—
327#97—94no—
328C23×C3—21no—
329C23×C3—61no—
330C23×C3—61no—
331C23×C3#23511yescharacteristic
332C23×C3#23711yescharacteristic
333C22×D3—21no—
334C22×D3—21no—
335C3×Q8—32no—
336C3×Q8—32no—
337C3×Q8—61no—
338C3×Q8—61no—
339C3×D4—32no—
340C3×D4—32no—
341C3×D4—62no—
342C3×D4—62no—
343C3×D4—62no—
344C3×D4—62no—
345C3×D4—61no—
346C3×D4—61no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4—122no—
361C2×C3×C4C2×C4×D311yescharacteristic
362C2×C3×C4C2×C4×D311yescharacteristic
363C2×C3×C4C2×C4×D311yescharacteristic
364C2×C3×C4C2×C4×D311yescharacteristic
365C2×C3×C4#23511yescharacteristic
366C2×C3×C4D3×D411yescharacteristic
367C2×C3×C4D3×D411yescharacteristic
368C2×C3×C4#23711yescharacteristic
369C2×C3×C4C2×#6711yescharacteristic
370C2×C3×C4C2×#6711yescharacteristic
371C2×C3×C4C23×D311yescharacteristic
372C2×C3×C4C23×D311yescharacteristic
373#67—22no—
374#67—22no—
375#67—22no—
376#67—22no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—22no—
380C2×Dic3—42no—
381D12—21no—
382D12—21no—
383D12C2×C4×D312yes—
384D12C2×C4×D312yes—
385C4×D3—22no—
386C4×D3—22no—
387C4×D3—22no—
388C4×D3—22no—
389Dic6—21no—
390Dic6—21no—
391Dic6C2×C4×D312yes—
392Dic6C2×C4×D312yes—
393C3×C8—22no—
394C3×C8—22no—
395C3×C8—32no—
396C3×C8—32no—
397C3×C8—42no—
398C3⋊C8—32no—
399C3⋊C8—32no—
400C3⋊C8—32no—
401C3⋊C8—32no—
402C3⋊C8—62no—
403C3⋊C8—62no—
404C3⋊C8—62no—
405C3⋊C8—62no—
406C3⋊C8—122no—
407C3⋊C8C2×#6712yes—
408C3⋊C8C2×#6712yes—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32#44412yescomplemented
412C2×C32C2×C4×D411yescharacteristic
413C3×D3—42no—
414C3×D3—42no—
415#41—32no—
416#41—32no—
417#41—32no—
418#41—32no—
419#41—61no—
420#41—61no—
421#41—61no—
422#41—61no—
423C2×Q8—31no—
424C2×Q8—31no—
425C2×D4—31no—
426C2×D4—31no—
427C2×D4—61no—
428C2×D4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4C2×D3211yescharacteristic
433Q16—94no—
434SD16—94no—
435SD16—94no—
436D8—94no—
437M16—21no—
438M16—64no—
439M16—92no—
440M16—184no—
441M16—181no—
442M16C2×D3212yes—
443C2×C8—21no—
444C2×C8—32no—
445C2×C8—32no—
446C2×C8—31no—
447C2×C8—31no—
448C2×C8—64no—
449C2×C8—92no—
450C2×C8—91no—
451C2×C8—91no—
452C2×C8—92no—
453C2×C8—184no—
454C2×C8—181no—
455C42—32no—
456C42—61no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—42no—
463C22×C3—62no—
464C22×C3—62no—
465C22×C3—122no—
466C22×C3—122no—
467C22×C3#67⋊C411yescharacteristic
468C22×C3#67⋊C411yescharacteristic
469C22×C3C4×#6711yescharacteristic
470C22×C3C4×#6711yescharacteristic
471C22×C3C2×#23511yescharacteristic
472C22×C3C2×#23711yescharacteristic
473C2×D3—22no—
474C2×D3—22no—
475C2×D3—42no—
476C2×D3—42no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—62no—
482C3×C4—62no—
483C3×C4—122no—
484C3×C4#67⋊C411yescharacteristic
485C3×C4#67⋊C411yescharacteristic
486C3×C4C4×#6711yescharacteristic
487C3×C4C4×#6711yescharacteristic
488C3×C4C22×C4×D311yescharacteristic
489C3×C4C22×C4×D311yescharacteristic
490C3×C4C2×D3×D411yescharacteristic
491C3×C4C22×#6711yescharacteristic
492Dic3—22no—
493Dic3—22no—
494Dic3—42no—
495C32C2×#44411yescharacteristic, complemented, Sylow
496C23—61no—
497C23—61no—
498C23#379511yescharacteristic
499Q8—32no—
500Q8—32no—
501Q8—61no—
502Q8—61no—
503D4—32no—
504D4—32no—
505D4—62no—
506D4—62no—
507D4—62no—
508D4—62no—
509D4—61no—
510D4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—122no—
519C2×C4#379511yescharacteristic
520C2×C4C4×D3211yescharacteristic
521C2×C4C4×D3211yescharacteristic
522C2×C4D3×#6711yescharacteristic
523C2×C4D3×#6711yescharacteristic
524C2×C4C22×D3211yescharacteristic
525C8—22no—
526C8—32no—
527C8—32no—
528C8—92no—
529C8—92no—
530C8—182no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—22no—
536C2×C3—21no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—122no—
540C2×C3—122no—
541C2×C3#482312yescomplemented
542C2×C3#514512yescomplemented
543C2×C3C2×#67⋊C411yescharacteristic
544C2×C3C2×C4×#6711yescharacteristic
545D3—42no—
546D3—42no—
547C22—22no—
548C22—62no—
549C22—62no—
550C22—122no—
551C22—122no—
552C22#6387911yescharacteristic
553C22#6387911yescharacteristic
554C22C2×#379511yescharacteristic
555C4—62no—
556C4—62no—
557C4—122no—
558C4#6387911yescharacteristic
559C4#6387911yescharacteristic
560C4C2×C4×D3211yescharacteristic
561C4C2×D3×#6711yescharacteristic
562C3—21no—
563C3C2×#482311yescharacteristic, complemented
564C3C2×#514511yescharacteristic, complemented
565C2—21no—
566C2—21no—
567C2—122no—
568C2—122no—
569C2#1059097612yesdirect product factor
570C2C2×#6387911yescharacteristic
subgroupG/H
centerC2×C4D3×#67
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32C4×D4
prime radicalC2×#63822C2
cosocleC2×C32×C4C24
hypercenterC2×M16D32
hypocenterC32C2×#444
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×C3×#907C2semidirect
3. C2×#63814C2semidirect
4. C2×#63649C2semidirect
5. C2×C3×#825C2semidirect
6. C2×#63569C2semidirect
7. C2×#63554C2semidirect
8. #10590976C2direct (N × H)
19. C3×#907C22semidirect
22. C2×#3718C4semidirect
23. C2×#3715C4semidirect
24. C2×#3715C4semidirect
25. #63822C4semidirect
27. C2×#3713C4semidirect
30. #63814C22semidirect
32. #63649C22semidirect
33. C3×#825C22semidirect
34. #63569C22semidirect
35. #63554C22semidirect
40. C2×#907D3semidirect
45. C2×#825D3semidirect
89. #3718C2×C4semidirect
90. #3715C2×C4semidirect
91. #3715C2×C4semidirect
92. #3713C2×C4semidirect
134. #907C2×D3semidirect
143. #825C2×D3semidirect
411. C2×C32#444semidirect
495. C32C2×#444semidirect
541. C2×C3#4823semidirect
542. C2×C3#5145semidirect
563. C3C2×#4823semidirect
564. C3C2×#5145semidirect
569. C2#10590976direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9072
C2×#638222
C2×#638142
C2×#636492
C2×C3×#8252
C2×#635692
C2×#635542
#105909762
C2×#51453
C2×#48233